term - a single number or variable, or numbers and variables multiplied together
$3.75 in quarters and nickles in her car. The number of nickles is fifteen more than the number of q$3.75 in quarters and nickels in her car. The number of nickels is fifteen more than the number of quarters. How many of each type of coin does she have?
Let the number of nickels be n, and the number of quarters be q. We know nickels are 0.05, and quarters are 0.25. We're given:
[LIST=1]
[*]n = q + 15
[*]0.05n + 0.25q = 3.75
[/LIST]
Substituting (1) into (2), we get:
0.05(q + 15) + 0.25q = 3.75
0.05q + 0.75 + 0.25q = 3.75
Combine like term:
0.3q + 0.75 = 3.75
[URL='https://www.mathcelebrity.com/1unk.php?num=0.3q%2B0.75%3D3.75&pl=Solve']Typing this equation into our calculator[/URL], we get:
[B]q = 10[/B]
Substituting q = 10 into Equation (1), we get:
n = 10 + 15
[B]n = 25[/B]
(2x + y) + (3x - 2y) = 80*80(2x + y) + (3x - 2y) = 80*80
Combine like terms:
(2 + 3)x + (1 - 2)y = 6,400
[B]5x - y = 6,400[/B]
(2x10^3) x (2x10^6) x (2x10^12) = ?Multiply the 2's together:
2 x 2 x 2 = 8
Multiply the 10 with exponent terms together:
10^3 x 10^6 * 10^12 = 10^(3 + 6 + 12) = 10^21
Group both results:
[B]8 x 10^21
[MEDIA=youtube]pt-jJc8bxGk[/MEDIA][/B]
(4 - a)/(a^2 - 16)(4 - a)/(a^2 - 16)
(4 - a)/(a + 4)(a - 4)
-1(a - 4)/(a + 4)(a - 4)
Cancel a- 4 terms
[B]-1/(a + 4)[/B]
[MEDIA=youtube]H0-pqGsPapE[/MEDIA]
+÷+(-)+÷+(-)
Parentheses first(
(-)
12 - 6y
So we have:
+÷+12 - 6y
2y + 5/6y + 12 - 6y
Combine like terms:
[B]5/6y - 4y + 12[/B]
-11, -9, -7, -5, -3 What is the next number? What is the 200th term in this sequence?-11, -9, -7, -5, -3 What is the next number? What is the 200th term in this sequence?
We see that Term 1 is -11, Term 2 is -9, so we do a point slope equation of (1,-11)(2,-9) to get the [URL='https://www.mathcelebrity.com/search.php?q=%281%2C-11%29%282%2C-9%29']nth term of the formula[/URL]
f(n) = 2n - 13
The next number is the 6th term:
f(6) = 2(6) - 13
f(6) = 12 - 13
f(6) = [B]-1
[/B]
The 200th term is:
f(200) = 2(200) - 13
f(200) = 400 - 13
f(200) = [B]387[/B]
-5n - 5n - 5 = 5-5n - 5n - 5 = 5
Solve for [I]n[/I] in the equation -5n - 5n - 5 = 5
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(-5 - 5)n = -10n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
-10n - 5 = + 5
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -5 and 5. To do that, we add 5 to both sides
-10n - 5 + 5 = 5 + 5
[SIZE=5][B]Step 4: Cancel 5 on the left side:[/B][/SIZE]
-10n = 10
[SIZE=5][B]Step 5: Divide each side of the equation by -10[/B][/SIZE]
-10n/-10 = 10/-10
n = [B]- 1
[URL='https://www.mathcelebrity.com/1unk.php?num=-5n-5n-5%3D5&pl=Solve']Source[/URL][/B]
-g + 3/4a = y for a-g + 3/4a = y for a
Add g to each side:
-g + g + 3/4a = y + g
Cancel the g terms on the left side:
3/4a = y + g
Cross multiply:
3a = 4(y + g)
Divide each side by 3 to isolate a:
3a/3 = 4(y + g)/3
a = [B]4(y + g)/3[/B]
0,7,14,21 What is the next number? What is the 1000th term?0,7,14,21
What is the next number?
What is the 1000th term?
We're adding 7 to the last term, so we get a next term of:
21 + 7 = [B]28
[/B]
For our nth term, we notice a pattern for the nth term of:
7n - 7
[LIST]
[*]n = 1 --> 7(1) - 7 = 0
[*]n = 2 --> 7(2) - 7 = 7
[*]n = 3 --> 7(3) - 7 = 14
[/LIST]
For n = 1000, we have:
7(1000) - 7 = 7000 - 7 = [B]6993[/B]
1, 1/2, 1/3, 1/4, 1/5 What is the next number? What is the 89th term of the sequence?1, 1/2, 1/3, 1/4, 1/5
What is the next number?
What is the 89th term of the sequence?
Formula for nth term is 1/n
Next number is n = 5, so we have [B]1/5[/B]
With n = 89, we have [B]1/89[/B]
1, 1/2, 1/4, 1/8, 1/16 The next number in the sequence is 1/32. What is the function machine you wou1, 1/2, 1/4, 1/8, 1/16
The next number in the sequence is 1/32. What is the function machine you would use to find the nth term of this sequence?
Hint: look at the denominators
We notice that
1/2^0 = 1/1 = 1
1/2^1 = 1/2
1/2^2 = 1/4
1/2^3 = 1/8
1/2^4 = 1/32
So we write our explicit formula for term n:
f(n) = [B]1/2^(n - 1)[/B]
1, 1/2, 1/4, 1/8, 1/16, ...At first glance, we see powers of 2 in the denominator of every term except the first one.
But if we remember 2^0 = 1, we get our breakthrough.
1/2^0 = 1/1 = 1
Therefore, we stagger the powers of 2 by 1 less than the term we are on:
a(n) = [B]1/2^(n - 1)
[MEDIA=youtube]Ua-arUukOew[/MEDIA][/B]
1, 4, 9, 16, 25 What is the next number? What is the 50th term?1, 4, 9, 16, 25
What is the next number?
What is the 50th term?
We see that 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25
We build a formula for the nth term:
f(n) = n^2
The next number means n = 6th term:
f(6) = 6^2 = [B]36
[/B]
The 50th term means n = 50:
f(50) = 50^2 = [B]2500[/B]
1, 8, 27, 64 What is the 10th term?1, 8, 27, 64
What is the 10th term?
We see the following pattern:
1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64
We build our sequence function using this pattern:
f(n) = n^3
With n = 10, we have:
f(10) = 10^3
f(10) = [B]1,000[/B]
1.25, 2, 2.75, 3.5 What is the 100th term?1.25, 2, 2.75, 3.5 What is the 100th term?
The formula of nth term is:
f(n) = 0.75n + 0.5
So the 100th term is:
f(100) = 0.75(100) + 0.5
f(100) = 75 + 0.5
f(100) = [B]75.5[/B]
1/2, 3, 5&1/2, 8......203 What term is the number 203?1/2, 3, 5&1/2, 8......203
What term is the number 203?
We see the following pattern:
1/2 = 2.5*1 - 2
3 = 2.5*2 - 2
5&1/2 = 2.5*3 - 2
8 = 2.5*4 - 2
We build our function
f(n) = 2.5n - 2
Set 2.5n - 2 = 203
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=2.5n-2%3D203&pl=Solve']equation solver[/URL], we get:
n = [B]82[/B]
10, 1,000, 100,000, 10,000,000 What power of 10 is the 80th term?10, 1,000, 100,000, 10,000,000
What power of 10 is the 80th term?
We see the following pattern
10^1 = 10
10^3 = 1000
10^5 = 100,000
10^7 = 10,000,000
f(n) = 10^(2n - 1)
We build the 80th term:
f(80) = 10^(2(80) - 1)
f(80) = 10^(160 - 1)
f(80) = 10^[B]159[/B]
100, 75, 50, 25, 0, -25 What is the next number? What is the 100th term?100, 75, 50, 25, 0, -25 What is the next number? What is the 100th term?
Using point slope, we get (1, 100)(2, 75)
Our [URL='https://www.mathcelebrity.com/search.php?q=%281%2C+100%29%282%2C+75%29&x=0&y=0']series function becomes[/URL]
f(n) = -25n + 125
The next term is the 7th term:
f(7) = -25(7) + 125
f(7) = -175 + 125
f(7) = [B]-50
[/B]
The 100th term is found by n = 100:
f(100) = -25(100) + 125
f(100) = -2500 + 125
f(100) = [B]-2375[/B]
10n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 610n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 6
Solve for [I]n[/I] in the equation 10n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 6
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(10 - 9 + 8 - 7 + 6)n = 8n
[SIZE=5][B]Step 2: Group the constant terms on the right hand side:[/B][/SIZE]
10 - 9 + 8 - 7 + 6 = 8
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
8n = + 8
[SIZE=5][B]Step 4: Divide each side of the equation by 8[/B][/SIZE]
8n/8 = 8/8
n = [B]1[/B]
11 to the power of 6 multiply 11 to the power of 311 to the power of 6 multiply 11 to the power of 3
Take this in parts.
[U]Step 1: 11 to the power of 6 means we raise 11 to the 6th power using exponents:[/U]
11^6
[U]Step 2: 11 to the power of 3 means we raise 11 to the 3rd power using exponents:[/U]
11^3
[U]Step 3: Multiply each term together:[/U]
11^6 * 11^3
[U]Step 4: Simplify[/U]
Because we have 2 numbers that are the same, in this case, 11, we can add the exponents together when multiplying:
11^(6 + 3)
[B]11^9
[MEDIA=youtube]gCxVq7LqyHk[/MEDIA][/B]
15y + 13/c = m for y15y + 13/c = m for y
Subtract 13/c from each side to isolate the y term:
15y + 13/c - 13/c = m - 13/c
Cancel the 13/c on the left side and we get
15y = m - 13/c
Now, divide each side by 15 to isolate y:
15y/15 = (m - 13/c)/15
Cancel the 15 on the left side, and we get:
y = [B](m - 13/c)/15[/B]
18 multiplied by the quantity of 11 plus r18 multiplied by the quantity of 11 plus r
The quantity of 11 plus r is written as:
11 + r
18 multiplied by the [I]quantity[/I] means we take 18 and multiply it by the term 11 + r
[B]18(11 + r)
[MEDIA=youtube]2GYjQTjt8qM[/MEDIA][/B]
18 seconds faster than Tina’s time18 seconds faster than Tina’s time
Let Tina's time be t. Speaking in terms of time, faster means less. So we have an algebraic expression of:
[B]t - 18[/B]
2 Asset PortfolioFree 2 Asset Portfolio Calculator - Given a portfolio with 2 assets, this determines the expected return (mean), variance, and volatility (standard deviation) of the portfolio.
2 consecutive even integers that equal 118Let x be the first even integer. That means the next consecutive even integer must be x + 2.
Set up our equation:
x + (x + 2) = 118
Group x terms
2x + 2 = 118
Subtract 2 from each side
2x = 116
Divide each side by 2
x = 58
Which means the next consecutive even integer is 58 + 2 = 60
So our two consecutive even integers are [B]58, 60[/B]
Check our work:
58 + 60 = 118
2 Lines IntersectionFree 2 Lines Intersection Calculator - Enter any 2 line equations, and the calculator will determine the following:
* Are the lines parallel?
* Are the lines perpendicular
* Do the lines intersect at some point, and if so, which point?
* Is the system of equations dependent, independent, or inconsistent
2 movie tickets and 3 snacks are $24. 3 movie tickets and 4 snacks are $35. How much is a movie tick2 movie tickets and 3 snacks are $24. 3 movie tickets and 4 snacks are $35. How much is a movie ticket and how much is a snack?
Let a movie ticket cost be m, and a snack cost be s. We have:
2m + 3s = 24.3
3m + 4s = 35
Using the [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=2m+%2B+3s+%3D+24.3&term2=3m+%2B+4s+%3D+35&pl=Cramers+Method']simultaneous equations calculator[/URL], we get:
m = $7.8
s = $2.9
2 numbers add to 200. The first is 20 less than the second.2 numbers add to 200. The first is 20 less than the second.
Let the first number be x and the second number be y. We're given:
[LIST=1]
[*]x + y = 200
[*]x = y - 20
[/LIST]
Plug (2) into (1)
(y - 20) + y = 200
Group like terms:
2y - 20 = 200
[URL='https://www.mathcelebrity.com/1unk.php?num=2y-20%3D200&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]y = 110[/B] <-- This is the larger number
Plug y = 110 into Equation (2) to get the smaller number:
x = 110 - 20
[B]x = 90[/B] <-- This is the smaller number
Let's check our work for Equation (1) using x = 90, and y = 110
90 + 110 ? 200
200 = 200 <-- Good, our solutions check out for equation (1)
Let's check our work for Equation (2) using x = 90, and y = 110
90 = 110 - 20
90 = 90 <-- Good, our solutions check out for equation (2)
2 numbers that are equal have a sum of 602 numbers that are equal have a sum of 60
Let's choose 2 arbitrary variables for the 2 numbers
x, y
Were given 2 equations:
[LIST=1]
[*]x = y <-- Because we have the phrase [I]that are equal[/I]
[*]x + y = 60
[/LIST]
Because x = y in equation (1), we can substitute equation (1) into equation (2) for x:
y + y = 60
Add like terms to get:
2y = 60
Divide each side by 2:
2y/2 = 60/2
Cancel the 2's and we get:
y = [B]30
[/B]
Since x = y, x = y = 30
x = [B]30[/B]
2 pens and 1 eraser cost $35 and 3 pens and 4 erasers cost $65. X represents the cost of 1 pen and Y2 pens and 1 eraser cost $35 and 3 pens and 4 erasers cost $65. X represents the cost of 1 pen and Y represents the cost of 1 eraser. Write the 2 simultaneous equations and solve.
Set up our 2 equations where cost = price * quantity:
[LIST=1]
[*]2x + y = 35
[*]3x + 4y = 65
[/LIST]
We can solve this one of three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2x+%2B+y+%3D+35&term2=3x+%2B+4y+%3D+65&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2x+%2B+y+%3D+35&term2=3x+%2B+4y+%3D+65&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2x+%2B+y+%3D+35&term2=3x+%2B+4y+%3D+65&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
[LIST]
[*][B]x (cost of 1 pen) = 15[/B]
[*][B]y (cost of 1 eraser) = 5[/B]
[/LIST]
2 times a number minus 4 times another number is 6. The sum of 2 numbers is 8. Find the 2 numbers2 times a number minus 4 times another number is 6. The sum of 2 numbers is 8. Find the 2 numbers.
Let the first number be x, and the second number be y. We're given two equations:
[LIST=1]
[*]2x - 4y = 6
[*]x + y = 8
[/LIST]
Using our simultaneous equation calculator, there are 3 ways to solve this:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2x+-+4y+%3D+6&term2=x+%2B+y+%3D+8&pl=Substitution']Substitution[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2x+-+4y+%3D+6&term2=x+%2B+y+%3D+8&pl=Elimination']Elimination[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2x+-+4y+%3D+6&term2=x+%2B+y+%3D+8&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
They all give the same answers:
(x, y) = [B](6.3333333, 1.6666667)[/B]
2, 4, 6, 8....1000. What term is the number 1000?2, 4, 6, 8....1000. What term is the number 1000?
Formula for nth term is 2n
If 2n = 1000, then dividing each side by 2, we see that:
2n/2 = 1000/2
n = [B]500[/B]
2-thirds of the sum of 5 and a plus the product of 3 and z2-thirds of the sum of 5 and a plus the product of 3 and z
The sum of 5 and a
5 + a
2-thirds of this sum:
2(5 + a)/3
The product of 3 and z:
3z
The word [I]plus[/I] means we add the two terms together:
[B]2(5 + a)/3 + 3z[/B]
2/3 x 3/4 x 4/5 x 5/6 x 6/7 x 7/8 x 8/9 x 9/10 x 10/11 x 11/12 x 12/13 x 13/14Do this without a calculator.
Notice, all numerators have a matching denominator except 2 in the first term and 14 in the last term.
So we have 2/14
Using 2 as a common factor, we have:
2/2 / 14/2
[B]1/7[/B]
24 coloring books, 60 crayons, and 84 markers can be packaged into at most how many identical packag24 coloring books, 60 crayons, and 84 markers can be packaged into at most how many identical packages? How many of each would each package contain?
First, determine the greatest common factor (GCF) of 24, 60, and 84 using our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=24&num2=60&num3=84&pl=GCF']GCF calculator[/URL].
GCF(24, 60, 84) = 12
So we have 12 identical packages.
Now, figure out how many coloring books, crayons, and markers for each package
[LIST]
[*]24/12 = 2 coloring books
[*]60/12 = 5 crayons
[*]84/12 = 7 markers
[/LIST]
[B]So we have 12 identical packages, each containing 2 coloring books, 5 crayons, and 7 markers[/B]
2ade➗(ae)2ade[IMG]https://fonts.gstatic.com/s/e/notoemoji/14.0/2797/72.png[/IMG](ae)
2ade/ae
the ae terms cancel, so we have:
[B]2d[/B]
2consecutiveevenintegerssuchthatthesmalleraddedto5timesthelargergivesasumof702 consecutive even integers such that the smaller added to 5 times the larger gives a sum of 70.
Let the first, smaller integer be x. And the second larger integer be y. Since they are both even, we have:
[LIST=1]
[*]x = y - 2 <-- Since they're consecutive even integers
[*]x + 5y = 70 <-- Smaller added to 5 times the larger gives a sum of 70
[/LIST]
Substitute (1) into (2):
(y - 2) + 5y = 70
Group like terms:
(1 + 5)y - 2 = 70
6y - 2 = 70
[URL='https://www.mathcelebrity.com/1unk.php?num=6y-2%3D70&pl=Solve']Typing 6y - 2 = 70 into our search engine[/URL], we get:
[B]y = 12 <-- Larger integer[/B]
Plugging this into Equation (1) we get:
x = 12 - 2
[B]x = 10 <-- Smaller Integer[/B]
So (x, y) = (10, 12)
2n + 8 - n = 202n + 8 - n = 20
Solve for [I]n[/I] in the equation 2n + 8 - n = 20
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(2 - 1)n = n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
n + 8 = + 20
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 8 and 20. To do that, we subtract 8 from both sides
n + 8 - 8 = 20 - 8
[SIZE=5][B]Step 4: Cancel 8 on the left side:[/B][/SIZE]
n = [B]12[/B]
2n - 1&1/2n = 592n - 1&1/2n = 59
1&1/2n = 3/2n or 1.5n
So we have:
2n - 1.5n = 59
Solve for [I]n[/I] in the equation 2n - 1.5n = 59
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(2 - 1.5)n = 0.5n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
0.5n = + 59
[SIZE=5][B]Step 3: Divide each side of the equation by 0.5[/B][/SIZE]
0.5n/0.5 = 59/0.5
n = [B]118[/B]
2x/5 - 9y = 6 for x2x/5 - 9y = 6 for x
Add 9y to each side to isolate the x term:
2x/5 - 9y + 9y = 9y + 6
Cancel the 9y's on the left side:
2x/5 = 9y + 6
Multiply each side by 5:
2x * 5/5 = 5(9y + 6)
Cancel the 5's on the left side and we get:
2x = 5(9y + 6)
Divide each side by 2 to isolate x:
2x/2 = 5/2(9y + 6)
Cancel the 2's on the left side and we get our final literal equation of:
x = [B]5/2(9y + 6)[/B]
3 adults and 4 children must pay $136. 2 adults and 3 children must pay $97.3a + 4c = 136
2a + 3c = 97
[URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=3a+%2B+4c+%3D+136&term2=2a+%2B+3c+%3D+97&pl=Cramers+Method']Using any of the 3 methods here[/URL]:
[B]a = 20
c = 19[/B]
3, 6, 12, 24, 48 What is the function machine for this sequence?3, 6, 12, 24, 48
What is the function machine for this sequence?
We see the following pattern:
3 * 2^0 = 3
3 * 2^1 = 6
3 * 2^2 = 12
3 * 2^3 = 24
3 * 2^4 = 48
Our function machine for term n is:
[B]f(n) = 3 * 2^(n - 1)[/B]
3, 8, 13, 18, .... , 5008 What term is the number 5008?3, 8, 13, 18, .... , 5008 What term is the number 5008?
For term n, we have a pattern:
f(n) = 5(n - 1) + 3
Set this equal to 5008
5(n - 1) + 3 = 5008
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=5%28n-1%29%2B3%3D5008&pl=Solve']equation solver,[/URL] we get:
n = [B]1002[/B]
31,29,24,22,17 what comes next31,29,24,22,17 what comes next
We see that each sequence term alternates between subtracting 2 and subtracting 5. Since the last term, 17, was found by subtracting 5, our next term is found by subtracting 2 from 17:
17 - 2 = [B]15[/B]
3abc^4/12a^3(b^3c^2)^2 * 8ab^-4c/4a^2b3abc^4/12a^3(b^3c^2)^2 * 8ab^-4c/4a^2b
Expand term 1:
3abc^4/12a^3(b^3c^2)^2
3abc^4/12a^3b^6c^4
Now simplify term 1:
3/12 = 1/4
c^4 terms cancel
Subtract powers from variables since the denominator powers are higher:
b^(6 - 1) = b^5
a^(3 - 1) = a^2
1/4a^2b^5
Now simplify term 2:
8ab^-4c/4a^2b
8/4 = 2
2c/a^(2 - 1)b^(1 - -4)
2c/ab^5
Now multiply simplified term 1 times simplified term 2:
1/4a^2b^5 * 2c/ab^5
(1 * 2c)/(4a^2b^5 * ab^5)
2c/4a^(2 + 1)b^(5 + 5)
2c/4a^3b^10
2/4 = 1/2, so we have:
[B]c/2a^3b^10[/B]
3x less than 2 times the sum of 2x and 1 is equal to the sum of 2 and 53x less than 2 times the sum of 2x and 1 is equal to the sum of 2 and 5
This is an algebraic expression. Let's take this algebraic expression in 5 parts:
[LIST=1]
[*]The sum of 2x and 1 means we add 1 to 2x: 2x + 1
[*]2 times the sum of 2x and 1: 2(2x + 1)
[*]3x less than the sum of 2x and 1 means we subtract 3x from 2(2x + 1): 2(2x + 1) - 3x
[*]The sum of 2 and 5 means we add 5 to 2: 2 + 5
[*]Finally, the phrase [I]equal[/I] means an equation, so we set #3 equal to #4
[/LIST]
Our algebraic expression is:
[B]2(2x + 1) - 3x = 2 + 5[/B]
Now, some problems may ask you to simplify. In this case, we multiply through and group like terms:
4x + 2 - 3x = 7
[B]x + 2 = 7 <-- This is our simplified algebraic expression
[/B]
Now, what if the problem asks you to solve for x, [URL='https://www.mathcelebrity.com/1unk.php?num=x%2B2%3D7&pl=Solve']you type this into our search engine[/URL] and get:
x =[B] 5
[MEDIA=youtube]3hzyc2NPCGI[/MEDIA][/B]
4 adults and 3 children cost $40. Two adults and 6 children cost $384 adults and 3 children cost $40. Two adults and 6 children cost $38
Givens and Assumptions:
[LIST]
[*]Let the number of adults be a
[*]Let the number of children be c
[*]Cost = Price * Quantity
[/LIST]
We're given 2 equations:
[LIST=1]
[*]4a + 3c = 40
[*]2a + 6c = 38
[/LIST]
We can solve this system of equations 3 ways
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=4a+%2B+3c+%3D+40&term2=2a+%2B+6c+%3D+38&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=4a+%2B+3c+%3D+40&term2=2a+%2B+6c+%3D+38&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=4a+%2B+3c+%3D+40&term2=2a+%2B+6c+%3D+38&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we use, we get:
[LIST]
[*][B]a = 7[/B]
[*][B]c = 4[/B]
[/LIST]
4 consecutive integers such that the sum of the first 3 integers is equal to the 4th4 consecutive integers such that the sum of the first 3 integers is equal to the 4th
Let n be our first consecutive integer.
[LIST=1]
[*]n
[*]n + 1
[*]n + 2
[*]n + 3
[/LIST]
The sum of the first 3 integers is equal to the 4th:
n + n + 1 + n + 2 = n + 3
Simplify by grouping like terms:
(n + n + n) + (1 + 2) = n + 3
3n + 3 = n + 3
3n = n
n = 0
n = 0
n + 1 = 1
n + 2 = 2
n + 3 = 3
Check our work:
0 + 1 +2 ? 3
3 = 3
Our final answer is [B](0, 1, 2, 3}[/B]
4 rectangular strips of wood, each 30 cm long and 3 cm wide, are arranged to form the outer section4 rectangular strips of wood, each 30 cm long and 3 cm wide, are arranged to form the outer section of a picture frame. Determine the area inside the wooden frame.
Area inside forms a square, with a length of 30 - 3 - 3 = 24. We subtract 3 twice, because we account for 2 rectangular strips with a width of 3.
Area of a square is side * side. So we have 24 * 24 = [B]576cm^2[/B]
401(k) BalanceFree 401(k) Balance Calculator - Determines your 401(k) balance given a salary history per year, contribution percentage rate, employer match percentage, and a rate of return.
414 people used public pool. Daily prices are $1.75 for children and $2.00 for adults. Total cost wa414 people used public pool. Daily prices are $1.75 for children and $2.00 for adults. Total cost was $755.25. How many adults and children used the pool
Let the number of children who used the pool be c, and the number of adults who used the pool be a. We're given two equations:
[LIST=1]
[*]a + c = 414
[*]2a + 1.75c = 755.25
[/LIST]
We have a simultaneous equations. You can solve this any of 3 ways below:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+c+%3D+414&term2=2a+%2B+1.75c+%3D+755.25&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+c+%3D+414&term2=2a+%2B+1.75c+%3D+755.25&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+c+%3D+414&term2=2a+%2B+1.75c+%3D+755.25&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
Whichever method you choose, you get the same answer:
[LIST]
[*][B]a = 123[/B]
[*][B]c = 291[/B]
[/LIST]
5, 14, 23, 32, 41....1895 What term is the number 1895?5, 14, 23, 32, 41....1895 What term is the number 1895?
Set up a point slope for the first 2 points:
(1, 5)(2, 14)
Using [URL='https://www.mathcelebrity.com/search.php?q=%281%2C+5%29%282%2C+14%29&x=0&y=0']point slope formula, our series function[/URL] is:
f(n) = 9n - 4
To find what term 1895 is, we set 9n - 4 = 1895 and solve for n:
9n - 4 = 1895
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=9n-4%3D1895&pl=Solve']equation solver[/URL], we get:
n = [B]211[/B]
5,10,15,20 What is the next number? What is the 100th term?5,10,15,20
What is the next number?
What is the 100th term?
Increment is by 5, so next number is 20 + 5 = [B]25[/B]
Formula for nth number is 5 * n
With n = 100, we have 5 * 100 = [B]500[/B]
52% of a town's households have children and 25% have pets. If 12% have both, what percent have neit52% of a town's households have children and 25% have pets. If 12% have both, what percent have neither
Let C represent households with children. Let P represents households with pets. We have the formula to determine households with Children or Pets as C U P (C Union P) or (C or P):
C U P = C + P - (C and P)
C U P = 52% + 25% - 12%
C U P = 65%
Now, if we want to find what percent have neither, we use (C U P)':
(C U P)' = 100% - (C U P)
(C U P)' = 100% - 65%
(C U P)' = [B]35%[/B]
5×5 squared5×5 squared
Determine index form
5^2 <-- index form
Evaluate:
5^2 = 5 * 5 = 25
6 times j squared minus twice j squared6 times j squared minus twice j squared
j squared means we raise the variable j to the power of 2:
j^2
6 times j squared means we multiply j^2 by 6:
6j^2
Twice j squared means we multiply j^2 by 2:
2j^2
The word [I]minus[/I] means we subtract 2j^2 from 6j^2
6j^2 - 2j^2
So if you must simplify, we group like terms and get:
(6 - 2)j^2
[B]4j^2[/B]
6 times the reciprocal of a number equals 2 times the reciprocal of 7. What is the number6 times the reciprocal of a number equals 2 times the reciprocal of 7. What is the number
We've got two algebraic expressions here. Let's take it in parts:
Term 1:
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The reciprocal is 1/x
Multiply this by 6: 6/x
Term 2:
Reciprocal of 7: 1/7
2 times this: 2/7
We set these terms equal to each other:
6/x = 2/7
[URL='https://www.mathcelebrity.com/prop.php?num1=6&num2=2&den1=x&den2=7&propsign=%3D&pl=Calculate+missing+proportion+value']Type this proportion into the search engine[/URL], and we get:
[B]x = 21[/B]
7 and 105 are successive terms in a geometric sequence. what is the term following 105?7 and 105 are successive terms in a geometric sequence. what is the term following 105?
Geometric sequences are set up such that the next term in the sequence equals the prior term multiplied by a constant. Therefore, we express the relationship in the following equation:
7k = 105 where k is the constant
[URL='https://www.mathcelebrity.com/1unk.php?num=7k%3D105&pl=Solve']Type this equation into our search engine[/URL] and we get:
k = 15
The next term in the geometric sequence after 105 is found as follows:
105*15 = [B]1,575[/B]
7, 10, 15, 22 What is the next number in the sequence? What is the 500th term?7, 10, 15, 22
What is the next number in the sequence?
What is the 500th term?
We see that:
1^2 + 6 = 7
2^2 + 6 = 10
3^3 + 6 = 15
4^2 + 6 = 22
We build our function as f(n) = n^2 + 6
Next term in the sequence is f(5)
f(5) = 5^2 + 6
f(5) = 25 + 6
f(5) = [B]31
[/B]
Calculate the 500th term:
f(500) = 500^2 + 6
f(500) = 250,000 + 6
f(500) = [B]250,006[/B]
7n + 4 + n - 5 = 637n + 4 + n - 5 = 63
Solve for [I]n[/I] in the equation 7n + 4 + n - 5 = 63
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(7 + 1)n = 8n
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
4 - 5 = -1
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
8n - 1 = + 63
[SIZE=5][B]Step 4: Group constants:[/B][/SIZE]
We need to group our constants -1 and 63. To do that, we add 1 to both sides
8n - 1 + 1 = 63 + 1
[SIZE=5][B]Step 5: Cancel 1 on the left side:[/B][/SIZE]
8n = 64
[SIZE=5][B]Step 6: Divide each side of the equation by 8[/B][/SIZE]
8n/8 = 64/8
n = [B]8[/B]
8,11,14,17,20 What is the next number? What is the 150th term?8,11,14,17,20
What is the next number?
What is the 150th term?
We're adding by 3 to the last number in the sequence, so we have the next number as:
20 + 3 = [B]23
[/B]
For the nth term, we have a formula of this:
3n + 5
3(1) + 5 = 8
3(2) + 5 = 11
3(3) + 5 = 14
With n = 150, we have:
3(150) + 5 = 450 + 5 = [B]455[/B]
9, 3, 1, 1/3, 1/9 What is the next number in this sequence? What is the function machine for this se9, 3, 1, 1/3, 1/9
What is the next number in this sequence?
What is the function machine for this sequence?
We see the following pattern in this sequence:
9 = 9/3^0
3 = 9/3^1
1 = 9/3^2
1/3 = 9/3^3
1/9 = 9/3^4
Our function machine formula is:
[B]f(n) = 9/3^(n - 1)
[/B]
Next term is the 6th term:
f(6) = 9/3^(6 - 1)
f(6) = 9/3^5
f(6) = 9/243
f(6) = [B]1/27[/B]
A 1.5 inch tall preying mantis will sometimes hold its ground and attempt to fight a person who is 6A 1.5 inch tall preying mantis will sometimes hold its ground and attempt to fight a person who is 6 feet tall. If a person who is 6 feet tall is engaged in a battle with an animal that was proportionally as tall as the person is to the preying mantis, how tall would the animal be?
In terms of inches, [URL='https://www.mathcelebrity.com/linearcon.php?quant=6&pl=Calculate&type=foot']6 feet = 72 inches[/URL]
Set up a proportion of height of smaller creature to larger creature where h is the heigh of the animal
1.5/72 = 72/h
Using our [URL='https://www.mathcelebrity.com/proportion-calculator.php?num1=1.5&num2=72&den1=72&den2=h&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator[/URL], we get:
h = 3456 inches
In terms of feet, we have [URL='https://www.mathcelebrity.com/linearcon.php?quant=3456&pl=Calculate&type=inch']3456 inches[/URL] = [B]288 feet[/B]
A 100 point test contains a total of 20 questions. The multiple choice questions are worth 3 pointsA 100 point test contains a total of 20 questions. The multiple choice questions are worth 3 points each and short response questions are worth 8 points each. Write a system of linear equations that represents this situation
Assumptions:
[LIST]
[*]Let m be the number of multiple choice questions
[*]Let s be the number of short response questions
[/LIST]
Since total points = points per problem * number of problems, we're given 2 equations:
[LIST=1]
[*][B]m + s = 20[/B]
[*][B]3m + 8s = 100[/B]
[/LIST]
We can solve this system of equations 3 ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=m+%2B+s+%3D+20&term2=3m+%2B+8s+%3D+100&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=m+%2B+s+%3D+20&term2=3m+%2B+8s+%3D+100&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=m+%2B+s+%3D+20&term2=3m+%2B+8s+%3D+100&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get:
[B]m = 12, s = 8[/B]
A 12% acid solution is made by mixing 8% and 20% solutions. If the 450 ml of the 12% solution is reqA 12% acid solution is made by mixing 8% and 20% solutions. If the 450 ml of the 12% solution is required, how much of each solution is required?
Component Unit Amount
8% Solution: 0.08 * x = 0.08x
20% Solution: 0.2 * y = 0.2y
12% Solution: 0.12 * 450 = 54
We add up the 8% solution and 20% solution to get two equations:
[LIST=1]
[*]0.08x + 0.2y = 54
[*]x + y = 450
[/LIST]
We have a simultaneous set of equations. We can solve it using three methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.08x+%2B+0.2y+%3D+54&term2=x+%2B+y+%3D+450&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.08x+%2B+0.2y+%3D+54&term2=x+%2B+y+%3D+450&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.08x+%2B+0.2y+%3D+54&term2=x+%2B+y+%3D+450&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
[LIST]
[*][B]x = 300 ml[/B]
[*][B]y = 150 ml[/B]
[/LIST]
A 124-inch length of ribbon is to be cut into three pieces. The longest piece is to be 36 inches loA 124-inch length of ribbon is to be cut into three pieces. The longest piece is to be 36 inches longer than the shortest piece, and the third piece is to be half the length of the longest piece. Find the length of each piece of ribbon.
[LIST]
[*]Let the longest piece be l.
[*]The shortest piece is s = l - 36
[*]The third medium piece m = 0.5l
[/LIST]
We know s + m + l = 124. Now substitute for s and m
(l - 36) + 0.5l + l = 124
Combine like terms:
2.5l - 36 = 124
Type [URL='http://www.mathcelebrity.com/1unk.php?num=2.5l-36%3D124&pl=Solve']2.5l - 36 = 124 into our search engine[/URL], we get l = [B]64[/B]
Shortest piece s = 64 - 36 = [B]28[/B]
Medium piece m = 0.5(64) = [B]32[/B]
A 50-pound bowling ball and an 8-pound bowling ball are dropped from a tall building. Which ball wilA 50-pound bowling ball and an 8-pound bowling ball are dropped from a tall building. Which ball will hit first?
[B]They will land at the same time[/B]
[B]How fast something falls due to gravity is determined by a number known as the "acceleration of gravity", which is 9.81 m/s^2 at the surface of our Earth. In one second, [I]any object[/I]’s downward velocity will increase by 9.81 m/s because of gravity. This is just the way gravity works - it accelerates everything at exactly the same rate.[/B]
a 9-foot rope is cut into two pieces one piece is x feet express the length of the other piece in tea 9-foot rope is cut into two pieces one piece is x feet express the length of the other piece in terms of x
Piece 1 + Piece 2 = 9
Piece 1 = x
x + Piece 2 = 9
Subtracting x from each side, we get:
x - x + Piece 2 = 9 - x
Cancel the x's on the left side, we get:
Piece 2 = [B]9 - x
[/B]
Check our work:
x + 9 - x ? 9
9 = 9
A 98-inch piece of wire must be cut into two pieces. One piece must be 10 inches shorter than the otA 98-inch piece of wire must be cut into two pieces. One piece must be 10 inches shorter than the other. How long should the pieces be?
The key phrase in this problem is [B]two pieces[/B].
Declare Variables:
[LIST]
[*]Let the short piece length be s
[*]Let the long piece length be l
[/LIST]
We're given the following
[LIST=1]
[*]s = l - 10
[*]s + l = 98 (Because the two pieces add up to 98)
[/LIST]
Substitute equation (1) into equation (2) for s:
l - 10+ l = 98
Group like terms:
2l - 10 = 98
Solve for [I]l[/I] in the equation 2l - 10 = 98
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -10 and 98. To do that, we add 10 to both sides
2l - 10 + 10 = 98 + 10
[SIZE=5][B]Step 2: Cancel 10 on the left side:[/B][/SIZE]
2l = 108
[SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE]
2l/2 = 108/2
l = [B]54[/B]
To solve for s, we substitute l = 54 into equation (1):
s = 54 - 10
s = [B]44[/B]
Check our work:
The shorter piece is 10 inches shorter than the longer piece since 54 - 44 = 10
Second check: Do both pieces add up to 98
54 + 44 ? 98
98 = 98
A bag contains 120 marbles. Some are red and the rest are black. There are 19 red marbles for everyA bag contains 120 marbles. Some are red and the rest are black. There are 19 red marbles for every black marble. How many red marbles are in the bag?
Let the red marbles be r
Let the black marbles be b.
A 19 to 1 red to black is written as:
r = 19b
We're also given:
b + r = 120
Substitute r = 19b into this equation and we get:
b + 19b = 120
Combine like terms:
20b = 120
To solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=20b%3D120&pl=Solve']we type it in our search engine [/URL]and we get:
b = 6
Since r = 19b, we substitute b = 6 into this equation to solve for r:
r = 19(6)
r = [B]114[/B]
A bag contains 6 red balls and 7 green balls. You plan to select 4 balls at random. Determine the prA bag contains 6 red balls and 7 green balls. You plan to select 4 balls at random. Determine the probability of selecting 4 green balls.
Assuming draw without replacement of the balls, we have:
[LIST=1]
[*]Selection 1: 7 green out of 13 balls
[*]Selection 2: 6 green out of 12 balls
[*]Selection 3: 5 green out of 11 balls
[*]Selection 4: 4 green out of 10 balls
[/LIST]
Since each draw is independent, we multiply each probability of green:
P(GGGG) = 7/13 * 6/12 * 5/11 * 4/10
P(GGGG) = 840/17,160
P(GGGG) = [B]0.05[/B]
A baker determined the annual profit in dollars from selling pies using p(n ) = 52n - 0.05n^2, whereA baker determined the annual profit in dollars from selling pies using p(n ) = 52n - 0.05n^2, where n is the number of pies sold. What is the annual profit if the baker sells 700 pies?
p(700) = 52(700) - 0.05(700)^2
p(700) = 36400 - 0.05 * 490000
p(700) = 36400 - 24500
p(700) = [B]11900[/B]
A baker determined the annual profit in dollars from selling pies using p(n) = 52n - 0.05n^2 , whereA baker determined the annual profit in dollars from selling pies using p(n) = 52n - 0.05n^2 , where n is the number of pies sold. What is the annual profit if the baker sells 400 pies?
p(400) = 52(400) - 0.05(400)^2
p(400) = 20800 - 0.05(160000)
p(400) = 20800 - 8000
p(400) = [B]12800[/B]
A bakery sells 5800 muffins in 2010. The bakery sells 7420 muffins in 2015. Write a linear model thaA bakery sells 5800 muffins in 2010. The bakery sells 7420 muffins in 2015. Write a linear model that represents the number y of muffins that the bakery sells x years after 2010.
Find the number of muffins sold after 2010 through 2015:
7,420 - 5,800 = 1,620
Now, since the problem states a linear sales model, we need to determine the sales per year:
1,620 muffins sold since 2010 / 5 years = 324 muffins per year.
Build our linear model:
[B]y = 5,800 + 324x
[/B]
Reading this out loud, we start with 5,800 muffins at the end of 2010, and we add 324 more muffins for each year after 2010.
A barn contains cows, ducks, and a 3-legged dog named Tripod. There are twice as many cows as ducksA barn contains cows, ducks, and a 3-legged dog named Tripod. There are twice as many cows as ducks in the barn and a total of 313 legs. How many ducks are there in the barn?
[LIST]
[*]Let the number of ducks be d. Duck legs = 2 * d = 2d
[*]Number of cows = 2d. Cow legs = 4 * 2d = 8d
[*]1 dog Tripod has 3 legs
[/LIST]
Total legs:
2d + 8d + 3 = 313
Solve for [I]d[/I] in the equation 2d + 8d + 3 = 313
[SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE]
(2 + 8)d = 10d
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
10d + 3 = + 313
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 3 and 313. To do that, we subtract 3 from both sides
10d + 3 - 3 = 313 - 3
[SIZE=5][B]Step 4: Cancel 3 on the left side:[/B][/SIZE]
10d = 310
[SIZE=5][B]Step 5: Divide each side of the equation by 10[/B][/SIZE]
10d/10 = 310/10
d = [B]31[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=2d%2B8d%2B3%3D313&pl=Solve']Source[/URL]
A bike is purchased for $200 and sold for $150. Determine the percentage of profit or loss.A bike is purchased for $200 and sold for $150. Determine the percentage of profit or loss.
[U]Since sale price is less than purchase price, we have a loss:[/U]
Loss = Sale Price - Purchase Price
Loss = 150 - 200
Loss = -50
[U]Calculate percent loss:[/U]
Percent Loss = 100% * Loss / Purchase Price
Percent Loss = 100% * -50/200
Percent Loss = 100% *- 1/4
Percent Loss = [B]-25%[/B]
a boat traveled 336 km downstream with the current. The trip downstream took 12 hours. write an equaa boat traveled 336 km downstream with the current. The trip downstream took 12 hours. write an equation to describe this relationship
We know the distance (d) equation in terms of rate (r) and time (t) as:
d = rt
We're given d = 336km and t = 12 hours, so we have:
[B]336 km = 12t [/B] <-- this is our equation
Divide each side by 12 to solve for t:
12t/12 = 336/12
t = [B]28 km / hour[/B]
A Bouquet of lillies and tulips has 12 flowers. Lillies cost $3 each, and tulips cost $2 each. The bA Bouquet of lillies and tulips has 12 flowers. Lillies cost $3 each, and tulips cost $2 each. The bouquet costs $32. Write and solve a system of linear equations to find the number of lillies and tulips in the bouquet.
Let l be the number of lillies and t be the number of tulips. We're given 2 equations:
[LIST=1]
[*]l + t = 12
[*]3l + 2t = 32
[/LIST]
With this system of equations, we can solve it 3 ways.
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=l+%2B+t+%3D+12&term2=3l+%2B+2t+%3D+32&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=l+%2B+t+%3D+12&term2=3l+%2B+2t+%3D+32&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=l+%2B+t+%3D+12&term2=3l+%2B+2t+%3D+32&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter which method we choose, we get:
[LIST]
[*][B]l = 8[/B]
[*][B]t = 4[/B]
[/LIST]
[B]Now Check Your Work For Equation 1[/B]
l + t = 12
8 + 4 ? 12
12 = 12
[B]Now Check Your Work For Equation 2[/B]
3l + 2t = 32
3(8) + 2(4) ? 32
24 + 8 ? 32
32 = 32
A box contains 22 red apples and 3 green apples. Three apples are selected at random, one after theA box contains 22 red apples and 3 green apples. Three apples are selected at random, one after the other, without replacement. please show the steps.
(a) The first two apples are green. What is the probability that the third apple is red?
(b) What is the probability that exactly two of the three apples are red?
a) You have 22 red apples left and 1 green left leaving 23 total apples left. Therefore, probability of red is
[B]P(R) = 22/23[/B]
b) Determine our sample space to select exactly two red apples in three picks.
[LIST=1]
[*]RRG
[*]RGR
[*]GRR
[/LIST]
[U]Now determine the probabilities of each event in the sample space[/U]
P(RRG) = 22/25 * 21/24 * 3/23 = 0.1004
P(RGR) = 22/25 * 3/24 * 21/23 = 0.1004
P(GRR) = 3/25 * 22/24 * 21/23 = 0.1004
[U]We want the sum of the three probabilities[/U]
P(RRG) + P(RGR) + P(GRR) = 0.1004 + 0.1004 + 0.1004
P(RRG) + P(RGR) + P(GRR) = 3(0.1004)
P(RRG) + P(RGR) + P(GRR) = [B]0.3012[/B]
A boy is 10 years older than his brother. In 4 years he will be twice as old as his brother. Find thA boy is 10 years older than his brother. In 4 years he will be twice as old as his brother. Find the present age of each?
Let the boy's age be b and his brother's age be c. We're given two equations:
[LIST=1]
[*]b = c + 10
[*]b + 4 = 2(c + 4)
[/LIST]
Substitute equation (1) into equation (2):
(c + 10) + 4 = 2(c + 4)
Simplify by multiplying the right side through and grouping like terms:
c + 14 = 2c + 8
[URL='https://www.mathcelebrity.com/1unk.php?num=c%2B14%3D2c%2B8&pl=Solve']Type this equation into our search engine[/URL] and we get:
c = [B]6[/B]
Now plug c = 6 into equation (1):
b = 6 + 10
b = [B]16[/B]
a boy purchased a party-length sandwich 57 inches long. he wants to cut it into three pieces so thata boy purchased a party-length sandwich 57 inches long. he wants to cut it into three pieces so that the middle piece is 6inches longer than the shortest piece and the shortest piece is 9 inches shorter than the longest price. how long should the three pieces be?
Let the longest piece be l. The middle piece be m. And the short piece be s. We have 2 equations in terms of the shortest piece:
[LIST=1]
[*]l = s + 9 (Since the shortest piece is 9 inches shorter, this means the longest piece is 9 inches longer)
[*]m = s + 6
[*]s + m + l = 57
[/LIST]
We substitute equations (1) and (2) into equation (3):
s + (s + 6) + (s + 9) = 57
Group like terms:
(1 + 1 + 1)s + (6 + 9) = 57
3s + 15 = 57
To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=3s%2B15%3D57&pl=Solve']type this equation into our search engine[/URL] and we get:
s = [B]14
[/B]
[U]Plug s = 14 into equation 2 to solve for m:[/U]
m = 14 + 6
m = [B]20
[/B]
[U]Plug s = 14 into equation 1 to solve for l:[/U]
l = 14 + 9
l = [B]23
[/B]
Check our work for equation 3:
14 + 20 + 23 ? 57
57 = 57 <-- checks out
[B][/B]
A bunny population is doubling every 2 years. There are currently 45 bunnies. How many will there beA bunny population is doubling every 2 years. There are currently 45 bunnies. How many will there be in 10 years?
Find the number of doubling periods:
Number of Doubling periods = Time / Doubling period
Number of Doubling periods = 10/2
Number of Doubling periods = 5
Create a function to determine the amount of bunnies after each doubling period:
B(n) = 45 * 2^n
Since we calculated 5 doubling periods, we want B(5):
B(5) = 45 * 2^5
B(5) = 45 * 32
B(5) = [B]1,440[/B]
a carnival charges $6 admission and $2.50 per ride. You have $50 to spend at the carnival. Which ofa carnival charges $6 admission and $2.50 per ride. You have $50 to spend at the carnival. Which of the following inequalities represents the situation if r is the number of rides?
We set up our inequality using less than or equal to, since our cash is capped at $50. We use S for our :
Cost per ride * r + Admission <= 50
Plugging in our numbers, we get:
2.50r + 6 <= 50
[B][/B]
Now, if the problem asks you to put this in terms of r, then [URL='https://www.mathcelebrity.com/1unk.php?num=2.50r%2B6%3C%3D50&pl=Solve']we plug this inequality into our search engine[/URL] and we get:
r <= 17.6
Since we cannot do fractional rides, we round down to 17:
[B]r <= 17[/B]
A carpet cleaner charges $75 to clean the first 180 sq ft of carpet. There is an additional chargeA carpet cleaner charges $75 to clean the first 180 sq ft of carpet. There is an additional charge of 25¢ per square foot for any footage that exceeds 180 sq ft and $1.30 per step for any carpeting on a staircase. A customers cleaning bill was $253.95. This included the cleaning of a staircase with 14 steps. In addition to the staircase, how many square feet of carpet did the customer have cleaned?
Calculate the cost of the staircase cleaning.
Staircase cost = $1.30 * steps
Staircase cost = $1.30 * 14
Staircase cost = $18.20
Subtract this from the cost of the total cleaning bill of $253.95. We do this to isolate the cost of the carpet.
Carpet cost = $253.95 - $18.20
Carpet cost = $235.75
Now, the remaining carpet cost can be written as:
75 + $0.25(s - 180) = $235.75 <-- were s is the total square foot of carpet cleaned
Multiply through and simplify:
75 + 0.25s - 45 = $235.75
Combine like terms:
0.25s + 30 = 235.75
[URL='https://www.mathcelebrity.com/1unk.php?num=0.25s%2B30%3D235.75&pl=Solve']Type this equation into our search engine[/URL] to solve for s, and we get:
s = [B]823[/B]
A car’s purchase price is $24,000. At the end of each year, the value of the car is three-quarters oA car’s purchase price is $24,000. At the end of each year, the value of the car is three-quarters of the value at the beginning of the year. Write the first four terms of the sequence of the car’s value at the end of each year.
three-quarters means 3/4 or 0.75. So we have the following function P(y) where y is the number of years since purchase price:
P(y) = 24000 * 0.75^y
First four terms:
P(1) = 24000 * 0.75 = [B]18000[/B]
P(2) = 18000 * 0.75 = [B]13500[/B]
P(3) = 13500 * 0.75 = [B]10125[/B]
P(4) = 10125 * 0.75 = [B]7593.75[/B]
A cash register contains $5 bills and $20 bills with a total value of $180 . If there are 15 bills tA cash register contains $5 bills and $20 bills with a total value of $180 . If there are 15 bills total, then how many of each does the register contain?
Let f be the number of $5 dollar bills and t be the number of $20 bills. We're given the following equations:
[LIST=1]
[*]f + t = 15
[*]5f + 20t = 180
[/LIST]
We can solve this system of equations 3 ways. We get [B]t = 7[/B] and [B]f = 8[/B].
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+15&term2=5f+%2B+20t+%3D+180&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+15&term2=5f+%2B+20t+%3D+180&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+15&term2=5f+%2B+20t+%3D+180&pl=Cramers+Method']Cramers Method[/URL]
[/LIST]
A cashier has 44 bills, all of which are $10 or $20 bills. The total value of the money is $730. HowA cashier has 44 bills, all of which are $10 or $20 bills. The total value of the money is $730. How many of each type of bill does the cashier have?
Let a be the amount of $10 bills and b be the amount of $20 bills. We're given two equations:
[LIST=1]
[*]a + b = 44
[*]10a + 20b = 730
[/LIST]
We rearrange equation 1 in terms of a. We subtract b from each side and we get:
[LIST=1]
[*]a = 44 - b
[*]10a + 20b = 730
[/LIST]
Now we substitute equation (1) for a into equation (2):
10(44 - b) + 20b = 730
Multiply through to remove the parentheses:
440 - 10b + 20b = 730
Group like terms:
440 + 10b = 730
Now, to solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=440%2B10b%3D730&pl=Solve']type this equation into our search engine[/URL] and we get:
b = [B]29
[/B]
To get a, we take b = 29 and substitute it into equation (1) above:
a = 44 - 29
a = [B]15
[/B]
So we have [B]15 ten-dollar bills[/B] and [B]29 twenty-dollar bills[/B]
A cashier has a total of 52 bills in her cash drawer. There are only $10 bills and $5 bills in herA cashier has a total of 52 bills in her cash drawer. There are only $10 bills and $5 bills in her drawer. The value of the bills is $320. How many $10 bills are in the drawer?
Let f be the amount of $5 bills in her drawer. Let t be the amount of $10 bills in her drawer. We're given two equations:
[LIST=1]
[*]f + t = 52
[*]5f + 10t = 320
[/LIST]
We have a system of equations. We can solve this 3 ways below:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+52&term2=5f+%2B+10t+%3D+320&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+52&term2=5f+%2B+10t+%3D+320&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+52&term2=5f+%2B+10t+%3D+320&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter what method we choose, we get:
f = 40 and t = 12
So the answer for how many $10 bills are in the drawer is [B]12[/B].
Let's check our work for equation 1:
40 + 12 ? 52
52 = 52 <-- Confirmed
Let's check our work for equation 2:
5(40) + 10(12) ? 320
200 + 120 ? 320
320 = 320 <-- Confirmed
A city’s population in the 1987 was 125,524. In 2007 the population was 436,884. Determine the populA city’s population in the 1987 was 125,524. In 2007 the population was 436,884. Determine the population rate of increase or decrease
[U]Find the population change:[/U]
Population Change = New Population - Old Population
Population Change = 436,884 - 125,524
Population Change = 311,360
[U]Since the population change increased, we calculate the rate of increase:[/U]
Rate of increase = 100% * Population Change / Starting Population
Rate of increase = 100% * 311,360 / 125,524
Rate of increase = 100% * 2.48
Rate of increase = [B]248%[/B]
A coffee franchise is opening a new store. The company estimates that there is a 75% chance the stoA coffee franchise is opening a new store. The company estimates that there is a 75% chance the store will have a profit of $45,000, a 10% chance the store will break even, and a 15% chance the store will lose $2,500. Determine the expected gain or loss for this store.
Calculate the expected value E(x). Expected value is the sum of each event probability times the payoff or loss:
E(x) = 0.75(45,000) + 0.1(0) + 0.15(-2,500) <-- Note, break even means no profit and no loss and a loss is denoted with a negative sign
E(x) = 33,750 + 0 - 375
E(x) = [B]33,375 gain[/B]
A company is planning to manufacture a certain product. The fixed costs will be $474778 and it willA company is planning to manufacture a certain product. The fixed costs will be $474778 and it will cost $293 to produce each product. Each will be sold for $820. Find a linear function for the profit, P , in terms of units sold, x .
[U]Set up the cost function C(x):[/U]
C(x) = Cost per product * x + Fixed Costs
C(x) = 293x + 474778
[U]Set up the Revenue function R(x):[/U]
R(x) = Sale Price * x
R(x) = 820x
[U]Set up the Profit Function P(x):[/U]
P(x) = Revenue - Cost
P(x) = R(x) - C(x)
P(x) = 820x - (293x + 474778)
P(x) = 820x - 293x - 474778
[B]P(x) = 527x - 474778[/B]
A company that manufactures lamps has a fixed monthly cost of $1800. It costs $90 to produce each lA company that manufactures lamps has a fixed monthly cost of $1800. It costs $90 to produce each lamp, and the selling price is $150 per lamp.
Set up the Cost Equation C(l) where l is the price of each lamp:
C(l) = Variable Cost x l + Fixed Cost
C(l) = 90l + 1800
Determine the revenue function R(l)
R(l) = 150l
Determine the profit function P(l)
Profit = Revenue - Cost
P(l) = 150l - (90l + 1800)
P(l) = 150l - 90l - 1800
[B]P(l) = 60l - 1800[/B]
Determine the break even point:
Breakeven --> R(l) = C(l)
150l = 90l + 1800
[URL='https://www.mathcelebrity.com/1unk.php?num=150l%3D90l%2B1800&pl=Solve']Type this into the search engine[/URL], and we get [B]l = 30[/B]
A crate contains 300 coins and stamps. The coins cost $3 each and the stamps cost $1.5 each. The totA crate contains 300 coins and stamps. The coins cost $3 each and the stamps cost $1.5 each. The total value of the items is $825. How many coins are there?
Let c be the number of coins, and s be the number of stamps. We're given:
[LIST=1]
[*]c + s = 300
[*]3c + 1.5s = 825
[/LIST]
We have a set of simultaneous equations, or a system of equations. We can solve this 3 ways:
[LIST=1]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+s+%3D+300&term2=3c+%2B+1.5s+%3D+825&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+s+%3D+300&term2=3c+%2B+1.5s+%3D+825&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+s+%3D+300&term2=3c+%2B+1.5s+%3D+825&pl=Cramers+Method']Cramers Method[/URL]
[/LIST]
No matter which way we pick, we get:
s = 50
c = [B]250[/B]
A dad gave his 3 sons each the same amount of money in an envelope. He took $20 from one son for getA dad gave his 3 sons each the same amount of money in an envelope. He took $20 from one son for getting a D on a math test and he gave another son an extra $35 for doing extra chores. Combined, the sons had $81. Figure out how much each son had.
Let x, y, and z be the money each son received.
To begin, x = y = z.
But then, Dad took 20 from son X and gave it to son Y.
So now, x = y - 20
Next, he gave Son Z an extra $35 for chores
So z is now y + 35 since y and z used to be equal
Combined, they all have 81.
x + y + z = 181
But with the changes, it is:
(y - 20) + y + (y + 35)
Combine like terms:
3y - 20 + 35 = 81
3y + 15 = 81
Subtract 15 from each side:
3y = 66
Divide each side by 3 to isolate y
y = 22
Since x = y - 20, x = 2
Since z = y + 35, we have z = 57
Checking our work, 2 + 22 + 57 = 81.
A dormitory manager buys 38 bed sheets and 61 towels for $791.50. The manager get another 54 bed sheA dormitory manager buys 38 bed sheets and 61 towels for $791.50. The manager get another 54 bed sheets and 50 towels for $923 from the same store. What is the cost of one bed sheet and one towel?
Let s be bed sheets and t be towels. We have two equations:
[LIST=1]
[*]38s + 61t = 791.50
[*]54s + 50t = 923
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=38s+%2B+61t+%3D+791.50&term2=54s+%2B+50t+%3D+923&pl=Cramers+Method']system of equations calculator,[/URL] we get:
[LIST]
[*]s = 12
[*]t = 5.5
[/LIST]
A family buys airline tickets online. Each ticket costs $167. The family buys travel insurance withA family buys airline tickets online. Each ticket costs $167. The family buys travel insurance with each ticket that costs $19 per ticket. The Web site charges a fee of $16 for the entire purchase. The family is charged a total of $1132. How many tickets did the family buy?
Let t be the number of tickets. We have the following equation with ticket price, insurance, and flat fee:
167t + 19t + 16 = 1132
Combine like terms:
186t + 16 = 1132
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=186t%2B16%3D1132&pl=Solve']equation calculator[/URL], we have:
[B]t = 6[/B]
A family is renting an apartment. For 2007, the rent is $1376 per month. The monthly rent in 2007A family is renting an apartment. For 2007, the rent is $1376 per month. The monthly rent in 2007 is 7.5% higher than the monthly rent in 2006. Determine the monthly rent in 2006.
7.5% as a decimal is 0.075
To increase a value by 7.5%, we multiply by 1.075
[U]Calculate Rent Increase[/U]
R(2007) = R(2006) * 1.075
R(2007) = 1376 * 1.075
R(2007) = [B]1,479.20[/B]
A father is K years old and his son is M years younger. The sum of their ages is 53.A father is K years old and his son is M years younger. The sum of their ages is 53.
Father's Age = K
Son's Age = K - M
and we know K + (K - M) = 53
Combine like terms:
2K - M = 53
Add M to each side:
2K - M + M = 53 + M
Cancel the M's on the left side, we get:
2K = 53+ M
Divide each side by 2:
2K/2 = (53 + M)/2
Cancel the 2's on the left side:
K = [B](53 + M)/2[/B]
A financial advisor has invested $7000 in two accounts. If one account contains x dollars, express tA financial advisor has invested $7000 in two accounts. If one account contains x dollars, express the amount in the second account in terms of x
The other account contains:
[B]7000 - x[/B]
A first number plus twice a second number is 10. Twice the first number plus the second totals 29. FA first number plus twice a second number is 10. Twice the first number plus the second totals 29. Find the numbers.
Let the first number be x. Let the second number be y. We are given the following two equations:
[LIST=1]
[*]x + 2y = 10
[*]2x + y = 29
[/LIST]
We can solve this 3 ways using:
[LIST=1]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+10&term2=2x+%2By+%3D+29&pl=Substitution']Substitution[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+10&term2=2x+%2By+%3D+29&pl=Elimination']Elimination[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+10&term2=2x+%2By+%3D+29&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
Using any of the 3 methods, we get the same answers of [B](x, y) = (16, -3)[/B]
A first number plus twice a second number is 10. Twice the first number plus the second totals 35. FA first number plus twice a second number is 10. Twice the first number plus the second totals 35. Find the numbers.
[U]The phrase [I]a number[/I] means an arbitrary variable[/U]
A first number is written as x
A second number is written as y
[U]Twice a second number means we multiply y by 2:[/U]
2y
[U]A first number plus twice a second number:[/U]
x + 2y
[U]A first number plus twice a second number is 10 means we set x + 2y equal to 10:[/U]
x + 2y = 10
[U]Twice the first number means we multiply x by 2:[/U]
2x
[U]Twice the first number plus the second:[/U]
2x + y
[U]Twice the first number plus the second totals 35 means we set 2x + y equal to 35:[/U]
2x + y = 35
Therefore, we have a system of two equations:
[LIST=1]
[*]x + 2y = 10
[*]2x + y = 35
[/LIST]
Since we have an easy multiple of 2 for the x variable, we can solve this by multiply the first equation by -2:
[LIST=1]
[*]-2x - 4y = -20
[*]2x + y = 35
[/LIST]
Because the x variables are opposites, we can add both equations together:
(-2 + 2)x + (-4 + 1)y = -20 + 35
The x terms cancel, so we have:
-3y = 15
To solve this equation for y, we [URL='https://www.mathcelebrity.com/1unk.php?num=-3y%3D15&pl=Solve']type it in our search engine[/URL] and we get:
y = [B]-5
[/B]
Now we substitute this y = -5 into equation 2:
2x - 5 = 35
To solve this equation for x, we[URL='https://www.mathcelebrity.com/1unk.php?num=2x-5%3D35&pl=Solve'] type it in our search engine[/URL] and we get:
x = [B]20[/B]
A first number plus twice a second number is 11. Twice the first number plus the second totals 34. FA first number plus twice a second number is 11. Twice the first number plus the second totals 34. Find the numbers.
Let the first number be x and the second number be y. We're given:
[LIST=1]
[*]x + 2y = 11
[*]2x + y = 34
[/LIST]
Using our simultaneous equations calculator, we have 3 methods to solve this:
[LIST=1]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+11&term2=2x+%2B+y+%3D+34&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+11&term2=2x+%2B+y+%3D+34&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+11&term2=2x+%2B+y+%3D+34&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
All 3 methods give the same solution:
[LIST]
[*][B]x = 19[/B]
[*][B]y = -4[/B]
[/LIST]
A first number plus twice a second number is 22. Twice the first number plus the second totals 28. FA first number plus twice a second number is 22. Twice the first number plus the second totals 28. Find the numbers.
Let the first number be x. Let the second number be y. We're given two equations:
[LIST=1]
[*]x + 2y = 22 <-- Since twice means multiply by 2
[*]2x + y = 28 <-- Since twice means multiply by 2
[/LIST]
We have a set of simultaneous equations. We can solve this three ways
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+22&term2=2x+%2B+y+%3D+28+&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+22&term2=2x+%2B+y+%3D+28&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+22&term2=2x+%2B+y+%3D+28&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter which method we use, we get the same answer:
[LIST]
[*][B]x = 11 & 1/3[/B]
[*][B]y = 5 & 1/3[/B]
[/LIST]
A first number plus twice a second number is 3. Twice the first number plus the second totals 24.A first number plus twice a second number is 3. Twice the first number plus the second totals 24.
Let the first number be x. Let the second number be y. We're given:
[LIST=1]
[*]x + 2y = 3 <-- Because [I]twice[/I] means multiply by 2
[*]2x + y = 24 <-- Because [I]twice[/I] means multiply by 2
[/LIST]
We have a system of equations. We can solve it any one of three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+3&term2=2x+%2B+y+%3D+24&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+3&term2=2x+%2B+y+%3D+24&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+3&term2=2x+%2B+y+%3D+24&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which way we choose, we get:
[LIST]
[*]x = [B]15[/B]
[*]y = [B]-6[/B]
[/LIST]
A first number plus twice a second number is 7. Twice the first number plus the second totals 23. FiA first number plus twice a second number is 7. Twice the first number plus the second totals 23. Find the numbers
Let the first number be a and the second number be b. We have:
[LIST=1]
[*]a + 2b = 7
[*]2a + b = 23
[/LIST]
Rearrange (1) into (3)
(3) a = 7 - 2b
Substitute (3) into (2):
2(7 - 2b) + b = 23
Multiply through:
14 - 4b + b = 23
Combine like terms:
14 - 3b = 23
Subtract 14 from each side:
-3b = 9
Divide each side by -3
[B]b = -3[/B]
Substitute this into (3)
a = 7 - 2b
a = 7 - 2(-3)
a = 7 + 6
[B]a = 13[/B]
[B](a, b) = (13, -3)[/B]
A food truck sells salads for $6.50 each and drinks for $2.00 each. The food trucks revenue from selA food truck sells salads for $6.50 each and drinks for $2.00 each. The food trucks revenue from selling a total of 209 salads and drinks in one day was $836.50. How many salads were sold that day?
Let the number of drinks be d. Let the number of salads be s. We're given two equations:
[LIST=1]
[*]2d + 6.50s = 836.50
[*]d + s = 209
[/LIST]
We can use substitution to solve this system of equations quickly. The question asks for the number of salads (s). Therefore, we want all expressions in terms of s. Rearrange Equation 2 by subtracting s from both sides:
d + s - s = 209 - s
Cancel the s's, we get:
d = 209 - s
So we have the following system of equations:
[LIST=1]
[*]2d + 6.50s = 836.50
[*]d = 209 - s
[/LIST]
Substitute equation (2) into equation (1) for d:
2(209 - s) + 6.50s = 836.50
Multiply through to remove the parentheses:
418 - 2s + 6.50s = 836.50
To solve this equation for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=418-2s%2B6.50s%3D836.50&pl=Solve']type it into our search engine and we get[/URL]:
s = [B]93[/B]
a football team won 3 more games than it lost.the team played 11 games.how many did it win?a football team won 3 more games than it lost.the team played 11 games.how many did it win?
Let wins be w. Let losses be l. We're given two equations:
[LIST=1]
[*]w = l + 3
[*]l + w = 11
[/LIST]
Plug equation (1) into equation (2) to solve for l:
l + (l + 3) = 11
Group like terms:
2l + 3 = 11
[URL='https://www.mathcelebrity.com/1unk.php?num=2l%2B3%3D11&pl=Solve']Typing this equation into our search engine[/URL], we get:
l = 4
To solve for w, we plug in l = 4 above into equation (1):
w = 4 + 3
w = [B]7[/B]
A gallon of milk sells for $3.09 and a quart of milk sells for $0.80.A gallon of milk sells for $3.09 and a quart of milk sells for $0.80.
Let's get this in terms of a common measurement for unit cost. Let's use gallons:
1 gallon = 4 quarts
Gallon of milk
$3.09 / gallon
Quart of milk
0.80 / 0.25 gallons
Multiply top and bottom by 4:
$3.20 / gallon
[B]This means the gallon is the better buy[/B]
A garden table and a bench cost $977 combined. The gardenA garden table and a bench cost $977 combined. The garden table costs $77 more than the bench. What is the cost of the bench?
Let the garden table cost be g and the bench cost be b. We're given
[LIST=1]
[*]b + g = 977
[*]g = b + 77 <-- The phrase [I]more than[/I] means we add
[/LIST]
Substitute (2) into (1):
b + (b + 77) = 977
Combine like terms:
2b + 77 = 977
[URL='https://www.mathcelebrity.com/1unk.php?num=2b%2B77%3D977&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]b = $450[/B]
A group of p people sold a car for $5000. Write an expression in terms of p for how much money eachA group of p people sold a car for $5000. Write an expression in terms of p for how much money each person gets.
Each person gets:
[B]5000/p[/B]
A group of people was surveyed to determine what newspaper they read. 80% of those interviewed readA group of people was surveyed to determine what newspaper they read. 80% of those interviewed read the New York Times, while 50% read U.S.A. Today. If 35% read both papers, what percent read neither paper?
New York Times:
80% - 35% for both = 45%
USA Today:
50% - 35% for both = 15%
45% + 15% + 35% = 95%
Which means 100% - 95% = [B]5% read neither[/B]
A helicopter is flying at an altitude of 785 feet. It descends 570 feet, and then ascends 595 feet.A helicopter is flying at an altitude of 785 feet. It descends 570 feet, and then ascends 595 feet. Write an expression to represent this situation. Then determine and interpret the sum.
[LIST]
[*]Start at +785 feet
[*]Descend 570 feet means using a minus sign -570
[*]Ascend 595 feet means using a plus sign +595
[/LIST]
[U]Calculate the sum:[/U]
+785 - 570 + 595
[B]+810[/B]
A high school graduating class is made up of 440 students. There are 168 more girls than boys. How mA high school graduating class is made up of 440 students. There are 168 more girls than boys. How many boys are in the class?
Let b be the number of boys and g be the number of girls. We're given 2 equations:
[LIST=1]
[*]b + g = 440
[*]g = b + 168
[/LIST]
Substitute (2) into (1)
b + (b + 168) = 440
Combine like terms:
2b + 168 = 440
[URL='https://www.mathcelebrity.com/1unk.php?num=2b%2B168%3D440&pl=Solve']Type this equation into the search engine[/URL], and we get:
[B]b = 136[/B]
A home is to be built on a rectangular plot of land with a perimeter of 800 feet. If the length is 2A home is to be built on a rectangular plot of land with a perimeter of 800 feet. If the length is 20 feet less than 3 times the width, what are the dimensions of the rectangular plot?
[U]Set up equations:[/U]
(1) 2l + 2w = 800
(2) l = 3w - 20
[U]Substitute (2) into (1)[/U]
2(3w - 20) + 2w = 800
6w - 40 + 2w = 800
[U]Group the w terms[/U]
8w - 40 = 800
[U]Add 40 to each side[/U]
8w = 840
[U]Divide each side by 8[/U]
[B]w = 105
[/B]
[U]Substitute w = 105 into (2)[/U]
l = 3(105) - 20
l = 315 - 20
[B]l = 295[/B]
A house costs 3.5 times as much as the lot. Together they sold for $135,000. Find the cost of eachA house costs 3.5 times as much as the lot. Together they sold for $135,000. Find the cost of each.
Let the house cost be h, and the lot cost be l. We have the following equations:
[LIST=1]
[*]h = 3.5l
[*]h + l = 135,000
[/LIST]
Substitute (1) into (2)
3.5l + l = 135,000
Combine like terms:
4.5l = 135,000
Divide each side by 4.5 to isolate l
[B]l = 30,000[/B]
Substitute this back into equation (1)
h = 3.5(30,000)
[B]h = 105,000[/B]
A house valued at 70,000 in 1989 increased in value to 125,000 in 2000. Find a function which givesA house valued at 70,000 in 1989 increased in value to 125,000 in 2000. Find a function which gives the value of the house, v, as a function of y, the number of years after 1989.
Let's determine the years:
2000 - 1989 = 11
Let's determine the change in value:
125,000 - 70,000 = 55,000
Assuming a linear progression, we have:
55,000/11 = 5,000 per year increase
[B]y = 70,000 + 5,000v[/B] where v is the number of years after 1989
Plug in 11 to check our work
y = 70,000 + 5,000(11)
y = 70,000 + 55,000
y = 125,000
A limo costs $85 to rent for 3 hours plus a 7% sales tax. What is the total cost to rent the limo foA limo costs $85 to rent for 3 hours plus a 7% sales tax. What is the total cost to rent the limo for 6 hours?
Determine the number of 3 hour blocks:
3 hour blocks = Total Rental Time / 3
3 hour blocks = 6 hours / 3
3 hour blocks = 2
With 7% = 0.07, we have:
Total Cost = $85 * / 3 hours * 2 (3 hour blocks) * 1.07
Total Cost = 85 * 2 * 1.07
Total Cost = [B]181.9[/B]
A local shop sold 499 hamburgers and cheese burgers. There were 51 fewer cheese burgers sold. How maA local shop sold 499 hamburgers and cheese burgers. There were 51 fewer cheese burgers sold. How many hamburgers were sold?
Let h = number of hamburgers sold and c be the number of cheeseburgers sold.
We have two equations:
(1) c = h - 51
(2) c + h = 499
Substitute (1) into (2)
h - 51 + h = 499
Combine like terms
2h - 51 = 499
Add 51 to both sides
2h = 550
Divide each side by 2 to isolate h
[B]h = 275[/B]
A man is 5 years older than his wife, and the daughter age is half of the mother, and if you add theA man is 5 years older than his wife, and the daughter age is half of the mother, and if you add their ages is equal 100
Let the man's age be m. Let the wife's age be w. Let the daughter's age be d. We're given:
[LIST=1]
[*]m = w + 5
[*]d = 0.5m
[*]d + m + w = 100
[/LIST]
Rearrange equation 1 in terms of w my subtracting 5 from each side:
[LIST=1]
[*]w = m - 5
[*]d = 0.5m
[*]d + m + w = 100
[/LIST]
Substitute equation (1) and equation (2) into equation (3)
0.5m + m + m - 5 = 100
We [URL='https://www.mathcelebrity.com/1unk.php?num=0.5m%2Bm%2Bm-5%3D100&pl=Solve']type this equation into our search engine[/URL] to solve for m and we get:
m = [B]42
[/B]
Now, substitute m = 42 into equation 2 to solve for d:
d = 0.5(42)
d = [B]21
[/B]
Now substitute m = 42 into equation 1 to solve for w:
w = 42 - 5
w = [B]37
[/B]
To summarize our ages:
[LIST]
[*]Man (m) = 42 years old
[*]Daughter (d) = 21 years old
[*]Wife (w) = 37 years old
[/LIST]
A man purchased 20 tickets for a total of $225. The tickets cost $15 for adults and $10 for childrenA man purchased 20 tickets for a total of $225. The tickets cost $15 for adults and $10 for children. What was the cost of each ticket?
Declare variables:
[LIST]
[*]Let a be the number of adult's tickets
[*]Let c be the number of children's tickets
[/LIST]
Cost = Price * Quantity
We're given two equations:
[LIST=1]
[*]a + c = 20
[*]15a + 10c = 225
[/LIST]
Rearrange equation (1) in terms of a:
[LIST=1]
[*]a = 20 - c
[*]15a + 10c = 225
[/LIST]
Now that I have equation (1) in terms of a, we can substitute into equation (2) for a:
15(20 - c) + 10c = 225
Solve for [I]c[/I] in the equation 15(20 - c) + 10c = 225
We first need to simplify the expression removing parentheses
Simplify 15(20 - c): Distribute the 15 to each term in (20-c)
15 * 20 = (15 * 20) = 300
15 * -c = (15 * -1)c = -15c
Our Total expanded term is 300-15c
Our updated term to work with is 300 - 15c + 10c = 225
We first need to simplify the expression removing parentheses
Our updated term to work with is 300 - 15c + 10c = 225
[SIZE=5][B]Step 1: Group the c terms on the left hand side:[/B][/SIZE]
(-15 + 10)c = -5c
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
-5c + 300 = + 225
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 300 and 225. To do that, we subtract 300 from both sides
-5c + 300 - 300 = 225 - 300
[SIZE=5][B]Step 4: Cancel 300 on the left side:[/B][/SIZE]
-5c = -75
[SIZE=5][B]Step 5: Divide each side of the equation by -5[/B][/SIZE]
-5c/-5 = -75/-5
c = [B]15[/B]
Recall from equation (1) that a = 20 - c. So we substitute c = 15 into this equation to solve for a:
a = 20 - 15
a = [B]5[/B]
A math test is worth 100 points and has 38 problems. Each problem is worth either 5 points or 2 poinA math test is worth 100 points and has 38 problems. Each problem is worth either 5 points or 2 points. How many problems of each point value are on the test?
Let's call the 5 point questions m for multiple choice. Let's call the 2 point questions t for true-false. We have two equations:
[LIST=1]
[*]m + t = 38
[*]5m + 2t = 100
[/LIST]
Rearrange (1) to solve for m - subtract t from each side:
3. m = 38 - t
Now, substitute (3) into (2)
5(38 - t) + 2t = 100
190 - 5t + 2t = 100
Combine like terms:
190 - 3t = 100
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=190-3t%3D100&pl=Solve']equation solver[/URL], we get [B]t = 30[/B].
Plugging t = 30 into (1), we get:
30 + t = 38
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=m%2B30%3D38&pl=Solve']equation solver[/URL] again, we get [B]m = 8[/B].
Check our work for (1)
8 + 30 = 38 <-- Check
Check our work for (2)
5(8) + 2(30) ? 100
40 + 60 ? 100
100 = 100 <-- Check
You could also use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=m+%2B+t+%3D+38&term2=5m+%2B+2t+%3D+100&pl=Cramers+Method']simultaneous equations calculator[/URL]
A movie theater charges $7 for adults and $3 for seniors on a particular day when 324 people paid anA movie theater charges $7 for adults and $3 for seniors on a particular day when 324 people paid an admission the total receipts were 1228 how many were seniors and how many were adults?
Let the number of adult tickets be a. Let the number of senior tickets be s. We're given two equations:
[LIST=1]
[*]a + s = 324
[*]7a + 3s = 1228
[/LIST]
We have a set of simultaneous equations we can solve using 3 methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+s+%3D+324&term2=7a+%2B+3s+%3D+1228&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+s+%3D+324&term2=7a+%2B+3s+%3D+1228&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+s+%3D+324&term2=7a+%2B+3s+%3D+1228&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we choose, we get:
[LIST]
[*][B]a = 64[/B]
[*][B]s = 260[/B]
[/LIST]
A movie theater charges 7.00 for adults and 2.00 for seniors citizens. On a day when 304 people paidA movie theater charges 7.00 for adults and 2.00 for seniors citizens. On a day when 304 people paid for admission, the total receipt were 1118. How many who paid were adults ? How many were senior citizens?
Let a be the number of adult tickets. Let s be the number of senior citizen tickets. We're given two equations:
[LIST=1]
[*]a + s = 304
[*]7a + 2s = 1118
[/LIST]
We can solve this system of equations three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+s+%3D+304&term2=7a+%2B+2s+%3D+1118&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+s+%3D+304&term2=7a+%2B+2s+%3D+1118&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+s+%3D+304&term2=7a+%2B+2s+%3D+1118&pl=Cramers+Method']Cramer's Method[/URL]
[/LIST]
No matter which way we choose, we end up with the same answer:
[LIST]
[*]a = [B]102[/B]
[*]s = [B]202[/B]
[/LIST]
A movie theater has a seating capacity of 143. The theater charges $5.00 for children, $7.00 for stuA movie theater has a seating capacity of 143. The theater charges $5.00 for children, $7.00 for students, and $12.00 of adults. There are half as many adults as there are children. If the total ticket sales was $ 1030, How many children, students, and adults attended?
Let c be the number of children's tickets, s be the number of student's tickets, and a be the number of adult's tickets. We have 3 equations:
[LIST=1]
[*]a + c + s = 143
[*]a = 0.5c
[*]12a + 5c + 7s =1030
[/LIST]
Substitute (2) into (1)
0.5c + c + s = 143
1.5c + s = 143
Subtract 1.5c from each side
4. s = 143 - 1.5c
Now, take (4) and (2), and plug it into (3)
12(0.5c) + 5c + 7(143 - 1.5c) = 1030
6c + 5c + 1001 - 10.5c = 1030
Combine like terms:
0.5c + 1001 = 1030
Use our [URL='http://www.mathcelebrity.com/1unk.php?num=0.5c%2B1001%3D1030&pl=Solve']equation calculator[/URL] to get [B]c = 58[/B].
Plug this back into (2)
a = 0.5(58)
[B]a = 29
[/B]
Now take the a and c values, and plug it into (1)
29 + 58 + s = 143
s + 87 = 143
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=s%2B87%3D143&pl=Solve']equation calculator[/URL] again, we get [B]s = 56[/B].
To summarize, we have:
[LIST]
[*]29 adults
[*]58 children
[*]56 students
[/LIST]
A number n is no less than 2 and no more than 49.A number n is no less than 2 and no more than 49.
This is a compound inequality. Let's break it into parts.
Step 1: No more than 49 means 49 or less. Or, less than or equal to 49
<= 49
Step 2: no less than 2 means 2 or greater. Or, greater than or equal to 2
>=2
Writing this in terms of the number n, we have:
[B]2 <= n <= 49[/B]
A number y increased by itselfA number y increased by itself
increased by itself means we add the variable y to itself to get our final algebraic expression of:
[B]y + y
[/B]
[I]If[/I] the problem asks you to simplify, we group like terms and get:
[B]2y[/B]
A party rental company has chairs and tables for rent. The total cost to rent 5 chairs and 3 tablesA party rental company has chairs and tables for rent. The total cost to rent 5 chairs and 3 tables is $37. The total cost to rent 2 chairs and 6 tables is $64. What is the cost to rent each chair and each table?
Let c be the cost of renting one chair and t be the cost of renting one table. We're given two equations:
[LIST=1]
[*]5c + 3t = 37
[*]2c + 6t =64
[/LIST]
We have a system of equations. Using our system of equations calculator, we can solve this problem any of 3 ways below:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=5c+%2B+3t+%3D+37&term2=2c+%2B+6t+%3D+64&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=5c+%2B+3t+%3D+37&term2=2c+%2B+6t+%3D+64&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=5c+%2B+3t+%3D+37&term2=2c+%2B+6t+%3D+64&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
All 3 methods give the same answer:
[LIST]
[*][B]Chairs (c) cost $1.25[/B]
[*][B]Tables (t) cost $10.25[/B]
[/LIST]
A pawn broker buys a tv and a computer for $600. He sells the computer at a markup of 30% and the tvA pawn broker buys a tv and a computer for $600. He sells the computer at a markup of 30% and the tv at a markup of 20%. If he makes a profit of $165 on the sale of the two items, what did he pay for the computer?
Let c be the price of the computer and t be the price of the tv. WE have:
[LIST=1]
[*]c + t = 600
[*]c(1.3) + t(1.2) = 765 <-- (600 + 165 profit)
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+t+%3D+600&term2=1.3c+%2B+1.2t+%3D+765&pl=Cramers+Method']simultaneous equation calculator[/URL], we get:
[B]c = 450[/B]
t = 150
A person has $13,000 invested in stock A and stock B. Stock A currently sells for $20 a share andA person has $13,000 invested in stock A and stock B. Stock A currently sells for $20 a share and stock B sells for $90 a share. If stock B triples in value and stock A goes up 50%, his stock will be worth $33,000. How many shares of each stock does he own?
Set up the given equations, where A is the number of shares for Stock A, and B is the number of shares for Stock B
[LIST=1]
[*]90A + 20B = 13000
[*]3(90A) + 1.5(20B) = 33000 <-- [I]Triple means multiply by 3, and 50% gain means multiply by 1.5[/I]
[/LIST]
Rewrite (2) by multiplying through:
270A + 30B = 33000
Using our simultaneous equations calculator, we get [B]A = 100 and B = 200[/B]. Click the links below to solve using each method:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=90A+%2B+20B+%3D+13000&term2=270A+%2B+30B+%3D+33000&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=90A+%2B+20B+%3D+13000&term2=270A+%2B+30B+%3D+33000&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=90A+%2B+20B+%3D+13000&term2=270A+%2B+30B+%3D+33000&pl=Cramers+Method']Cramers Method[/URL]
[/LIST]
Check our work using equation (1)
90(100) + 20(200) ? 13,000
9000 + 4000 ? 13,000
13000 = 13000
A person invested 30,000 in stocks and bonds. Her investment in bonds is 2000 more than 1-third herA person invested 30,000 in stocks and bonds. Her investment in bonds is 2000 more than 1-third her investments in stocks. How much did she invest in stocks? How much did she invest in bonds?
Let the stock investment be s, and the bond investment be b. We're given:
[LIST=1]
[*]b + s = 30000
[*]b = 1/3s + 2000
[/LIST]
Plug in (2) to (1):
1/3s + 2000 + s = 30000
Group like terms:
(1/3 + 1)s + 2000 = 30000
Since 1 = 3/3, we have:
4/3s + 2000 = 30000
Subtract 2000 from each side:
4/3s + 2000 - 2000 = 30000 - 2000
Cancel the 2000's on the left side, we get:
4/3s = 28000
[URL='https://www.mathcelebrity.com/1unk.php?num=4%2F3s%3D28000&pl=Solve']Typing this equation into our calculator[/URL], we get:
s = [B]21,000[/B]
A person places $230 in an investment account earning an annual rate of 6.8%, compounded continuouslA person places $230 in an investment account earning an annual rate of 6.8%, compounded continuously. Using the formula V = Pe^{rt}V=Pe^rt, where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 20 years
Using our [URL='http://www.mathcelebrity.com/simpint.php?av=&p=230&int=6.8&t=20&pl=Continuous+Interest']continuous compounding calculator[/URL], we get:
V = [B]896.12[/B]
A person places $96300 in an investment account earning an annual rate of 2.8%, compounded continuouA person places $96300 in an investment account earning an annual rate of 2.8%, compounded continuously. Using the formula V=PertV = Pe^{rt} V=Pe rt , where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 7 years.
Substituting our given numbers in where P = 96,300, r = 0.028, and t = 7, we get:
V = 96,300 * e^(0.028 * 7)
V = 96,300 * e^0.196
V = 96,300 * 1.21652690533
V = [B]$117,151.54[/B]
A person will devote 31 years to be sleeping and watching tv. The number of years sleeping will exceA person will devote 31 years to be sleeping and watching tv. The number of years sleeping will exceed the number of years watching tv by 19. How many years will the person spend on each of these activities
Let s be sleeping years and t be tv years, we have two equations:
[LIST=1]
[*]s + t = 31
[*]s = t + 19
[/LIST]
Substitute (2) into (1)
(t + 19) + t = 31
Combine like terms:
2t + 19 = 31
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2t%2B19%3D31&pl=Solve']equation solver[/URL], we get [B]t = 6[/B]. Using equation (2), we have
s = 6 + 19
s = [B]25[/B]
A Petri dish contains 2000. The number of bacteria triples every 6 hours. How many bacteria will exiA Petri dish contains 2000. The number of bacteria triples every 6 hours. How many bacteria will exist after 3 days?
Determine the amount of tripling periods:
[LIST]
[*]There are 24 hours in a day.
[*]24 hours in a day * 3 days = 72 hours
[*]72 hours / 6 hours tripling period = 12 tripling periods
[/LIST]
Our bacteria population function B(t) where t is the amount of tripling periods. Tripling means we multiply by 3, so we have:
B(t) = 2000 * 3^t
with t = 12 tripling periods, we have:
B(12) = 2000 * 3^12
B(12) = 2000 * 531441
B(12) = [B]1,062,882,000[/B]
A pile of coins, consisting of quarters and half dollars, is worth 11.75. If there are 2 more quarteA pile of coins, consisting of quarters and half dollars, is worth 11.75. If there are 2 more quarters than half dollars, how many of each are there?
Let h be the number of half-dollars and q be the number of quarters. Set up two equations:
(1) q = h + 2
(2) 0.25q + 0.5h = 11.75
[U]Substitute (1) into (2)[/U]
0.25(h + 2) + 0.5h = 11.75
0.25h + 0.5 + 0.5h = 11.75
[U]Group h terms[/U]
0.75h + 0.5 = 11.75
[U]Subtract 0.5 from each side[/U]
0.75h = 11.25
[U]Divide each side by h[/U]
[B]h = 15[/B]
[U]Substitute h = 15 into (1)[/U]
q = 15 + 2
[B]q = 17[/B]
A plumber charges $45 for a house call plus $25 for each hour worked.Let h represent the number of hA plumber charges $45 for a house call plus $25 for each hour worked.Let h represent the number of hours worked. Write the expression that shows how much a plumber charges for a job. Then find how much the plumbers charges for a job lasting 4 hours
[U]Set up the cost function C(h) where h is the number of hours:[/U]
C(h) = Hours worked * hourly rate + house call fee
[B]C(h) = 25h + 45 <-- This is the expression for how much the plumber charges for a job
[/B]
[U]Now determine how much the plumber charges for a job lasting 4 hours[/U]
We want C(4)
C(4) = 25(4) + 45
C(4) = 100 + 45
C(4) = [B]$145[/B]
A plumber charges $50 to visit a house plus $40 for every hour of work.A plumber charges $50 to visit a house plus $40 for every hour of work.
Set up the cost function in terms of hours (h) using the flat fee of $50
[B]C(h) = 40h + 50[/B]
A problem states: "There are 9 more children than parents in a room. There are 25 people in the roomA problem states: "There are 9 more children than parents in a room. There are 25 people in the room in all. How many children are there in the room?"
Let the number of children be c. Let the number of parents be p
We're given:
[LIST=1]
[*]c = p + 9 [I](9 more children than parents)[/I]
[*]c + p = 25
[/LIST]
to solve this system of equations, we plug equation (1) into equation (2) for c:
(p + 9) + p = 25
Group like terms:
2p + 9 = 25
To solve this equation for p, we [URL='https://www.mathcelebrity.com/1unk.php?num=2p%2B9%3D25&pl=Solve']type it in our search engine[/URL] and we get:
p = [B]8[/B]
A random variable X follows the uniform distribution with a lower limit of 670 and an upper limitA random variable X follows the uniform distribution with a lower limit of 670 and an upper limit
a. Calculate the mean and standard deviation of this distribution. (Round intermediate calculation for standard deviation to 4 decimal places and final answer to 2 decimal places.)
Using our [URL='http://www.mathcelebrity.com/uniform.php?a=+670&b=+770&x=+680&t=+3&pl=PDF']uniform distribution calculator[/URL], we get:
[B]Mean = 720
Standard deviation = 28.87
[/B]
b. What is the probability that X is less than 730? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
Using our [URL='http://www.mathcelebrity.com/uniform.php?a=+670&b=+770&x=+730&t=+3&pl=CDF']uniform distribution calculator[/URL], we get:
[B]0.6[/B]
a rectangle has a length of x-7 and a width of x + 5. Write an expression that represents the area oa rectangle has a length of x-7 and a width of x + 5. Write an expression that represents the area of the rectangle in terms of x.
Area of a rectangle (A) with length(l) and width (w) is expressed as follows:
A = lw
Plugging in our values given above, we have:
[B]A = (x - 7)(x + 5)[/B]
A rectangular field is to be enclosed with 1120 feet of fencing. If the length of the field is 40 feA rectangular field is to be enclosed with 1120 feet of fencing. If the length of the field is 40 feet longer than the width, then how wide is the field?
We're given:
[LIST=1]
[*]l = w + 40
[/LIST]
And we know the perimeter of a rectangle is:
P = 2l + 2w
Substitute (1) into this formula as well as the given perimeter of 1120:
2(w + 40) + 2w = 1120
Multiply through and simplify:
2w + 80 + 2w = 1120
Group like terms:
4w + 80 = 1120
[URL='https://www.mathcelebrity.com/1unk.php?num=4w%2B80%3D1120&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]w = 260[/B]
A rectangular parking lot has a perimeter of 152 yards. If the length of the parking lot is 12 yardsA rectangular parking lot has a perimeter of 152 yards. If the length of the parking lot is 12 yards greater than the width. What is the width of the parking lot?
The perimeter of a rectangle is: 2l + 2w = P.
We're given 2 equations:
[LIST=1]
[*]2l + 2w = 152
[*]l = w + 12
[/LIST]
Substitute equation (2) into equation (1) for l:
2(w + 12) + 2w = 152
2w + 24 + 2w = 152
Combine like terms:
4w + 24 = 152
To solve for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=4w%2B24%3D152&pl=Solve']type this equation into our search engine[/URL] and we get:
w =[B] 32[/B]
A rental truck costs $49.95+$0.59 per mile and another costs $39.95 plus $0.99, set up an equation tA rental truck costs $49.95+$0.59 per mile and another costs $39.95 plus $0.99, set up an equation to determine the break even point?
Set up the cost functions for Rental Truck 1 (R1) and Rental Truck 2 (R2) where m is the number of miles
R1(m) = 0.59m + 49.95
R2(m) = 0.99m + 39.95
Break even is when we set the cost functions equal to one another:
0.59m + 49.95 = 0.99m + 39.95
[URL='https://www.mathcelebrity.com/1unk.php?num=0.59m%2B49.95%3D0.99m%2B39.95&pl=Solve']Typing this equation into the search engine[/URL], we get [B]m = 25[/B].
A roof drops 4 feet for every 12 feet forward. Determine the slope of the roof.A roof drops 4 feet for every 12 feet forward. Determine the slope of the roof.
Slope = Rise or Drop / Run
Slope = 4/12
We can simplify this fraction. We [URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F12&frac2=3%2F8&pl=Simplify']type 4/12 into our search engine[/URL] and get:
Slope. = [B]1/3[/B]
A set of 4 consecutive integers adds up to 314. What is the least of the 4 integers?A set of 4 consecutive integers adds up to 314. What is the least of the 4 integers?
First integer is x. The next 3 are x + 1, x + 2, and x + 3. Set up our equation:
x + (x + 1) + (x + 2) + (x + 3) = 314
Group x terms and group constnats
(x + x + x + x) + (1 + 2 + 3) = 314
Simplify and combine
4x + 6 = 314
[URL='http://www.mathcelebrity.com/1unk.php?num=4x%2B6%3D314&pl=Solve']Enter this in the equation solver[/URL]
[B]x = 77[/B]
A soccer team has picked its five best players to take part in penalty kicks to determine the winnerA soccer team has picked its five best players to take part in penalty kicks to determine the winner of a soccer match that is tied. Each of the five players will get one shot against the opposing team's goalie. The coach needs to decide the order in which the five players will take their shots. How many possible ways are there to arrange the five players?
First shot, 5 players can take the shot. Next shot is 4, then 3, then 2, then 1
5! = 5 x 4 x 3 x 2 x 1 = [B]120 ways[/B]
A store sells small notebooks for $6 and large notebooks for $12. If a student buys 6 notebooks andA store sells small notebooks for $6 and large notebooks for $12. If a student buys 6 notebooks and spends $60, how many of each did he buy?
Let the amount of small notebooks be s. Let the amount of large notebooks be l. We're given two equations:
[LIST=1]
[*]l + s = 6
[*]12l + 6s = 60
[/LIST]
Multiply equation (1) by -6
[LIST=1]
[*]-6l - 6s = -36
[*]12l + 6s = 60
[/LIST]
Now add equation 1 to equation 2:
12l - 6l + 6s - 6s = 60 - 36
Cancel the 6s terms, and we get:
6l = 24
To solve for l, we [URL='https://www.mathcelebrity.com/1unk.php?num=6l%3D24&pl=Solve']type this equation into our search engine[/URL] and we get:
l = [B]4
[/B]
Now substitute this into equation 1:
4 + s = 6
To solve for s, [URL='https://www.mathcelebrity.com/1unk.php?num=4%2Bs%3D6&pl=Solve']we type this equation into our search engine[/URL] and we get:
s = [B]2[/B]
A straight line has the equation ax + by=23. The points (5,-2) and (1,-5) lie on the line. Find theA straight line has the equation ax + by=23. The points (5,-2) and (1,-5) lie on the line. Find the values of a and b.
plug in both points and form 2 equations:
[LIST=1]
[*]5a - 2b = 23
[*]1x - 5b = 23
[/LIST]
We can solve this simultaneous equations any one of three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=5a+-+2b+%3D+23&term2=1a+-+5b+%3D+23&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=5a+-+2b+%3D+23&term2=1a+-+5b+%3D+23&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=5a+-+2b+%3D+23&term2=1a+-+5b+%3D+23&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
[LIST]
[*][B]a = 3[/B]
[*][B]b = -4[/B]
[/LIST]
a string measures 20 inches is cut into pieces. Let z represent the length of one of the resulting pa string measures 20 inches is cut into pieces. Let z represent the length of one of the resulting pieces. express length of the second piece in terms of the length z of the first pice
Second piece length = [B]20 - z[/B]
A student was trying to determine a formula for changing speeds that are written in feet per secondA student was trying to determine a formula for changing speeds that are written in feet per second into miles per hour. If a sprinter runs at a speed of n feet per second, what is her speed in miles per hour?
3600 seconds per hour = 3600n feet per hour
5280 feet per mile so we have:
3600n feet per hour / 5280 feet per mile = [B]0.6818n feet per second[/B]
A suitcase contains nickels, dimes and quarters. There are 2&1/2 times as many dimes as nickels andA suitcase contains nickels, dimes and quarters. There are 2&1/2 times as many dimes as nickels and 5 times the number of quarters as the number of nickels. If the coins have a value of $24.80, how many nickels are there in the suitcase?
Setup number of coins:
[LIST]
[*]Number of nickels = n
[*]Number of dimes = 2.5n
[*]Number of quarters = 5n
[/LIST]
Setup value of coins:
[LIST]
[*]Value of nickels = 0.05n
[*]Value of dimes = 2.5 * 0.1n = 0.25n
[*]Value of quarters = 5 * 0.25n = 1.25n
[/LIST]
Add them up:
0.05n + 0.25n + 1.25n = 24.80
Solve for [I]n[/I] in the equation 0.05n + 0.25n + 1.25n = 24.80
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(0.05 + 0.25 + 1.25)n = 1.55n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
1.55n = + 24.8
[SIZE=5][B]Step 3: Divide each side of the equation by 1.55[/B][/SIZE]
1.55n/1.55 = 24.80/1.55
n = [B]16[/B]
[B]
[URL='https://www.mathcelebrity.com/1unk.php?num=0.05n%2B0.25n%2B1.25n%3D24.80&pl=Solve']Source[/URL][/B]
A survey of 950 college students found that 85% of the men and 90% of the women identified math as tA survey of 950 college students found that 85% of the men and 90% of the women identified math as their favorite subject. If altogether 834 students reported math to be their favorite subject how many men and women participated in the survey
Let m be the number of men and w be the number of women. We are given 2 equations
[LIST=1]
[*]m + w = 950
[*]0.85m + 0.90w = 834
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=m+%2B+w+%3D+950&term2=0.85m+%2B+0.90w+%3D+834&pl=Cramers+Method']simultaneous equations calculator[/URL], we get:
[LIST]
[*]m = [B]420[/B]
[*]w = [B]530[/B]
[/LIST]
A test has twenty questions worth 100 points . The test consist of true/false questions worth 3 poinA test has twenty questions worth 100 points . The test consist of true/false questions worth 3 points each and multiple choice questions worth 11 points each . How many multiple choice questions are on the test?
Set up equations where t = true false and m = multiple choice:
[LIST=1]
[*]t + m = 20
[*]3t + 11m = 100
[/LIST]
Use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=t+%2B+m+%3D+20&term2=3t+%2B+11m+%3D+100&pl=Cramers+Method']simultaneous equation calculator[/URL]:
[B]t = 15, m = 5[/B]
A test has twenty questions worth 100 points total. the test consists of true/false questions worthA test has twenty questions worth 100 points total. the test consists of true/false questions worth 3 points each and multiple choice questions worth 11 points each. How many true/false questions are on the test?
Let m be the number of multiple choice questions and t be the number of true/false questions. We're given:
[LIST=1]
[*]m + t = 20
[*]11m + 3t = 100
[/LIST]
We can solve this system of equations 3 ways below:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=m+%2B+t+%3D+20&term2=11m+%2B+3t+%3D+100&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=m+%2B+t+%3D+20&term2=11m+%2B+3t+%3D+100&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=m+%2B+t+%3D+20&term2=11m+%2B+3t+%3D+100&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the following answers:
[LIST]
[*][B]m = 5[/B]
[*][B]t = 15[/B]
[/LIST]
Check our work in equation 1:
5 + 15 ? 20
[I]20 = 20[/I]
Check our work in equation 2:
11(5) + 3(15) ? 100
55 + 45 ? 100
[I]100 = 100[/I]
A test has twenty questions worth 100 points. The test consists of True/False questions worth 3 poinA test has twenty questions worth 100 points. The test consists of True/False questions worth 3 points each and multiple choice questions worth 11 points each. How many multiple choice questions are on the test?
Let the number of true/false questions be t. Let the number of multiple choice questions be m. We're given two equations:
[LIST=1]
[*]m + t = 20
[*]11m + 3t = 100
[/LIST]
We have a set of simultaneous equations. We can solve this using 3 methods:
[LIST=1]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=1m+%2B+t+%3D+20&term2=11m+%2B+3t+%3D+100&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=1m+%2B+t+%3D+20&term2=11m+%2B+3t+%3D+100&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=1m+%2B+t+%3D+20&term2=11m+%2B+3t+%3D+100&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we pick, we get the same answer:
[LIST]
[*][B]m = 5[/B]
[*][B]t = 15[/B]
[/LIST]
A textbook store sold a combined total of 307 biology and chemistry textbooks in a week. The numberA textbook store sold a combined total of 307 biology and chemistry textbooks in a week. The number of chemistry textbooks sold was 71 less than the number of biology textbooks sold. How many textbooks of each type were sold?
Let b be the number of biology books and c be the number of chemistry books. We have two equations:
[LIST=1]
[*]b + c = 307
[*]c = b - 71
[/LIST]
Substitute (2) into (1) for c
b + (b - 71) = 307
Combine like terms:
2b - 71 = 307
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2b-71%3D307&pl=Solve']equation solver[/URL], we get:
[B]b = 189[/B]
Now substitute that into (2):
c = 189 - 71
[B]c = 118[/B]
A theatre contains 459 seats and the ticket prices for a recent play were $53 for adults and $16 forA theatre contains 459 seats and the ticket prices for a recent play were $53 for adults and $16 for children. If the total proceeds were $13,930 for a sold- out matinee, how many of each type of ticket were sold?
Let a be the number of adult tickets and c be the number of children tickets. We have the following equations:
[LIST=1]
[*]a + c =459
[*]53a + 16c = 13930
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=a%2Bc%3D459&term2=53a+%2B+16c+%3D+13930&pl=Cramers+Method']simultaneous equation calculator[/URL], we have:
[B]a = 178
c = 281[/B]
A tire repair shop charges $5 for tool cost and $2 for every minute the worker spends on the repair.A tire repair shop charges $5 for tool cost and $2 for every minute the worker spends on the repair. A) Write an equation of the total cost of repair, $y, in terms of a total of x minutes of repair.
y = Variable Cost + Fixed Cost
y = Cost per minute of repair * minutes of repair + Tool Cost
[B]y = 2x + 5[/B]
A trapezoid has one base that is 120% of the length of the other base. The two sides are each 1/2 thA trapezoid has one base that is 120% of the length of the other base. The two sides are each 1/2 the length of the smaller base. If the perimeter of the trapezoid is 54.4 inches, what is the length of the smaller base of the trapezoid?
Setup measurements:
[LIST]
[*]Small base = n
[*]Large base = 1.2n
[*]sides = n/2
[*]Perimeter = n + 1.2n + 0.5n + 0.5n = 54.4
[/LIST]
Solve for [I]n[/I] in the equation n + 1.2n + 0.5n + 0.5n = 54.4
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 1.2 + 0.5 + 0.5)n = 3.2n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
3.2n = + 54.4
[SIZE=5][B]Step 3: Divide each side of the equation by 3.2[/B][/SIZE]
3.2n/3.2 = 54.4/3.2
n = [B]17[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=n%2B1.2n%2B0.5n%2B0.5n%3D54.4&pl=Solve']Source[/URL]
A traveler is walking on a moving walkway in an airport. the traveler must walk back on the walkwayA traveler is walking on a moving walkway in an airport. the traveler must walk back on the walkway to get a bag he forgot. the traveler's ground speed is 2 ft/s against the walkway and 6 ft/s with the walkway. what is the traveler's speed off the walkway? What is the speed of the moving walkway.
We have two equations, where w is the speed of the walkway and t is the speed of the traveler.
[LIST=1]
[*]t - w = 2
[*]t + w = 6
[*]Rearrange (1) to solve for t: t = w + 2
[/LIST]
Now plug (3) into (2)
(w + 2) + w = 6
Combine like terms:
2w + 2 = 6
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2w%2B2%3D6&pl=Solve']equation solver[/URL], we get [B]w = 2[/B]
Plug this into (1)
t - 2 = 2
Add 2 to each side
[B]t = 4[/B]
A tree grows 35 cm in 2 years. If it continues to grow at the same rate determine how long it wouldA tree grows 35 cm in 2 years. If it continues to grow at the same rate determine how long it would take to grow 85 cm
We set up a proportion of cm to years where y is the number of years it takes to grow 85 cm:
35/2 = 85/y
To solve this proportion for y, [URL='https://www.mathcelebrity.com/prop.php?num1=35&num2=85&den1=2&den2=y&propsign=%3D&pl=Calculate+missing+proportion+value']we type it in our search engine[/URL] and we get:
[B]y = 4.86[/B]
A turtle and rabbit are in a race to see who is the first to reach a point 100 feet away. The turtleA turtle and rabbit are in a race to see who is the first to reach a point 100 feet away. The turtle travels at a constant speed of 20 feet per minute for the entire 100 feet. The rabbit travels at a constant speed of 40 feet per minute for the first 50 feet, stops for 3 minutes, and then continuous at a constant speed of 40 feet per minute for the last 50 feet.
(i) Determine which animal won the race.
(ii). By how much time the animal won the race.
(iii) Explain one life lesson from the race.
We know the distance formula is:
d = rt
For the turtle, he has a rate (r) of 20 feet / minute and distance (d) of 100. We want to solve for time:
[URL='https://www.mathcelebrity.com/drt.php?d=+100&r=+20&t=&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator solving for t[/URL], we get:
t = 5
The rabbit has 3 parts of the race:
Rabbit Part 1: Distance (d) = 50 and rate (r) = 40
[URL='https://www.mathcelebrity.com/drt.php?d=50&r=40&t=+&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator solving for t[/URL], we get:
t = 1.25
Rabbit Part 2: The rabbit stops for 3 minutes (t = 3)
Rabbit Part 3: Distance (d) = 50 and rate (r) = 40
[URL='https://www.mathcelebrity.com/drt.php?d=50&r=40&t=+&pl=Calculate+the+missing+Item+from+D%3DRT']Using our distance rate time calculator solving for t[/URL], we get:
t = 1.25
Total time for the rabbit from the 3 parts is (t) = 1.25 + 3 + 1.25
Total time for the rabbit from the 3 parts is (t) = 5.5
[LIST]
[*](i) The [B]turtle won[/B] the race because he took more time to finish and they both started at the same time
[*](ii) We subtract the turtles time from the rabbit's time: 5.5 - 5 = [B]0.5 minutes which is also 30 seconds[/B]
[*](iii) [B]Slow and Steady wins the race[/B]
[/LIST]
A used book store also started selling used CDs and videos. In the first week, the store sold a combA used book store also started selling used CDs and videos. In the first week, the store sold a combination of 40 CDs and videos. They charged $4 per CD and $6 per video and the total sales were $180. Determine the total number of CDs and videos sold
Let c be the number of CDs sold, and v be the number of videos sold. We're given 2 equations:
[LIST=1]
[*]c + v = 40
[*]4c + 6v = 180
[/LIST]
You can solve this system of equations three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter what method we choose, we get [B]c = 30, v = 10[/B].
Now let's check our work for both given equations for c = 30 and v = 10:
[LIST=1]
[*]30 + 10 = 40 <-- This checks out
[*]4c + 6v = 180 --> 4(30) + 6(10) --> 120 + 60 = 180 <-- This checks out
[/LIST]
A used book store also started selling used CDs and videos. In the first week, the store sold a combA used book store also started selling used CDs and videos. In the first week, the store sold a combination of 40 CDs and videos. They charged $4 per CD and $6 per video and the total sales were $180. Determine the total number of CDs and videos sold.
Let the number of cd's be c and number of videos be v. We're given two equations:
[LIST=1]
[*]c + v = 40
[*]4c + 6v = 180
[/LIST]
We can solve this system of equations using 3 methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+v+%3D+40&term2=4c+%2B+6v+%3D+180&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
[B]c = 30
v = 10[/B]
A vehicle purchased for $25,000 depreciates at a constant rate of 5%. Determine the approximate valuA vehicle purchased for $25,000 depreciates at a constant rate of 5%. Determine the approximate value of the vehicle 11 years after purchase. Round to the nearest whole dollar.
Depreciation at 5% means it retains 95% of the value. Set up the depreciation equation to get Book Value B(t) at time t.
B(t) = $25,000 * (1 - 0.05)^t
Simplifying, this is:
B(t) = $25,000 * (0.95)^t
The problem asks for B(11)
B(11) = $25,000 * (0.95)^11
B(11) = $25,000 * 0.5688
B(11) = [B]$14,220[/B]
A woman walked for 5 hours, first along a level road, then up a hill, and then she turned around andA woman walked for 5 hours, first along a level road, then up a hill, and then she turned around and walked back to the starting point along the same path. She walks 4mph on level ground, 3 mph uphill, and 6 mph downhill. Find the distance she walked.
Hint: Think about d = rt, which means that t = d/r. Think about each section of her walk, what is the distance and the rate. You know that the total time is 5 hours, so you know the sum of the times from each section must be 5.
Let Level distance = L and hill distance = H. Add the times it took for each section of the walk:
L/4 + H /3 + H/6 + L/4 = 5
The LCD of this is 12 from our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=4&num2=3&num3=6&pl=LCM']LCD Calculator[/URL]
[U]Multiply each side through by our LCD of 12[/U]
3L + 4H + 2H + 3L = 60
[U]Combine like terms:[/U]
6L + 6H = 60
[U]Divide each side by 3:[/U]
2L + 2H = 20
The woman walked [B]20 miles[/B]
a writer can write a novel at a rate of 3 pages per 5 hour work. if he wants to finish the novel ina writer can write a novel at a rate of 3 pages per 5 hour work. if he wants to finish the novel in x number of pages, determine a function model that will represent the accumulated writing hours to finish his novel
if 3 pages = 5 hours, then we divide each side by 3 to get:
1 page = 5/3 hours per page
So x pages takes:
5x/3 hours
Our function for number of pages x is:
[B]f(x) = 5x/3[/B]
A+B+D=255 B+15=A D+12=B A=A+B+D=255 B+15=A D+12=B A=
[LIST=1]
[*]A + B + D = 255
[*]B + 15 = A
[*]D + 12 = B
[*]A = ?
[*]Rearrange (3) to solve for D by subtracting 12 from each side: D = B - 12
[/LIST]
Substitute (2) and (5) into 1
(B + 15) + B + (B - 12) = 255
Combine like terms:
3B + 3 = 255
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=3b%2B3%3D255&pl=Solve']equation solver[/URL], b = 84
Substitute b = 84 into equation (2):
A = 84 + 15
[B]A = 99[/B]
ab/d + c = e for dab/d + c = e for d
I know this is a literal equation because we are asked to solve for a variable [U]in terms of[I] another variable
[/I][/U]
Subtract c from each side to isolate the d term:
ab/d + c - c = e - c
Cancel the c's on the left side and we get:
ab/d = e - c
Cross multiply:
ab = d(e - c)
Divide each side of the equation by (e - c):
ab/(e - c)= d(e - c)/(e - c)
Cancel the (e - c) on the right side, and we get:
d = [B]ab/(e - c)[/B]
Accuracy and PrecisionFree Accuracy and Precision Calculator - Given an integer or decimal, this determines the precision and accuracy (scale)
Add 5 and 6 and then multiply by 3Add 5 and 6 and then multiply by 3
Add 5 and 6:
(5 + 6)
Then multiply by 3:
[B]3(5 + 6)
[/B]
If you want to evaluate this term, then we [URL='https://www.mathcelebrity.com/distributive-property.php?a=3&b=5&c=6&pl=Distributive']type it into the math engine[/URL] and we get:
[B]33[/B]
add p to 7, add the result to 10, then multiply 4 by what you haveadd p to 7, add the result to 10, then multiply 4 by what you have
Add p to 7:
p + 7
Add the result to 10:
p + 7 + 10
p + 17 <-- combine like terms
Then multiply 4 by what you have:
[B]4(p + 17)[/B]
ADG,BEH,CFI,___,___,___ADG,BEH,CFI,___,___,___
Looking at this pattern, we see:
[LIST=1]
[*]the first term starts with A and increments by 1 letter
[*]the second term starts with D and increments by 1 letter
[*]the third term starts with G and increments by 1 letter
[/LIST]
So terms 4, 5, and 6 are:
[LIST]
[*][B]DGJ[/B]
[*][B]EHK[/B]
[*][B]FIL[/B]
[/LIST]
Admission to a baseball game is $2.00 for general admission and $5.50 for reserved seats. The receiAdmission to a baseball game is $2.00 for general admission and $5.50 for reserved seats. The receipts were $3577.00 for 1197 paid admissions. How many of each ticket were sold?
Let g be the number of tickets for general admission
Let r be the number of tickets for reserved seats
We have two equations:
[LIST=1]
[*]g + r = 1197
[*]2g + 5.50r = 3577
[/LIST]
We can solve this a few ways, but let's use substitution using our [URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=g+%2B+r+%3D+1197&term2=2g+%2B+5.50r+%3D+3577&pl=Substitution']simultaneous equations calculator[/URL]:
[LIST]
[*][B]r = 338[/B]
[*][B]g = 859[/B]
[/LIST]
Age DifferenceFree Age Difference Calculator - Determines the ages for an age difference word problem.
Age now problemsLet f be the age of the father and d be the age of the daughter and s be the age of the son. We have:
[LIST=1]
[*]f = 3s
[*]d = s - 3
[*]d - 3 + f - 3 + s - 3 = 63
[/LIST]
Simplify (3)
d + f + s - 9 = 63
d + f + s = 72
Now, substitute (1) and (2) into the modified (3)
(s - 3) + 3s + s = 72
Combine like terms:
5s - 3 = 72
Add 3 to each side
5s = 75
Divide each side by 5
s = 15
We want f, so we substitute s = 15 into (1)
f = 3(15)
[B]f = 45[/B]
Age Word ProblemsFree Age Word Problems Calculator - Determines age in age word problems
Ahmad has a jar containing only 5-cent and 20-cent coins. In total there are 31 coins with a total vAhmad has a jar containing only 5-cent and 20-cent coins. In total there are 31 coins with a total value of $3.50. How many of each type of coin does Ahmad have?
Let the number of 5-cent coins be f.
Let the number of 20-cent coins be t.
We're given two equations:
[LIST=1]
[*]f + t = 31
[*]0.05f + 0.2t = 3.50
[/LIST]
We can solve this simultaneous system of equations 3 ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+31&term2=0.05f+%2B+0.2t+%3D+3.50&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+31&term2=0.05f+%2B+0.2t+%3D+3.50&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+31&term2=0.05f+%2B+0.2t+%3D+3.50&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter which method we choose, we get:
[LIST]
[*][B]f = 18[/B]
[*][B]t = 13[/B]
[/LIST]
Alberto and Willie each improved their yards by planting daylilies and ivy. They bought their suppliAlberto and Willie each improved their yards by planting daylilies and ivy. They bought their supplies from the same store. Alberto spent $64 on 3 daylilies and 8 pots of ivy. Willie spent $107 on 9 daylilies and 7 pots of ivy. What is the cost of one daylily and the cost of one pot of ivy?
Assumptions:
[LIST]
[*]Let d be the cost of one daylily
[*]Let i be the cost of one pot of ivy
[/LIST]
Givens:
[LIST=1]
[*]3d + 8i = 64
[*]9d + 7i = 107
[/LIST]
To solve this system of equations, you can use any of three methods below:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=3d+%2B+8i+%3D+64&term2=9d+%2B+7i+%3D+107&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=3d+%2B+8i+%3D+64&term2=9d+%2B+7i+%3D+107&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=3d+%2B+8i+%3D+64&term2=9d+%2B+7i+%3D+107&pl=Cramers+Method']Cramer's Method[/URL]
[/LIST]
No matter what method we use, we get the same answer:
[LIST]
[*][B]d = 8[/B]
[*][B]i = 5[/B]
[/LIST]
[B][MEDIA=youtube]K1n3niERg-U[/MEDIA][/B]
Alberto’s salary was $1500 greater than 5 times Nick’s salary. Write an equation stating Alberto’s aAlberto’s salary was $1500 greater than 5 times Nick’s salary. Write an equation stating Alberto’s and Nick’s salaries in terms of x and y.
Let x be Alberto's salary. Let y be Nick's salary. We have:
Let's break this down:
[LIST=1]
[*]5 times Nick's salary (y), means we multiply the variable y by 5
[*]$1500 greater means we add $1500 to 5y
[/LIST]
[B]x = 5y - 1500[/B]
Alberto’s salary was $2000 greater than 4 times Nick’s salary. Write an equation stating Alberto’s aAlberto’s salary was $2000 greater than 4 times Nick’s salary. Write an equation stating Alberto’s and Nick’s salaries in terms of x and y.
If Alberto's salary is x and Nick's salary is y, we have:
[B]x = 4y + 2000[/B]
Alberto’s salary was $2000 greater than 4 times Nick’s salary. Write an equation stating Alberto’s aAlberto’s salary was $2000 greater than 4 times Nick’s salary. Write an equation stating Alberto’s and Nick’s salaries in terms of x and y.
Let Alberto's salary be x, and Nick's salary be y.
We have:
[B]x = 4y + 2000[/B]
Alberto’s salary was $2000 greater than 4 times Nick’s salary. Write an equation stating Alberto’s aAlberto’s salary was $2000 greater than 4 times Nick’s salary. Write an equation stating Alberto’s and Nick’s salaries in terms of x and y.
If Alberto's salary is x and Nick's is y, we have:
[B]x = 4y + 2000 [/B](since greater than means we add)
Alexis is working at her schools bake sale. Each mini cherry pie sells for $4 and each mini peach piAlexis is working at her schools bake sale. Each mini cherry pie sells for $4 and each mini peach pie sells for $3. Alexis sells 25 pies and collects $84. How many pies of each kind does she sell?
Let each cherry pie be c and each peach pie be p. We have the following equations:
[LIST=1]
[*]c + p = 25
[*]4c + 3p = 84
[/LIST]
You can solve this system of equations 3 ways.
[URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c%2Bp%3D25&term2=4c+%2B+3p+%3D+84&pl=Substitution']Substitution Rule[/URL]
[URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c%2Bp%3D25&term2=4c+%2B+3p+%3D+84&pl=Elimination']Elimination Rule[/URL]
[URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c%2Bp%3D25&term2=4c+%2B+3p+%3D+84&pl=Cramers+Method']Cramers Rule[/URL]
No matter which way you choose, you get [B]c = 9 and p = 16[/B].
algexpress: letthefirstnumberequalx.thesecondnumberis3morethantwicethefirstnumber.expressthesecondnuLet the first number equal x. The second number is 3 more than twice the first number. Express the second number in terms of the first number x.
[LIST]
[*]Let the second number be y.
[*]Twice means multiply by 2
[*]3 more than means we add 3
[/LIST]
So we have the following algebraic expression:
[B]y = 2x + 3[/B]
Alice is 3 years younger than Barbara, and Barbara is 5 years younger than Carol. Together the sisteAlice is 3 years younger than Barbara, and Barbara is 5 years younger than Carol. Together the sisters are 68 years old. How old is Barbara?
Let a be Alice's age, b be Barbara's age, and c be Carol's age. We have 3 given equations:
[LIST=1]
[*]a = b - 3
[*]b = c - 5
[*]a + b + c = 68
[/LIST]
Rearrange (2)
c = b + 5
Now plug in (1) and (2) revised into (3). We want to isolate for b.
a + b + c = 68
(b - 3) + b + (b + 5) = 68
Combine like terms:
(b + b + b) + (5 - 3) = 68
3b + 2 = 68
Run this through our [URL='https://www.mathcelebrity.com/1unk.php?num=3b%2B2%3D68&pl=Solve']equation calculator[/URL], and we get b = [B]22[/B]
Alisha is 5 years younger than her brother. If the age of her brother is y years then age of AlishaAlisha is 5 years younger than her brother. If the age of her brother is y years then age of Alisha in terms of her brother
Younger means we subtract. If her brother is y years of age, then Alisha is:
[B]y - 5[/B]
Alvin is 12 years younger than Elga. The sum of their ages is 60 . What is Elgas age?Alvin is 12 years younger than Elga. The sum of their ages is 60 . What is Elgas age?
Let a be Alvin's age and e be Elga's age. We have the following equations:
[LIST=1]
[*]a = e - 12
[*]a + e = 60
[/LIST]
Plugging in (1) to (2), we get:
(e - 12) + e = 60
Grouping like terms:
2e - 12 = 60
Add 12 to each side:
2e = 72
Divide each side by 2
[B]e = 36[/B]
An ancient Greek was said to have lived 1/4 of his live as a boy, 1/5 as a youth, 1/3 as a man, andAn ancient Greek was said to have lived 1/4 of his live as a boy, 1/5 as a youth, 1/3 as a man, and spent the last 13 years as an old man. How old was he when he died?
Set up his life equation per time lived as a boy, youth, man, and old man
1/4 + 1/5 + 1/3 + x = 1.
Using our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=4&num2=3&num3=5&pl=LCM']LCM Calculator[/URL], we see the LCM of 3,4,5 is 60. This is our common denominator.
So we have 15/60 + 12/60 + 20/60 + x/60 = 60/60
[U]Combine like terms[/U]
x + 47/60 = 60/60
[U]Subtract 47/60 from each side:[/U]
x/60 = 13/60
x = 13 out of the 60 possible years, so he was [B]60 when he died[/B].
An auto repair bill is $126 for parts and $35 for each hour of labor. If h is the number of hours ofAn auto repair bill is $126 for parts and $35 for each hour of labor. If h is the number of hours of labor, express the amount of the repair bill in terms of number of hours of labor.
Set up cost function, where h is the number of hours of labor:
[B]C(h) = 35h + 136[/B]
An employee earns $7.00 an hour for the first 35 hours worked in a week and $10.50 for any hours oveAn employee earns $7.00 an hour for the first 35 hours worked in a week and $10.50 for any hours over 35. One weeks paycheck (before deductions) was for $308.00. How many hours did the employee work?
Let's first check to see if the employee worked overtime:
Regular Hours: 35 * 7 = 245
Since the employee made $308, they worked overtime. Let's determine how much overtime money was made:
308 - 245 = 63
Now, to calculate the overtime hours, we divide overtime pay by overtime rate
63/10.50 = 6
Now figure out the total hours worked in the week:
Total Hours = Regular Pay Hours + Overtime Hours
Total Hours = 35 + 6
[B]Total Hours = 41[/B]
An executive invests $23,000, some at 8% and some at 4% annual interest. If he receives an annual reAn executive invests $23,000, some at 8% and some at 4% annual interest. If he receives an annual return of $1,560, how much is invested at each rate?
Let x be the amount invested at 8% and y be the amount invested at 4%. We have two equations:
[LIST=1]
[*]x + y = 23,000
[*]0.08x + 0.04y = 1,560
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+y+%3D+23000&term2=0.08x+%2B+0.04y+%3D+1560&pl=Cramers+Method']system of equations calculator[/URL], we get:
[B]x = 16,000
y = 7,000[/B]
An experienced accountant can balance the books twice as fast as a new accountant. Working togetherAn experienced accountant can balance the books twice as fast as a new accountant. Working together it takes the accountants 10 hours. How long would it take the experienced accountant working alone?
Person A: x/2 job per hour
Person B: 1/x job per hour
Set up our equation:
1/x + 1/(2x) = 1/10
Multiply the first fraction by 2/2 to get common denominators;
2/(2x) + 1/(2x) = 1/10
Combine like terms
3/2x = 1/10
Cross multiply:
30 = 2x
Divide each side by 2:
[B]x = 15[/B]
An initial deposit of $50 is now worth $400. The account earns 5.2% interest compounded continuouslyAn initial deposit of $50 is now worth $400. The account earns 5.2% interest compounded continuously. Determine how long the money has been in the account.
[URL='https://www.mathcelebrity.com/simpint.php?av=400&p=50&int=5.2&t=&pl=Continuous+Interest']Using our continuous interest compound calculator solving for t[/URL], we get:
t =[B] 39.99 periods[/B]
An isosceles triangles non-congruent angle is 16 more than twice the congruent ones. What is the meaAn isosceles triangles non-congruent angle is 16 more than twice the congruent ones. What is the measure of all 3 angles?
Let the congruent angles measurement be c. And the non-congruent angle measurement be n. We're given:
[LIST=1]
[*]n = 2c + 16 <-- Twice means we multiply by 2, and more than means we add 16
[*]2c + n = 180 <-- Since the sum of angles in an isosceles triangle is 180
[/LIST]
Substitute (1) into (2):
2c + (2c + 16) = 180
Group like terms:
4c + 16 = 180
[URL='https://www.mathcelebrity.com/1unk.php?num=4c%2B16%3D180&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]c = 41[/B]
Substituting this value into Equation 1, we get
n = 2(41) + 16
n = 82 + 16
[B]n = 98[/B]
Angelo is paid double time for each hour he works over 40 hours in a week. Last week he worked 46 hoAngelo is paid double time for each hour he works over 40 hours in a week. Last week he worked 46 hours and earned $624. What is his normal hourly rate?
Let h be Angelo's hourly rate. We have:
40h + (46 - 40) * 2 * h = 624
40h + 6 * 2 * h = 624
40h + 12h = 624
Combine like terms:
52h = 624
[URL='https://www.mathcelebrity.com/1unk.php?num=52h%3D624&pl=Solve']Typing this equation into our search engine[/URL], we get [B]h = 12[/B].
Angie and Kenny play online video games. Angie buy 2 software packages and 4 months of game play. KeAngie and Kenny play online video games. Angie buy 2 software packages and 4 months of game play. Kenny buys 1 software package and 1 month of game play. Each software package costs $25. If their total cost is $155, what is the cost of one month of game play.
Let s be the cost of software packages and m be the months of game play. We have:
[LIST]
[*]Angie: 2s + 4m
[*]Kenny: s + m
[/LIST]
We are given each software package costs $25. So the revised equations above become:
[LIST]
[*]Angie: 2(25) + 4m = 50 + 4m
[*]Kenny: 25 + m
[/LIST]
Finally, we are told their combined cost is 155. So we add Angie and Kenny's costs together:
4m + 50 + 25 + m = 155
Combine like terms:
5m + 75 = 155
[URL='http://www.mathcelebrity.com/1unk.php?num=5m%2B75%3D155&pl=Solve']Typing this into our search engine[/URL], we get [B]m = 16[/B]
Angle Ratio for a TriangleFree Angle Ratio for a Triangle Calculator - Given an angle ratio for a triangle of a:b:c, this determines the angle measurements of the triangle.
Anna’s age increased by 3 times her age, the result is 72Anna’s age increased by 3 times her age, the result is 72.
Let a be Anna's age. We have:
a + 3a = 72
Combine like terms:
(1 + 3)a = 72
4a = 72
[URL='https://www.mathcelebrity.com/1unk.php?num=4a%3D72&pl=Solve']Type 4a = 72 into our calculator[/URL], and we get [B]a = 18[/B].
Arithmetic and Geometric and Harmonic SequencesFree Arithmetic and Geometric and Harmonic Sequences Calculator - This will take an arithmetic series or geometric series or harmonic series, and an optional amount (n), and determine the following information about the sequence
1) Explicit Formula
2) The remaining terms of the sequence up to (n)
3) The sum of the first (n) terms of the sequence
Also known as arithmetic sequence, geometric sequence, and harmonic sequence
Arnie bought some bagels at 20 cents each. He ate 4, and sold the rest at 30 cents each. His profitArnie bought some bagels at 20 cents each. He ate 4, and sold the rest at 30 cents each. His profit was $2.40. How many bagels did he buy?
Let x be the number of bagels Arnie sold. We have the following equation:
0.30(x - 4) - 0.20(4) = 2.40
Distribute and simplify:
0.30x - 1.20 - 0.8 = 2.40
Combine like terms:
0.30x - 2 = 2.40
Add 2 to each side:
0.30x = 4.40
Divide each side by 0.3
[B]x = 14.67 ~ 15[/B]
Ashley age is 2 times Johns age. The sum of their ages is 63. What is Johns age?Ashley age is 2 times Johns age. The sum of their ages is 63. What is Johns age?
Let Ashley's age be a.
Let John's age be j.
We have two equations:
[LIST=1]
[*]a = 2j
[*]a + j = 63
[/LIST]
Now substitute (1) into (2)
(2j) + j = 63
Combine like terms:
3j = 63
[URL='http://www.mathcelebrity.com/1unk.php?num=3j%3D63&pl=Solve']Typing 3j = 63 into our search engine[/URL], we get
[B]j = 21[/B]
At 1:00 pm you have 24 megabytes of a movie and at 1:15 pm you have 96 megabytes of a movie. What isAt 1:00 pm you have 24 megabytes of a movie and at 1:15 pm you have 96 megabytes of a movie. What is the download rate in megabytes per minute?
First, find the number of minutes:
1:15 - 1:00 = 15 minutes
Next, determine the difference in megabytes
96 - 24 = 72
Finally, determine the download rate:
72 megabytes / 15 minutes = [B]4.8 megabytes per minute
[MEDIA=youtube]RCvs3TQMzdM[/MEDIA][/B]
at a bakery the cost of one cupcake and 2 slices of pie is $12.40. the cost of 2 cupcakes and 3 slicat a bakery the cost of one cupcake and 2 slices of pie is $12.40. the cost of 2 cupcakes and 3 slices of pie costs $20.20. what is the cost of one cupcake?
Let the number of cupcakes be c
Let the number of pie slices be p
Total Cost = Unit cost * quantity
So we're given two equations:
[LIST=1]
[*]1c + 2p = 12.40
[*]2c + 3p = 20.20
[/LIST]
We can solve this system of equations any one of three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=1c+%2B+2p+%3D+12.40&term2=2c+%2B+3p+%3D+20.20&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=1c+%2B+2p+%3D+12.40&term2=2c+%2B+3p+%3D+20.20&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=1c+%2B+2p+%3D+12.40&term2=2c+%2B+3p+%3D+20.20&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
[LIST]
[*][B]c = 3.2[/B]
[*]p = 4.6
[/LIST]
At a festival, Cherly bought 5 ride tokens and 9 game tokens. She spent $59. Let x represent the cosAt a festival, Cherly bought 5 ride tokens and 9 game tokens. She spent $59. Let x represent the cost of ride tokens and let y represent the cost of game tokens. Write an equation in standard for that can be used to determine how much each type of token costs.
We know that:
Token Cost + Game Cost = Total Cost
Since cost = price * quantity, we have:
[B]5x + 9y = 59[/B]
At a football game, a vender sold a combined total of 117 sodas and hot dogs. The number of hot dogsAt a football game, a vender sold a combined total of 117 sodas and hot dogs. The number of hot dogs sold was 59 less than the number of sodas sold. Find the number of sodas sold and the number of hot dogs sold.
[U]Let h = number of hot dogs and s = number of sodas. Set up our given equations:[/U]
[LIST=1]
[*]h + s = 117
[*]h = s - 59
[/LIST]
[U]Substitute (2) into (1)[/U]
(s - 59) + s = 117
[U]Combine s terms[/U]
2s - 59 = 117
[U]Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2s-59%3D117&pl=Solve']equation solver[/URL], we find:[/U]
[B]s = 88
[/B]
[U]Plug s = 88 into (2)[/U]
h = 88 - 59
[B]h = 29[/B]
Austin deposited $4000 into an account with 4.8% interest,compounded monthly. Assuming that noAustin deposited $4000 into an account with 4.8% interest, compounded monthly. Assuming that no withdrawals are made, how much will he have in the account after 4 years? Do not round any intermediate computations, and round your answer to the nearest cent.
Using our [URL='http://www.mathcelebrity.com/compoundint.php?bal=40000&nval=4&int=4.8&pl=Annually']balance calculator[/URL], we get:
[B]$48,250.87[/B]
Automorphic NumberFree Automorphic Number Calculator - This calculator determines the nth automorphic number
ax - mn = mn + bx for xax - mn = mn + bx for x
Add mn to each side:
ax - mn + mn = mn + bx + mn
Cancel the mn terms on the left side and we get:
ax = bx + 2mn
Subtract bx from each side:
ax - bx = bx - bx + 2mn
Cancel the bx terms on the right side:
ax - bx = 2mn
Factor out x on the left side:
x (a - b) = 2mn
Divide each side of the equation by (a - b):
x (a - b)/(a - b) = 2mn/(a - b)
Cancel the (a - b) on the left side and we get:
x = [B]2mn/(a - b)[/B]
b/3d - h = 343 for bb/3d - h = 343 for b
A literal equation means we solve for one variable in terms of another variable or variables
Add h to each side to isolate the b term:
b/3d - h + h = 343 + h
Cancel the h's on the left side, we get:
b/3d = 343 + h
Cross multiply:
b = [B]3d(343 + h)[/B]
Babylonian MethodFree Babylonian Method Calculator - Determines the square root of a number using the Babylonian Method.
Balance SheetFree Balance Sheet Calculator - Given various asset and liability entries, this determines various calculations that can be made from the balance sheet.
Before Barry Bonds, Mark McGwire, and Sammy Sosa, Roger Maris held the record for the most home runsBefore Barry Bonds, Mark McGwire, and Sammy Sosa, Roger Maris held the record for the most home runs in one season. Just behind Maris was Babe Ruth. The numbers of home runs hit by these two athletes in their record-breaking seasons form consecutive integers. Combined, the two athletes hit 121 home runs. Determine the number of home runs hit by Maris and Ruth in their record-breaking seasons.
We want [URL='https://www.mathcelebrity.com/consecintwp.php?num=121&pl=Sum']the sum of 2 consecutive integers equals 121[/URL].
[B]We get Maris at 61 and Ruth at 60[/B]
Ben has $4.50 in quarters(Q) and dimes(D). a)Write an equation expressing the total amount of moneyBen has $4.50 in quarters(Q) and dimes(D). a)Write an equation expressing the total amount of money in terms of the number of quarters and dimes. b)Rearrange the equation to isolate for the number of dimes (D)
a) The equation is:
[B]0.1d + 0.25q = 4.5[/B]
b) Isolate the equation for d. We subtract 0.25q from each side of the equation:
0.1d + 0.25q - 0.25q = 4.5 - 0.25q
Cancel the 0.25q on the left side, and we get:
0.1d = 4.5 - 0.25q
Divide each side of the equation by 0.1 to isolate d:
0.1d/0.1 = (4.5 - 0.25q)/0.1
d = [B]45 - 2.5q[/B]
Beth is 5 years younger than celeste. Next year, their ages will have a sum equal to 57. How old isBeth is 5 years younger than celeste. Next year, their ages will have a sum equal to 57. How old is each now?
Let b = Beth's age
Let c = Celeste's age
We are given:
[LIST=1]
[*]b = c - 5
[*]b + 1 + c + 1 = 57
[/LIST]
Substitute (1) into (2)
(c - 5) + 1 + c + 1 = 57
Group like terms:
2c - 3 = 57
[URL='https://www.mathcelebrity.com/1unk.php?num=2c-3%3D57&pl=Solve']Type 2c - 3 = 57 into our search engine[/URL], we get [B]c = 30[/B]
Substitute c = 30 into Equation (1), we get:
b = 30 - 5
[B]b = 25
[/B]
Therefore, Beth is 25 and Celeste is 30.
Better Buy ComparisonFree Better Buy Comparison Calculator - Given two items with a price and quantity, this determines which is the better buy by comparing unit prices. Finds the better deal.
Bob has half as many quarters as dimes. He has $3.60. How many of each coin does he have?Bob has half as many quarters as dimes. He has $3.60. How many of each coin does he have?
Let q be the number of quarters. Let d be the number of dimes. We're given:
[LIST=1]
[*]q = 0.5d
[*]0.25q + 0.10d = 3.60
[/LIST]
Substitute (1) into (2):
0.25(0.5d) + 0.10d = 3.60
0.125d + 0.1d = 3.6
Combine like terms:
0.225d = 3.6
[URL='https://www.mathcelebrity.com/1unk.php?num=0.225d%3D3.6&pl=Solve']Typing this equation into our search engine[/URL], we're given:
[B]d = 16[/B]
Substitute d = 16 into Equation (1):
q = 0.5(16)
[B]q = 8[/B]
Bond Price FormulasFree Bond Price Formulas Calculator - Given a face value, coupon percent, yield percent, term, and redemption value, this calculates the price of a bond using the four price formulas for bonds
1) Basic
2) Premium/Discount
3) Base
4) Makeham
Boolean Algebra MultiplicationFree Boolean Algebra Multiplication Calculator - Determines the product of two expressions using boolean algebra.
Brian's age is 3/4 of Marcus'. The sum of their ages is 14. How old are Marcus and Brian?Brian's age is 3/4 of Marcus'. The sum of their ages is 14. How old are Marcus and Brian?
Let Marcus's age be m.
Then Brian's age = 3/4m
The sum is:
m + 3m/4 = 14
Combine like terms
7m/4 = 14
Cross multiply:
7m = 56
[URL='http://www.mathcelebrity.com/1unk.php?num=7m%3D56&pl=Solve']Plugging this into the search engine[/URL], we get m = 8.
So Brian's age = 3(8)/4 = 24/4 = 6
C varies directly as the cube of a and inversely as the 4th power of BC varies directly as the cube of a and inversely as the 4th power of B
The cube of a means we raise a to the 3rd power:
a^3
The 4th power of B means we raise b to the 4th power:
b^4
Varies directly means there exists a constant k such that:
C = ka^3
Also, varies inversely means we divide by the 4th power of B
C = [B]ka^3/b^4[/B]
Varies [I]directly [/I]as means we multiply by the constant k.
Varies [I]inversely [/I]means we divide k by the term which has inverse variation.
[MEDIA=youtube]fSsG1OB3qdk[/MEDIA]
c/a=db/r for ac/a=db/r for a
Cross multiply the proportion:
cr = adb
Divide each side of the equation by db to isolate a:
cr/db = adb/db
Cancel the db terms on the left side and we get:
a = [B]cr/db[/B]
Caleb earns points on his credit card that he can use towards future purchases.Let f = dollars spent on flights, h dollars spent on hotels, and p dollars spent on all other purchases.
[U]Set up our equations:[/U]
(1) 4f + 2h + p = 14660
(2) f + h + p = 9480
(3) f = 2h + 140
[U]First, substitute (3) into (2)[/U]
(2h + 140) + h + p = 9480
3h + p + 140 = 9480
3h + p = 9340
[U]Subtract 3h to isolate p to form equation (4)[/U]
(4) p = 9340 - 3h
[U]Take (3) and (4), and substitute into (1)[/U]
4(2h + 140) + 2h + (9340 - h) = 14660
[U]Multiply through[/U]
8h + 560 + 2h + 9340 - 3h = 14660
[U]Combine h terms and constants[/U]
(8 + 2 - 3)h + (560 + 9340) = 14660
7h + 9900 = 14660
[U]Subtract 9900 from both sides:[/U]
7h = 4760
[U]Divide each side by 7[/U]
[B]h = 680[/B]
[U]Substitute h = 680 into equation (3)[/U]
f = 2(680) + 140
f = 1360 + 140
[B]f = 1,500[/B]
[U]
Substitute h = 680 and f = 1500 into equation (2)[/U]
1500 + 680 + p = 9480
p + 2180 = 9480
[U]Subtract 2180 from each side:[/U]
[B]p = 7,300[/B]
Calls-Puts-Option ΔFree Calls-Puts-Option Δ Calculator - Calculates the call price, put price, and option Δ based on an option under the risk neutral scenario with a 1 year term.
Cam is 3 years older than Lara. If their combined age is 63, determine their ages by solving an apprCam is 3 years older than Lara. If their combined age is 63, determine their ages by solving an appropriate pair of equations.
Let Cam's age be c. Let Lara's age be l. We're given two equations:
[LIST=1]
[*]c = l + 3 <-- older means we add
[*]c + l = 63 <-- combined ages mean we add
[/LIST]
Substitute equation (1) into equation (2):
l + 3 + l = 63
Combine like terms to simplify our equation:
2l + 3 = 63
To solve for l, [URL='https://www.mathcelebrity.com/1unk.php?num=2l%2B3%3D63&pl=Solve']we type this equation into our search engine[/URL] and we get:
l = [B]30[/B]
Now, we plug l = 30 into equation (1) to solve for c:
c = 30 + 3
c = [B]33[/B]
Cam is 3 years older than Lara. If their combined age is 63, determine their ages by solving an apprCam is 3 years older than Lara. If their combined age is 63, determine their ages by solving an appropriate pair of equations.
Let Cam's age be c.
Let Lara's age be l.
We're given two equations:
[LIST=1]
[*]c = l + 3 (Since older means we add)
[*]c + l = 63
[/LIST]
To solve this system of equations, we substitute equation (1) into equation (2) for c:
l + 3 + l = 63
To solve this equation for l, we [URL='https://www.mathcelebrity.com/1unk.php?num=l%2B3%2Bl%3D63&pl=Solve']type it in our search engine [/URL]and we get:
l = [B]30
[/B]
Now, we take l = 30 and substitute it in equation (1) to solve for c:
c = 30 + 3
c = [B]33[/B]
Can a coefficient of determination be negative? Why or why not?Can a coefficient of determination be negative? Why or why not?
[B]Yes, reasons below[/B]
[LIST]
[*] predictions that are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data
[*] where linear regression is conducted without including an intercept
[*] Yes, negative values of R2 may occur when fitting non-linear functions to data
[/LIST]
can you continue this pattern 1,5,13,29can you continue this pattern 1,5,13,29
Looking at the numbers, we see a pattern of the next number as the prior number * 2 and then add 3
With each term as t(n), we find t(n + 1) as:
t(n + 1) = [B]2*t(n) + 3[/B]
t(2) = 2(1) + 3 = 2 + 3 = 5
t(3) = 2(5) + 3 = 10 + 3 = 13
t(4) = 2(13) + 3 = 26 + 3 = 29
t(5) = 2(29) + 3 = 58 + 3 = [B]61[/B]
Carlos was asked to write an equivalent equation to 2x/5 = 1 - x. he wrote it as 2x = 1 - 5x. do youCarlos was asked to write an equivalent equation to 2x/5 = 1 - x. he wrote it as 2x = 1 - 5x. do you agree with his conclusion? explain your answer for x
Cross multiply
2x/5 = 1 - x
2x = 5(1 - x)
2x = 5 - 5x
I disagree with his conclusion. He forgot to multiply the 5 through to [B]both terms[/B]
Chi-Square χ2 TestFree Chi-Square χ2 Test Calculator - This calculator determines a χ2 chi-square test on a test statistic and determines if it is outside an accepted range with critical value test and conclusion.
Chris, Alex and Jesse are all siblings in the same family. Alex is 5 years older than chris. Jesse iChris, Alex and Jesse are all siblings in the same family. Alex is 5 years older than chris. Jesse is 6 years older than Alex. The sum of their ages is 31 years. How old is each one of them?
Set up the relational equations where a = Alex's age, c = Chris's aged and j = Jesse's age
[LIST=1]
[*]a = c + 5
[*]j = a + 6
[*]a + c + j = 31
[*]Rearrange (1) in terms of c: c = a - 5
[/LIST]
[U]Plug in (4) and (2) into (3)[/U]
a + (a - 5) + (a + 6) = 31
[U]Combine like terms:[/U]
3a + 1 = 31
[U]Subtract 1 from each side[/U]
3a = 30
[U]Divide each side by 3[/U]
[B]a = 10[/B]
[U]Plug in 1 = 10 into Equation (4)[/U]
c = 10 - 5
[B]c = 5[/B]
Now plug 1 = 10 into equation (2)
j = 10 + 6
[B]j = 16[/B]
Circle EquationFree Circle Equation Calculator - This calculates the standard equation of a circle and general equation of a circle from the following given items:
* A center (h,k) and a radius r
* A diameter A(a1,a2) and B(b1,b2)
This also allows you to enter a standard or general form equation so that the center (h,k) and radius r can be determined.
Classify FractionFree Classify Fraction Calculator - Determines the if a fraction is proper, improper, or whole.
Closest FractionFree Closest Fraction Calculator - Determines the closest fraction in a list to a target fraction
Coin CombinationsFree Coin Combinations Calculator - Given a selection of coins and an amount, this determines the least amount of coins needed to reach that total.
Coin Toss ProbabilityFree Coin Toss Probability Calculator - This calculator determines the following coin toss probability scenarios
* Coin Toss Sequence such as HTHHT
* Probability of x heads and y tails
* Probability of at least x heads in y coin tosses
* Probability of at least x tails in y coin tosses
* Probability of no more than x heads in y coin tosses
* Probability of no more than x tails in y coin tosses
* (n) Coin Tosses with a list of scenario results displayed
* Monte Carlo coin toss simulation
Colin was thinking of a number. Colin divides by 8, then adds 1 to get an answer of 2. What was theColin was thinking of a number. Colin divides by 8, then adds 1 to get an answer of 2. What was the original number?
Let the number be n.
Divide by 8:
n/8
Then add 1:
n/8 + 1
The phrase [I]get an answer[/I] of means an equation, so we set n/8 + 1 equal to 2:
n/8 + 1 = 2
To solve for n, we subtract 1 from each side to isolate the n term:
n/8 + 1 - 1 = 2 - 1
Cancel the 1's on the left side, we get:
n/8 = 1
Cross multiply:
n = 8*1
n = [B]8[/B]
Combined RatioFree Combined Ratio Calculator - Given a ratio a:b and a ratio b:c, this determines the combined ratio a:c
Company a charges $25 plus $0.10 a mile. Company b charges $20 plus $0.15 per mile. How far would yoCompany a charges $25 plus $0.10 a mile. Company b charges $20 plus $0.15 per mile. How far would you need to travel to get each charge to be the same?
Let x be the number of miles traveled
Company A charge: C = 25 + 0.10x
Company B charge: C = 20 + 0.15x
Set up an equation find out when the charges are the same.
25 + 0.10x = 20 + 0.15x
Combine terms and simplify
0.05x = 5
Divide each side of the equation by 0.05 to isolate x
x = [B]100[/B]
Compare RaisesFree Compare Raises Calculator - Given two people with a salary and annual raise amount, this determines how long it takes for the person with the lower salary to catch the person with the higher salary.
Complementary and Supplementary AnglesFree Complementary and Supplementary Angles Calculator - This calculator determines the complementary and supplementary angle of a given angle that you enter OR it checks to see if two angles that you enter are complementary or supplementary.
Complex Number OperationsFree Complex Number Operations Calculator - Given two numbers in complex number notation, this calculator:
1) Adds (complex number addition), Subtracts (complex number subtraction), Multiplies (complex number multiplication), or Divides (complex number division) any 2 complex numbers in the form a + bi and c + di where i = √-1.
2) Determines the Square Root of a complex number denoted as √a + bi
3) Absolute Value of a Complex Number |a + bi|
4) Conjugate of a complex number a + bi
Composite NumberFree Composite Number Calculator - This calculator determines the nth composite number. Helps you generate composite numbers.
Compute a 75% Chebyshev interval around the mean for x values and also for y values.Compute a 75% Chebyshev interval around the mean for [I]x[/I] values and also for [I]y[/I] values.
[B][U]Grid E: [I]x[/I] variable[/U][/B]
11.92 34.86 26.72 24.50 38.93 8.59 29.31
23.39 24.13 30.05 21.54 35.97 7.48 35.97
[B][U]Grid H: [I]y[/I] variable[/U][/B]
27.86 13.29 33.03 44.31 16.58 42.43
39.61 25.51 39.14 16.58 47.13 14.70 57.47 34.44
According to Chebyshev's Theorem,
[1 - (1/k^2)] proportion of values will fall between Mean +/- (k*SD)
k in this case equal to z
z = (X-Mean)/SD
X = Mean + (z*SD)
1 - 1/k^2 = 0.75
- 1/k^2 = 0.75 - 1= - 0.25
1/k^2 = 0.25
k^2 = 1/0.25
k^2 = 4
k = 2
Therefore, z = k = 2
First, [URL='http://www.mathcelebrity.com/statbasic.php?num1=11.92%2C34.86%2C26.72%2C24.50%2C38.93%2C8.59%2C29.31%2C23.39%2C24.13%2C30.05%2C21.54%2C35.97%2C7.48%2C35.97&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']determine the mean and standard deviation of x[/URL]
Mean(x) = 25.24
SD(x) = 9.7873
Required Interval for x is:
Mean - (z * SD) < X < Mean + (z * SD)
25.24 - (2 * 9.7873) < X < 25.24 - (2 * 9.7873)
25.24 - 19.5746 < X < 25.24 + 19.5746
5.6654 < X < 44.8146
Next, [URL='http://www.mathcelebrity.com/statbasic.php?num1=27.86%2C13.29%2C33.03%2C44.31%2C16.58%2C42.43%2C39.61%2C25.51%2C39.14%2C16.58%2C47.13%2C14.70%2C57.47%2C34.44&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']determine the mean and standard deviation of y[/URL]
Mean(y) = 32.29
SD(y) = 9.7873
Required Interval for y is:
Mean - (z * SD) < Y < Mean + (z * SD)
32.29 - (2 * 13.1932) < Y < 32.29 - (2 * 13.1932)
32.29 - 26.3864 < Y < 32.29 + 26.3864
5.9036 < X < 58.6764
Confidence Interval/Hypothesis Testing for the Difference of MeansFree Confidence Interval/Hypothesis Testing for the Difference of Means Calculator - Given two large or two small distriutions, this will determine a (90-99)% estimation of confidence interval for the difference of means for small or large sample populations.
Also performs hypothesis testing including standard error calculation.
Congratulations!! You are hired at Roof and Vinyl Housing Systems. Your starting salary is $45,600 fCongratulations!! You are hired at Roof and Vinyl Housing Systems. Your starting salary is $45,600 for the year. Each year you stay employed with them your salary will increase by 3.5%. Determine what your salary would be if you worked for the company for 12 years.
Set up a function S(y) where y is the number of years after you start at the Roof and Vinyl place.
S(y) = 45600 * (1.035)^y <-- Since 3.5% = 0.035
The question asks for S(12):
S(12) = 45600 * (1.035)^12
S(12) = 45600 * 1.51106865735
S(12) = [B]68,904.73[/B]
Congruence Modulo nFree Congruence Modulo n Calculator - Given a possible congruence relation a ≡ b (mod n), this determines if the relation holds true (b is congruent to c modulo n).
Conner earned these scores on the first three tests in science this term: 86, 88, and 78. What is thConner earned these scores on the first three tests in science this term: 86, 88, and 78. What is the lowest that Conner can earn on the fourth and final test of the term if he wants to have an average of at least 83?
Using our [URL='https://www.mathcelebrity.com/missingaverage.php?num=86%2C+88%2C78&avg=83&pl=Calculate+Missing+Score']missing average calculator[/URL], we find that the fourth score must be [B]80[/B]
Consider the case of a manufacturer who has an automatic machine that produces an important part. PaConsider the case of a manufacturer who has an automatic machine that produces an important part. Past records indicate that at the beginning of the data the machine is set up correctly 70 percent of the time. Past experience also shows that if the machine is set up correctly it will produce good parts 90 percent of the time. If it is set up incorrectly, it will produce good parts 40 percent of the time. Since the machine will produce 60 percent bad parts, the manufacturer is considering using a testing procedure. If the machine is set up and produces a good part, what is the revised probability that it is set up correctly?
[U]Determine our events:[/U]
[LIST]
[*]C = Correctly Set Machine = 0.7
[*]C|G = Correctly Set Machine And Good Part = 0.9
[*]I = Incorrectly Set Machine = 1 - 0.7 = 0.3
[*]I|G = Incorrectly Set Machine And Good Part = 0.4
[*]B< = BAD PARTS = 0.60
[/LIST]
P[correctly set & part ok] = P(C) * P(C|G)
P[correctly set & part ok] = 70% * 90% = 63%
P[correctly set & part ok] = P(I) * P(I|G)
P[incorrectly set & part ok] = 30% *40% = 12%
P[correctly set | part ok] = P[correctly set & part ok]/(P[correctly set & part ok] + P[incorrectly set & part ok])
P[correctly set | part ok] = 63/(63+12) = [B]0.84 or 84%[/B]
Consider the following 15 numbers 1, 2, y - 4, 4, 5, x, 6, 7, 8, y, 9, 10, 12, 3x, 20 - The mean oConsider the following 15 numbers
1, 2, y - 4, 4, 5, x, 6, 7, 8, y, 9, 10, 12, 3x, 20
- The mean of the last 10 numbers is TWICE the mean of the first 10 numbers
- The sum of the last 2 numbers is FIVE times the sum of the first 3 numbers
(i) Calculate the values of x and y
We're given two equations:
[LIST=1]
[*](x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10 = 2(1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/10
[*]3x - 20 = 5(1 + 2 + y - 4)
[/LIST]
Let's evaluate and simplify:
[LIST=1]
[*](x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10 = (1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/5
[*]3x - 20 = 5(y - 1)
[/LIST]
Simplify some more:
[URL='https://www.mathcelebrity.com/polynomial.php?num=x%2B6%2B7%2B8%2By%2B9%2B10%2B12%2B3x%2B20&pl=Evaluate'](x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10[/URL] = (4x + y + 72)/10
[URL='https://www.mathcelebrity.com/polynomial.php?num=1%2B2%2By-4%2B4%2B5%2Bx%2B6%2B7%2B8%2By&pl=Evaluate'](1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/5[/URL] = (2y + x + 29)/5
5(y - 1) = 5y - 5
So we're left with:
[LIST=1]
[*](4x + y + 72)/10 = (2y + x + 29)/5
[*]3x - 20 = 5y - 5
[/LIST]
Cross multiply equations in 1, we have:
5(4x + y + 72) = 10(2y + x + 29)
20x + 5y + 360 = 20y + 10x + 290
We have:
[LIST=1]
[*]20x + 5y + 360 = 20y + 10x + 290
[*]3x - 20 = 5y - 5
[/LIST]
Combining like terms:
[LIST=1]
[*]10x - 15y = -70
[*]3x - 5y = 15
[/LIST]
Now we have a system of equations which we can solve any of three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10x+-+15y+%3D+-70&term2=3x+-+5y+%3D+15&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10x+-+15y+%3D+-70&term2=3x+-+5y+%3D+15&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10x+-+15y+%3D+-70&term2=3x+-+5y+%3D+15&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
(x, y) = [B](-115, -72)[/B]
Continuous AnnuityFree Continuous Annuity Calculator - Determines the Present Value and Accumulated Value of a Continuous Annuity
convert 5 minutes to hours expressing your answer as a fraction in its lowest termsconvert 5 minutes to hours expressing your answer as a fraction in its lowest terms
1 hour has 60 minutes, so we have:
5/60
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=5%2F60&frac2=3%2F8&pl=Simplify']fraction simplifier[/URL], we get:
[B]1/12[/B]
Cost Recovery MethodFree Cost Recovery Method Calculator - Given a sales price, cost, and set of payments, this determines the gross profit per year based on the cost recovery method.
Country A produces about 7 times the amount of diamonds in carats produce in Country B. If the totalCountry A produces about 7 times the amount of diamonds in carats produce in Country B. If the total produced in both countries is 40,000,000 carats, find the amount produced in each country.
Set up our two given equations:
[LIST=1]
[*]A = 7B
[*]A + B = 40,000,000
[/LIST]
Substitute (1) into (2)
(7B) + B = 40,000,000
Combine like terms
8B = 40,000,000
Divide each side by 8
[B]B = 5,000,000[/B]
Substitute this into (1)
A = 7(5,000,000)
[B]A = 35,000,000[/B]
Coupon ComparisonFree Coupon Comparison Calculator - Given a cost of goods, a dollar off coupon, and a percentage off coupon, this calculator will compare the two deals and determine which one is of more value. If the dollar coupon wins, the calculator will project the break even price where the dollar coupon would surpass the percentage coupon
Covariance and Correlation coefficient (r) and Least Squares Method and Exponential FitFree Covariance and Correlation coefficient (r) and Least Squares Method and Exponential Fit Calculator - Given two distributions X and Y, this calculates the following:
* Covariance of X and Y denoted Cov(X,Y)
* The correlation coefficient r.
* Using the least squares method, this shows the least squares regression line (Linear Fit) and Confidence Intervals of α and Β (90% - 99%)
Exponential Fit
* Coefficient of Determination r squared r2
* Spearmans rank correlation coefficient
* Wilcoxon Signed Rank test
Cross PartitionsFree Cross Partitions Calculator - Given a set of partitions, this determines the cross partitions.
Cross ProductFree Cross Product Calculator - Given two vectors A and B in R3, this calculates the cross product A × B as well as determine if the two vectors are parallel
cx+b/d=y for bcx+b/d=y for b
Subtract cx from each side to isolate b/d:
cx - cx + b/d = y - cx
Cancel the cx terms on each side:
b/d = y - cx
Cross multiply:
b = [B]d(y - cx)[/B]
Dad is (y) years old. Mom is 5 years younger than Dad. What is the total of their agesDad is (y) years old. Mom is 5 years younger than Dad. What is the total of their ages
Dad's age:
y
Mom's age (younger means we subtract):
y - 5
The total of their ages is found by adding them together:
y + y - 5
Group like terms, and we get:
[B]2y - 5[/B]
Dave has a savings account that pays interest at 3 1/2% per year. His opening balance for May was $1Dave has a savings account that pays interest at 3 1/2% per year. His opening balance for May was $1374.67. He did not deposit or withdraw money during the month. The interest is calculated daily. How much interest did the account earn in May?
First, determine n, which is 31, since May has 31 days.
We use our [URL='http://www.mathcelebrity.com/compoundint.php?bal=1374.67&nval=31&int=3.5&pl=Daily']compound interest balance calculator[/URL] to get:
[B]1,378.76[/B]
Decagonal NumberFree Decagonal Number Calculator - This calculator determines the nth decagonal number
DecayFree Decay Calculator - Determines decay based on an initial mass and decay percentage and time.
Determine a conversion ratio that could be used to convert miles to inchesDetermine a conversion ratio that could be used to convert miles to inches.
We know that 1 mile equals 5,280 feet.
We know that 1 foot equals 12 inches.
So 1 miles = 5,280 feet * 12 inches per foot= [B]63,360 inches[/B]
Determine if the statement below is True or FalseDetermine if the statement below is True or False
If B ⊂ A, then A ∩ B = B
Is this statement True or False?
[B]True:[/B] If B ⊂ A, then B ∈ A
So A ∩ B is the similar elements of both. B contains itself as a subset.
So this is [U]true[/U]
Determine the area under the standard normal curve that lies between:Determine the area under the standard normal curve that lies between:
(a) Z = -0.38 and Z = 0.38
(b) Z = -2.66 and Z = 0
(c) Z = -1.04 and Z - 1.67
[B](a) 0.2961 using our [URL='http://www.mathcelebrity.comzscore.php?z=+p%28-0.38%3Cz%3C0.38%29&pl=Calculate+Probability']z score calculator[/URL]
(b) 0.4961 using our [URL='http://www.mathcelebrity.com/zscore.php?z=+p%28-2.66%3Cz%3C0%29&pl=Calculate+Probability']z score calculator[/URL]
(c) 0.8034 using our [URL='http://www.mathcelebrity.com/zscore.php?z=+p%28-1.04%3Cz%3C1.67%29&pl=Calculate+Probability']z score calculator[/URL][/B]
Determine the formula of the given statement by following the procedures. Choose any number then addDetermine the formula of the given statement by following the procedures. Choose any number then add 2. Multiply your answer to 3 and minus 2
For the phrase [I]choose any number[/I] we can use an arbitrary variable, let's call it x.
Add 2:
x + 2
Multiply your answer to 3:
3(x + 2)
And minus 2 which means we subtract:
[B]3(x + 2) - 2[/B]
Determine ux and sigma(x) from the given parameters of the population and sample size u = 76, sigmaDetermine ux and sigma(x) from the given parameters of the population and sample size
u = 76, sigma = 28, n = 49
ux = ?
sigma(x) = ?
[B]u = ux = 76[/B]
sigma(x) = sigma/sqrt(n) so we have
28/sqrt(49) = 28/7 = [B]4[/B]
Determine whether the random variable is discrete or continuous. In each case, state the possible vDetermine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable.
(a) The number of customers arriving at a bank between noon and 1:00 P.M.
(i) The random variable is continuous. The possible values are x >= 0.
(ii) The random variable is discrete. The possible values are x = 0, 1, 2,...
(iii) The random variable is continuous. The possible values are x = 0, 1, 2,...
(iv) The random variable is discrete. The possible values are x >= 0.
(b) The amount of snowfall
(i) The random variable is continuous. The possible values are s = 0, 1, 2,...
(ii) The random variable is discrete. The possible values are s >= 0.
(iii) The random variable is discrete. The possible values are s = 0, 1, 2,...
(iv) The random variable is continuous. The possible values are s >= 0.
[B](a) (ii) The random variable is discrete. The possible values are x = 0, 1, 2,...
Discrete variables are limited in the values they can take between 9 and ∞
(b) (iv) The random variable is continuous. The possible values are s >= 0. Snowfall can be a decimal and can vary between 0 and ∞[/B]
Determine whether the statement is true or false. If 0 < a < b, then Ln a < Ln bDetermine whether the statement is true or false. If 0 < a < b, then Ln a < Ln b
We have a logarithmic property that states:
ln(a) - ln(b) = ln (a / b)
We're given a < b, so (a / b) < 1
Therefore:
ln (a / b) < 0
And since ln(a) - ln(b) = ln (a / b)
Then Ln(a) - Ln(b) < 0
Add Ln(b) to each side and we get:
Ln(a) - Ln(b) + Ln(b) < 0 + Ln(b)
Cancel the Ln(b) on the left side and we get:
Ln(a)
Determine whether the statement is true or false. If y = e^2, then y’ = 2eDetermine whether the statement is true or false. If y = e^2, then y’ = 2e
e^2 is a constant, and the derivative of a constant is 0. So y' = 0
So this is [B]FALSE[/B]
Determine whether the statement is true or false. You can always divide by e^xDetermine whether the statement is true or false. You can always divide by e^x
[B]True. As x --> infinity, 1/e^x approaches 0 but never touches it.[/B]
Dewey Decimal System ClassificationFree Dewey Decimal System Classification Calculator - Given a 3 digit code, this will determine the class, division, and section of the library book using the Dewey Decimal System.
difference between 2 positive numbers is 3 and the sum of their squares is 117difference between 2 positive numbers is 3 and the sum of their squares is 117
Declare variables for each of the two numbers:
[LIST]
[*]Let the first variable be x
[*]Let the second variable be y
[/LIST]
We're given 2 equations:
[LIST=1]
[*]x - y = 3
[*]x^2 + y^2 = 117
[/LIST]
Rewrite equation (1) in terms of x by adding y to each side:
[LIST=1]
[*]x = y + 3
[*]x^2 + y^2 = 117
[/LIST]
Substitute equation (1) into equation (2) for x:
(y + 3)^2 + y^2 = 117
Evaluate and simplify:
y^2 + 3y + 3y + 9 + y^2 = 117
Combine like terms:
2y^2 + 6y + 9 = 117
Subtract 117 from each side:
2y^2 + 6y + 9 - 117 = 117 - 117
2y^2 + 6y - 108 = 0
This is a quadratic equation:
Solve the quadratic equation 2y2+6y-108 = 0
With the standard form of ax2 + bx + c, we have our a, b, and c values:
a = 2, b = 6, c = -108
Solve the quadratic equation 2y^2 + 6y - 108 = 0
The quadratic formula is denoted below:
y = -b ± sqrt(b^2 - 4ac)/2a
[U]Step 1 - calculate negative b:[/U]
-b = -(6)
-b = -6
[U]Step 2 - calculate the discriminant Δ:[/U]
Δ = b2 - 4ac:
Δ = 62 - 4 x 2 x -108
Δ = 36 - -864
Δ = 900 <--- Discriminant
Since Δ is greater than zero, we can expect two real and unequal roots.
[U]Step 3 - take the square root of the discriminant Δ:[/U]
√Δ = √(900)
√Δ = 30
[U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U]
Numerator 1 = -b + √Δ
Numerator 1 = -6 + 30
Numerator 1 = 24
[U]Step 5 - find numerator 2 which is -b - the square root of the Discriminant:[/U]
Numerator 2 = -b - √Δ
Numerator 2 = -6 - 30
Numerator 2 = -36
[U]Step 6 - calculate your denominator which is 2a:[/U]
Denominator = 2 * a
Denominator = 2 * 2
Denominator = 4
[U]Step 7 - you have everything you need to solve. Find solutions:[/U]
Solution 1 = Numerator 1/Denominator
Solution 1 = 24/4
Solution 1 = 6
Solution 2 = Numerator 2/Denominator
Solution 2 = -36/4
Solution 2 = -9
[U]As a solution set, our answers would be:[/U]
(Solution 1, Solution 2) = (6, -9)
Since one of the solutions is not positive and the problem asks for 2 positive number, this problem has no solution
Digit ProblemsFree Digit Problems Calculator - Determines how many (n) digit numbers can be formed based on a variety of criteria.
Digraph ItemsFree Digraph Items Calculator - Given a digraph, this determines the leader, and symmetric matrix.
divide the sum of the square of a and b by thrice cdivide the sum of the square of a and b by thrice c
Sum of the squares of a and b is found as follows:
[LIST]
[*]a squared means we raise a to the power of 2: a^2
[*]b squared means we raise b to the power of 2: b^2
[*]Sum of the squares means we add both terms: a^2 + b^2
[*]Thrice c means we multiply c by 3: 3c
[/LIST]
Divide means we have a quotient:
[B](a^2 + b^2)/3c[/B]
Dividend Discount ModelFree Dividend Discount Model Calculator - This calculator determines the present value of dividends using the Dividend Discount Model.
During a recent season Miguel Cabrera and Mike Jacobs hit a combined total of 46 home runs. CabreraDuring a recent season Miguel Cabrera and Mike Jacobs hit a combined total of 46 home runs. Cabrera hit 6 more home runs than Jacobs how many home runs did each player hit
Let c be Miguel Cabrera's home runs and j be Mike Jacobs home runs. We are given two equations:
[LIST=1]
[*]c + j = 46
[*]c = j + 6
[/LIST]
Substitute (2) into (1)
(j + 6) + j = 46
Combine like terms:
2j + 6 = 46
[URL='https://www.mathcelebrity.com/1unk.php?num=2j%2B6%3D46&pl=Solve']Plugging this into our equation calculator[/URL], we get [B]j = 20[/B].
Substitute this into equation (2), we have:
c = 20 + 6
[B]c = 26
[/B]
Therefore, Mike Jacobs hit 20 home runs and Miguel Cabrera hit 26 home runs.
Earnings Before Interest and Taxes (EBIT) and Net IncomeFree Earnings Before Interest and Taxes (EBIT) and Net Income Calculator - Given inputs of sales, fixed costs, variable costs, depreciation, and taxes, this will determine EBIT and Net Income and Profit Margin
Elapsed TimeFree Elapsed Time Calculator - This determines the elapsed time between two clock readings.
Ellen reads 23 pages in 40 minutes. Rob reads 9 pages in 16 minutes. Who is the faster reader? JuEllen reads 23 pages in 40 minutes. Rob reads 9 pages in 16 minutes. Who is the faster reader? Justify your answer.
Compare in terms of pages per minute.
Ellen = 23 pages / 40 minutes =0.575 pages per minute
Rob = 9 pages / 16 minutes = 0.5625 pages per minute
[B]Ellen reads faster.[/B]
Emil bought a camera for $268.26, including tax. He made a down payment of $12.00 and paid the balanEmil bought a camera for $268.26, including tax. He made a down payment of $12.00 and paid the balance in 6 equal monthly payments. What was Emil’s monthly payment for this camera?
Calculate remaining balance
268.26 - 12 = 256.26
Determine monthly payment:
256.26/6 = [B]21.36[/B]
Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30.Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30. What is the greatest age Mary could be?
Let e = Emily's age and m = Mary's age.
We have the equation e = 2m + 3 and the inequality e + m < 30
Substitute the equation for e into the inequality:
2m + 3 + m < 30
Add the m terms
3m + 3 < 30
Subtract 3 from each side of the inequality
3m < 27
Divide each side of the inequality by 3 to isolate m
m < 9
Therefore, the [B]greatest age[/B] Mary could be is 8, since less than 9 [U]does not include[/U] 9.
Equivalent FractionsFree Equivalent Fractions Calculator - Given a fraction, this will determine equivalent fractions
Eric is taking a trip of 245 miles. If he has traveled x miles, represent the remainder of the tripEric is taking a trip of 245 miles. If he has traveled x miles, represent the remainder of the trip in terms of x.
Remaining distance = [B]245 - x[/B]
Erin has 72 stamps in her stamp drawer along with a quarter, two dimes and seven pennies. She has 3Erin has 72 stamps in her stamp drawer along with a quarter, two dimes and seven pennies. She has 3 times as many 3-cent stamps as 37-cent stamps and half the number of 5-cent stamps as 37-cent stamps. The value of the stamps and coins is $8.28. How many 37-cent stamps does Erin have?
Number of stamps:
[LIST]
[*]Number of 37 cent stamps = s
[*]Number of 3-cent stamps = 3s
[*]Number of 5-cent stamps = 0.5s
[/LIST]
Value of stamps and coins:
[LIST]
[*]37 cent stamps = 0.37s
[*]3-cent stamps = 3 * 0.03 = 0.09s
[*]5-cent stamps = 0.5 * 0.05s = 0.025s
[*]Quarter, 2 dime, 7 pennies = 0.52
[/LIST]
Add them up:
0.37s + 0.09s + 0.025s + 0.52 = 8.28
Solve for [I]s[/I] in the equation 0.37s + 0.09s + 0.025s + 0.52 = 8.28
[SIZE=5][B]Step 1: Group the s terms on the left hand side:[/B][/SIZE]
(0.37 + 0.09 + 0.025)s = 0.485s
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
0.485s + 0.52 = + 8.28
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 0.52 and 8.28. To do that, we subtract 0.52 from both sides
0.485s + 0.52 - 0.52 = 8.28 - 0.52
[SIZE=5][B]Step 4: Cancel 0.52 on the left side:[/B][/SIZE]
0.485s = 7.76
[SIZE=5][B]Step 5: Divide each side of the equation by 0.485[/B][/SIZE]
0.485s/0.485 = 7.76/0.485
s = [B]16[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=0.37s%2B0.09s%2B0.025s%2B0.52%3D8.28&pl=Solve']Source[/URL]
Event LikelihoodFree Event Likelihood Calculator - Given a probability, this determines how likely that event is
Expected FrequencyFree Expected Frequency Calculator - Given a contingency table (two-way table), this will calculate expected frequencies and then determine a conclusion based on a Χ2 test with critical value test and conclusion.
Express cos4θ and sin4θ in terms of sines and cosines of multiples of θExpress cos4θ and sin4θ in terms of sines and cosines of multiples of θ.
Using a trignometric identity:
cos (2θ) = cos^2(θ) - sin^2(θ)
Since 4θ = 2*2θ, so we have:
[B]cos(4θ) = cos^2(2θ) - sin^2(2θ)[/B]
Using another trignometric identity, we have:
sin(2θ) = 2 sin(θ) cos(θ)
Since 4θ = 2*2θ, so we have:
[B]sin(4θ) = 2 sin(2θ) cos(2θ)[/B]
F varies directly as g and inversely as r^2F varies directly as g and inversely as r^2
[U]Givens and assumptions[/U]
[LIST]
[*]We take a constant of variation called k.
[*][I]Varies directly means we multiply our variable term by k[/I]
[*][I]Varies inversely means we divide k by our variable term[/I]
[/LIST]
The phrase varies directly or varies inversely means we have a constant k such that:
[B]F = kg/r^2[/B]
Factor the following expression: 4a(9b + 7) - 3(9b + 7)Factor the following expression: 4a(9b + 7) - 3(9b + 7)
The common factor in each term is (9b + 7). Factor this out and we get:
[B](4a - 3)(9b + 7)
[MEDIA=youtube]0thkumkEie0[/MEDIA][/B]
Factoring and Root FindingFree Factoring and Root Finding Calculator - This calculator factors a binomial including all 26 variables (a-z) using the following factoring principles:
* Difference of Squares
* Sum of Cubes
* Difference of Cubes
* Binomial Expansions
* Quadratics
* Factor by Grouping
* Common Term
This calculator also uses the Rational Root Theorem (Rational Zero Theorem) to determine potential roots
* Factors and simplifies Rational Expressions of one fraction
* Determines the number of potential positive and negative roots using Descarte’s Rule of Signs
Find 2 consecutive numbers such that the sum of twice the smaller number and 3 times the larger numbFind 2 consecutive numbers such that the sum of twice the smaller number and 3 times the larger number is 73.
Let x be the smaller number and y be the larger number. We are given:
2x + 3y = 73
Since the numbers are consecutive, we know that y = x + 1. Substitute this into our given equation:
2x + 3(x + 1) = 73
Multiply through:
2x + 3x + 3 = 73
Group like terms:
5x + 3 = 73
[URL='https://www.mathcelebrity.com/1unk.php?num=5x%2B3%3D73&pl=Solve']Type 5x + 3 = 73 into the search engine[/URL], and we get [B]x = 14[/B].
Our larger number is 14 + 1 = [B]15
[/B]
Therefore, our consecutive numbers are[B] (14, 15)[/B]
Find 3 consecutive integers such that the sum of twice the smallest and 3 times the largest is 126Find 3 consecutive integers such that the sum of twice the smallest and 3 times the largest is 126.
Let the first integer be n, the second integer be n + 1, and the third integer be n + 2. We have:
Sum of the smallest and 3 times the largest is 126:
n + 3(n + 2) = 126
Multiply through:
n + 3n + 6 = 126
Group like terms:
4n + 6 = 126
[URL='https://www.mathcelebrity.com/1unk.php?num=4n%2B6%3D126&pl=Solve']Type 4n + 6 = 126 into our calculator[/URL], we get n = 30. Which means the next two integers are 31 and 32.
[B]{30, 31, 32}[/B]
Find 3 Even Integers with a sum of 198Find 3 Even Integers with a sum of 198
Let x be the first even integer. Then y is the next, and z is the third even integer.
[LIST=1]
[*]y = x + 2
[*]z = x + 4
[*]x + y + z = 198
[/LIST]
Substituting y and z into (3):
x + x + 2 + x + 4 = 198
Group x terms
3x + 6 = 198
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=3x%2B6%3D198&pl=Solve']equation solver[/URL], we get:
[B]x = 64[/B]
y = 64 + 2
[B]y= 66[/B]
z = 64 + 4
[B]z = 68[/B]
Find four consecutive odd numbers which add to 64Find four consecutive odd numbers which add to 64.
Let the first number be x. The next three numbers are:
x + 2
x + 4
x + 6
Add them together to get 64:
x + (x + 2) + (x + 4) + (x + 6) = 64
Group like terms:
4x + 12 = 64
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=4x%2B12%3D64&pl=Solve']equation calculator[/URL], we get:
[B]x = 13[/B]
The next 3 odd numbers are:
x + 2 = 13 + 2 = 15
x + 4 = 13 + 4 = 17
x + 6 = 13 + 6 = 19
So the 4 consecutive odd numbers which add to 64 are:
[B](13, 15, 17, 19)[/B]
Find Mean 106 and standard deviation 10 of the sample mean which is 25mean of 106 inches and a standard deviation of 10 inches and for sample of size is 25. Determine the mean and the standard deviation of /x
Find Mean 106 and standard deviation 10 of the sample mean which is 25Do you mean x bar?
mean of 106 inches and a standard deviation of 10 inches and for sample of size is 25. Determine the mean and the standard deviation of /x
If so, x bar equals the population mean. So it's [B]106[/B].
Sample standard deviation = Population standard deviation / square root of n
10/Sqrt(25)
10/5
[B]2[/B]
Find two consecutive odd integers such that the sum of their squares is 290Find two consecutive odd integers such that the sum of their squares is 290.
Let the first odd integer be n.
The next odd integer is n + 2
Square them both:
n^2
(n + 2)^2 = n^2 + 4n + 4 from our [URL='https://www.mathcelebrity.com/expand.php?term1=%28n%2B2%29%5E2&pl=Expand']expansion calculator[/URL]
The sum of the squares equals 290
n^2 + n^2 + 4n + 4 = 290
Group like terms:
2n^2 + 4n + 4 = 290
[URL='https://www.mathcelebrity.com/quadratic.php?num=2n%5E2%2B4n%2B4%3D290&pl=Solve+Quadratic+Equation&hintnum=+0']Enter this quadratic into our search engine[/URL], and we get:
n = 11, n = -13
Which means the two consecutive odd integer are:
11 and 11 + 2 = 13. [B](11, 13)[/B]
-13 and -13 + 2 = -11 [B](-13, -11)[/B]
Find two consecutive positive integers such that the sum of their squares is 25Find two consecutive positive integers such that the sum of their squares is 25.
Let the first integer be x. The next consecutive positive integer is x + 1.
The sum of their squares equals 25. We write this as::
x^2 + (x + 1)^2
Expanding, we get:
x^2 + x^2 + 2x + 1 = 25
Group like terms:
2x^2 + 2x + 1 = 25
Subtract 25 from each side:
2x^2 + 2x - 24 = 0
Simplify by dividing each side by 2:
x^2 + x - 12 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-12%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get x = 3 or x = -4. The problem asks for positive integers, so we discard -4, and use 3.
This means, our next positive integer is 3 + 1 = 4. So we have [B](3, 4) [/B]as our answers.
Let's check our work:
3^2 + 4^2 = 9 + 16 = 25
Find two numbers word problemsFree Find two numbers word problems Calculator - Given two numbers with a sum of s where one number is n greater than another, this calculator determines both numbers.
FloorFree Floor Calculator - Determines the floor of a number
For g(x) = 4-5x, determine the input for x when the output of g(x) is -6For g(x) = 4-5x, determine the input for x when the output of g(x) is -6
We want to know when:
4 - 5x = 6
To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=4-5x%3D6&pl=Solve']type it in our search engine[/URL] and we get:
x = [B]-0.4 or -2/5[/B]
For the first 10 seconds of the ride, the height of the coaster can be determined by h(t) = 0.3t^3 -For the first 10 seconds of the ride, the height of the coaster can be determined by h(t) = 0.3t^3 - 5t^2 + 21t, where t is the time in seconds and h is the height in feet. classify this polynomial by degree and by number of terms.
[URL='http://www.mathcelebrity.com/polynomial.php?num=0.3t%5E3-5t%5E2%2B21t&pl=Evaluate']Using our polynomial calculator, we determine[/URL]:
[LIST]
[*]The degree of the polynomial is 3
[*]There are 3 terms
[/LIST]
Foster is centering a photo that is 9/1/2 inches wide on a scrapbook pages that is 10 inches wide. HFoster is centering a photo that is 9/1/2 inches wide on a scrapbook pages that is 10 inches wide. How far from each side of the pages should he put the picture? Enter your answer as a mixed number.
First, determine your margins, which is the difference between the width and photo width, divided by 2.
10 - 9 & 1/2 = 1/2
1/2 / 2 = [B]1/4[/B]
Frequency and Wavelength and Photon EnergyFree Frequency and Wavelength and Photon Energy Calculator - Provides the following 3 items using the speed of light and Plancks constant (h):
- Given a frequency of centimeters, feet, meters, or miles the calculator will determine wavelength in Hz, KHz, MHz, GHz
-
Given a wavelength of Hz, KHz, MHz, GHz, the calculator will determine frequency in centimeters, feet, meters, or miles
- Calculates photon energy
Frequency Distribution TableFree Frequency Distribution Table Calculator - Determines the classes and frequency distribution using the 2 to k rule.
g=1+2a/aCancel the a's in the last term:
g = 1 + 2
g = 3
Gary has three less pets than Abe. If together they own 15 pets, how many pets does Gary own?Let g = Gary's pets and a = Abe's pets.
We are given two equations:
(1) g = a - 3
(2) a + g = 15
Substitute (1) into (2)
a + (a - 3) = 15
Combine Like Terms:
2a - 3 = 15
Add 3 to each side:
2a = 18
Divide each side by 2 to isolate a:
a = 9 --> Abe has 9 pets
Substitute a = 9 into Equation (1)
g = 9 - 3
g = 6 --> Gary has 6 pets
Gayle has 36 coins, all nickels and dimes, worth $2.40. How many dimes does she have?Gayle has 36 coins, all nickels and dimes, worth $2.40. How many dimes does she have?
Set up our given equations using n as the number of nickels and d as the number of dimes:
[LIST=1]
[*]n + d = 36
[*]0.05n + 0.1d = 2.40
[/LIST]
Use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=n+%2B+d+%3D+36&term2=0.05n+%2B+0.1d+%3D+2.40&pl=Cramers+Method']simultaneous equations calculator[/URL] to get:
n = 24
[B]d = 12[/B]
Germany and Austria have a total of 25 states. Germany has 7 more states than Austria has. Create 2Germany and Austria have a total of 25 states. Germany has 7 more states than Austria has. Create 2 equations.
Let g be the number of German states. Let a be the number of Austrian states. We're given two equations:
[LIST=1]
[*]a + g = 25
[*]g = a + 7
[/LIST]
To solve this system of equations, we substitute equation (2) into equation (1) for g:
a + (a + 7) = 25
Combine like terms:
2a + 7 = 25
To solve for a, we[URL='https://www.mathcelebrity.com/1unk.php?num=2a%2B7%3D25&pl=Solve'] type this equation into our search engine[/URL] and we get:
[B]a = 9[/B]
To solve for g, we plug in a = 9 into equation (2):
g = 9 + 7
[B]g = 16[/B]
Given g(a)=a² - 2a - 1 and f(x)=x² - 2x, Find: a) f(a+2)-f(a)/2 b) g(a+h)-g(a)/hGiven g(a)=a² - 2a - 1 and f(x)=x² - 2x:
Find:
a) f(a+2) - f(a)/2
b) g(a+h) - g(a)/h
a) f(a + 2) = (a + 2)^2 - 2(a + 2)
f(a + 2) = a^2 + 2a + 4 - 2a - 4
Simplify and combine like terms:
the 2a and 4's cancel, so we have:
f(a + 2) = a^2
f(a)/2 = (a^2 - 2a)/2
Subtract one from the other, we get:
a^2 - a^2/2 - a
[B]a) a^2/2 - a
------------------------[/B]
b) g(a + h) = (a + h)^2 - 2(a + h) - 1
g(a + h) = a^2 +2ah + h^2 - 2a - 2h - 1
g(a)/2 = (a^2 - 2a - 1)/h
g(a)/2 = (a^2 - 2a - 1)/h
Subtract one from the other:
g(a+h) - g(a)/h
a^2 +2ah + h^2 - 2a - 2h - 1 - (a^2 - 2a - 1)/h
Multiply through by h
[B]a^2h + 2ah^2 + h^3 - 2ah - 2h^2 - h - a^2 + 2a + 1[/B]
Given that E[Y]=2 and Var [Y] =3, find E[(2Y + 1)^2]Given that E[Y]=2 and Var [Y] =3, find E[(2Y + 1)^2]
Multiply through
E[(2Y + 1)^2] = E[4y^2 + 4y + 1]
We can take the expected value of each term
E[4y^2] + E[4y] + E[1]
For the first term, we have:
4E[Y^2]
We define the Var[Y] = E[Y^2] - (E[Y])^2
Rearrange this term, we have E[Y^2] = Var[Y] + (E[Y])^2
E[Y^2] = 3+ 2^2
E[Y^2] = 3+ 4
E[Y^2] = 7
So our first term is 4(7) = 28
For the second term using expected value rules of separating out a constant, we have
4E[Y] = 4(2) = 8
For the third term, we have:
E[1] = 1
Adding up our three terms, we have:
E[4y^2] + E[4y] + E[1] = 28 + 8 + 1
E[4y^2] + E[4y] + E[1] = [B]37[/B]
Given w(x) = 3x + 8, find w(2b + 6).Given w(x) = 3x + 8, find w(2b + 6).
Plug the value of 2b + 6 in for x
w(2b + 6) = 3(2b + 6) + 8
Multiply through:
w(2b + 6) = 6b + 18 + 8
Group like terms:
w(2b + 6) = [B]6b + 26[/B]
Grandmother, mother and daughter are celebrating 150 years of life. The Mother is 25 years older thaGrandmother, mother and daughter are celebrating 150 years of life. The Mother is 25 years older than her daughter, but 31 years younger than her mother (the grandmother). How old are the three
Let grandmother's age be g. Let mother's age be m. Let daughter's age be d. We're given 3 equations:
[LIST=1]
[*]m = d + 25
[*]m = g - 31
[*]d + g + m = 150
[/LIST]
This means the daughter is:
d = 25 + 31 = 56 years younger than her grandmother. So we have:
4. d = g - 56
Plugging in equation (2) and equation(4) into equation (3) we get:
g - 56 + g + g - 31
Combine like terms:
3g - 87 = 150
[URL='https://www.mathcelebrity.com/1unk.php?num=3g-87%3D150&pl=Solve']Typing this equation into the search engine[/URL], we get:
[B]g = 79[/B]
Plug this into equation (2):
m = 79 - 31
[B]m = 48[/B]
Plug this into equation (4):
d = 79 - 56
[B]d = 23[/B]
Gravitational ForceFree Gravitational Force Calculator - Using Sir Isaac Newtons Law of Gravitational Force, this calculator determines the force between two objects with mass in kilograms at a distance apart in meters using the constant of gravity.
Greatest Common Factor and Least Common MultipleFree Greatest Common Factor and Least Common Multiple Calculator - Given 2 or 3 numbers, the calculator determines the following:
* Greatest Common Factor (GCF) using Factor Pairs
* Rewrite Sum using the Distributive Property and factoring out the GCF
* Least Common Multiple (LCM) / Least Common Denominator (LCD) using Factor Pairs
* GCF using the method of Successive Division
* GCF using the Prime Factorization method
* Determine if the numbers are coprime and twin prime
Greatest Common Factors of MonomialsFree Greatest Common Factors of Monomials Calculator - This calculator will determine the Greatest Common Factors of a set of Monomials
Group CombinationsFree Group Combinations Calculator - Given an original group of certain types of member, this determines how many groups/teams can be formed using a certain condition.
Half of a pepperoni pizza plus 3/4ths of a ham and pineapple pizza has 765 calories. 1/4th of a peppHalf of a pepperoni pizza plus 3/4ths of a ham and pineapple pizza has 765 calories. 1/4th of a pepperoni pizza and a whole ham and pineapple pizza contains 745 calories. How many calories are each of the 2 kinds of pizzas individually?
Let p be the pepperoni pizza calories and h be the ham and pineapple pizza calories. We're given
[LIST=1]
[*]0.5p + 0.75h = 765
[*]0.25p + h = 745
[/LIST]
With this system of equations, we can solve using 3 methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.5p+%2B+0.75h+%3D+765&term2=0.25p+%2B+h+%3D+745&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.5p+%2B+0.75h+%3D+765&term2=0.25p+%2B+h+%3D+745&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.5p+%2B+0.75h+%3D+765&term2=0.25p+%2B+h+%3D+745&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we choose, we get:
[B]h = 580
p = 660[/B]
Half-Life of a SubstanceFree Half-Life of a Substance Calculator - Given a half-life (h) of a substance at time t, this determines the new substance size at time tn, otherwise known as decay.
Help PlzNick's age: x
John's age: x/2
Pip's age = 2/3 * x/2 = x/3
The sum is 26, so we have:
x + x/2 + x/3 = 26
Common denominator is (1 * 2 * 3) = 6
6x/6 + 3x/6 + 2x/6 = 26
Combine like terms:
11x/6 = 26
Cross multiply:
11x = 156
x = 14.1818
This doesn't make sense for age. Are you sure you wrote out the problem right?
HELP SOLVEPerform a one-sample z-test for a population mean. Be sure to state the hypotheses and the significance level, to compute the value of the test statistic, to obtain the P-value, and to state your conclusion.
Five years ago, the average math SAT score for students at one school was 475. A teacher wants to perform a hypothesis test to determine whether the mean math SAT score of students at the school has changed. The mean math SAT score for a random sample of 40 students from this school is 469. Do the data provide sufficient evidence to conclude that the mean math SAT score for students at the school has changed from the previous mean of 475? Perform the appropriate hypothesis test using a significance level of 10%. Assume that Η = 73.
Heptagonal NumberFree Heptagonal Number Calculator - This calculator determines the nth heptagonal number
Herfindahl IndexFree Herfindahl Index Calculator - Given a market share of a set of companies, this determines the Herfindahl Index and Normalized Herfindahl Index.
Hexagonal NumberFree Hexagonal Number Calculator - This calculator determines the nth hexagonal number
I bought four candles that cost 7.00, 8.00, 9.00, and 20.00. If I have a budget of $50, how much canI bought four candles that cost 7.00, 8.00, 9.00, and 20.00. If I have a budget of $50, how much can I spend on the last candle?
[U]Add up your total spending:[/U]
7 + 8 + 9 + 20 = 44
[U]Determine your remaining budget:[/U]
Remaining Budget = Total Budget - Spending
Remaining Budget = 50 - 44
Remaining Budget = [B]$6.00[/B]
I had a brother but my brother had no brothers. how can this beI had a brother but my brother had no brothers. how can this be
Because "I" is a female.
To solve trick questions like this, you must expand your theory of constraints.
Most people look at this problem and see the word [I]brother [/I]twice and limit themselves to thinking in terms of men.
I HAVE $11.60, all dimes and quarters, in my pocket. I have 32 more dimes than quarters. how many diI HAVE $11.60, all dimes and quarters, in my pocket. I have 32 more dimes than quarters. how many dimes, and how many quarters do i have?
Let d = dimes and q = quarters.
We have two equations:
[LIST=1]
[*]0.10d + 0.25q = 11.60
[*]d - q = 32
[/LIST]
Set up a [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=0.10d+%2B+0.25q+%3D+11.60&term2=d+-+q+%3D+32&pl=Cramers+Method']system of equations[/URL] to solve for d and q.
[B]dimes (d) = 56 and quarters (q) = 24[/B]
Check our work:
56 - 24 = 32
0.10(56) + 0.25(24) = $5.60 + $6.00 = $11.60
I have 20 bills consisting of $5 and $10. If the total amount of my money is $130, how many of eachI have 20 bills consisting of $5 and $10. If the total amount of my money is $130, how many of each bill do i have?
Let f be $5 bills and t be $10 bills, we have:
f + t = 20
5f + 10t = 130
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=f%2Bt%3D20&term2=5f+%2B+10t+%3D+130&pl=Cramers+Method']system of equation solver[/URL], we get:
[LIST]
[*][B]f = 14[/B]
[*][B]t = 6[/B]
[/LIST]
If 100 runners went with 4 bicyclists and 5 walkers, how many bicyclists would go with 20 runners anIf 100 runners went with 4 bicyclists and 5 walkers, how many bicyclists would go with 20 runners and 2 walkers?
[U]Set up a joint variation equation, for the 100 runners, 4 bicyclists, and 5 walkers:[/U]
100 = 4 * 5 * k
100 = 20k
[U]Divide each side by 20[/U]
k = 5 <-- Coefficient of Variation
[U]Now, take scenario 2 to determine the bicyclists with 20 runners and 2 walkers[/U]
20 = 2 * 5 * b
20 = 10b
[U]Divide each side by 10[/U]
[B]b = 2[/B]
If 11 times a number is added to twice the number, the result is 104If 11 times a number is added to twice the number, the result is 104
Let [I]the number[/I] be an arbitrary variable we call x.
11 times a number:
11x
Twice the number (means we multiply x by 2):
2x
The phrase [I]is added to[/I] means we add 2x to 11x:
11x + 2x
Simplify by grouping like terms:
(11 + 2)x = 13x
The phrase [I]the result is[/I] means an equation, so we set 13x equal to 104:
13x = 104 <-- This is our algebraic expression
To solve this equation for x, [URL='https://www.mathcelebrity.com/1unk.php?num=13x%3D104&pl=Solve']we type it in our search engine[/URL] and we get:
x = [B]8[/B]
If 12 times a number is added to twice the number, the result is 112If 12 times a number is added to twice the number, the result is 112.
Let the number be n, so we have:
12n + 2n = 112
Combine like terms
14n = 112
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=14n%3D112&pl=Solve']equation solver[/URL], we get [B]n = 8[/B].
if 200 is divided in the ratio of 1:3:4 , what is the greatest numberif 200 is divided in the ratio of 1:3:4 , what is the greatest number
Determine the ratio denominator by adding up the ratio amounts:
1 + 3 + 4 = 8
So we have the following ratios and ratio amounts with our greatest number in bold:
[LIST]
[*]1/8 * 200 = 25
[*]3/8 * 200 = 75
[*]4/8 * 200 = [B]100[/B]
[/LIST]
If 4(x-9)=3x-8x, what is x?[SIZE=5]If 4(x-9)=3x-8x, what is x?
[/SIZE]
[SIZE=4]Multiply through:
4x - 36 = 3x - 8x
Group like terms:
4x - 36 = -5x
[/SIZE]
[URL='https://www.mathcelebrity.com/1unk.php?num=4x-36%3D-5x&pl=Solve'][SIZE=4]Typing this equation into the search[/SIZE][/URL][SIZE=4][URL='https://www.mathcelebrity.com/1unk.php?num=4x-36%3D-5x&pl=Solve'] engine[/URL], we get:
[B]x = 4[/B][/SIZE]
If 4x+7=xy-6, then what is the value of x, in terms of yIf 4x+7=xy-6, then what is the value of x, in terms of y
Subtract xy from each side:
4x + 7 - xy = -6
Add 7 to each side:
4x - xy = -6 - 7
4x - xy = -13
Factor out x:
x(4 - y) = -13
Divide each side of the equation by (4 - y)
[B]x = -13/(4 - y)[/B]
If 800 feet of fencing is available, find the maximum area that can be enclosed.If 800 feet of fencing is available, find the maximum area that can be enclosed.
Perimeter of a rectangle is:
2l + 2w = P
However, we're given one side (length) is bordered by the river and the fence length is 800, so we have:
So we have l + 2w = 800
Rearranging in terms of l, we have:
l = 800 - 2w
The Area of a rectangle is:
A = lw
Plug in the value for l in the perimeter into this:
A = (800 - 2w)w
A = 800w - 2w^2
Take the [URL='https://www.mathcelebrity.com/dfii.php?term1=800w+-+2w%5E2&fpt=0&ptarget1=0&ptarget2=0&itarget=0%2C1&starget=0%2C1&nsimp=8&pl=1st+Derivative']first derivative[/URL]:
A' = 800 - 4w
Now set this equal to 0 for maximum points:
4w = 800
[URL='https://www.mathcelebrity.com/1unk.php?num=4w%3D800&pl=Solve']Typing this equation into the search engine[/URL], we get:
w = 200
Now plug this into our perimeter equation:
l = 800 - 2(200)
l = 800 - 400
l = 400
The maximum area to be enclosed is;
A = lw
A = 400(200)
A = [B]80,000 square feet[/B]
if a and b are odd then a + b is evenif a and b are odd then a + b is even
Let a and b be positive odd integers of the form:
[LIST]
[*]a = 2n + 1
[*]b = 2m + 1
[/LIST]
a + b = 2n + 1 + 2m + 1
a + b = 2n + 2m + 1 + 1
Combing like terms, we get:
a + b = 2n + 2m + 2
a + b = 2(n + m) + 2
Let k = n + m
a + b = 2k + 2
[B]Therefore a + b is even[/B]
If a is an even integer and b is an odd integer then prove a − b is an odd integerIf a is an even integer and b is an odd integer then prove a − b is an odd integer
Let a be our even integer
Let b be our odd integer
We can express a = 2x (Standard form for even numbers) for some integer x
We can express b = 2y + 1 (Standard form for odd numbers) for some integer y
a - b = 2x - (2y + 1)
a - b = 2x - 2y - 1
Factor our a 2 from the first two terms:
a - b = 2(x - y) - 1
Since x - y is an integer, 2(x- y) is always even. Subtracting 1 makes this an odd number.
[MEDIA=youtube]GDVuQ7bGHx8[/MEDIA]
If Distance equals Speed times Time (D = S x T), then what does time equal in terms of speed and disIf Distance equals Speed times Time (D = S x T), then what does time equal in terms of speed and distance?
Divide each side by S to isolate T:
D/S = S x T/S
Cancel the S's on the right side:
[B]T = D/S[/B]
If EF = 9x - 17, FG = 17x - 14, and EG = 20x + 17, what is FG?If EF = 9x - 17, FG = 17x - 14, and EG = 20x + 17, what is FG?
By segment addition, we know that:
EF + FG = EG
Substituting in our values for the 3 segments, we get:
9x - 17 + 17x - 14 = 20x + 17
Group like terms and simplify:
(9 + 17)x + (-17 - 14) = 20x - 17
26x - 31 = 20x - 17
Solve for [I]x[/I] in the equation 26x - 31 = 20x - 17
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 26x and 20x. To do that, we subtract 20x from both sides
26x - 31 - 20x = 20x - 17 - 20x
[SIZE=5][B]Step 2: Cancel 20x on the right side:[/B][/SIZE]
6x - 31 = -17
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -31 and -17. To do that, we add 31 to both sides
6x - 31 + 31 = -17 + 31
[SIZE=5][B]Step 4: Cancel 31 on the left side:[/B][/SIZE]
6x = 14
[SIZE=5][B]Step 5: Divide each side of the equation by 6[/B][/SIZE]
6x/6 = 14/6
x = [B]2.3333333333333[/B]
if i = square root of -1 what is the sum (7 + 3i) + (-8 + 9i)if i = square root of -1 what is the sum (7 + 3i) + (-8 + 9i)
We group like terms, and we get:
7 - 8 + (3 + 9)i
Simplifying, we get:
[B]-1 + 12i[/B]
If Jody had $3 more she would have twice as much as Lars together they have $60If Jody had $3 more she would have twice as much as Lars together they have $60.
Let j be Jody's money and l be Lars's money. We have two equations:
[LIST=1]
[*]j + l = 60
[*]j + 3 = 2l
[/LIST]
Rearrange (2) to solve for j by subtracting 3
j = 2l - 3
Now substitute this into (1)
(2l - 3) + l = 60
Combine like terms
3l - 3 = 60
Enter this into our [URL='http://www.mathcelebrity.com/1unk.php?num=3l-3%3D60&pl=Solve']equation calculator[/URL], and we get:
[B]l = 21[/B]
Now plug l = 21 into our rearranged equation above:
j = 2(21) - 3
j = 42 - 3
[B]j = 39[/B]
If Mr hernandez has 5 times as many students as Mr daniels and together they have 150 students how mIf Mr hernandez has 5 times as many students as Mr daniels and together they have 150 students how many students do each have?
Let h = Mr. Hernandez's students and d = Mr. Daniels students.
We are given two equations:
(1) h = 5d
(2) d + h = 150
Substitute equation (1) into equation (2)
d + (5d) = 150
Combine like terms:
6d = 150
Divide each side of the equation by 6 to isolate d
d = 25 <-- Mr. Daniels Students
Now, plug the value for d into equation (1)
h = 5(25)
h = 125 <-- Mr. Hernandez students
If p = log2(x), what is the value of log2(2x^3) in terms of p?If p = log2(x), what is the value of log2(2x^3) in terms of p?
A. 6p
B. 2p^3
C. 1 + 3p
D. 3 + 3p
E. 1 + p^3
log2(2x^3) = log2(2) + log2(x^3)
log2(2) = 1, so we have:
log2(2x^3) = 1 + 3log2(x)
Since we're given log2(x) = p, we have:
log2(2x^3) = [B]1 + 3p - Answer C
[MEDIA=youtube]-fEkVno3bxs[/MEDIA][/B]
If p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equIf p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equal to 2.
We set up the variation equation with a constant k such that:
p = k/q^2 [I](inversely proportional means we divide)
[/I]
When q is 4 and p is 2, we have:
2 = k/4^2
2 = k/16
Cross multiply:
k = 2 * 16
k = 32
Now, the problem asks for p when q = 2:
p = 32/2^2
p = 32/4
p = [B]8
[MEDIA=youtube]Mro0j-LxUGE[/MEDIA][/B]
If the circumference of a circular rug is 16π feet, then what is the area of the rug in terms of piIf the circumference of a circular rug is 16π feet, then what is the area of the rug in terms of pi
C = 2pir, so we have:
C = 16π
16π = 2πr
Divide each side by 2π:
r = 16π/2π
r = 8
Now, the area of a circle A is denoted below:
A = πr^2
Given r = 8 from above, we have:
A = π(8)^2
A = [B]64π[/B]
If the cost of a bat and a baseball combined is $1.10 and the bat cost $1.00 more than the ball howLet a be the cost of the ball and b be the cost of the bat:
We're given 2 equations:
[LIST=1]
[*]a + b = 1.10
[*]b = a + 1
[/LIST]
Substitute equation (2) into equation (1) for b:
a + a + 1 = 1.10
Combine like terms:
2a + 1 = 1.10
Subtract 1 from each side:
2a + 1 - 1 = 1.10 - 1
2a = 0.10
Divide each side by 2:
2a/2 = 0.10/2
a = [B]0.05[/B]
[MEDIA=youtube]79q346Hy7R8[/MEDIA]
If the perimeter of a rectangular sign is 44cm and the width is 2cm shorter than half the length, thIf the perimeter of a rectangular sign is 44cm and the width is 2cm shorter than half the length, then what are the length and width?
The perimeter (P) of a rectangle is:
2l + 2w = P
We're given P = 44, so we substitute this into the rectangle perimeter equation:
2l + 2w = 44
We're also given w = 0.5l - 2. Substitute the into the Perimeter equation:
2l + 2(0.5l - 2) = 44
Multiply through and simplify:
2l + l - 4 = 44
Combine like terms:
3l - 4 = 44
[URL='https://www.mathcelebrity.com/1unk.php?num=3l-4%3D44&pl=Solve']Type this equation into the search engine[/URL], and we get:
[B]l = 16[/B]
Substitute this back into the equation w = 0.5l - 2
w = 0.5(16) - 2
w = 8 - 2
[B]w = 6[/B]
If two consecutive even numbers are added, the sum is equal to 226. What is the smaller of the two nIf two consecutive even numbers are added, the sum is equal to 226. What is the smaller of the two numbers?
Let the smaller number be n.
The next consecutive even number is n + 2.
Add them together to equal 226:
n + n + 2 = 226
Solve for [I]n[/I] in the equation n + n + 2 = 226
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 1)n = 2n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2n + 2 = + 226
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 2 and 226. To do that, we subtract 2 from both sides
2n + 2 - 2 = 226 - 2
[SIZE=5][B]Step 4: Cancel 2 on the left side:[/B][/SIZE]
2n = 224
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2n/2 = 224/2
n = [B]112
[URL='https://www.mathcelebrity.com/1unk.php?num=n%2Bn%2B2%3D226&pl=Solve']Source[/URL][/B]
If V is the volume of a cube whose side is s, express s in terms of V:If V is the volume of a cube whose side is s, express s in terms of V:
We know the Volume (V) of a cube with side length s is:
V = s^3
Take the cube root of each side:
V^1/3 = (s^3)^1/3
s = [B]V^1/3[/B]
If x represents the first, or the smaller, of two consecutive odd integers, express the sum of theIf x represents the first, or the smaller, of two consecutive odd integers, express the sum of the two integers in terms of x
If x is the first of two consecutive odd integers, then we find the next consecutive odd integer by adding 2 to x:
x + 2
The sum of the two consecutive odd integers is expressed by
x + (x + 2)
Simplify by grouping like terms, we get:
[B]2x + 2[/B]
If you buy a computer directly from the manufacturer for $3,509 and agree to repay it in 36 equal inIf you buy a computer directly from the manufacturer for $3,509 and agree to repay it in 36 equal installments at 1.73% interest per month on the unpaid balance, how much are your monthly payments? How much total interest will be paid?
[U]Determine the monthly payment[/U]
The monthly payment is [B]$114.87[/B] using our [URL='http://www.mathcelebrity.com/annimmpv.php?pv=3059&av=&pmt=&n=36&i=1.73&check1=1&pl=Calculate']annuity calculator[/URL]
[U]Determine the total payments made[/U]
Total payment is 36 times $114.87 = $4,135.37
[U]Now determine the total interest paid[/U]
Take the total payments of $4,135.37 and subtract the original loan of $3,059 to get interest paid of [B]$1,076.37[/B]
if you worked for 3 hours and earned a total of $24, determine your hourly pay rateif you worked for 3 hours and earned a total of $24, determine your hourly pay rate
Hourly Pay = Total Pay / Hours Worked
Hourly Pay = $24 /3
Hourly Pay = [B]$8 per hour[/B]
In a bike shop they sell bicycles & tricycles. I counted 80 wheels & 34 seats. How many bicycles & tIn a bike shop they sell bicycles & tricycles. I counted 80 wheels & 34 seats. How many bicycles & tricycles were in the bike shop?
Let b be the number or bicycles and t be the number of tricycles. Since each bicycle has 2 wheels and 1 seat and each tricycle has 3 wheels and 1 seat, we have the following equations:
[LIST=1]
[*]2b + 3t = 80
[*]b + t = 34
[/LIST]
We can solve this set of simultaneous equations 3 ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2b+%2B+3t+%3D+80&term2=b+%2B+t+%3D+34&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2b+%2B+3t+%3D+80&term2=b+%2B+t+%3D+34&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2b+%2B+3t+%3D+80&term2=b+%2B+t+%3D+34&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
[LIST]
[*][B]b = 22[/B]
[*][B]t = 12[/B]
[/LIST]
In a class there are 5 more boys than girls. There are 13 students in all. How many boys are there iIn a class there are 5 more boys than girls. There are 13 students in all. How many boys are there in the class?
We start by declaring variables for boys and girls:
[LIST]
[*]Let b be the number of boys
[*]Let g be the number of girls
[/LIST]
We're given two equations:
[LIST=1]
[*]b = g + 5
[*]b + g = 13
[/LIST]
Substitute equation (1) for b into equation (2):
g + 5 + g = 13
Grouping like terms, we get:
2g + 5 = 13
Subtract 5 from each side:
2g + 5 - 5 = 13 - 5
Cancel the 5's on the left side and we get:
2g = 8
Divide each side of the equation by 2 to isolate g:
2g/2 = 8/2
Cancel the 2's on the left side and we get:
g = 4
Substitute g = 4 into equation (1) to solve for b:
b = 4 + 5
b = [B]9[/B]
In Super Bowl XXXV, the total number of points scored was 41. The winning team outscored the losingIn Super Bowl XXXV, the total number of points scored was 41. The winning team outscored the losing team by 27 points. What was the final score of the game? In Super Bowl XXXV, the total number of points scored was 41. The winning team outscored the losing team by 27 points. What was the final score of the game?
Let w be the winning team's points, and l be the losing team's points. We have two equations:
[LIST=1]
[*]w + l = 41
[*]w - l = 27
[/LIST]
Add the two equations:
2w = 68
Divide each side by 2
[B]w = 34[/B]
Substitute this into (1)
34 + l = 41
Subtract 34 from each side
[B]l = 7[/B]
Check your work:
[LIST=1]
[*]34 + 7 = 41 <-- check
[*]34 - 7 = 27 <-- check
[/LIST]
The final score of the game was [B]34 to 7[/B].
You could also use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=w+%2B+l+%3D+41&term2=w+-+l+%3D+27&pl=Cramers+Method']simultaneous equation solver[/URL].
In the year 1980, Rick was twice as old as Nancy who was twice as old as Michael. In the year 1992 RIn the year 1980, Rick was twice as old as Nancy who was twice as old as Michael. In the year 1992 Ric, Nancy, and Michael ages added up to 78 years. How old was Ric in 1980?
Age in 1980:
[LIST]
[*]Let Michael's age be m
[*]Nancy's age is 2m
[*]Rick's age is 2 * 2m = 4m
[/LIST]
Age in 1992:
[LIST]
[*]Michael's age = m + 12
[*]Nancy's age is 2m + 12
[*]Rick's age is 2 * 2m = 4m + 12
[/LIST]
Total them up:
m + 12 + 2m + 12 + 4m + 12 = 78
Solve for [I]m[/I] in the equation m + 12 + 2m + 12 + 4m + 12 = 78
[SIZE=5][B]Step 1: Group the m terms on the left hand side:[/B][/SIZE]
(1 + 2 + 4)m = 7m
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
12 + 12 + 12 = 36
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
7m + 36 = + 78
[SIZE=5][B]Step 4: Group constants:[/B][/SIZE]
We need to group our constants 36 and 78. To do that, we subtract 36 from both sides
7m + 36 - 36 = 78 - 36
[SIZE=5][B]Step 5: Cancel 36 on the left side:[/B][/SIZE]
7m = 42
[SIZE=5][B]Step 6: Divide each side of the equation by 7[/B][/SIZE]
7m/7 = 42/7
m = 6
Rick's age = 6 * 4 = [B]24
[URL='https://www.mathcelebrity.com/1unk.php?num=m%2B12%2B2m%2B12%2B4m%2B12%3D78&pl=Solve']Source[/URL]
[/B]
In x years time, Peter will be 23 years old. How old is he now?In x years time, Peter will be 23 years old. How old is he now?
Let Peter's current age be a. In x years time means we add x to a, so we're given:
a + x = 23
We want to find a, s we subtract x from each side to get:
a + x - x = 23 - x
Cancel the x terms on the left side and we get:
a = [B]23 - x[/B]
Incremental Cash FlowFree Incremental Cash Flow Calculator - Given cash inflows, outflows, depreciable amounts, and tax rates, this determines the incremental cash flows.
Input TableFree Input Table Calculator - Given an input table with input and output values, this will determine the operator and rule used to populate the missing values.
Installment Sales Method of AccountingFree Installment Sales Method of Accounting Calculator - Given a sales price, cost amount, installment payment amount and term, this will show the accounting for the Installment Payment method.
Int FunctionFree Int Function Calculator - Determines the integer of a number
Integers BetweenFree Integers Between Calculator - This calculator determines all integers between two numbers (Decimals)
Irrational Numbers BetweenFree Irrational Numbers Between Calculator - This calculator determines all irrational numbers between two numbers
is 6x a monomial?[B]Yes[/B]. It's an algebraic expression consisting of one term.
The constant is 6, and the variable is x.
Isosceles TriangleFree Isosceles Triangle Calculator - Given a long side (a) and a short side (b), this determines the following items of the isosceles triangle:
* Area (A)
* Semi-Perimeter (s)
* Altitude a (ha)
* Altitude b (hb)
* Altitude c (hc)
Jack has 34 bills and coins in 5’s and 2’s. The total value is $116. How many 5 dollar bills does heJack has 34 bills and coins in 5’s and 2’s. The total value is $116. How many 5 dollar bills does he have?
Let the number of 5 dollar bills be f. Let the number of 2 dollar bills be t. We're given two equations:
[LIST=1]
[*]f + t = 34
[*]5f + 2t = 116
[/LIST]
We have a system of equations, which we can solve 3 ways:
[LIST=1]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+34&term2=5f+%2B+2t+%3D+116&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+34&term2=5f+%2B+2t+%3D+116&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=f+%2B+t+%3D+34&term2=5f+%2B+2t+%3D+116&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answers:
[LIST]
[*][B]f = 16[/B]
[*][B]t = 18[/B]
[/LIST]
James is four time as old as peter if their combined age is 30 how old is James.James is four time as old as peter if their combined age is 30 how old is James.
Let j be Jame's age. Let p be Peter's age. We're given:
[LIST=1]
[*]j = 4p
[*]j + p = 30
[/LIST]
Substitute (1) into (2)
4p + p = 30
Combine like terms:
5p = 30
[URL='https://www.mathcelebrity.com/1unk.php?num=5p%3D30&pl=Solve']Type 5p = 30 into our search engine[/URL], and we get p = 6.
Plug p = 6 into equation (1) to get James's age, we get:
j = 4(6)
j = [B]24[/B]
Jason has an equal number of nickels and dimes. The total value of his nickels and dimes is $2.25. HJason has an equal number of nickels and dimes. The total value of his nickels and dimes is $2.25. How many nickels does Jason have?
Let the number of nickels be n
Let the number of dimes be d
We're given two equations:
[LIST=1]
[*]d = n
[*]0.05n + 0.1d = 2.25
[/LIST]
Substitute equation (1) for d into equation (2):
0.05n + 0.1n = 2.25
Solve for [I]n[/I] in the equation 0.05n + 0.1n = 2.25
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(0.05 + 0.1)n = 0.15n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
0.15n = + 2.25
[SIZE=5][B]Step 3: Divide each side of the equation by 0.15[/B][/SIZE]
0.15n/0.15 = 2.25/0.15
n = [B]15[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=0.05n%2B0.1n%3D2.25&pl=Solve']Source[/URL]
Jenny threw the javelin 4 metres further than Angus but 5 metres less than Cameron. if the combinedJenny threw the javelin 4 metres further than Angus but 5 metres less than Cameron. if the combined distance thrown by the 3 friends is 124 metres, how far did Angus throw the javelin?
Assumptions and givens:
[LIST]
[*]Let a be the distance Angus threw the javelin
[*]Let c be the distance Cameron threw the javelin
[*]Let j be the distance Jenny threw the javelin
[/LIST]
We're given 3 equations:
[LIST=1]
[*]j = a + 4
[*]j = c - 5
[*]a + c + j = 124
[/LIST]
Since j is the common variable in all 3 equations, let's rearrange equation (1) and equation (2) in terms of j as the dependent variable:
[LIST=1]
[*]a = j - 4
[*]c = j + 5
[*]a + c + j = 124
[/LIST]
Now substitute equation (1) and equation (2) into equation (3) for a and c:
j - 4 + j + 5 + j = 124
To solve this equation for j, we [URL='https://www.mathcelebrity.com/1unk.php?num=j-4%2Bj%2B5%2Bj%3D124&pl=Solve']type it in our math engine[/URL] and we get:
j = 41
The question asks how far Angus (a) threw the javelin. Since we have Jenny's distance j = 41 and equation (1) has j and a together, let's substitute j = 41 into equation (1):
a = 41 - 4
a = [B]37 meters[/B]
Jenny went shoe shopping. Now she has 5 more pairs than her brother. Together they have 25 pairs. HoJenny went shoe shopping. Now she has 5 more pairs than her brother. Together they have 25 pairs. How many pairs does Jenny have and how many pairs does her brother have?
[U]Let j be the number of shoes Jenny has and b be the number of s hoes her brother has. Set up 2 equations:[/U]
(1) b + j = 25
(2) j = b + 5
[U]Substitute (2) into (1)[/U]
b + (b + 5) = 25
[U]Group the b terms[/U]
2b + 5 = 25
[U]Subtract 5 from each side[/U]
2b = 20
[U]Divide each side by b[/U]
[B]b = 10
[/B]
[U]Substitute b = 10 into (2)[/U]
j = 10 + 5
[B]j = 15[/B]
Jill and Jack are getting vegetables from the Farmer's Market. Jill buys 12 carrots and 8 tomatoes fJill and Jack are getting vegetables from the Farmer's Market. Jill buys 12 carrots and 8 tomatoes for $34. Jack buys 10 carrots and 7 tomatoes for $29. How much does each carrot and each tomato cost?
Let the cost of carrots be c and the cost of tomatoes be t. Since the total cost is price times quantity, We're given two equations:
[LIST=1]
[*]12c + 8t = 34 <-- Jill
[*]10c + 7t = 29 <-- Jack
[/LIST]
We have a system of equations. We can solve this one of three ways:
[LIST=1]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=12c+%2B+8t+%3D+34&term2=10c+%2B+7t+%3D+29&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=12c+%2B+8t+%3D+34&term2=10c+%2B+7t+%3D+29&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=12c+%2B+8t+%3D+34&term2=10c+%2B+7t+%3D+29&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we choose, we get:
[LIST]
[*][B]t = 2[/B]
[*][B]c = 1.5[/B]
[/LIST]
Jim is 9 years older than June. Alex is 8 years younger than June. If the total of their ages is 82,Jim is 9 years older than June. Alex is 8 years younger than June. If the total of their ages is 82, how old is the eldest of them
Let j be Jim's age, a be Alex's age, and u be June's age. We have 3 given equations:
[LIST=1]
[*]j + a + u = 82
[*]j = u + 9
[*]a = u - 8
[/LIST]
Substitute (2) and (3) into (1)
(u + 9) + (u - 8) + u = 82
Combine Like Terms:
3u + 1 = 82
[URL='https://www.mathcelebrity.com/1unk.php?num=3u%2B1%3D82&pl=Solve']Type this equation into the search engine[/URL], and we get u = 27.
The eldest (oldest) of the 3 is Jim. So we have from equation (2)
j = u + 9
j = 27 + 9
[B]j = 36[/B]
Jim works for his dad and earns $400 every week plus $22 for every chair (c) he sells. Write an equaJim works for his dad and earns $400 every week plus $22 for every chair (c) he sells. Write an equation that can be used to determine jims weekly salary (S) given the number of chairs (c) he sells.
[B]S(c) = 400 + 22c[/B]
Joe earns $9 per hour. He worked x hours on both Wednesday and Friday, and 8 hours on both Tuesday aJoe earns $9 per hour. He worked x hours on both Wednesday and Friday, and 8 hours on both Tuesday and Saturday. Write an expression to represent how much joe earned.
Earnings = Hourly Rate * hours worked, so we have:
[LIST]
[*]Wednesday: 9x
[*]Friday: 9x
[*]Tuesday: 9(8) = 72
[*]Saturday: 9(8) = 72
[/LIST]
Joe's total earnings come from adding up all 4 days:
9x + 9x + 72 + 72
Combine like terms:
(9 + 9)x + (72 + 72)
[B]18x + 144[/B]
Joel bought 88 books. Some books cost $13 each and some cost $17 each. In all, he spent $128. WhichJoel bought 88 books. Some books cost $13 each and some cost $17 each. In all, he spent $128. Which system of linear equations represents the given situation?
Let a be the number of the $13 book, and b equal the number of $17 books. We have the following system of linear equations:
[LIST=1]
[*][B]a + b = 88[/B]
[*][B]13a + 17b = 128[/B]
[/LIST]
To solve this system, use our calculator for the following methods:
[LIST]
[*][URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+b+%3D+88&term2=13a+%2B+17b+%3D+128&pl=Substitution']Substitution[/URL]
[*][URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+b+%3D+88&term2=13a+%2B+17b+%3D+128&pl=Elimination']Elimination[/URL]
[*][URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+b+%3D+88&term2=13a+%2B+17b+%3D+128&pl=Cramers+Method']Cramers Method[/URL]
[/LIST]
John spent $10.40 on 5 notebooks and 5 pens. Ariana spent $7.00 on 4 notebooks and 2 pens. What is tJohn spent $10.40 on 5 notebooks and 5 pens. Ariana spent $7.00 on 4 notebooks and 2 pens. What is the ost of 1 notebook and what is the cost of 1 pen?
Let the number of notebooks be n and the number of pens be p. We have two equations:
[LIST=1]
[*]5n + 5p = 10.40
[*]4n + 2p = 7
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=5n+%2B+5p+%3D+10.40&term2=4n+%2B+2p+%3D+7&pl=Cramers+Method']simultaneous equation calculator[/URL], we have:
[LIST]
[*][B]n = 1.42[/B]
[*][B]p = 0.66[/B]
[/LIST]
John took 20,000 out of his retirement and reinvested it. He earned 4% for one investment and 5% onJohn took 20,000 out of his retirement and reinvested it. He earned 4% for one investment and 5% on the other. How much did he invest in each if the total amount earned was 880?
The first principal portion is x. Which means the second principal portion is 20,000 - x. We have:
0.04x + 0.05(20,000 - x) = 880
0.04x + 1,000 - 0.05x = 880
Group like terms:
-0.01x + 1000 = 880
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=-0.01x%2B1000%3D880&pl=Solve']equation solver[/URL], we get x = [B]12,000[/B]. Which means the other fund has 20,000 - 12,000 = [B]8,000[/B].
Jonathan earns a base salary of $1500 plus 10% of his sales each month. Raymond earns $1200 plus 15%Jonathan earns a base salary of $1500 plus 10% of his sales each month. Raymond earns $1200 plus 15% of his sales each month. How much will Jonathan and Raymond have to sell in order to earn the same amount each month?
[U]Step 1: Set up Jonathan's sales equation S(m) where m is the amount of sales made each month:[/U]
S(m) = Commission percentage * m + Base Salary
10% written as a decimal is 0.1. We want decimals to solve equations easier.
S(m) = 0.1m + 1500
[U]Step 2: Set up Raymond's sales equation S(m) where m is the amount of sales made each month:[/U]
S(m) = Commission percentage * m + Base Salary
15% written as a decimal is 0.15. We want decimals to solve equations easier.
S(m) = 0.15m + 1200
[U]The question asks what is m when both S(m)'s equal each other[/U]:
The phrase [I]earn the same amount [/I]means we set Jonathan's and Raymond's sales equations equal to each other
0.1m + 1500 = 0.15m + 1200
We want to isolate m terms on one side of the equation.
Subtract 1200 from each side:
0.1m + 1500 - 1200 = 0.15m + 1200 - 1200
Cancel the 1200's on the right side and we get:
0.1m - 300 = 0.15m
Next, we subtract 0.1m from each side of the equation to isolate m
0.1m - 0.1m + 300 = 0.15m - 0.1m
Cancel the 0.1m terms on the left side and we get:
300 = 0.05m
Flip the statement since it's an equal sign to get the variable on the left side:
0.05m = 300
To solve for m, we divide each side of the equation by 0.05:
0.05m/0.05 = 300/0.05
Cancelling the 0.05 on the left side, we get:
m = [B]6000[/B]
Julia owes 18.20 for the month of November. Her plan costs 9.00 for the first 600 text messages andJulia owes 18.20 for the month of November. Her plan costs 9.00 for the first 600 text messages and .10 cents for additional texts. How many texts did she send out?
Let m be the number of messages. We have a cost function of:
C(m) = 9 + 0.1(m - 600)
We are given C(m) = 18.20
18.20 = 9 + 0.1(m - 600)
18.20 = 9 + 0.1m - 60
Combine like terms:
18.20 = 0.1m - 51
Add 51 to each side
0.1m = 69.20
Divide each side by 0.1
[B]m = 692[/B]
k=g-a/5 for gk=g-a/5 for g
Add a/5 to each side;
k + a/5 = g - a/5 + a/5
Cancel the a/5 terms on the right side, and we get:
g = [B]k + a/5[/B]
Kate spent 1 more than Lauren, and together they spent 5Kate spent 1 more than Lauren, and together they spent 5.
Let k be the amount Kate spent, and l be the amount Lauren spent. We're given:
[LIST=1]
[*]k = l + 1
[*]k + l = 5
[/LIST]
Substitute (1) into (2):
(l + 1) + l = 5
Group like terms
2l + 1 = 5
[URL='https://www.mathcelebrity.com/1unk.php?num=2l%2B1%3D5&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]l = 2[/B]
Plug this into Equation (1), we get:
k = 2 + 1
[B]k = 3
[/B]
Kate Spent 3, and Lauren spent 2
Katie is twice as old as her sister Mara. The sum of their age is 24.Let k = Katie's age and m = Mara's age.
We have 2 equations:
(1) k = 2m
(2) k + m = 24
Substitute (1) into (2)
(2m) + m = 24
Combine like terms:
3m = 24
Divide each side of the equation by 3 to isolate m
m = 8
If m = 8, substituting into (1) or (2), we get k = 16.
[MEDIA=youtube]Cu7gSgNkQPg[/MEDIA]
Kelly took clothes to the cleaners 3 times last month. First, she brought 4 shirts and 1 pair of slaKelly took clothes to the cleaners 3 times last month. First, she brought 4 shirts and 1 pair of slacks and paid11.45. Then she brought 5 shirts, 3 pairs of slacks, and 1 sports coat and paid 27.41. Finally, she brought 5 shirts and 1 sports coat and paid 16.94. How much was she charged for each shirt, each pair of slacks, and each sports coat?
Let s be the cost of shirts, p be the cost of slacks, and c be the cost of sports coats. We're given:
[LIST=1]
[*]4s + p = 11.45
[*]5s + 3p + c = 27.41
[*]5s + c = 16.94
[/LIST]
Rearrange (1) by subtracting 4s from each side:
p = 11.45 - 4s
Rearrange (3)by subtracting 5s from each side:
c = 16.94 - 5s
Take those rearranged equations, and plug them into (2):
5s + 3(11.45 - 4s) + (16.94 - 5s) = 27.41
Multiply through:
5s + 34.35 - 12s + 16.94 - 5s = 27.41
[URL='https://www.mathcelebrity.com/1unk.php?num=5s%2B34.35-12s%2B16.94-5s%3D27.41&pl=Solve']Group like terms using our equation calculator [/URL]and we get:
[B]s = 1.99 [/B] <-- Shirt Cost
Plug s = 1.99 into modified equation (1):
p = 11.45 - 4(1.99)
p = 11.45 - 7.96
[B]p = 3.49[/B] <-- Slacks Cost
Plug s = 1.99 into modified equation (3):
c = 16.94 - 5(1.99)
c = 16.94 - 9.95
[B]c = 6.99[/B] <-- Sports Coat cost
Kendra has $5.70 in quarters and nickels. If she has 12 more quarters than nickels, how many of eachKendra has $5.70 in quarters and nickels. If she has 12 more quarters than nickels, how many of each coin does she have?
Let n be the number of nickels and q be the number of quarters. We have:
[LIST=1]
[*]q = n + 12
[*]0.05n + 0.25q = 5.70
[/LIST]
Substitute (1) into (2)
0.05n + 0.25(n + 12) = 5.70
0.05n + 0.25n + 3 = 5.70
Combine like terms:
0.3n + 3 = 5.70
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=0.3n%2B3%3D5.70&pl=Solve']equation calculator[/URL], we get [B]n = 9[/B].
Substituting that back into (1), we get:
q = 9 + 12
[B]q = 21[/B]
Kendra is half as old as Morgan and 3 years younger than Lizzie. The total of their ages is 39. HowKendra is half as old as Morgan and 3 years younger than Lizzie. The total of their ages is 39. How old are they?
Let k be Kendra's age, m be Morgan's age, and l be Lizzie's age. We're given:
[LIST=1]
[*]k = 0.5m
[*]k = l - 3
[*]k + l + m = 39
[/LIST]
Rearranging (1) by multiplying each side by 2, we have:
m = 2k
Rearranging (2) by adding 3 to each side, we have:
l = k + 3
Substituting these new values into (3), we have:
k + (k + 3) + (2k) = 39
Group like terms:
(k + k + 2k) + 3 = 39
4k + 3 = 39
[URL='https://www.mathcelebrity.com/1unk.php?num=4k%2B3%3D39&pl=Solve']Type this equation into the search engine[/URL], and we get:
[B]k = 9
[/B]
Substitute this back into (1), we have:
m = 2(9)
[B]m = 18
[/B]
Substitute this back into (2), we have:
l = (9) + 3
[B][B]l = 12[/B][/B]
Kerry asked a bank teller to cash 390 check using 20 bills and 50 bills. If the teller gave her a toKerry asked a bank teller to cash 390 check using 20 bills and 50 bills. If the teller gave her a total of 15 bills, how many of each type of bill did she receive?
Let t = number of 20 bills and f = number of 50 bills. We have two equations.
(1) 20t + 50f = 390
(2) t + f = 15
[U]Rearrange (2) into (3) for t, by subtracting f from each side:[/U]
(3) t = 15 - f
[U]Now substitute (3) into (1)[/U]
20(15 - f) + 50f = 390
300 - 20f + 50f = 390
[U]Combine f terms[/U]
300 + 30f = 390
[U]Subtract 300 from each side[/U]
30f = 90
[U]Divide each side by 30[/U]
[B]f = 3[/B]
[U]Substitute f = 3 into (3)[/U]
t = 15 - 3
[B]t = 12[/B]
Kevin and randy have a jar containing 41 coins, all of which are either quarters or nickels. The totKevin and randy have a jar containing 41 coins, all of which are either quarters or nickels. The total value of the jar is $7.85. How many of each type?
Let d be dimes and q be quarters. Set up two equations from our givens:
[LIST=1]
[*]d + q = 41
[*]0.1d + 0.25q = 7.85
[/LIST]
[U]Rearrange (1) by subtracting q from each side:[/U]
(3) d = 41 - q
[U]Now, substitute (3) into (2)[/U]
0.1(41 - q) + 0.25q = 7.85
4.1 - 0.1q + 0.25q = 7.85
[U]Combine q terms[/U]
0.15q + 4.1 = 7.85
[U]Using our [URL='http://www.mathcelebrity.com/1unk.php?num=0.15q%2B4.1%3D7.85&pl=Solve']equation calculator[/URL], we get:[/U]
[B]q = 25[/B]
[U]Substitute q = 25 into (3)[/U]
d = 41 - 25
[B]d = 16[/B]
Kevin and Randy Muise have a jar containing 52 coins, all of which are either quarters or nickels.Kevin and Randy Muise have a jar containing 52 coins, all of which are either quarters or nickels. The total value of the coins in the jar is $6.20. How many of each type of coin do they have?
Let q be the number of quarters, and n be the number of nickels. We have:
[LIST=1]
[*]n + q = 52
[*]0.05n + 0.25q = 6.20
[/LIST]
We can solve this system of equations three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=n+%2B+q+%3D+52&term2=0.05n+%2B+0.25q+%3D+6.20&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=n+%2B+q+%3D+52&term2=0.05n+%2B+0.25q+%3D+6.20&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=n+%2B+q+%3D+52&term2=0.05n+%2B+0.25q+%3D+6.20&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter what method we choose, we get the same answer:
[LIST]
[*][B]n = 34[/B]
[*][B]q = 18[/B]
[/LIST]
Kevin is 4 times old as Daniel and is also 6 years older than DanielKevin is 4 times old as Daniel and is also 6 years older than Daniel.
Let k be Kevin's age and d be Daniel's age. We have 2 equations:
[LIST=1]
[*]k = 4d
[*]k = d + 6
[/LIST]
Plug (1) into (2):
4d = d + 6
Subtract d from each side:
4d - d = d - d + 6
Cancel the d terms on the right side and simplify:
3d = 6
Divide each side by 3:
3d/3 = 6/3
Cancel the 3 on the left side:
d = 2
Plug this back into equation (1):
k = 4(2)
k = 8
So Daniel is 2 years old and Kevin is 8 years old
Kyle can walk ½ mile in ¼ of an hour. What is Kyle’s speed in miles per hour?Kyle can walk ½ mile in ¼ of an hour. What is Kyle’s speed in miles per hour?
We write this in terms of miles per hour as:
1/2 / 1/4
We want 1 for the denominator to represent an hour, so we multiply top and bottom of the fraction by 4:
4/2 / 4/4
2 / 1
[B]2 miles per hour[/B]
larger of 2 numbers is 12 more than the smaller number. if the sum of the 2 numbers is 74 find the 2larger of 2 numbers is 12 more than the smaller number. if the sum of the 2 numbers is 74 find the 2 numbers
Declare Variables for each number:
[LIST]
[*]Let l be the larger number
[*]Let s be the smaller number
[/LIST]
We're given two equations:
[LIST=1]
[*]l = s + 12
[*]l + s = 74
[/LIST]
Equation (1) already has l solved for. Substitute equation (1) into equation (2) for l:
s + 12 + s = 74
Solve for [I]s[/I] in the equation s + 12 + s = 74
[SIZE=5][B]Step 1: Group the s terms on the left hand side:[/B][/SIZE]
(1 + 1)s = 2s
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2s + 12 = + 74
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 12 and 74. To do that, we subtract 12 from both sides
2s + 12 - 12 = 74 - 12
[SIZE=5][B]Step 4: Cancel 12 on the left side:[/B][/SIZE]
2s = 62
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2s/2 = 62/2
s = [B]31[/B]
To solve for l, we substitute in s = 31 into equation (1):
l = 31 + 12
l = [B]43[/B]
larger of 2 numbers is 4 more than the smaller. the sum of the 2 is 40. what is the larger number?larger of 2 numbers is 4 more than the smaller. the sum of the 2 is 40. what is the larger number?
Declare variables for the 2 numbers:
[LIST]
[*]Let l be the larger number
[*]Let s be the smaller number
[/LIST]
We're given two equations:
[LIST=1]
[*]l = s + 4
[*]l + s = 40
[/LIST]
To get this problem in terms of the larger number l, we rearrange equation (1) in terms of l.
Subtract 4 from each side in equation (1)
l - 4 = s + 4 - 4
Cancel the 4's and we get:
s = l - 4
Our given equations are now:
[LIST=1]
[*]s = l - 4
[*]l + s = 40
[/LIST]
Substitute equation (1) into equation (2) for s:
l + l - 4 = 40
Grouping like terms for l, we get:
2l - 4 = 40
Add 4 to each side:
2l - 4 + 4 = 40 + 4
Cancelling the 4's on the left side, we get
2l = 44
Divide each side of the equation by 2 to isolate l:
2l/2 = 44/2
Cancel the 2's on the left side and we get:
l = [B]22[/B]
Larry Mitchell invested part of his $31,000 advance at 6% annual simple interest and the rest at 7%Larry Mitchell invested part of his $31,000 advance at 6% annual simple interest and the rest at 7% annual simple interest. If the total yearly interest from both accounts was $2,090, find the amount invested at each rate.
Let x be the amount invested at 6%. Then 31000 - x is invested at 7%.
We have the following equation:
0.06x + (31000 - x)0.07 = 2090
Simplify:
0.06x + 2170 - 0.07x = 2090
Combine like Terms
-0.01x + 2170 = 2090
Subtract 2170 from each side
-0.01x = -80
Divide each side by -0.01
x = [B]8000 [/B]at 6%
Which means at 7%, we have:
31000 - 8000 = [B]23,000[/B]
Last December at Dubai International Airport 1,309,738 passengers travelled through terminal 1 and 2Last December at Dubai International Airport 1,309,738 passengers travelled through terminal 1 and 2,516,989 passengers through terminal 2. How many passengers travelled through terminal 1 and terminal 2 altogether?
The word [I]altogether[/I] means we add Terminal 1 to Terminal 2:
1,309,738 + 2,516,989 = [B]3,826,727[/B]
Laura weighs 45 pounds more than her pet dog. When they are on the scale together, they weigh 85 pouLaura weighs 45 pounds more than her pet dog. When they are on the scale together, they weigh 85 pounds. How much does Laura weigh?
Let Laura weigh l and her dog weigh d. WE have:
[LIST=1]
[*]l = d + 45
[*]d + l = 85
[/LIST]
Substitute equation (1) into Equation (2) for l:
d + d + 45 = 85
Solve for [I]d[/I] in the equation d + d + 45 = 85
[SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE]
(1 + 1)d = 2d
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2d + 45 = + 85
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 45 and 85. To do that, we subtract 45 from both sides
2d + 45 - 45 = 85 - 45
[SIZE=5][B]Step 4: Cancel 45 on the left side:[/B][/SIZE]
2d = 40
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2d/2 = 40/2
d = 20
From equation (1), we substitute d = 20:
l = d + 45
l = 20 + 45
l = [B]65 pounds
[URL='https://www.mathcelebrity.com/1unk.php?num=d%2Bd%2B45%3D85&pl=Solve']Source[/URL][/B]
Lena purchased a prepaid phone card for $15. Long distance calls cost 24 cents a minute using thisLena purchased a prepaid phone card for $15. Long distance calls cost 24 cents a minute using this card. Lena used her card only once to make a long distance call. If the remaining credit on her card is $4.92, how many minutes did her call last?
[U]Figure out how many minutes Lena used:[/U]
Lena spent $15 - $4.92 = $10.08.
[U]Now determine the amount of minutes[/U]
$10.08/0.24 cents per minute = [B]42 minutes[/B]
Let f(x) = 3x - 6 and g(x)= -2x + 5. Which of the following is f(x) - g(x)?Let f(x) = 3x - 6 and g(x)= -2x + 5. Which of the following is f(x) - g(x)?
f(x) - g(x) = 3x - 6 - (-2x + 5)
Distribute the negative sign where double negative equals a plus:
f(x) - g(x) = 3x - 6 + 2x - 5
Combine like terms:
f(x) - g(x) = (3 + 2)x - 6 - 5
f(x) - g(x) = [B]5x - 11[/B]
Letter Arrangements in a WordFree Letter Arrangements in a Word Calculator - Given a word, this determines the number of unique arrangements of letters in the word.
Liz harold has a jar in her office that contains 47 coins. Some are pennies and the rest are dimes.Liz harold has a jar in her office that contains 47 coins. Some are pennies and the rest are dimes. If the total value of the coins is 2.18, how many of each denomination does she have?
[U]Set up two equations where p is the number of pennies and d is the number of dimes:[/U]
(1) d + p = 47
(2) 0.1d + 0.01p = 2.18
[U]Rearrange (1) into (3) by solving for d[/U]
(3) d = 47 - p
[U]Substitute (3) into (2)[/U]
0.1(47 - p) + 0.01p = 2.18
4.7 - 0.1p + 0.01p = 2.18
[U]Group p terms[/U]
4.7 - 0.09p = 2.18
[U]Add 0.09p to both sides[/U]
0.09p + 2.18 = 4.7
[U]Subtract 2.18 from both sides[/U]
0.09p = 2.52
[U]Divide each side by 0.09[/U]
[B]p = 28[/B]
[U]Now substitute that back into (3)[/U]
d =47 - 28
[B]d = 19[/B]
Luke and Dan's total debt is $72. If Luke's debt is three times Dan's debt, what is Dan's debt?Luke and Dan's total debt is $72. If Luke's debt is three times Dan's debt, what is Dan's debt?
Let Dan's debt be d.
Let Luke's debt be l.
We're given two equations:
[LIST=1]
[*]d + l = 72
[*]l = 3d
[/LIST]
Substitute equation (2) for l into equation (1):
d + 3d = 72
Solve for [I]d[/I] in the equation d + 3d = 72
[SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE]
(1 + 3)d = 4d
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
4d = + 72
[SIZE=5][B]Step 3: Divide each side of the equation by 4[/B][/SIZE]
4d/4 = 72/4
d = [B]18[/B]
Lunch meat A is 10.00 for 2 lbs or meat B for 6.00 for 1lbLunch meat A is 10.00 for 2 lbs or meat B for 6.00 for 1lb
Determine unit cost:
Unit Cost A = 10/2 lbs = 5 per lb
Unit Cost B = 6/1lb = 6 per lb
[B]Unit Cost A is less, so that is the better buy.[/B]
M/n = p-6 for mM/n = p-6 for m
Solve this literal equation by multiplying each side by n to isolate M:
Mn/n = n(p - 6)
Cancelling the n terms on the left side, we get:
[B]M = n(p - 6)[/B]
m=y-b/x-t for ym=y-b/x-t for y
Add b/x + t to each side:
m + b/x + t = y - b/x - t + b/x + t
Cancel b/x terms and t terms on the right side to get:
y = [B]m + b/x + t[/B]
MAPE - MPE - MAPDFree MAPE - MPE - MAPD Calculator - Given a time series of actual and forecasted values, this determines the following:
* Mean Absolute Percentage Error (MAPE) also known as the Mean Absolute Percentage Deviation (MAPD)
* Symmetric Mean Absolute Percentage Error (sMAPE)
* Mean Absolute Percentage Error (MPE)
Margin of Error from Confidence IntervalFree Margin of Error from Confidence Interval Calculator - Given a confidence interval, this determines the margin of error and sample mean.
Marissa has 24 coins in quarters and nickels. She has 3 dollars. How many of the coins are quarters?Let n be the number of nickels and q be the number of quarters.
We have two equations:
(1) n + q = 24
(2) 0.05n + 0.25q = 3
Rearrange (1) to solve for n in terms of q for another equation (3)
(3) n = 24 - q
Plug (3) into (2)
0.05(24 - q) + 0.25q = 3
Multiply through:
1.2 - 0.05q + 0.25q = 3
Combine q terms
0.2q + 1.2 = 3
Subtract 1.2 from each side:
0.2q = 1.8
Divide each side by 0.2
[B]q = 9[/B]
Marty is 3 years younger than 6 times his friend Warrens age. The sum of their ages is greater thanMarty is 3 years younger than 6 times his friend Warrens age. The sum of their ages is greater than 11. What is the youngest age Warren can be?
Let m be Marty's age and w be Warren's age. We have two equations:
(1) m = 6w - 3
(2) m + w > 11
Plug (1) into (2)
6w - 3 + w > 11
Combine w terms
7w - 3 > 11
Add 3 to each side
7w > 14
Divide each side by 7
w > 2 which means [B]w = 3[/B] as the youngest age.
Match each variable with a variable by placing the correct letter on each line.Match each variable with a variable by placing the correct letter on each line.
a) principal
b) interest
c) interest rate
d) term/time
2 years
1.5%
$995
$29.85
[B]Principal is $995
Interest is $29.85 since 995 * .0.15 * 2 = 29.85
Interest rate is 1.5%
Term/time is 2 year[/B]s
Matrix PropertiesFree Matrix Properties Calculator - Given a matrix |A|, this calculates the following items if they exist:
* Determinant = det(A)
* Inverse = A-1
* Transpose = AT
* Adjoint = adj(A)
* Eigen equation (characteristic polynomial) = det|λI - A|
* Trace = tr(A)
* Gauss-Jordan Elimination using Row Echelon and Reduced Row Echelon Form
* Dimensions of |A| m x n
* Order of a matrix
* Euclidean Norm ||A||
* Magic Sum if it exists
* Determines if |A| is an Exchange Matrix
Matthew's cat weighs 10 pounds more than his pet hamster. His dog weighs the same as his cat. If theMatthew's cat weighs 10 pounds more than his pet hamster. His dog weighs the same as his cat. If the weight of all three pets is 35 pounds, ow much does his hamster weigh?
Setup weights and relations:
[LIST]
[*]Hamster weight: w
[*]Cat weight: w + 10
[*]Dog weight:w + 10
[/LIST]
Add all the weights up:
w + w + 10 + w + 10 = 35
Solve for [I]w[/I] in the equation w + w + 10 + w + 10 = 35
[SIZE=5][B]Step 1: Group the w terms on the left hand side:[/B][/SIZE]
(1 + 1 + 1)w = 3w
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
10 + 10 = 20
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
3w + 20 = + 35
[SIZE=5][B]Step 4: Group constants:[/B][/SIZE]
We need to group our constants 20 and 35. To do that, we subtract 20 from both sides
3w + 20 - 20 = 35 - 20
[SIZE=5][B]Step 5: Cancel 20 on the left side:[/B][/SIZE]
3w = 15
[SIZE=5][B]Step 6: Divide each side of the equation by 3[/B][/SIZE]
3w/3 = 15/3
w =[B] 5[/B]
[B]
[URL='https://www.mathcelebrity.com/1unk.php?num=w%2Bw%2B10%2Bw%2B10%3D35&pl=Solve']Source[/URL][/B]
Matthew's pay increases by 20% each month. If his first pay is $450, determine the amount of his payMatthew's pay increases by 20% each month. If his first pay is $450, determine the amount of his pay in month 5.
Let me be the number of months. We have a pay functionalists P(m) as:
P(m) = Initial Pay * (1 + Increase %/100)^m
With m = 5, initial pay = 450, and Increase % = 20, we have
P(5) = 450 * (1.2)^5
P(5) = 450 * 2.48832
P(5) = [B]1,119.74[/B]
Max and Bob went to the store. Max bought 2 burgers and 2 drinks for $5.00 bob bought 3 burgers andMax and Bob went to the store. Max bought 2 burgers and 2 drinks for $5.00. Bob bought 3 burgers and 1 drink for $5.50. How much is each burger and drink?
[U]Set up the givens where b is the cost of a burger and d is the cost of a drink:[/U]
Max: 2b + 2d = 5
Bob: 3b + d = 5.50
[U]Rearrange Bob's equation by subtracting 3b from each side[/U]
(3) d = 5.50 - 3b
[U]Now substitute that d equation back into Max's Equation[/U]
2b + 2(5.50 - 3b) = 5
2b + 11 - 6b = 5
[U]Combine b terms:[/U]
-4b + 11 = 5
[U]Subtract 11 from each side[/U]
-4b = -6
[U]Divide each side by -4[/U]
b = 3/2
[B]b = $1.50[/B]
[U]Now plug that back into equation (3):[/U]
d = 5.50 - 3(1.50)
d = 5.50 - 4.50
[B]d = $1.00[/B]
Max is 23 years younger than his father.Together their ages add up to 81.Max is 23 years younger than his father.Together their ages add up to 81.
Let Max's age be m, and his fathers' age be f. We're given:
[LIST=1]
[*]m = f - 23 <-- younger means less
[*]m + f = 81
[/LIST]
Substitute Equation (1) into (2):
(f - 23) + f = 81
Combine like terms to form the equation below:
2f - 23 = 81
[URL='https://www.mathcelebrity.com/1unk.php?num=2f-23%3D81&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]f = 52[/B]
Substitute this into Equation (1):
m = 52 - 23
[B]m = 29[/B]
Mcnemar TestFree Mcnemar Test Calculator - Given a 2 x 2 contingency table and a significance level, this will determine the test statistic, critical value, and hypothesis conclusion using a Mcnemar test.
Michelle and Julie sold 65 cupcakes. If Julie sold 9 more cupcakes than Michelle, how many cupcakesMichelle and Julie sold 65 cupcakes. If Julie sold 9 more cupcakes than Michelle, how many cupcakes did each of them sell?
Let m = Michelle's cupcakes and j = Julie's cupcakes.
We have two equations:
m + j = 65
j = m + 9
Substituting, we get:
m + (m + 9) = 65
Combine like terms, we get:
2m + 9 = 65
Subtract 9 from each side:
2m = 56
Divide each side by 2 to isolate m
m = 28
If m = 28, then j = 28 + 9 = 37
So (m, j) = (28, 37)
Mike cut 2 acres of grass in 30 minutes on his tractor. Which proportion would determine how many acMike cut 2 acres of grass in 30 minutes on his tractor. Which proportion would determine how many acres of grass Mike cut in 60 minutes?
Let a be the number of acres of grass cut by Mike in 60 minutes. We have the following proportion:
2/30 = a/60
[URL='https://www.mathcelebrity.com/prop.php?num1=2&num2=a&den1=30&den2=60&propsign=%3D&pl=Calculate+missing+proportion+value']Typing this problem into our search engine[/URL], we get [B]a = 4[/B].
Mindy and troy combined ate 9 pieces of the wedding cake. Mindy ate 3 pieces of cake and troy had 1Mindy and troy combined ate 9 pieces of the wedding cake. Mindy ate 3 pieces of cake and troy had 1/4 of the total cake. Write an equation to determine how many pieces of cake (c) that were in total
Let c be the total number of pieces of cake. Let m be the number of pieces Mindy ate. Let t be the number of pieces Troy ate. We have the following given equations:
[LIST]
[*]m + t = 9
[*]m = 3
[*]t = 1/4c
[/LIST]
Combining (2) and (3) into (1), we have:
3 + 1/4c = 9
Subtract 3 from each side:
1/4c = 6
Cross multiply:
[B]c = 24
[MEDIA=youtube]aeqWQXr5f_Y[/MEDIA][/B]
ModulusFree Modulus Calculator - Given 2 integers a and b, this modulo calculator determines a mod b or simplifies modular arithmetic such as 7 mod 3 + 5 mod 8 - 32 mod 5
Molly is making strawberry infused water. For each ounce of strawberry juice, she uses two times asMolly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water as juice. How many ounces of strawberry juice and how many ounces of water does she need to make 40 ounces of strawberry infused water?
Let j be the ounces of strawberry juice and w be the ounces of water. We're given:
[LIST=1]
[*]j + w = 40
[*]w = 3j
[/LIST]
Substitute (2) into (1):
j + 3j = 40
Combine like terms:
4j = 40
[URL='https://www.mathcelebrity.com/1unk.php?num=4j%3D40&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]j = 10[/B]
From equation (2), we substitute j = 2:
w = 3(10)
[B]w = 30
[/B]
This means we have [B]10 ounces of juice[/B] and [B]30 ounces of water[/B] for a 40 ounce mix.
Mr. Jimenez has a pool behind his house that needs to be fenced in. The backyard is an odd quadrilatMr. Jimenez has a pool behind his house that needs to be fenced in. The backyard is an odd quadrilateral shape and the pool encompasses the entire backyard. The four sides are 1818a, 77b, 1111a, and 1919b in length. How much fencing? (the length of the perimeter) would he need to enclose the pool?
The perimeter P is found by adding all 4 sides:
P = 1818a + 77b + 1111a + 1919b
Group the a and b terms
P = (1818 + 1111)a + (77 + 1919b)
[B]P = 2929a + 1996b[/B]
Ms. Jeffers is splitting $975 among her three sons. If the oldest gets twice as much as the youngestMs. Jeffers is splitting $975 among her three sons. If the oldest gets twice as much as the youngest and the middle son gets $35 more than the youngest, how much does each boy get?
Let 0 be the oldest son, m be the middle sun, and y be the youngest son. Set up our given equations
[LIST]
[*]o = 2y
[*]m = y + 35
[*]o + m + y = 975
[/LIST]
[U]Substitute the first and second equations into Equation 3[/U]
2y + y + 35 + y = 975
[U]Combine the y terms[/U]
4y + 35 = 975
Subtract 35 using our [URL='http://www.mathcelebrity.com/1unk.php?num=4y%2B35%3D975&pl=Solve']equation calculator[/URL] to solve and get [B]y = 235[/B]
[U]Plug y = 235 into equation 2[/U]
m = 235 + 35
[B]m = 270[/B]
[U]Plug y = 235 into equation 2[/U]
o = 2(235)
[B]o = 470[/B]
My brother is x years old. I am 5 years older than him. Our combined age is 30 years old. How old isMy brother is x years old. I am 5 years older than him. Our combined age is 30 years old. How old is my brother
Brother's age is x:
I am 5 years older, meaning I'm x + 5:
The combined age is found by adding:
x + (x + 5) = 30
Group like terms:
2x + 5 = 30
To solve for x, [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B5%3D30&pl=Solve']type this equation into our search engine[/URL] and we get:
x = [B]12.5[/B]
n + .07n = $90.95n + .07n = $90.95
Group like terms:
1.07n = $90.95
Solve for [I]n[/I] in the equation 1.07n = 90.95
[SIZE=5][B]Step 1: Divide each side of the equation by 1.07[/B][/SIZE]
1.07n/1.07 = 90.95/1.07
n = [B]85
[URL='https://www.mathcelebrity.com/1unk.php?num=1.07n%3D90.95&pl=Solve']Source[/URL][/B]
n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6
Solve for [I]n[/I] in the equation n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 2 + 3 + 4)n = 10n
[SIZE=5][B]Step 2: Group the constant terms on the right hand side:[/B][/SIZE]
2 + 3 + 4 + 5 + 6 = 20
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
10n = + 20
[SIZE=5][B]Step 4: Divide each side of the equation by 10[/B][/SIZE]
10n/10 = 20/10
n = [B]2[/B]
n + 9n - 8 - 5 = 2n + 3n + 9n - 8 - 5 = 2n + 3
Solve for [I]n[/I] in the equation n + 9n - 8 - 5 = 2n + 3
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 9)n = 10n
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
-8 - 5 = -13
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
10n - 13 = 2n + 3
[SIZE=5][B]Step 4: Group variables:[/B][/SIZE]
We need to group our variables 10n and 2n. To do that, we subtract 2n from both sides
10n - 13 - 2n = 2n + 3 - 2n
[SIZE=5][B]Step 5: Cancel 2n on the right side:[/B][/SIZE]
8n - 13 = 3
[SIZE=5][B]Step 6: Group constants:[/B][/SIZE]
We need to group our constants -13 and 3. To do that, we add 13 to both sides
8n - 13 + 13 = 3 + 13
[SIZE=5][B]Step 7: Cancel 13 on the left side:[/B][/SIZE]
8n = 16
[SIZE=5][B]Step 8: Divide each side of the equation by 8[/B][/SIZE]
8n/8 = 16/8
n = [B]2[/B]
n + 9n - 90 = 0n + 9n - 90 = 0
Solve for [I]n[/I] in the equation n + 9n - 90 = 0
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 9)n = 10n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
10n - 90 =
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -90 and 0. To do that, we add 90 to both sides
10n - 90 + 90 = 0 + 90
[SIZE=5][B]Step 4: Cancel 90 on the left side:[/B][/SIZE]
10n = 90
[SIZE=5][B]Step 5: Divide each side of the equation by 10[/B][/SIZE]
10n/10 = 90/10
n = [B]9[/B]
n - n = 10 - nn - n = 10 - n
Solve for [I]n[/I] in the equation n - n = 10 - n
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 - 1)n = 0n = 0
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
= - n + 10
[SIZE=5][B]Step 3: Group variables:[/B][/SIZE]
We need to group our variables and -n. To do that, we add n to both sides
+ n = -n + 10 + n
[SIZE=5][B]Step 4: Cancel -n on the right side:[/B][/SIZE]
n = [B]10[/B]
n = b + d^2a for an = b + d^2a for a
Let's start by isolating the one term with the a variable.
Subtract b from each side:
n - b = b - b + d^2a
Cancel the b terms on the right side and we get:
n - b = d^2a
With the a term isolated, let's divide each side of the equation by d^2:
(n - b)/d^2 = d^2a/d^2
Cancel the d^2 on the right side, and we'll display this with the variable to solve on the left side:
a = [B](n - b)/d^2
[MEDIA=youtube]BCEVsZmoKoQ[/MEDIA][/B]
N squared multiplied by the difference of n and 3N squared multiplied by the difference of n and 3
n squared means we raise n to the power of 2:
n^2
The difference of n and 3 means we subtract 3 from n:
n - 3
Now we multiply both terms together:
[B]n^2(n - 3)[/B]
nandita earned $224 last month. she earned $28 by selling cards at a craft fair and the rest of thenandita earned $224 last month. she earned $28 by selling cards at a craft fair and the rest of the money by babysitting. Complete an equation that models the situation and can be used to determine x, the number of dollars nandita earned last month by babysitting.
We know that:
Babysitting + Card Sales = Total earnings
Set up the equation where x is the dollars earned from babysitting:
[B]x + 28 = 224[/B]
To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=x%2B28%3D224&pl=Solve']type it in our math engine[/URL] and we get:
x = [B]196[/B]
Natalie made a deal with a farmer. She agreed to work for an entire year and in return, the farmer wNatalie made a deal with a farmer. She agreed to work for an entire year and in return, the farmer would give her $10,200 plus a prize pig.
After working for 5 months, Natalie decided to quit. The farmer determined that 5 months of work was equal to $3375 plus the pig. How much money was the pig worth?
The value of a year's work is $10,200 plus a pig of unknown value. The farmer took away $6825 because Natalie worked 5 months. If Natalie worked 7 more months, she would have been paid the additional $6825.
6825/7 months work = $975 per month
A full year's work is $975 * 12 = $11,700
Pig value = Full years work - payout
Pig value = 11,700 - 10,200
Pig value = [B]1,500[/B]
Nava is 17 years older than Edward. the sum of Navas age and Edwards ages id 29. How old is Nava?Nava is 17 years older than Edward. the sum of Navas age and Edwards ages id 29. How old is Nava?
Let Nava's age be n and Edward's age be e. We have 2 equations:
[LIST=1]
[*]n = e + 17
[*]n + e = 29
[/LIST]
Substitute (1) into (2)
(e + 17) + e = 29
Group like terms:
2e + 17 = 29
Running this equation [URL='http://www.mathcelebrity.com/1unk.php?num=2e%2B17%3D29&pl=Solve']through our search engine[/URL], we get:
e = 6
Substitute this into equation (1)
n = 6 + 17
[B]n = 23[/B]
Net Present Value (NPV) - Internal Rate of Return (IRR) - Profitability IndexFree Net Present Value (NPV) - Internal Rate of Return (IRR) - Profitability Index Calculator - Given a series of cash flows Ct at times t and a discount rate of (i), the calculator will determine the Net Present Value (NPV) at time 0, also known as the discounted cash flow model.
Profitability Index
Also determines an Internal Rate of Return (IRR) based on a series of cash flows. NPV Calculator
Nominal YieldFree Nominal Yield Calculator - Given an effective annual rate of interest based on a compounding period, this determines the nominal yield.
Nonagonal NumberFree Nonagonal Number Calculator - This calculator determines the nth nonagonal number
Number PropertyFree Number Property Calculator - This calculator determines if an integer you entered has any of the following properties:
* Even Numbers or Odd Numbers (Parity Function or even-odd numbers)
* Evil Numbers or Odious Numbers
* Perfect Numbers, Abundant Numbers, or Deficient Numbers
* Triangular Numbers
* Prime Numbers or Composite Numbers
* Automorphic (Curious)
* Undulating Numbers
* Square Numbers
* Cube Numbers
* Palindrome Numbers
* Repunit Numbers
* Apocalyptic Power
* Pentagonal
* Tetrahedral (Pyramidal)
* Narcissistic (Plus Perfect)
* Catalan
* Repunit
Oakdale School is sponsoring a canned food drive. In the first week of the drive, the students colleOakdale School is sponsoring a canned food drive. In the first week of the drive, the students collected 638 cans. They collected 698 cans in the second week and 758 cans in the third week. If the students continue to collect cans at this rate, in which week will they collect more than 1,000 cans?
We have an arithmetic sequence where each successive term increases by 50.
[URL='https://www.mathcelebrity.com/sequenceag.php?num=638%2C698%2C758&n=10&pl=Calculate+Series&a1=5&d=3']Using our sequence calculator[/URL], we find that week #8 is when the students cross 1,000 cans.
Octagonal NumberFree Octagonal Number Calculator - This calculator determines the nth octagonal number
Odds RatioFree Odds Ratio Calculator - This calculator determines the odds ratio for 2 groups X and Y with success and failure for an outcome.
On an algebra test, the highest grade was 42 points higher than the lowest grade. The sum of the twoOn an algebra test, the highest grade was 42 points higher than the lowest grade. The sum of the two grades was 138. Find the lowest grade.
[U]Let h be the highest grade and l be the lowest grade. Set up the given equations:[/U]
(1) h = l + 42
(2) h + l = 138
[U]Substitute (1) into (2)[/U]
l + 42 + l = 138
[U]Combine l terms[/U]
2l + 42 = 138
[U]Enter that equation into our [URL='http://www.mathcelebrity.com/1unk.php?num=2l%2B42%3D138&pl=Solve']equation calculator[/URL] to get[/U]
[B]l = 48
[/B]
[U]Substitute l = 48 into (1)[/U]
h = 48 + 42
[B]h = 90[/B]
On Monday the office staff at your school paid 8.77 for 4 cups of coffee and 7 bagels. On WednesdayOn Monday the office staff at your school paid 8.77 for 4 cups of coffee and 7 bagels. On Wednesday they paid 15.80 for 8 cups of coffee and 14 bagels. Can you determine the cost of a bagel
Let the number of cups of coffee be c
Let the number of bagels be b.
Since cost = Price * Quantity, we're given two equations:
[LIST=1]
[*]7b + 4c = 8.77
[*]14b + 8c = 15.80
[/LIST]
We have a system of equations. We can solve this 3 ways:
[LIST=1]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=7b+%2B+4c+%3D+8.77&term2=14b+%2B+8c+%3D+15.80&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=7b+%2B+4c+%3D+8.77&term2=14b+%2B+8c+%3D+15.80&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=7b+%2B+4c+%3D+8.77&term2=14b+%2B+8c+%3D+15.80&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we use, we get the same answer
[LIST]
[*]The system is inconsistent. Therefore, we have no answer.
[/LIST]
On the day of a child's birth, a deposit of $25,000 is made in a trust fund that pays 8.5% interest.On the day of a child's birth, a deposit of $25,000 is made in a trust fund that pays 8.5% interest. Determine that balance in this account on the child's 25th birthday.
Using our [URL='https://www.mathcelebrity.com/compoundint.php?bal=25000&nval=25&int=8.5&pl=Annually']compound interest calculator[/URL], we get:
[B]192,169.06 [/B]
On the first day of ticket sales the school sold 3 senior citizen tickets and 10 child tickets for aOn the first day of ticket sales the school sold 3 senior citizen tickets and 10 child tickets for a total of $82. The school took in $67 on the second day by selling 8 senior citizen tickets And 5 child tickets. What is the price of each ticket?
Let the number of child tickets be c
Let the number of senior citizen tickets be s
We're given two equations:
[LIST=1]
[*]10c + 3s = 82
[*]5c + 8s = 67
[/LIST]
We have a system of simultaneous equations. We can solve it using any one of 3 ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10c+%2B+3s+%3D+82&term2=5c+%2B+8s+%3D+67&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10c+%2B+3s+%3D+82&term2=5c+%2B+8s+%3D+67&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10c+%2B+3s+%3D+82&term2=5c+%2B+8s+%3D+67&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we choose, we get:
[LIST]
[*][B]c = 7[/B]
[*][B]s = 4[/B]
[/LIST]
One number exceeds another by 15. The sum of the numbers is 51. What are these numbersOne number exceeds another by 15. The sum of the numbers is 51. What are these numbers?
Let the first number be x, and the second number be y. We're given two equations:
[LIST=1]
[*]x = y + 15
[*]x + y = 51
[/LIST]
Plug (1) into (2)
(y + 15) + y = 51
Combine like terms:
2y + 15 = 51
[URL='https://www.mathcelebrity.com/1unk.php?num=2y%2B15%3D51&pl=Solve']Plug this equation into the search engine[/URL] and we get:
[B]y = 18[/B]
Now plug this into (1) to get:
x = 18 + 15
[B]x = 33[/B]
Ordered and Unordered PartitionsFree Ordered and Unordered Partitions Calculator - Given a population size (n) and a group population of (m), this calculator determines how many ordered or unordered groups of (m) can be formed from (n)
Ordinal NumberFree Ordinal Number Calculator - This calculator determines the ordinal number of an integer
p/q = f/q- f for fp/q = f/q- f for f
Isolate f in this literal equation.
Factor out f on the right side:
p/q = f(1/q - 1)
Rewriting the term in parentheses, we get:
p/q = f(1 - q)/q
Cross multiply:
f = pq/q(1 - q)
Cancelling the q/q on the right side, we get:
f = [B]p/(1 - q)[/B]
Pam has two part-time jobs. At one job, she works as a cashier and makes $8 per hour. At the secondPam has two part-time jobs. At one job, she works as a cashier and makes $8 per hour. At the second job, she works as a tutor and makes$12 per hour. One week she worked 30 hours and made$268 . How many hours did she spend at each job?
Let the cashier hours be c. Let the tutor hours be t. We're given 2 equations:
[LIST=1]
[*]c + t = 30
[*]8c + 12t = 268
[/LIST]
To solve this system of equations, we can use 3 methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+t+%3D+30&term2=8c+%2B+12t+%3D+268&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+t+%3D+30&term2=8c+%2B+12t+%3D+268&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+t+%3D+30&term2=8c+%2B+12t+%3D+268&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we use, we get the same answer:
[LIST]
[*]c = [B]23[/B]
[*]t = [B]7[/B]
[/LIST]
ParabolasFree Parabolas Calculator - Determines the focus, directrix, and other related items for a parabola.
Payback PeriodFree Payback Period Calculator - Given a set of cash inflows and cash outflows at certain times, this determines the net cash flow, cumulative cash flow, and payback period
Penelope and Owen work at a furniture store. Penelope is paid $215 per week plus 3.5% of her total sPenelope and Owen work at a furniture store. Penelope is paid $215 per week plus 3.5% of her total sales in dollars, xx, which can be represented by g(x)=215+0.035x. Owen is paid $242 per week plus 2.5% of his total sales in dollars, xx, which can be represented by f(x)=242+0.025x. Determine the value of xx, in dollars, that will make their weekly pay the same.
Set the pay functions of Owen and Penelope equal to each other:
215+0.035x = 242+0.025x
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=215%2B0.035x%3D242%2B0.025x&pl=Solve']equation calculator[/URL], we get:
[B]x = 2700[/B]
Pentagonal NumberFree Pentagonal Number Calculator - This calculator determines the nth pentagonal number
Percentage of CompletionFree Percentage of Completion Calculator - Given a sales price, total costs, and costs per period, this determines the gross profit to date using the percentage of completion method.
Percentage-Decimal-Fraction RelationsFree Percentage-Decimal-Fraction Relations Calculator - Calculates the relational items between a fraction, a decimal (including repeating decimal and terminating decimal), a percentage, and the numerator and denominator piece of that fraction. Also calculates the percentage change going from one number to another or the amount increase or decrease of a percentage above/below a number. Round decimals. decimals into fractions
PercentilesFree Percentiles Calculator - Given a set of scores and a target score, this will determine the percentile of the target score using two different formulas.
Peter is buying office supplies. He is able to buy 3 packages of paper and 4 staplers for $40, or hePeter is buying office supplies. He is able to buy 3 packages of paper and 4 staplers for $40, or he is able to buy 5 packages of paper and 6 staplers for $62. How much does a package of paper cost? How much does a stapler cost?
Let the cost of paper packages be p and the cost of staplers be s. We're given two equations:
[LIST=1]
[*]3p + 4s = 40
[*]5p + 6s = 62
[/LIST]
We have a system of equations. We can solve this three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=3p+%2B+4s+%3D+40&term2=5p+%2B+6s+%3D+62&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=3p+%2B+4s+%3D+40&term2=5p+%2B+6s+%3D+62&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=3p+%2B+4s+%3D+40&term2=5p+%2B+6s+%3D+62&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we choose, we get the same answer:
[LIST]
[*][B]p = 4[/B]
[*][B]s = 7[/B]
[/LIST]
Phone Number TranslatorFree Phone Number Translator Calculator - Given a phone number with letters in it, this calculator will determine the numeric phone number for you to dial.
Pick's TheoremFree Pick's Theorem Calculator - This calculator determines the area of a simple polygon using interior points and boundary points using Pick's Theorem
Pixels Per Inch PPIFree Pixels Per Inch PPI Calculator - This calculator determines the PPI from width, height, and diagonal in inches
please solve the fourth word problemLet x be the first number, y be the second number, and z be the number. We have the following equations:
[LIST=1]
[*]x + y + z = 305
[*]x = y - 5
[*]z = 3y
[/LIST]
Substitute (2) and (3) into (1)
(y - 5) + y + (3y) = 305
Combine like terms:
5y - 5 = 305
Use our [URL='http://www.mathcelebrity.com/1unk.php?num=5y-5%3D305&pl=Solve']equation solver[/URL]
[B]y = 62
[/B]
Substitute y = 62 into (3)
z = 3(62)
[B]z = 186
[/B]
x = (62) - 5
[B]x = 57[/B]
please solve the third word problemA Web music store offers two versions of a popular song. The size of the standard version is
2.7
megabytes (MB). The size of the high-quality version is
4.7
MB. Yesterday, the high-quality version was downloaded three times as often as the standard version. The total size downloaded for the two versions was
4200
MB. How many downloads of the standard version were there?
Let s be the standard version downloads and h be the high quality downloads. We have two equations:
[LIST=1]
[*]h = 3s
[*]2.7s + 4.7h = 4200
[/LIST]
Substitute (1) into (2)
2.7s + 4.7(3s) = 4200
2.7s + 14.1s = 4200
Combine like terms:
16.8s = 4200
Divide each side by 16.8
[B]s = 250[/B]
Polar ConicsFree Polar Conics Calculator - Given eccentricity (e), directrix (d), and angle θ, this determines the vertical and horizontal directrix polar equations.
Polygon SideFree Polygon Side Calculator - Determines the sides of a polygon given an interior angle sum.
PolygonsFree Polygons Calculator - Using various input scenarios of a polygon such as side length, number of sides, apothem, and radius, this calculator determines Perimeter or a polygon and Area of the polygon.
This also determines interior angles of a polygon and diagonals of a polygon as well as the total number of 1 vertex diagonals.
PolynomialFree Polynomial Calculator - This calculator will take an expression without division signs and combine like terms.
It will also analyze an polynomial that you enter to identify constant, variables, and exponents. It determines the degree as well.
Population Doubling TimeFree Population Doubling Time Calculator - Determines population growth based on a doubling time.
Population GrowthFree Population Growth Calculator - Determines population growth based on an exponential growth model.
Positivity RateFree Positivity Rate Calculator - This calculator determines the positivity rate using positive tests and total tests
power set for S= {b,c,f}power set for S= {b,c,f}
The [I]power set[/I] P is the set of all subsets of S including S and the empty set ∅.
Since S contains 3 terms, our Power Set should contain 2^3 = 8 items
[URL='https://www.mathcelebrity.com/powerset.php?num=b%2Cc%2Cf&pl=Show+Power+Set+and+Partitions']Link to power set for this problem[/URL]
P = [B]{{}, {b}, {c}, {f}, {b,c}, {b,f}, {c,f}, {b,c,f}}[/B]
Power Sets and Set PartitionsFree Power Sets and Set Partitions Calculator - Given a set S, this calculator will determine the power set for S and all the partitions of a set.
Powers OfFree Powers Of Calculator - Determines the powers of a number from 1 to n.
Primitive RootFree Primitive Root Calculator - Given a prime number p and a potential root of b, this determines if b is a primitive root of p.
Prizes hidden on a game board with 10 spaces. One prize is worth $100, another is worth $50, and twImagine you are in a game show. Prizes hidden on a game board with 10 spaces. One prize is worth $100, another is worth $50, and two are worth $10. You have to pay $20 to the host if your choice is not correct. Let the random variable x be the winning
(a) What is your expected winning in this game?
(b) Determine the standard deviation of x. (Round the answer to two decimal places)
(a) 100(0.1) + 50(0.1) + 10(0.2) - 20 = 10 + 5 + 2 - 20 = [B]-3[/B]
(b) 3.3 using our [URL='http://www.mathcelebrity.com/statbasic.php?num1=+100,50,10&num2=+0.1,0.1,0.2&usep=usep&pl=Number+Set+Basics']standard deviation calculator[/URL]
Profit EquationFree Profit Equation Calculator - Using the Profit Equation with inputs (Revenue-Cost-Profit-Tax), this determines the relevant output including gross proft, gross profit margin, net profit, and net profit margin.
ProportionFree Proportion Calculator - 1) Calculates the missing link of 2 equivalent proportions or ratios.
2) Also determines if two numerical proportions that you entered such as 1/10=6/12 are equivalent or not equivalent.
Note: You can use all allowable operators such as =,<,≤,>,≥
Proportion Sample SizeFree Proportion Sample Size Calculator - This calculator determines a sample size to select to meet certain criteria related to a confidence percentage, reliability percentage, and a p value proportion. Simply enter your values not using percentage signs. This works whether p^ is known or not known.
Prove sqrt(2) is irrationalUse proof by contradiction. Assume sqrt(2) is rational.
This means that sqrt(2) = p/q for some integers p and q, with q <>0.
We assume p and q are in lowest terms.
Square both side and we get:
2 = p^2/q^2
p^2 = 2q^2
This means p^2 must be an even number which means p is also even since the square of an odd number is odd.
So we have p = 2k for some integer k. From this, it follows that:
2q^2 = p^2 = (2k)^2 = 4k^2
2q^2 = 4k^2
q^2 = 2k^2
q^2 is also even, therefore q must be even.
So both p and q are even.
This contradicts are assumption that p and q were in lowest terms.
So sqrt(2) [B]cannot be rational.
[MEDIA=youtube]tXoo9-8Ewq8[/MEDIA][/B]
Prove that the difference between alternate consecutive squares as always evenTake an integer n. The next alternate consecutive integer is n + 2
Subtract the difference of the squares:
(n + 2)^2 - n^2
n^2 + 4n + 4 - n^2
n^2 terms cancel, we get:
4n + 4
Factor out a 4:
4(n + 1)
If n is odd, n + 1 is even. 4 * even is always even
If n is even, n + 1 is odd. 4 * odd is always odd
Since both cases are even, we've proven our statement.
[MEDIA=youtube]J_E9lR5qFY0[/MEDIA]
Prove that the difference of two consecutive cubes is never divisible by 3Take two consecutive integers:
n, n + 1
The difference of their cubes is:
(n + 1)^3 - n^3
n^3 + 3n^2 + 3n + 1 - n^3
Cancel the n^3
3n^2 + 3n + 1
Factor out a 3 from the first 2 terms:
3(n^2 + n) + 1
The first two terms are always divisible by 3 but then the + 1 makes this expression not divisible by 3:
3(n^2 + n) + 1 = 1 (mod 3)
[MEDIA=youtube]hFvJ3epqmyE[/MEDIA]
Prove the difference between two consecutive square numbers is always oddTake an integer n. The next consecutive integer is n + 1
Subtract the difference of the squares:
(n + 1)^2 - n^2
n^2 + 2n + 1 - n^2
n^2 terms cancel, we get:
2n + 1
2 is even. For n, if we use an even:
we have even * even = Even
Add 1 we have Odd
2 is even. For n, if we use an odd:
we have even * odd = Even
Add 1 we have Odd
Since both cases are odd, we've proven our statement.
[MEDIA=youtube]RAi0HbH5bqc[/MEDIA]
pv/t = ab/c for cpv/t = ab/c for c
Cross multiply:
cpv = abt
Divide each side of the equation by pv to isolate c:
cpv/pv = abt/pv
Cancel the pv terms on the left side and we get:
c = [B]abt/pv[/B]
Pythagorean TheoremFree Pythagorean Theorem Calculator - Figures out based on user entry the missing side or missing hypotenuse of a right triangle. In addition, the calculator shows the proof of the Pythagorean Theorem and then determines by numerical evaluation if the 2 sides and hypotenuse you entered are a right triangle using the Pythagorean Theorem
q increased by the difference between 18 times q and 5q increased by the difference between 18 times q and 5
Take this algebraic expression in parts.
18 times q:
18q
The difference between 18 times q and 5 means we subtract 5 from 18q:
18q - 5
q increased by the difference between 18 times q and 5 means we add 18q - 5 to q:
q + (18q - 5)
[B]q + 18q - 5[/B]
IF we want to simplify, we group like terms:
[B]19q - 5[/B]
Quadratic Equations and InequalitiesFree Quadratic Equations and Inequalities Calculator - Solves for quadratic equations in the form ax2 + bx + c = 0. Also generates practice problems as well as hints for each problem.
* Solve using the quadratic formula and the discriminant Δ
* Complete the Square for the Quadratic
* Factor the Quadratic
* Y-Intercept
* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)2 + k
* Concavity of the parabola formed by the quadratic
* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.
QuadrilateralFree Quadrilateral Calculator - Given 4 points entered, this determines the area using Brahmaguptas Formula and perimeter of the quadrilateral formed by the points as well as checking to see if the quadrilateral (quadrangle) is a parallelogram.
Rachel works at a bookstore. On Tuesday, she sold twice as many books as she did on Monday. On WedneRachel works at a bookstore. On Tuesday, she sold twice as many books as she did on Monday. On Wednesday, she sold 6 fewer books than she did on Tuesday. During the 3 days Rachel sold 19 books. Create an equation that can be used to find m, a number of books Rachel sold on Monday.
Let me be the number of books Rachel sold on Monday. We're given Tuesday's book sales (t) and Wednesday's books sales (w) as:
[LIST=1]
[*]t = 2m
[*]w = t - 6
[*]m + t + w = 19
[/LIST]
Plug (1) and (2) into (3):
Since t = 2m and w = t - 6 --> 2m - 6, we have:
m + 2m + 2m - 6 = 19
Combine like terms:
5m - 6 = 19
[URL='https://www.mathcelebrity.com/1unk.php?num=5m-6%3D19&pl=Solve']Plugging this equation into our search engine[/URL], we get:
[B]m = 5[/B]
Random TestFree Random Test Calculator - Given a set of data and an α value, this determines the test statistic and accept/reject hypothesis based on randomness of a dataset.
Rational Numbers BetweenFree Rational Numbers Between Calculator - This calculator determines all rational numbers between two numbers
RatiosFree Ratios Calculator - * Simplifies a ratio of a:b
* Given a ratio in the form a:b or a to b, and a total population amount, this calculator will determine the expected value of A and B from the ratio.
Receivables RatiosFree Receivables Ratios Calculator - Given Net Sales, Beginning Accounts Receivable, and Ending Accounts Receivable, this determines Average Accounts Receivable, Receivables turnover ratio, and Average Collection Period.
Rectangular NumberFree Rectangular Number Calculator - This calculator determines the nth rectangular number
Relative CoordinatesFree Relative Coordinates Calculator - Given a starting point (x1,y1), this will determine your relative coordinates after moving up, down, left, and right.
Rhonda raised $245 for her softball team's fundraiser.She wants to raise no less than $455.Write andRhonda raised $245 for her softball team's fundraiser.She wants to raise no less than $455.Write and solve an inequality to determine how much more money Rhonda must raise to reach her goal. Let d represent the amount of money in dollars Rhonda must raise to reach her goal.
The phrase [I]no less than[/I] is an inequality using the greater than or equal sign:
d >= 455 - 245
d >= [B]210[/B]
Richard is thrice as old as Alvin. The sum of their ages is 52 years. Find their agesRichard is thrice as old as Alvin. The sum of their ages is 52 years. Find their ages.
Let r be Richard's age. And a be Alvin's age. We have:
[LIST=1]
[*]r = 3a
[*]a + r = 52
[/LIST]
Substitute (1) into (2)
a + 3a = 52
Group like terms:
4a = 52
[URL='https://www.mathcelebrity.com/1unk.php?num=4a%3D52&pl=Solve']Typing this into the search engine[/URL], we get [B]a = 13[/B].
This means Richard is 3(13) = [B]39[/B]
Rigby and Eleanor's combined score on the systems of equations test was 181. Rigby scored 9 more poiRigby and Eleanor's combined score on the systems of equations test was 181. Rigby scored 9 more points than Eleanor. What were Eleanor and Rigby's scores?
Let Rigby's score be r
Let Eleanor's score be e
We're given two equations:
[LIST=1]
[*]r = e + 9
[*]e + r = 181
[/LIST]
Substitute equation (1) into equation (2):
e + (e + 9) = 181
Group like terms:
2e + 9 = 181
To solve this equation for e, we [URL='https://www.mathcelebrity.com/1unk.php?num=2e%2B9%3D181&pl=Solve']type it in our search engine[/URL] and we get:
e = [B]86[/B]
Rob has 40 coins, all dimes and quarters, worth $7.60. How many dimes and how many quarters does heRob has 40 coins, all dimes and quarters, worth $7.60. How many dimes and how many quarters does he have?
We have two equations where d is the number of dimes and q is the number of quarters:
[LIST=1]
[*]d + q = 40
[*]0.1d + 0.25q = 7.60
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=d+%2B+q+%3D+40&term2=0.1d+%2B+0.25q+%3D+7.60&pl=Cramers+Method']simultaneous equation calculator[/URL], we get:
[B]d = 16
q = 24[/B]
Robert and Robert go to the movie theater and purchase refreshments for their friends. Robert spendRobert and Robert go to the movie theater and purchase refreshments for their friends. Robert spends a total of $65.25 on 4 drinks and 9 bags of popcorn. Robert spends a total of $51.75 on 8 drinks and 3 bags of popcorn. Write a system of equations that can be used to find the price of one drink and the price of one bag of popcorn. Using these equations, determine and state the price of a bag of popcorn, to the nearest cent.
Let d be the cost of each drink, and p be the price of each popcorn bag. We have 2 equations for our system of equations:
[LIST=1]
[*][B]4d + 9p = 65.25[/B]
[*][B]8d + 3p = 51.75[/B]
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=4d+%2B+9p+%3D+65.25&term2=8d+%2B+3p+%3D+51.75&pl=Cramers+Method']system of equations calculator[/URL], we get:
[LIST]
[*]d = 4.5
[*][B]p = 5.25 <-- Since the problem asks for the cost of each popcorn bag[/B]
[/LIST]
Run Length EncodingFree Run Length Encoding Calculator - Given a string, this will determine the run length encoding using repeating patterns of characters.
Sally and Adam works a different job. Sally makes $5 per hour and Adam makes $4 per hour. They eachSally and Adam works a different job. Sally makes $5 per hour and Adam makes $4 per hour. They each earn the same amount per week but Adam works 2 more hours. How many hours a week does Adam work?
[LIST]
[*]Let [I]s[/I] be the number of hours Sally works every week.
[*]Let [I]a[/I] be the number of hours Adam works every week.
[*]We are given: a = s + 2
[/LIST]
Sally's weekly earnings: 5s
Adam's weekly earnings: 4a
Since they both earn the same amount each week, we set Sally's earnings equal to Adam's earnings:
5s = 4a
But remember, we're given a = s + 2, so we substitute this into Adam's earnings:
5s = 4(s + 2)
Multiply through on the right side:
5s = 4s + 8 <-- [URL='https://www.mathcelebrity.com/expand.php?term1=4%28s%2B2%29&pl=Expand']multiplying 4(s + 2)[/URL]
[URL='https://www.mathcelebrity.com/1unk.php?num=5s%3D4s%2B8&pl=Solve']Typing this equation into the search engine[/URL], we get s = 8.
The problem asks for Adam's earnings (a). We plug s = 8 into Adam's weekly hours:
a = s + 2
a = 8 + 2
[B]a = 10[/B]
Sally is 4 years older than Mark. Twice Sally's age plus 5 times Mark's age is equal to 64.Sally is 4 years older than Mark. Twice Sally's age plus 5 times Mark's age is equal to 64.
Let Sally's age be s. Let Mark's age be m. We're given two equations:
[LIST=1]
[*]s = m + 4
[*]2s + 5m = 64 <-- [I]Since Twice means we multiply by 2[/I]
[/LIST]
Substitute equation (1) into equation (2):
2(m + 4) + 5m = 64
Multiply through:
2m + 8 + 5m = 64
Group like terms:
(2 + 5)m + 8 = 64
7m + 8 = 64
[URL='https://www.mathcelebrity.com/1unk.php?num=7m%2B8%3D64&pl=Solve']Type this equation into the search engine[/URL] and we get:
m = [B]8[/B]
Sam has $2.25 in quarters and dimes, and the total number of coins is 12. How many quarters and howSam has $2.25 in quarters and dimes, and the total number of coins is 12. How many quarters and how many dimes?
Let d be the number of dimes. Let q be the number of quarters. We're given two equations:
[LIST=1]
[*]0.1d + 0.25q = 2.25
[*]d + q = 12
[/LIST]
We have a simultaneous system of equations. We can solve this 3 ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.1d+%2B+0.25q+%3D+2.25&term2=d+%2B+q+%3D+12&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.1d+%2B+0.25q+%3D+2.25&term2=d+%2B+q+%3D+12&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=0.1d+%2B+0.25q+%3D+2.25&term2=d+%2B+q+%3D+12&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
[LIST]
[*][B]d = 5[/B]
[*][B]q = 7[/B]
[/LIST]
Sam has x model planes. Anton has 8 more planes than Sam does. How many model planes Does Anton haveSam has x model planes. Anton has 8 more planes than Sam does. How many model planes Does Anton have? how many planes do they have together?
Sam has x
Anton has [B]x + 8[/B] since the word [I]more[/I] means we add
The word [I]together[/I] means we add, so we have:
Sam + Anton = x + x + 8
Grouping like terms, we have:
Sam + Anton = [B]2x + 8[/B]
Sara wants to arrange the seven scrabble letters she has in every possible way so she can determineSara wants to arrange the seven scrabble letters she has in every possible way so she can determine if she has a 7-letter word. how many different ways are there for Sara to arrange all seven letters?
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = [B]5,040 ways[/B]
Scientific NotationFree Scientific Notation Calculator - * Converts a number into scientific notation and determines order of magnitude
* converts scientific notation to a number (standard notation). Also handles scientific notation operations.
Scientists are studying a cell that divides in half every 15 minutes. How many cells will there by aScientists are studying a cell that divides in half every 15 minutes. How many cells will there by after 2.5 hours?
Divide 2.5 hours into 15 minute blocks.
2.5 hours = 2(60) + 0.5(60) minutes
2.5 hours = 120 + 30 minutes
2.5 hours = 150 minutes
Now determine the amount of 15 minute blocks
150 minutes/15 minutes = 10 blocks or divisions
[LIST]
[*]We start with 1 cell at time 0, and double it every 15 minutes
[*]We have A(0) = 1, we want A(10).
[*]Our accumulation function is A(t) = A(0) * 2^t
[/LIST]
A(10) = 1 * 2^10
A(10) = [B]1024[/B]
SequencesFree Sequences Calculator - Given a function a(n) and a count of sequential terms you want to expand (n), this calcuator will determine the first (n) terms of your sequence, {a1, a2, ..., an}
Set NotationFree Set Notation Calculator - Given two number sets A and B, this determines the following:
* Union of A and B, denoted A U B
* Intersection of A and B, denoted A ∩ B
* Elements in A not in B, denoted A - B
* Elements in B not in A, denoted B - A
* Symmetric Difference A Δ B
* The Concatenation A · B
* The Cartesian Product A x B
* Cardinality of A = |A|
* Cardinality of B = |B|
* Jaccard Index J(A,B)
* Jaccard Distance Jσ(A,B)
* Dice's Coefficient
* If A is a subset of B
* If B is a subset of A
She ordered 6 large pizzas. Luckily, she had a coupon for 3 off each pizza. If the bill came to 38.9She ordered 6 large pizzas. Luckily, she had a coupon for 3 off each pizza. If the bill came to 38.94, what was the price for a large pizza?
[U]Determine additional amount the pizzas would have cost without the coupon[/U]
6 pizzas * 3 per pizza = 18
[U]Add 18 to our discount price of 38.94[/U]
Full price for 6 large pizzas = 38.94 + 18
Full price for 6 large pizzas = 56.94
Let x = full price per pizza before the discount. Set up our equation:
6x = 56.94
Divide each side by 6
[B]x = $9.49[/B]
Sierra borrows $310 from her brother to buy a lawn mower. She will repay $85 to start, and then anotSierra borrows $310 from her brother to buy a lawn mower. She will repay $85 to start, and then another $25 per week. A. Write an equation that can be used to determine w, the number of weeks it will take for Sierra to repay the entire amount.
Let w be the number of weeks. We have the equation:
25w + 85 = 310
[URL='https://www.mathcelebrity.com/1unk.php?num=25w%2B85%3D310&pl=Solve']Type this equation into the search engine[/URL], and we get:
w = [B]9[/B]
Sign TestFree Sign Test Calculator - This will determine whether to accept or reject a null hypothesis based on a number set, mean value, alternative hypothesis, and a significance level using the Sign Test.
Simplify 7sqrt(3) - sqrt(12)7sqrt(3) is broken down.
sqrt(12) is not broken down. Let's find all the factors of 12 and see if we stumble on a perfect square:
[LIST]
[*]1 * 12
[*]2 * 6
[*]3 * 4
[/LIST]
4 is a perfect square, since sqrt(4) = 2.
So sqrt(12) = sqrt(3 * 4)
We pull the sqrt(4) = 2 outside the radical and rewrite our problem as:
7sqrt(3) - 2sqrt(3)
These are like terms, so we have:
(7 - 2)sqrt(3)
[B]5sqrt(3) [/B]
[MEDIA=youtube]ljXVXWnKiWY[/MEDIA]
sin(x)cot(x)sin(x)cot(x)
We know that cot(x) = cos(x)/sin(x), so we rewrite this as:
sin(x)cos(x)/sin(x)
The sin(x) terms cancel and we get:
[B]cos(x)[/B]
Sinking Fund Depreciation MethodFree Sinking Fund Depreciation Method Calculator - Using the Sinking Fund method of Depreciation, this calculator determines the following:
* Depreciation at time t (Dt)
* Asset Value (A)
* Salvage Value (S)
* Book Value at time t (Bt)
Six Less than the total of three times a number and negative eightSix Less than the total of three times a number and negative eight.
Let's take this in pieces:
Three times a number = 3x
The total of this and negative eight means we subtract eight
3x - 8
Six Less than this total means we subtract 6
3x - 8 - 6
Simplify by combining like terms:
[B]3x - 14[/B]
Small pizzas were $3 and large pizzas were $5. To feed the throng, it was necessary to spend $475 foSmall pizzas were $3 and large pizzas were $5. To feed the throng, it was necessary to spend $475 for 125 pizzas. How many small pizzas were purchased?
Let s be the number of small pizzas and l be the number of large pizzas. We have two given equations:
[LIST=1]
[*]l + s = 125
[*]3s + 5l = 475
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=l+%2B+s+%3D+125&term2=3s+%2B+5l+%3D+475&pl=Cramers+Method']simultaneous equation calculator[/URL], we get [B]s = 75[/B]:
Solution MixtureFree Solution Mixture Calculator - Determines a necessary amount of a Solution given two solution percentages and 1 solution amount.
Solve 11 - 1/2y = 3 + 6x for ySolve 11 - 1/2y = 3 + 6x for y
Subtract 11 from each side so we can isolate the y term:
11 -11 - 1/2y = 3 + 6x - 11
Cancelling the 11's on the left side, we get:
-1/2y = 6x - 8 <-- Since 3 - 11 = -8
Multiply both sides of the equation by -2 to remove the -1/2 on the left side:
-2(-1/2)y = -2(6x - 8)
Simplifying, we get:
y = [B]-12x + 16
[MEDIA=youtube]38uwIaj88Lw[/MEDIA][/B]
Solve for h. rs + h^2 = lSolve for h. rs + h^2 = l
[U]Subtract rs from each side to isolate h:[/U]
rs - rs + h^2 = l - rs
[U]Cancel the rs terms on the left side, and we get:[/U]
h^2 = l - rs
[U]Take the square root of each side:[/U]
h = [B]sqrt(l - rs)[/B]
Solve mgh=1/2mv^2+1/2(2/5)mr^2(v^2/r^2) for vSolve mgh=1/2mv^2+1/2(2/5)mr^2(v^2/r^2) for v
1/2(2/5) = 1/5 since the 2's cancel
r^2/r^2 = 1
So we simplify, and get:
mgh=1/2mv^2+1/5(mv^2) for v
Divide each side by m, so m's cancel in each term on the left and right side:
gh = 1/2v^2 + 1/5(v^2)
Combine like terms for v^2 on the right side:
1/2 + 1/5 = 7/10 from our [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F2&frac2=1%2F5&pl=Add']fraction calculator[/URL]
So we have:
gh = 7v^2/10
Multiply each side by 10:
10gh = 7v^2
Now divide each side by 7
10gh/7 = v^2
Take the square root of each side:
[B]v = sqrt(10gh/7)[/B]
Solve the problema confidence interval for a population mean has a margin of error of 0.081. Determine the length of the confidence interval
Solving word problems with the matrix method?Hello everyone.
I am stuck on a work question that we are required to solve using the matrix (or Gauss-Jordan) method.
[CENTER]"A car rental company wants to buy 100 new cars. Compact cars cost $12,000 each,
intermediate size cars cost $18,000 each, full size cars cost $24,000 each, and the company
has a budget of $1,500,000. If they purchase twice as many compact cars as intermediate
sized cars, determine the number of cars of each type that they buy, assuming they
spend the entire budget."
[/CENTER]
I am fairly certain that I could solve this easily, except I cannot figure out the proper three equations that correspond to this question. I someone could help me figure them out, it would be greatly appreciated!
Solving word problems with the matrix method?Let c be the cost of compact cars, i be the cost of intermediate cards, and f be the cost of full-size cars. We have the following equations:
[LIST]
[*]c + i + f = 100
[*]12,000c + 18,000i + 24,000 f = 1,500,000
[*]c = 2i
[/LIST]
Some History teachers at Richmond High School are purchasing tickets for students and their adult chSome History teachers at Richmond High School are purchasing tickets for students and their adult chaperones to go on a field trip to a nearby museum. For her class, Mrs. Yang bought 30 student tickets and 30 adult tickets, which cost a total of $750. Mr. Alexander spent $682, getting 28 student tickets and 27 adult tickets. What is the price for each type of ticket?
Let the number of adult tickets be a
Let the number of student tickets be s
We're given two equations:
[LIST=1]
[*]30a + 30s = 750
[*]27a + 28s = 682
[/LIST]
To solve the simultaneous equations, we can use any of three methods below:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=30a+%2B+30s+%3D+750&term2=27a+%2B+28s+%3D+682&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=30a+%2B+30s+%3D+750&term2=27a+%2B+28s+%3D+682&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=30a+%2B+30s+%3D+750&term2=27a+%2B+28s+%3D+682&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we use, we get the same answers:
[LIST]
[*][B]a = 18[/B]
[*][B]s = 7[/B]
[/LIST]
Sonia visited a park in California that had redwood trees. When Sonia asked how tall a certain largeSonia visited a park in California that had redwood trees. When Sonia asked how tall a certain large redwood tree was, the ranger said that he wouldn't tell its height, but would give Sonia a clue. How tall is the redwood tree Sonia asked about?
Sonia said the tree is 64 times my height. The tree is also 112 feet taller than the tree next to it. The two trees plus my height total 597.5 feet.
[LIST]
[*]Rangers's height = n
[*]Tree height = 64n
[*]Smaller tree height = 64n - 112
[*]Total height = 64n - 112 + 64n = 597.5
[/LIST]
Solve for [I]n[/I] in the equation 64n - 112 + 64n = 597.5
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(64 + 64)n = 128n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
128n - 112 = + 597.5
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -112 and 597.5. To do that, we add 112 to both sides
128n - 112 + 112 = 597.5 + 112
[SIZE=5][B]Step 4: Cancel 112 on the left side:[/B][/SIZE]
128n = 709.5
[SIZE=5][B]Step 5: Divide each side of the equation by 128[/B][/SIZE]
128n/128 = 709.5/128
n = 5.54296875
Tree height = 64 * ranger height
Tree height = 64 * 5.54296875
Tree height = [B]354.75 feet[/B]
Square NumberFree Square Number Calculator - This calculator determines the nth square number
Square Roots and ExponentsFree Square Roots and Exponents Calculator - Given a number (n), or a fraction (n/m), and/or an exponent (x), or product of up to 5 radicals, this determines the following:
* The square root of n denoted as √n
* The square root of the fraction n/m denoted as √n/m
* n raised to the xth power denoted as nx (Write without exponents)
* n raised to the xth power raised to the yth power denoted as (nx)y (Write without exponents)
* Product of up to 5 square roots: √a√b√c√d√e
* Write a numeric expression such as 8x8x8x8x8 in exponential form
Stanley bought a ruler and a yardstick for $1.25. If the yardstick cost 45 cents more than the rulerStanley bought a ruler and a yardstick for $1.25. If the yardstick cost 45 cents more than the ruler, what was the cost of the yardstick?
Let r be the cost of the ruler
Let y be the cost of the yardstick
We're given 2 equations:
[LIST=1]
[*]r + y = 1.25
[*]y = r + 0.45
[/LIST]
Substitute equation (2) into equation (1) for y
r + r + 0.45 = 1.25
Solve for [I]r[/I] in the equation r + r + 0.45 = 1.25
[SIZE=5][B]Step 1: Group the r terms on the left hand side:[/B][/SIZE]
(1 + 1)r = 2r
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2r + 0.45 = + 1.25
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 0.45 and 1.25. To do that, we subtract 0.45 from both sides
2r + 0.45 - 0.45 = 1.25 - 0.45
[SIZE=5][B]Step 4: Cancel 0.45 on the left side:[/B][/SIZE]
2r = 0.8
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2r/2 = 0.8/2
r = 0.4
Substitute r = 0.4 into equation (2) above:
y = r + 0.45
y = 0.4 + 0.45
r = [B]0.85
[URL='https://www.mathcelebrity.com/1unk.php?num=r%2Br%2B0.45%3D1.25&pl=Solve']Source[/URL][/B]
Static Determinacy and StabilityFree Static Determinacy and Stability Calculator - Given a number of joints (j) and a number of members (m), this determines if a truss is statically determinate, statically indeterminate, or unstable
Stock A is worth 4.5. Stock B is worth 8.0. Stock C is worth 10.0. She purchased half as many sharesStock A is worth 4.5. Stock B is worth 8.0. Stock C is worth 10.0. She purchased half as many shares of B as A and half as many shares of C as B. If her investments are worth 660, how many shares of each stock does she own?
Let s be the number of shares in Stock A. We have:
[LIST=1]
[*]A: 4.5s
[*]B: 8s/2 = 4s
[*]C: 10s/4 = 2.5s
[/LIST]
Value equation: 4.5s + 4s + 2.5s = 660
Combining like terms:
11s = 660
Using the [URL='http://www.mathcelebrity.com/1unk.php?num=11s%3D660&pl=Solve']equation calculator[/URL], we get [B]s = 60[/B] for Stock A
Stock B shares is equal to 1/2A = [B]30[/B]
Stock C shares is equal to 1/2B = [B]15[/B]
Students stuff envelopes for extra money. Their initial cost to obtain the information for the job wStudents stuff envelopes for extra money. Their initial cost to obtain the information for the job was $140. Each envelope costs $0.02 and they get paid $0.03per envelope stuffed. Let x represent the number of envelopes stuffed. (a) Express the cost C as a function of x. (b) Express the revenue R as a function of x. (c) Determine analytically the value of x for which revenue equals cost.
a) Cost Function
[B]C(x) = 140 + 0.02x[/B]
b) Revenue Function
[B]R(x) = 0.03x[/B]
c) Set R(x) = C(x)
140 + 0.02x = 0.03x
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=140%2B0.02x%3D0.03x&pl=Solve']equation solver[/URL], we get x = [B]14,000[/B]
Subtract 4 from the sum of 2x and 5ySubtract 4 from the sum of 2x and 5y.
The sum of 2x and 5y means we add both terms:
2x + 5y
Subtract 4 from this sum to get our algebraic expression:
[B](2x + 5y) - 4[/B]
sum of a number and 7 is subtracted from 15 the result is 6.Sum of a number and 7 is subtracted from 15 the result is 6.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
We take this expression in pieces. Sum of a number and 7
x + 7
Subtracted from 15
15 - (x + 7)
The result is means an equation, so we set this expression above equal to 6
[B]15 - (x + 7) = 6 <-- This is our algebraic expression[/B]
If the problem asks you to solve for x, we Group like terms
15 - x - 7 = 6
8 - x = 6
[URL='https://www.mathcelebrity.com/1unk.php?num=8-x%3D6&pl=Solve']Type 8 - x = 6 into the search engine[/URL], and we get [B]x = 2[/B]
Sum of N and its next consecutive even integer is 65Sum of N and its next consecutive even integer is 65
Next even consecutive integer is N + 2.
We have N + (N + 2) = 65.
Combine like terms, we have 2N + 2 = 65
[URL='http://www.mathcelebrity.com/1unk.php?num=2n%2B2%3D65&pl=Solve']Running this problem through the search engine[/URL], we get n = 31.5. Meaning this problem is impossible, it cannot be done. n is not an integer, and neither is the next consecutive even integer.
Sum of the First (n) NumbersFree Sum of the First (n) Numbers Calculator - Determines the sum of the first (n)
* Whole Numbers
* Natural Numbers
* Even Numbers
* Odd Numbers
* Square Numbers
* Cube Numbers
* Fourth Power Numbers
Sum to Product and Product to Sum FormulasFree Sum to Product and Product to Sum Formulas Calculator - Given two angles in degrees of u and v, this determines the following:
* Sin(u) ± Sin(v)
* Cos(u) ± Cos(v)
* Sin(u)Sin(v)
* Cos(u)Cos(v)
* Sin(u)Cos(v)
* Cos(u)Sin(v)
* Sin(u + v)
* Sin(u - v)
* Cos(u + v)
* Cos(u - v)
* Tan(u + v)
* Tan(u - v)
Suppose Briley has 10 coins in quarters and dimes and has a total of 1.45. How many of each coin doeSuppose Briley has 10 coins in quarters and dimes and has a total of 1.45. How many of each coin does she have?
Set up two equations where d is the number of dimes and q is the number of quarters:
(1) d + q = 10
(2) 0.1d + 0.25q = 1.45
Rearrange (1) into (3) to solve for d
(3) d = 10 - q
Now plug (3) into (2)
0.1(10 - q) + 0.25q = 1.45
Multiply through:
1 - 0.1q + 0.25q = 1.45
Combine q terms
0.15q + 1 = 1.45
Subtract 1 from each side
0.15q = 0.45
Divide each side by 0.15
[B]q = 3[/B]
Plug our q = 3 value into (3)
d = 10 - 3
[B]d = 7[/B]
Suppose that Sn = 3 + 1/3 + 1/9 + ... + 1/3(n-2)Suppose that Sn = 3 + 1/3 + 1/9 + ... + 1/3(n-2)
a) Find S10 and S∞
b) If the common difference in an arithmetic sequence is twice the first term, show that Sn/Sm = n^2/m^2
a) Sum of the geometric sequence is
a = 3 and r = 1/3
(a(1 - r)^n)/(1 - r)
(3(1 - 1/3)^9)/(1 - 1/3)
[B]S10 = 4.499771376[/B]
For infinity, as n goes to infinity, the numerator goes to 1
so we have [B]S∞ = 3(1)/2/3 = 4.5[/B]
b) Sum of an arithmetic sequence formula is below:
n(a1 + an)/2
an = a1 + (n - 1)2a1 since d = 2a1
n(a1 + a1 + (n - 1)2a1)/2
(2a1n + n^2 - 2a1n)/2
n^2/2
For Sm
m(a1 + am)/2
am = a1 + (m - 1)2a1 since d = 2a1
m(a1 + 1 + (m - 1)2a1)/2
(2a1m + m^2 - 2a1m)/2
m^2/2
Sn/Sm = n^2/m^2 (cancel the 2's)
S10/S1 = 10^2/1^2
We know S1 = 3
So we have 100(3)/1
[B]S10 = 300[/B]
Suppose that the manager of the Commerce Bank at Glassboro determines that 40% of all depositors havSuppose that the manager of the Commerce Bank at Glassboro determines that 40% of all depositors have a multiple accounts at the bank. If you, as a branch manager, select a random sample of 200 depositors, what is the probability that the sample proportion of depositors with multiple accounts is between 35% and 50%?
[URL='http://www.mathcelebrity.com/proportion_hypothesis.php?x=50&n=+100&ptype==&p=+0.4&alpha=+0.05&pl=Proportion+Hypothesis+Testing']50% proportion probability[/URL]: z = 2.04124145232
[URL='http://www.mathcelebrity.com/proportion_hypothesis.php?x=+35&n=+100&ptype==&p=+0.4&alpha=+0.05&pl=Proportion+Hypothesis+Testing']35% proportion probability[/URL]: z = -1.02062072616
Now use the [URL='http://www.mathcelebrity.com/zscore.php?z=p%28-1.02062072616
Survival RatesFree Survival Rates Calculator - Given a set of times and survival population counts, the calculator will determine the following:
Survival Population lx
Mortality Population dx
Survival Probability px
Mortality Probability qx
In addition, the calculator will determine the probability of survival from tx to tx + n
Synthetic DivisionFree Synthetic Division Calculator - Using Ruffinis Rule, this performs synthetic division by dividing a polynomial with a maximum degree of 6 by a term (x ± c) where c is a constant root using the factor theorem. The calculator returns a quotient answer that includes a remainder if applicable. Also known as the Rational Zero Theorem
T-shirts sell for $19.97 and cost $14.02 to produce. Which equation represents p, the profit, in terT-shirts sell for $19.97 and cost $14.02 to produce. Which equation represents p, the profit, in terms of x, the number of t-shirts sold?
A) p = $19.97x - $14.02
B) p = x($19.97 - $14.02)
C) p = $19.97 + $14.02x
D) p = x($19.97 + $14.02)
[B]B) p = x($19.97 - $14.02)[/B]
[B][/B]
[LIST]
[*]Profit is Revenue - Cost
[*]Each shirt x generates a profit of 19.97 - 14.02
[/LIST]
Target Heart RateFree Target Heart Rate Calculator - Given an age, this calculator determines the following 5 target heart rate zones:
Healthy Heart Zone (Warm up) 50 - 60%
Fitness Zone (Fat Burning) 60 - 70%
Aerobic Zone (Endurance Training) 70 - 80%
Anaerobic Zone (Performance Training) 80 - 90%
Red Line (Maximum Effort) 90 - 100%
Temperature ChangeFree Temperature Change Calculator - This calculator determines the total temperature change
The admission fee at an amusement park is $1.50 for children and $4 for adults. On a certain day, 32The admission fee at an amusement park is $1.50 for children and $4 for adults. On a certain day, 327 people entered the park , and the admission fee collected totaled 978.00 dollars . How many children and how many adults were admitted?
Let the number of children's tickets be c. Let the number of adult tickets be a. We're given two equations:
[LIST=1]
[*]a + c = 327
[*]4a + 1.50c = 978
[/LIST]
We can solve this system of equation 3 ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+c+%3D+327&term2=4a+%2B+1.50c+%3D+978&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+c+%3D+327&term2=4a+%2B+1.50c+%3D+978&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+c+%3D+327&term2=4a+%2B+1.50c+%3D+978&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answers:
[LIST]
[*][B]a = 195[/B]
[*][B]c = 132[/B]
[/LIST]
The admission fee at an amusement park is $1.50 for children and $4.00 for adults. On a certain day,The admission fee at an amusement park is $1.50 for children and $4.00 for adults. On a certain day, 281 people entered the park, and the admission fees collected totaled $784 . How many children and how many adults were admitted?
Let c be the number of children and a be the number of adults. We have two equations:
[LIST=1]
[*]a + c = 281
[*]4a + 1.5c = 784
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=a%2Bc%3D281&term2=4a+%2B+1.5c+%3D+784&pl=Cramers+Method']simultaneous equations calculator[/URL], we get:
[LIST]
[*][B]a = 145[/B]
[*][B]c = 136[/B]
[/LIST]
The average of a number and double the number is 25.5Let x equal "a number".
Double the number is 2x.
The average is (x + 2x)/2
Combine the terms in the numerator:
3x/2
The word [I]is[/I] means equal to, so we set 3x/2 equal to 25.5
3x/2 = 25.5
Cross multiply the 2:
3x = 51
Divide each side by 3
[B]x = 17[/B]
The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognitThe brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the bran. How many adults must he survey in order to be 90% confident that his estimate is within seven percentage points of the true population percentage?
[IMG]https://ci5.googleusercontent.com/proxy/kc6cjrLvUq64guMaArhfiSR0mOnTrBwB9iFM9u9VaZ5YYn86CSDWXr1FNyqxylwytHdbQ3iYsUDnavt-zvt-OK0=s0-d-e1-ft#http://latex.codecogs.com/gif.latex?%5Chat%20p[/IMG] = 0.5
1 - [IMG]https://ci5.googleusercontent.com/proxy/kc6cjrLvUq64guMaArhfiSR0mOnTrBwB9iFM9u9VaZ5YYn86CSDWXr1FNyqxylwytHdbQ3iYsUDnavt-zvt-OK0=s0-d-e1-ft#http://latex.codecogs.com/gif.latex?%5Chat%20p[/IMG] = 0.5
margin of error (E) = 0.07
At 90% confidence level the t is,
alpha = 1 - 90%
alpha = 1 - 0.90
alpha = 0.10
alpha / 2 = 0.10 / 2 = 0.05
Zalpha/2 = Z0.05 = 1.645
sample size = n = (Z[IMG]https://ci4.googleusercontent.com/proxy/mwhpkw3aM19oMNA4tbO_0OdMXEHt9juW214BnNpz4kjXubiVJgwolO7CLbmWXXoSVjDPE_T0CGeUxNungBjN=s0-d-e1-ft#http://latex.codecogs.com/gif.latex?%5Calpha[/IMG] / 2 / E )2 * [IMG]https://ci5.googleusercontent.com/proxy/kc6cjrLvUq64guMaArhfiSR0mOnTrBwB9iFM9u9VaZ5YYn86CSDWXr1FNyqxylwytHdbQ3iYsUDnavt-zvt-OK0=s0-d-e1-ft#http://latex.codecogs.com/gif.latex?%5Chat%20p[/IMG] * (1 - [IMG]https://ci5.googleusercontent.com/proxy/kc6cjrLvUq64guMaArhfiSR0mOnTrBwB9iFM9u9VaZ5YYn86CSDWXr1FNyqxylwytHdbQ3iYsUDnavt-zvt-OK0=s0-d-e1-ft#http://latex.codecogs.com/gif.latex?%5Chat%20p[/IMG] )
= (1.645 / 0.07)^2 *0.5*0.5
23.5^2 * 0.5 * 0.5
552.25 * 0.5 * 0.5
= 138.06
[B]sample size = 138[/B]
[I]He must survey 138 adults in order to be 90% confident that his estimate is within seven percentage points of the true population percentage.[/I]
The club uses the function S(t) = -4,500t + 54,000 to determine the salvage S(t) of a fertilizer bleThe club uses the function S(t) = -4,500t + 54,000 to determine the salvage S(t) of a fertilizer blender t years after its purchase. How long will it take the blender to depreciate completely?
Complete depreciation means the salvage value is 0.
So S(t) = 0. We need to find t to make S(t) = 0
-4,500t + 54,000 = 0
Subtract 54,000 from each side
-4,500t = -54,000
Divide each side by -4,500
[B]t = 12[/B]
The coefficient of determination is found by taking the square root of the coefficient of correlatioThe coefficient of determination is found by taking the square root of the coefficient of correlation. True or False
[B]FALSE[/B] - It is found by squaring the coefficient of correlation
the cost of a buffet at a restaurant is different for adults and kids. the bill for 2 adults and 3 kthe cost of a buffet at a restaurant is different for adults and kids. the bill for 2 adults and 3 kids is $51. the bill for 3 adults and 1 kid is $45. what is the cost per adult and per kid?
Let the cost for each adult be a
Let the cost for each kid be k
We're given two equations:
[LIST=1]
[*]2a + 3k = 51
[*]3a + k = 45
[/LIST]
To solve this simultaneous set of equations, we can use three methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2a+%2B+3k+%3D+51&term2=3a+%2B+k+%3D+45&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2a+%2B+3k+%3D+51&term2=3a+%2B+k+%3D+45&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2a+%2B+3k+%3D+51&term2=3a+%2B+k+%3D+45&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we use, we get the same answer:
[LIST]
[*]a = [B]12[/B]
[*]k = [B]9[/B]
[/LIST]
The cube of the difference of 5 times the square of y and 7 divided by the square of 2 times yThe cube of the difference of 5 times the square of y and 7 divided by the square of 2 times y
Take this in algebraic expression in parts:
[U]Term 1[/U]
[LIST]
[*]The square of y means we raise y to the 2nd power: y^2
[*]5 times the square of y: 5y^2
[/LIST]
[U]Term 2[/U]
[LIST]
[*]2 times y: 2y
[*]The square of 2 times y: (2y)^2 = 4y^2
[*]7 divide by the square of 2 times y: 7/4y^2
[/LIST]
[U]The difference of these terms is written as Term 1 minus Term 2:[/U]
[LIST]
[*]5y^2/4y^2
[/LIST]
[U]The cube of the difference means we raise the difference to the power of 3:[/U]
[B](5y^2/4y^2)^3[/B]
The difference between 2 numbers is 108. 6 times the smaller is equal to 2 more than the larger. Wh?The difference between 2 numbers is 108. 6 times the smaller is equal to 2 more than the larger. What are the numbers?
Let the smaller number be x. Let the larger number be y. We're given:
[LIST=1]
[*]y - x = 108
[*]6x = y + 2
[/LIST]
Rearrange (1) by adding x to each side:
[LIST=1]
[*]y = x + 108
[/LIST]
Substitute this into (2):
6x = x + 108 + 2
Combine like terms
6x = x +110
Subtract x from each side:
5x = 110
[URL='https://www.mathcelebrity.com/1unk.php?num=5x%3D110&pl=Solve']Plugging this equation into our search engine[/URL], we get:
x = [B]22[/B]
the difference between 7 times a number and 9 less than a numberthe difference between 7 times a number and 9 less than a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
7 times a number means we multiply x by 7
7x
9 less than a number means we subtract 9 from x
x - 9
The difference between the two expressions means we subtract (x - 9) from 7x
7x - (x - 9)
Simplifying this, we have:
7x - x + 9
Grouping like terms, we get:
[B]6x + 9[/B]
the difference between triple a number and double a numberthe difference between triple a number and double a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
Triple a number means we multiply x by 3:
3x
Double a number means we multiply x by 2:
2x
The difference means we subtract 2x from 3x:
3x - 2x
Simplifying like terms, we have:
(3 - 2)x = [B]x[/B]
The difference of two numbers is 12 and their mean is 15. Find the two numbersThe difference of two numbers is 12 and their mean is 15. Find the two numbers.
Let the two numbers be x and y. We're given:
[LIST=1]
[*]x - y = 12
[*](x + y)/2 = 15. <-- Mean is an average
[/LIST]
Rearrange equation 1 by adding y to each side:
x - y + y = y + 12
Cancelling the y's on the left side, we get:
x = y + 12
Now substitute this into equation 2:
(y + 12 + y)/2 = 15
Cross multiply:
y + 12 + y = 30
Group like terms for y:
2y + 12 = 30
[URL='https://www.mathcelebrity.com/1unk.php?num=2y%2B12%3D30&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]y = 9[/B]
Now substitute this into modified equation 1:
x = y + 12
x = 9 + 12
[B]x = 21[/B]
The dimensions of a rectangle are 30 cm and 18 cm. When its length decreased by x cm and its width iThe dimensions of a rectangle are 30 cm and 18 cm. When its length decreased
by x cm and its width is increased by x cm, its area is increased by 35 sq. cm.
a. Express the new length and the new width in terms of x.
b. Express the new area of the rectangle in terms of x.
c. Find the value of x.
Calculate the current area. Using our [URL='https://www.mathcelebrity.com/rectangle.php?l=30&w=18&a=&p=&pl=Calculate+Rectangle']rectangle calculator with length = 30 and width = 18[/URL], we get:
A = 540
a) Decrease length by x and increase width by x, and we get:
[LIST]
[*]length = [B]30 - x[/B]
[*]width = [B]18 + x[/B]
[/LIST]
b) Our new area using the lw = A formula is:
(30 - x)(18 + x) = 540 + 35
Multiplying through and simplifying, we get:
540 - 18x + 30x - x^2 = 575
[B]-x^2 + 12x + 540 = 575[/B]
c) We have a quadratic equation. To solve this, [URL='https://www.mathcelebrity.com/quadratic.php?num=-x%5E2%2B12x%2B540%3D575&pl=Solve+Quadratic+Equation&hintnum=+0']we type it in our search engine, choose solve[/URL], and we get:
[B]x = 5 or x = 7[/B]
Trying x = 5, we get:
A = (30 - 5)(18 + 5)
A = 25 * 23
A = 575
Now let's try x = 7:
A = (30 - 7)(18 + 7)
A = 23 * 25
A = 575
They both check out.
So we can have
The first group orders 3 pizzas and 4 drinks for $33.50. The second group orders 5 pizzas and 6 drinThe first group orders 3 pizzas and 4 drinks for $33.50. The second group orders 5 pizzas and 6 drinks for $54. Find the cost for each pizza and each drink
Assumptions:
[LIST]
[*]Let the cost of each pizza be p
[*]Let the cost of each drink be d
[/LIST]
Givens:
[LIST=1]
[*]4d + 3p = 33.50
[*]6d + 5p = 54
[/LIST]
We have a simultaneous group of equations. To solve this, we can use 3 methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=4d+%2B+3p+%3D+33.50&term2=6d+%2B+5p+%3D+54&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=4d+%2B+3p+%3D+33.50&term2=6d+%2B+5p+%3D+54&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=4d+%2B+3p+%3D+33.50&term2=6d+%2B+5p+%3D+54&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we use, we get the same answer:
[LIST]
[*]d = [B]$2.75[/B]
[*]p = [B]$7.5[/B]
[/LIST]
The function f(x) = e^x(x - 3) has a critical point at x =The function f(x) = e^x(x - 3) has a critical point at x =
The critical point is where the derivative equals 0.
We multiply through for f(x) to get:
f(x) = xe^x - 3e^x
Using the product rule on the first term f'g + fg', we get:
f'(x) = xe^x + e^x - 3e^x
f'(x) = xe^x -2e^x
f'(x) = e^x(x - 2)
We want f'(x) = 0
e^x(x - 2) = 0
When [B]x = 2[/B], then f'(x) = 0
The function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the baThe function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the basis of a ticket price, x. Both the profit and the ticket price are in dollars. What is the maximum profit, and how much should the tickets cost?
Take the [URL='http://www.mathcelebrity.com/dfii.php?term1=-30x%5E2+%2B+360x+%2B+785&fpt=0&ptarget1=0&ptarget2=0&itarget=0%2C1&starget=0%2C1&nsimp=8&pl=1st+Derivative']derivative of the profit function[/URL]:
P'(x) = -60x + 360
We find the maximum when we set the profit derivative equal to 0
-60x + 360 = 0
Subtract 360 from both sides:
-60x = -360
Divide each side by -60
[B]x = 6 <-- This is the ticket price to maximize profit[/B]
Substitute x = 6 into the profit equation:
P(6) = -30(6)^2 + 360(6) + 785
P(6) = -1080 + 2160 + 785
[B]P(6) = 1865[/B]
The Lakers recently scored 81 points. Their points came from 2 and 3 point baskets. If they made 39The Lakers recently scored 81 points. Their points came from 2 and 3 point baskets. If they made 39 baskets total, how many of each basket did they make?
Let x = 2 point baskets and y = 3 point baskets. We have the following given equations:
[LIST=1]
[*]x + y = 39
[*]2x + 3y = 81
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=x%2By%3D39&term2=2x+%2B+3y+%3D+81&pl=Cramers+Method']simultaneous equations calculator[/URL], we get:
[B]x = 36 <-- 2 point baskets
y = 3 <-- 3[B] point baskets
[/B][/B]
To confirm our work:
[LIST=1]
[*]36 + 3 = 39
[*]2(36) + 3(3) = 72 + 9 = 81
[/LIST]
The left and right page numbers of an open book are two consecutive integers whose sum is 403. FindThe left and right page numbers of an open book are two consecutive integers whose sum is 403. Find these page numbers.
Page numbers left and right are consecutive integers. So we want to find a number n and n + 1 where:
n + n + 1 = 403
Combining like terms, we get:
2n + 1 = 403
Typing that equation into our search engine, we get:
[B]n = 201[/B]
This is our left hand page. Our right hand page is:
201 + 1 = [B]202[/B]
The length of a rectangular building is 6 feet less than 3 times the width. The perimeter is 120 feeThe length of a rectangular building is 6 feet less than 3 times the width. The perimeter is 120 feet. Find the width and length of the building.
P = 2l + 2w
Since P = 120, we have:
(1) 2l + 2w = 120
We are also given:
(2) l = 3w - 6
Substitute equation (2) into equation (1)
2(3w - 6) + 2w = 120
Multiply through:
6w - 12 + 2w = 120
Combine like terms:
8w - 12 = 120
Add 12 to each side:
8w = 132
Divide each side by 8 to isolate w:
w =16.5
Now substitute w into equation (2)
l = 3(16.5) - 6
l = 49.5 - 6
l = 43.5
So (l, w) = (43.5, 16.5)
The margarita is one of the most common tequila-based cocktails, made with tequila, triple sec, andThe margarita is one of the most common tequila-based cocktails, made with tequila, triple sec, and lime
juice, often served with salt on the glass rim. A manager at a local bar is concerned that the bartender is
not using the correct amounts of the three ingredients in more than 50% of margaritas. He secretly
observed the bartender and found that he used the CORRECT amounts in only 9 out of the 39
margaritas in the sample. Use the critical value approach to test if the manager's suspicion is justified
at α = 0.10. Let p represent the proportion of all margaritas made by the bartender that have
INCORRECT amounts of the three ingredients. Use Table 1.
a. Select the null and the alternative hypotheses.
[B]H0: p ≤ 0.50; HA: p > 0.50[/B]
[B][/B]
b. Calculate the sample proportion. (Round your answer to 3 decimal places.)
9/39 = [B]0.231
[/B]
c. Calculate the value of test statistic. (Round your intermediate calculations to 4 decimal places and final answer to 2 decimal places.)
Using our [URL='http://www.mathcelebrity.com/proportion_hypothesis.php?x=9&n=39&ptype=%3C&p=+0.5&alpha=+0.10&pl=Proportion+Hypothesis+Testing']proportion hypothesis calculator[/URL], we get:
[B]Test Stat = -3.36[/B]
[B][/B]
d. Calculate the critical value. (Round your answer to 2 decimal places.)
Using the link above, we get a critical value of [B]1.2816
[/B]
e. What is the conclusion?
[B]The manager’s suspicion is not justified since the value of the test statistic does not fall in the rejection region. Do not reject H0[/B]
[B][/B]
The next number in the series 2,5,11,20,32,47, isThe next number in the series 2,5,11,20,32,47, is
[LIST]
[*]2 + 3 = 5
[*]5 + 6 = 11
[*]11 + 9 = 20
[*]20 + 12 = 32
[*]32 + 15 = 47
[/LIST]
Notice the addition pattern:
3, 6, 9, 12, 15
This means our next term is:
47 + (15 + 3)
47 + 18
[B]65
[MEDIA=youtube]mAj3tqXUbZs[/MEDIA][/B]
The noise level of an ambulance siren is 10 decibels louder than that of a car horn. If d representsThe noise level of an ambulance siren is 10 decibels louder than that of a car horn. If d represents the noise level, in decibels, of a car horn, express the noise level of an ambulance siren in terms of the noise level of a car horn.
Let s be the level of the ambulance siren.
[B]s = d + 10[/B]
The perimeter of a garden is 70 meters. Find its actual dimensions if its length is 5 meters longerThe perimeter of a garden is 70 meters. Find its actual dimensions if its length is 5 meters longer than twice its width.
Let w be the width, and l be the length. We have:
P = l + w. Since P = 70, we have:
[LIST=1]
[*]l + w = 70
[*]l = 2w + 5
[/LIST]
Plug (2) into (1)
2w + 5 + w = 70
Group like terms:
3w + 5 = 70
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=3w%2B5%3D70&pl=Solve']equation calculator[/URL], we get [B]w = 21.66667[/B]. Which means length is:
l = 2(21.6667) + 5
l = 43.33333 + 5
[B]l = 48.3333[/B]
The perimeter of a rectangular field is 220 yd. the length is 30 yd longer than the width. Find theThe perimeter of a rectangular field is 220 yd. the length is 30 yd longer than the width. Find the dimensions
We are given the following equations:
[LIST=1]
[*]220 = 2l + 2w
[*]l = w + 30
[/LIST]
Plug (1) into (2)
2(w + 30) + 2w = 220
2w + 60 + 2w = 220
Combine like terms:
4w + 60 = 220
[URL='https://www.mathcelebrity.com/1unk.php?num=4w%2B60%3D220&pl=Solve']Plug 4w + 60 = 220 into the search engine[/URL], and we get [B]w = 40[/B].
Now plug w = 40 into equation (2)
l = 40 + 30
[B]l = 70[/B]
The perimeter of a rectangular outdoor patio is 54 ft. The length is 3 ft greater than the width. WhThe perimeter of a rectangular outdoor patio is 54 ft. The length is 3 ft greater than the width. What are the dimensions of the patio?
Perimeter of a rectangle is:
P = 2l + 2w
We're given l = w + 3 and P = 54. So plug this into our perimeter formula:
54= 2(w + 3) + 2w
54 = 2w + 6 + 2w
Combine like terms:
4w + 6 = 54
[URL='https://www.mathcelebrity.com/1unk.php?num=4w%2B6%3D54&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]w = 12[/B]
Plug this into our l = w + 3 formula:
l = 12 + 3
[B]l = 15[/B]
The price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she paThe price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she paid $75. What is the cost of the cheap backpack?
backpack cost = b
Cheap backpack = b - 15
The total of both items equals 75:
b + b - 15 = 75
Solve for [I]b[/I] in the equation b + b - 15 = 75
[SIZE=5][B]Step 1: Group the b terms on the left hand side:[/B][/SIZE]
(1 + 1)b = 2b
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2b - 15 = + 75
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -15 and 75. To do that, we add 15 to both sides
2b - 15 + 15 = 75 + 15
[SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE]
2b = 90
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2b/2 = 90/2
b = 45
Cheap backpack = 45 - 15 = [B]30
[URL='https://www.mathcelebrity.com/1unk.php?num=b%2Bb-15%3D75&pl=Solve']Source[/URL][/B]
The principal randomly selected six students to take an aptitude test. Their scores were: 87.4 86.9The principal randomly selected six students to take an aptitude test.
Their scores were: 87.4 86.9 89.9 78.3 75.1 70.6
Determine a 90% confidence interval for the mean score for all students.
The principal randomly selected six students to take an aptitude test. Their scores were: 87.4 86.9First, determine the [URL='http://www.mathcelebrity.com/statbasic.php?num1=87.4%2C86.9%2C89.9%2C78.3%2C75.1%2C70.6&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']mean and standard deviation[/URL] for the [I]sample[/I]
Mean = 81.3667
SD = 7.803
Next, use our [URL='http://www.mathcelebrity.com/normconf.php?n=6&xbar=81.3667&stdev=7.803&conf=90&rdig=4&pl=Small+Sample']confidence interval for the mean calculator[/URL] with these values and n = 6
[B]74.9478 < u < 87.7856[/B]
The product of two positive numbers is 96. Determine the two numbers if one is 4 more than the otherThe product of two positive numbers is 96. Determine the two numbers if one is 4 more than the other.
Let the 2 numbers be x and y.
We have:
[LIST=1]
[*]xy = 96
[*]x = y - 4
[/LIST]
[U]Substitute (2) into (1)[/U]
(y - 4)y = 96
y^2 - 4y = 96
[U]Subtract 96 from both sides:[/U]
y^2 - 4y - 96 = 0
[U]Factoring using our quadratic calculator, we get:[/U]
(y - 12)(y + 8)
So y = 12 and y = -8. Since the problem states positive numbers, we use [B]y = 12[/B].
Substituting y = 12 into (2), we get:
x = 12 - 4
[B]x = 8[/B]
[B]We have (x, y) = (8, 12)[/B]
The Radio City Music Hall is selling tickets to Kayla’s premiere at the Rockettes. On the first dayThe Radio City Music Hall is selling tickets to Kayla’s premiere at the Rockettes. On the first day of ticket sales they sold 3 senior citizen tickets and 9 child tickets for a total of $75. It took in $67 on the second day by selling 8 senior citizen tickets and 5 child tickets. What is the price of each senior citizen ticket and each child ticket?
Let the cost of child tickets be c
Let the cost of senior tickets be s
Since revenue = cost * quantity, we're given two equations:
[LIST=1]
[*]9c + 3s = 75
[*]5c + 8s = 67
[/LIST]
To solve this simultaneous group of equations, we can use 3 methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=9c+%2B+3s+%3D+75&term2=5c+%2B+8s+%3D+67&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=9c+%2B+3s+%3D+75&term2=5c+%2B+8s+%3D+67&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=9c+%2B+3s+%3D+75&term2=5c+%2B+8s+%3D+67&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we use, we get the same answer:
[LIST]
[*][B]c = 7[/B]
[*][B]s = 4[/B]
[/LIST]
The regular price for a television is Q dollars. Each Saturday televisions are 20% off (The discountThe regular price for a television is Q dollars. Each Saturday televisions are 20% off (The discount is .2Q). What is the price of a television on Saturday in terms of Q?
Q = Regular Price
.2Q = Discount
Discounted Price = Q - .2Q = [B]0.8Q[/B]
The relief time provided by a standard dose of a popular children’s allergy medicine averages 7.9The relief time provided by a standard dose of a popular children’s allergy medicine averages 7.9 hours with a standard deviation of 2.2 hours. Use Table 1.
a. Determine the percentage of children who experience relief for less than 6.4 hours if the relief time follows a normal distribution. (Round your answer to 2 decimal places.)
Using our [URL='http://www.mathcelebrity.com/probnormdist.php?xone=6.4&mean=7.9&stdev=2.2&n=1&pl=P%28X+%3C+Z%29']normal distribution calculator[/URL], we get
Answer = [B]0.25[/B]
The school is selling potted plants as a fundraiser. Kara sold 12 ferns and 8 ivy plants for 260.00.The school is selling potted plants as a fundraiser. Kara sold 12 ferns and 8 ivy plants for 260.00. Paul sold 15 ivy plants and 6 ferns for 240. What’s the selling price of each plant.
Let the cost of each fern be f
Let the cost of each ivy plant be I
We're given:
[LIST=1]
[*]12f + 8i = 260
[*]15i + 6f = 240
[/LIST]
To solve this system of equations, we can use 3 methods:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=12f+%2B+8i+%3D+260&term2=15f+%2B+6i+%3D+240&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=12f+%2B+8i+%3D+260&term2=15f+%2B+6i+%3D+240&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=12f+%2B+8i+%3D+260&term2=15f+%2B+6i+%3D+240&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
[LIST]
[*][B]f = 7.5[/B]
[*][B]i= 21.25[/B]
[/LIST]
The school yearbook costs $15 per book to produce with an overhead of $5500. The yearbook sells forThe school yearbook costs $15 per book to produce with an overhead of $5500. The yearbook sells for $40. Write a cost and revenue function and determine the break-even point.
[U]Calculate cost function C(b) with b as the number of books:[/U]
C(b) = Cost per book * b + Overhead
[B]C(b) = 15b + 5500[/B]
[U]Calculate Revenue Function R(b) with b as the number of books:[/U]
R(b) = Sales Price per book * b
[B]R(b) = 40b[/B]
[U]Calculate break even function E(b):[/U]
Break-even Point = Revenue - Cost
Break-even Point = R(b) - C(b)
Break-even Point = 40b - 15b - 5500
Break-even Point = 25b - 5500
[U]Calculate break even point:[/U]
Break-even point is where E(b) = 0. So we set 25b - 5500 equal to 0
25b - 5500 = 0
To solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=25b-5500%3D0&pl=Solve']type this equation into our search engine[/URL] and we get:
[B]b = 220[/B]
The senior class at high school A and high school B planned separate trips to the state fair. ThereThe senior class at high school A and high school B planned separate trips to the state fair. There senior class and high school A rented and filled 10 vans and 6 buses with 276 students. High school B rented and filled 5 vans and 2 buses with 117 students. Every van had the same number of students in them as did the buses. How many students can a van carry?? How many students can a bus carry??
Let b be the number of students a bus can carry. Let v be the number of students a van can carry. We're given:
[LIST=1]
[*]High School A: 10v + 6b = 276
[*]High School B: 5v + 2b = 117
[/LIST]
We have a system of equations. We can solve this 3 ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10v+%2B+6b+%3D+276&term2=5v+%2B+2b+%3D+117&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10v+%2B+6b+%3D+276&term2=5v+%2B+2b+%3D+117&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10v+%2B+6b+%3D+276&term2=5v+%2B+2b+%3D+117&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get:
[LIST]
[*][B]b = 21[/B]
[*][B]v = 15[/B]
[/LIST]
the square of the sum of x and y is less than 20the square of the sum of x and y is less than 20
The sum of x and y means we add y to x:
x + y
the square of the sum of x and y means we raise the term x + y to the 2nd power:
(x + y)^2
The phrase [I]is less than[/I] means an inequality, so we write this as follows:
[B](x + y)^2 < 20[/B]
The sum of -4x^2 - 5x + 7 and 2x^2 + 8x - 11 can be written in the form ax^2 + bx + c, where a, b, aThe sum of -4x^2 - 5x + 7 and 2x^2 + 8x - 11 can be written in the form ax^2 + bx + c, where a, b, and c are constants. What is the value of a + b + c?
The sum means we add the polynomials together. We do this by adding the like terms:
-4x^2 - 5x + 7 + 2x^2 + 8x - 11
(-4 +2)x^2 + (-5 + 8)x +(7 - 11)
-2x^2 + 3x - 4
We have (a, b, c) = (-2, 3, -4)
The question asks for a + b + c
a + b + c = -2 + 3 - 4
a + b + c = [B]-3[/B]
The sum of 2 consecutive numbers is 3 less than 3 times the first number. What are the numbers?The sum of 2 consecutive numbers is 3 less than 3 times the first number. What are the numbers?
Let the first number be x. And the second number be y. We're given:
[LIST=1]
[*]y = x + 1
[*]x + y = 3x - 3 (less 3 means subtract 3)
[/LIST]
Substitute (1) into (2):
x + x + 1 = 3x - 3
Combine like terms:
2x + 1 = 3x - 3
[URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B1%3D3x-3&pl=Solve']Type this equation into the search engine[/URL], we get:
x = 4
Substituting x = 4 into equation 1:
y = 4 + 1
y = 5
So (x, y) = [B](4, 5)[/B]
The sum of 2 numbers is 18. 3 times the greater number exceeds 4 times the smaller number by 5. FindThe sum of 2 numbers is 18. 3 times the greater number exceeds 4 times the smaller number by 5. Find the numbers.
Let the first number be x. The second number is y. We have:
[LIST=1]
[*]x + y = 18
[*]3x = 4y + 5
[/LIST]
Rearrange (2), by subtracting 4y from each side:
3x - 4y = 5
So we have a system of equations:
[LIST=1]
[*]x + y = 18
[*]3x - 4y = 5
[/LIST]
Using our [URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+y+%3D+18&term2=3x+-+4y+%3D+5&pl=Cramers+Method']simultaneous equations calculator[/URL], we get:
[B]x = 11
y = 7[/B]
the sum of 2 numbers is 5. 5 times the larger number plus 4 times the smaller number is 37. Find thethe sum of 2 numbers is 5. 5 times the larger number plus 4 times the smaller number is 37. Find the numbers
Let the first small number be x. Let the second larger number be y. We're given:
[LIST=1]
[*]x + y = 5
[*]5y + 4x = 37
[/LIST]
We can solve this 3 ways, using the following methods:
[LIST=1]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+y%3D5&term2=5y+%2B+4x+%3D+37&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+y%3D5&term2=5y+%2B+4x+%3D+37&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+y%3D5&term2=5y+%2B+4x+%3D+37&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we choose, we get:
[B]x = -12
y = 17
[/B]
Let's check our work using equation 1:
-12 + 17 ? 5
5 = 5 <-- Check
Let's check our work using equation 2:
5(17) + 4(-12) ? 37
85 - 48 ? 37
37 = 37 <-- Check
The sum of 2 numbers is 60. The larger number is thrice the smallerThe sum of 2 numbers is 60. The larger number is thrice the smaller.
Let the 2 numbers be x and y, where x is the smaller number and y is the larger number. We are given:
[LIST=1]
[*]x + y = 60
[*]y = 3x
[/LIST]
Substitute (2) into (1):
x + (3x) = 60
Combine like terms:
4x = 60
[URL='https://www.mathcelebrity.com/1unk.php?num=4x%3D60&pl=Solve']Type 4x = 60 into our search engine[/URL], and you get [B]x = 15[/B].
Substituting x = 15 into Equation (2) above, we get:
y = 3(15)
[B]y = 45
[/B]
Check our work in Equation (1):
15 + 45 ? 60
60 = 60
Check our work in Equation (2):
45 ? 15(3)
45 = 45
The numbers check out, so our answer is [B](x, y) = (15, 45)[/B]
The sum of 2 numbers is 70. The difference of these numbers is 24. Write and solve a system of equatThe sum of 2 numbers is 70. The difference of these numbers is 24. Write and solve a system of equations to determine the numbers.
Let the two numbers be x and y. We have the following equations:
[LIST=1]
[*]x + y = 70
[*]x - y = 24
[/LIST]
Add (1) to (2):
2x = 94
Divide each side by 2
[B]x = 47[/B]
Plug this into (1)
47 + y = 70
Subtract 47 from each side, we have:
[B]y = 23[/B]
the sum of 2 times a number and -2, added to 4 times a numberthe sum of 2 times a number and -2, added to 4 times a number.
The phrase, [I]a number[/I], means an arbitrary variable, let's call it x.
2 times a number
2x
The sum of means add, so we add -2, which is the same as subtracting 2
2x - 2
Now, we add 4 times x
2x - 2 + 4x
Combining like terms, we have:
(2 + 4)x - 2
[B]6x - 2[/B]
The sum of 3 consecutive integers is greater than 30.The sum of 3 consecutive integers is greater than 30.
Let the first consecutive integer be n
The second consecutive integer is n + 1
The third consecutive integer is n + 2
The sum is written as:
n + n + 1 + n + 2
Combine like terms:
(n + n + n) + (1 + 2)
3n + 3
The phrase [I]greater than[/I] means an inequality, which we write as:
[B]3n + 3 > 30[/B]
the sum of 3 consecutive natural numbers, the first of which is nthe sum of 3 consecutive natural numbers, the first of which is n
Natural numbers are counting numbers, so we the following expression:
n + (n + 1) + (n + 2)
Combine n terms and constants:
(n + n + n) + (1 + 2)
[B]3n + 3
Also expressed as 3(n + 1)[/B]
the sum of 3 consecutive natural numbers, the first of which is nthe sum of 3 consecutive natural numbers, the first of which is n
We have:
n + (n + 1) + (n + 2)
Grouping like terms, we have:
[B]3n + 3[/B]
The sum of 3 consecutive natural numbers, the first of which is nThe sum of 3 consecutive natural numbers, the first of which is n.
We have 3 numbers:
n, n + 1, and n + 2
Add them up:
n + (n + 1) + (n + 2)
Group like terms:
[B]3n + 3[/B]
The sum of 3, 7, and a number amounts to 16The sum of 3, 7, and a number amounts to 16
Let the number be n. A sum means we add. We're given:
3 + 7 + n = 16
Grouping like terms, we get:
n + 10 = 16
[URL='https://www.mathcelebrity.com/1unk.php?num=n%2B10%3D16&pl=Solve']Typing this equation into our search engine[/URL], we get:
n = [B]6 [/B]
The sum of 5 odd consecutive numbers is 145The sum of 5 odd consecutive numbers is 145.
Let the first odd number be n. We have the other 4 odd numbers denoted as:
[LIST]
[*]n + 2
[*]n + 4
[*]n + 6
[*]n + 8
[/LIST]
Add them all together
n + (n + 2) + (n + 4) + (n + 6) + (n + 8)
The sum of the 5 odd consecutive numbers equals 145
n + (n + 2) + (n + 4) + (n + 6) + (n + 8) = 145
Combine like terms:
5n + 20 = 145
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=5n%2B20%3D145&pl=Solve']equation solver[/URL], we get [B]n = 25[/B]. Using our other 4 consecutive odd numbers above, we get:
[LIST]
[*]27
[*]29
[*]31
[*]33
[/LIST]
Adding the sum up, we get: 25 + 27 + 29 + 31 + 33 = 145.
So our 5 odd consecutive number added to get 145 are [B]{25, 27, 29, 31, 33}[/B].
[MEDIA=youtube]0T2PDuQIIwI[/MEDIA]
The sum of 5x and 2x is at least 70[I]Is at least [/I]means greater than or equal to:
5x + 2x >= 70
If we combine like terms, we have:
7x >=70
We can further simplify by dividing each side of the inequality by 7
x >=10
If you want the interval notation for that, use the [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3E%3D10&pl=Show+Interval+Notation']interval notation calculator[/URL].
The sum of 6 times a number and -8, added to 3 times a numberThe sum of 6 times a number and -8, added to 3 times a number
The phrase "a number", means an arbitrary variable, let's call it x.
6 times a number:
6x
And means we add, so we have
6x - 8
Added to 3 times a number
6x - 8 + 3x
Combine like terms:
[B]9x - 8[/B]
The sum of a number and 34 times the numberThe sum of a number and 34 times the number
The phrase [I]a number[/I] means an arbitrary variable. Let's call it x.
x
34 times the number:
34x
The sum of a number and 34 times the number means we add both terms together:
x + 34x
The sum of Jocelyn and Joseph's age is 40. In 5 years, Joseph will be twice as Jocelyn's present ageThe sum of Jocelyn and Joseph's age is 40. In 5 years, Joseph will be twice as Jocelyn's present age. How old are they now?
Let Jocelyn's age be a
Let Joseph's age be b.
We're given two equations:
[LIST=1]
[*]a + b = 40
[*]2(a + 5) = b + 5
[/LIST]
We rearrange equation (1) in terms of a to get:
[LIST=1]
[*]a = 40 - b
[*]2a = b + 5
[/LIST]
Substitute equation (1) into equation (2) for a:
2(40 - b) = b + 5
80 - 2b = b + 5
To solve this equation for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=80-2b%3Db%2B5&pl=Solve']type it in our search engine[/URL] and we get:
[B]b (Joseph's age) = 25[/B]
Now, substitute b = 25 into equation (1) to solve for a:
a = 40 - 25
[B]a (Jocelyn's age) = 15[/B]
The sum of Mr. Adams and Mrs. Benson's age is 55. The difference is 3. What are their ages?The sum of Mr. Adams and Mrs. Benson's age is 55. The difference is 3. What are their ages?
[U]Givens[/U]
[LIST]
[*]Let Mr. Adam's age be a
[*]Let Mrs. Benson's age be b
[*]We're given two equations where [I]sum[/I] means we add and [I]difference[/I] means we subtract:
[/LIST]
[LIST=1]
[*]a + b = 55
[*]a - b = 3
[/LIST]
Since we have opposite coefficients for b, we can take a shortcut and add equation 1 to equation 2:
(a + a) + (b - b) = 55 + 3
Combining like terms and simplifying, we get:
2a = 58
To solve this equation for a, we [URL='https://www.mathcelebrity.com/1unk.php?num=2a%3D58&pl=Solve']type it in our search engine[/URL] and we get:
a = [B]29
[/B]
If a = 29, then we plug this into equation (1) to get:
29 + b = 55
b = 55 - 29
b = [B]26
[MEDIA=youtube]WwkpNqPvHs8[/MEDIA][/B]
the sum of n and twice n is 12Twice n means we multiply n by 2
2n
The sum of n and twice n means we add
n + 2n
The word [I]is[/I] means equal to, so we set that expression above equal to 12
n + 2n = 12
Combine like terms:
3n = 12
Divide each side of the equation by 3 to isolate n
n = 4
Check our work
Twice n is 2*4 = 8
Add that to n = 4
8 + 4
12
The sum of the digits of a certain two-digit number is 16. Reversing its digits increases the numberThe sum of the digits of a certain two-digit number is 16. Reversing its digits increases the number by 18. What is the number?
Let x and (16-x) represent the original ten and units digits respectively
Reversing its digits increases the number by 18
Set up the relational equation
[10x + (16-x)] + 18 = 10(16 - x) + x
Solving for x
9x + 34 = 160 - 9x
Combine like terms
18x = 126
Divide each side of the equation by 18
18x/18 = 126/18
x = 7
So 16 - x = 16 - 7 = 9
The first number is 79, the other number is 97. and 97 - 79 = 18 so we match up.
The number in our answer is [B]79[/B]
the sum of the squares of a and bthe sum of the squares of a and b
Square of a means we raise a to the 2nd power:
a^2
Square of b means we raise b to the 2nd power:
b^2
The sum of squares means we add these terms together to get our algebraic expression:
[B]a^2 + b^2[/B]
The sum of the sum of x and z and the difference of y and zThe sum of the sum of x and z and the difference of y and z
Take this algebraic expression in 3 parts:
Step 1: The sum of x and z is written as:
x + z
Step 2: The difference of y and z is written as:
y - z
Step 3: the sum of the sum and difference is written as:
x + z + (y - z)
x + z + y - z
Cancelling the z terms, we get:
[B]x + y
[MEDIA=youtube]bmoZXImYCrg[/MEDIA][/B]
The sum of three numbers is 171. The second number is 1/2 of the first and the third is 3/4 of the fThe sum of three numbers is 171. The second number is 1/2 of the first and the third is 3/4 of the first. Find the numbers.
We have three numbers, x, y, and z.
[LIST=1]
[*]x + y + z = 171
[*]y = 1/2x
[*]z = 3/4x
[/LIST]
Substitute (2) and (3) into (1)
x + 1/2x + 3/4x = 171
Use a common denominator of 4 for each x term
4x/4 + 2x/4 + 3x/4 = 171
(4 + 2 + 3)x/4 = 171
9x/4 = 171
[URL='https://www.mathcelebrity.com/prop.php?num1=9x&num2=171&den1=4&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']Plug this equation into our search engine[/URL], and we get [B]x = 76[/B]
So y = 1/2(76) --> [B]y = 38[/B]
Then z = 3/4(76) --> [B]z = 57[/B]
The sum of twice an integer and 3 times the next consecutive integer is 48The sum of twice an integer and 3 times the next consecutive integer is 48
Let the first integer be n
This means the next consecutive integer is n + 1
Twice an integer means we multiply n by 2:
2n
3 times the next consecutive integer means we multiply (n + 1) by 3
3(n + 1)
The sum of these is:
2n + 3(n + 1)
The word [I]is[/I] means equal to, so we set 2n + 3(n + 1) equal to 48:
2n + 3(n + 1) = 48
Solve for [I]n[/I] in the equation 2n + 3(n + 1) = 48
We first need to simplify the expression removing parentheses
Simplify 3(n + 1): Distribute the 3 to each term in (n+1)
3 * n = (3 * 1)n = 3n
3 * 1 = (3 * 1) = 3
Our Total expanded term is 3n + 3
Our updated term to work with is 2n + 3n + 3 = 48
We first need to simplify the expression removing parentheses
Our updated term to work with is 2n + 3n + 3 = 48
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(2 + 3)n = 5n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
5n + 3 = + 48
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 3 and 48. To do that, we subtract 3 from both sides
5n + 3 - 3 = 48 - 3
[SIZE=5][B]Step 4: Cancel 3 on the left side:[/B][/SIZE]
5n = 45
[SIZE=5][B]Step 5: Divide each side of the equation by 5[/B][/SIZE]
5n/5 = 45/5
Cancel the 5's on the left side and we get:
n = [B]9[/B]
The sum of two consecutive integers if n is the first integer.The sum of two consecutive integers if n is the first integer.
consecutive means immediately after, so we have:
n
n + 1
[U]The sum is written as:[/U]
n + n + 1
[U]Grouping like terms, we have:[/U]
(n + n) + 1
[B]2n + 1[/B]
The sum of two consecutive integers plus 18 is 123The sum of two consecutive integers plus 18 is 123.
Let our first integer be n and our next integer be n + 1. We have:
n + (n + 1) + 18 = 123
Group like terms to get our algebraic expression:
2n + 19 = 123
If we want to solve the algebraic expression using our [URL='http://www.mathcelebrity.com/1unk.php?num=2n%2B19%3D123&pl=Solve']equation solver[/URL], we get n = 52. This means the next integer is 52 + 1 = 53
The sum of two y and the quantity of three plus y plus twice the quantity two y minus two equals fifThe sum of two y and the quantity of three plus y plus twice the quantity two y minus two equals fifteen
The sum of two y and the quantity of three plus y
2y + (3 + y)
twice the quantity two y minus two
2(2y - 2)
The sum of two y and the quantity of three plus y plus twice the quantity two y minus two
2y + (3 + y) + 2(2y - 2)
Equals 15 to get our algebraic expression
[B]2y + (3 + y) + 2(2y - 2) = 15[/B]
[B][/B]
If the problem asks you to solve for yL
2y + 3 + y + 4y - 4 = 15
Group like terms:
7y - 1 = 15
Add 1 each side:
7y = 16
Divide each side by 7:
y = [B]16/7[/B]
The sum of x and 10 equals the sum of 2 times x and 12The sum of x and 10 equals the sum of 2 times x and 12
The sum of x and 10 means we add 10 to x:
x + 10
2 times x means we multiply x by 2:
2x
the sum of 2 times x and 12 means we add 12 to 2x:
2x + 12
The sum of x and 10 equals the sum of 2 times x and 12:
x + 10 + (2x + 12)
Distribute the parentheses, and we get:
x + 10 + 2x + 12
Group like terms:
(1 + 2)x + 10 + 12
[B]3x + 22[/B]
the sum of x squared plus y squaredthe sum of x squared plus y squared
x squared means we raise x to the power of 2:
x^2
y squared means we raise y to the power of 2:
y^2
The sum means we add both terms together:
[B]x^2 + y^2[/B]
The team A scored 13 more points than Team B. The total of their score was 47. How many points did tThe team A scored 13 more points than Team B. The total of their score was 47. How many points did team A score?
Let a be the amount of points A scored, and b be the amount of points B scored. We're given:
[LIST=1]
[*]a = b + 13
[*]a + b = 47
[/LIST]
Plug (1) into (2)
(b + 13) + b = 47
Combine like terms:
2b + 13 = 47
[URL='https://www.mathcelebrity.com/1unk.php?num=2b%2B13%3D47&pl=Solve']Typing this equation into our search engine[/URL], we get:
b = 17
Now plug this into (1):
a = 17 + 13
a = [B]30[/B]
The total age of three cousins is 48. Suresh is half as old as Hakima and 4 years older than Saad. hThe total age of three cousins is 48. Suresh is half as old as Hakima and 4 years older than Saad. How old are the cousins?
Let a be Suresh's age, h be Hakima's age, and c be Saad's age. We're given:
[LIST=1]
[*]a + h + c = 48
[*]a = 0.5h
[*]a = c + 4
[/LIST]
To isolate equations in terms of Suresh's age (a), let's do the following:
[LIST]
[*]Rewriting (3) by subtracting 4 from each side, we get c = a - 4
[*]Rewriting (2) by multiply each side by 2, we have h = 2a
[/LIST]
We have a new system of equations:
[LIST=1]
[*]a + h + c = 48
[*]h = 2a
[*]c = a - 4
[/LIST]
Plug (2) and (3) into (1)
a + (2a) + (a - 4) = 48
Group like terms:
(1 + 2 + 1)a - 4 = 48
4a - 4 = 48
[URL='https://www.mathcelebrity.com/1unk.php?num=4a-4%3D48&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]a = 13 [/B]<-- Suresh's age
This means that Hakima's age, from modified equation (2) above is:
h = 2(13)
[B]h = 26[/B] <-- Hakima's age
This means that Saad's age, from modified equation (3) above is:
c = 13 - 4
[B]c = 9[/B] <-- Saad's age
[B]
[/B]
There are 13 animals in the barn. some are chickens and some are pigs. there are 40 legs in all. HowThere are 13 animals in the barn. some are chickens and some are pigs. there are 40 legs in all. How many of each animal are there?
Chickens have 2 legs, pigs have 4 legs. Let c be the number of chickens and p be the number of pigs. Set up our givens:
(1) c + p = 13
(2) 2c + 4p = 40
[U]Rearrange (1) to solve for c by subtracting p from both sides:[/U]
(3) c = 13 - p
[U]Substitute (3) into (2)[/U]
2(13 - p) + 4p = 40
26 - 2p + 4p = 40
[U]Combine p terms[/U]
2p + 26 = 40
[U]Subtract 26 from each side:[/U]
2p = 14
[U]Divide each side by 2[/U]
[B]p = 7[/B]
[U]Substitute p = 7 into (3)[/U]
c = 13 - 7
[B]c = 6[/B]
For a shortcut, you could use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+p+%3D+13&term2=2c+%2B+4p+%3D+40&pl=Cramers+Method']simultaneous equations calculator[/URL]
There are 2 consecutive integers. Twice the first increased by the second yields 16. What are the nuThere are 2 consecutive integers. Twice the first increased by the second yields 16. What are the numbers?
Let x be the first integer. y = x + 1 is the next integer. We have the following givens:
[LIST=1]
[*]2x + y = 16
[*]y = x + 1
[/LIST]
Substitute (2) into (1)
2x + (x + 1) = 16
Combine x terms
3x + 1 = 16
Subtract 1 from each side
3x = 15
Divide each side by 3
[B]x = 5[/B]
So the other integer is 5 + 1 = [B]6[/B]
There are 320 pupils there are 24 more girls than boys how many boys are thereLet b = boys and g = girls. We have two equations:
(1) b + g = 320
(2) g = b + 24
Substitute (2) into (1) for g
b + (b + 24) = 320
Combine b terms:
2b + 24 = 320
Use our [URL='http://www.mathcelebrity.com/1unk.php?num=2b%2B24%3D320&pl=Solve']equation calculator[/URL]:
[B]b = 148
[/B]
Substitute b = 148 into (2)
g = 148 + 24
[B]g = 172[/B]
There are 33 students in an Algebra I class. There are 7 fewer girls than boys. How many girls are iThere are 33 students in an Algebra I class. There are 7 fewer girls than boys. How many girls are in the class?
Let b be the number of boys and g be the number of girls. We are given 2 equations:
[LIST=1]
[*]g = b - 7
[*]b + g = 33
[/LIST]
Substitute (1) into (2):
b + (b - 7) = 33
Combine like terms:
2b - 7 = 33
[URL='https://www.mathcelebrity.com/1unk.php?num=2b-7%3D33&pl=Solve']Typing this equation into our search engine[/URL], we get b = 20.
Since the problem asks for how many girls (g) we have, we substitute b = 20 into Equation (1):
g = 20 - 7
[B]g = 13[/B]
There are 5 pencil-cases on the desk. Each pencil-case contains at least 10 pencils, but not more th[SIZE=4]There are 5 pencil-cases on the desk. Each pencil-case contains at least 10 pencils, but not more than 14 pencils. Which of the following could be the total number of pencils in all 5 cases?
A) 35
B) 45
C) 65
D) 75
[U]Determine the minimum amount of pencils (At least means greater than or equal to):[/U]
Minimum Amount of pencils = Cases * Min Quantity
Minimum Amount of pencils = 5 * 10
Minimum Amount of pencils = 50
[SIZE=4][U]Determine the maximum amount of pencils (Not more than means less than or equal to):[/U]
Maximum Amount of pencils = Cases * Min Quantity
Maximum Amount of pencils = 5 * 14
Maximum Amount of pencils = 70[/SIZE]
So our range of pencils (p) is:
50 <= p <= 70
Now take a look at our answer choices. The only answer which fits in this inequality range is [B]C, 65[/B].
[B][/B][/SIZE]
There are 50 pairs of pants. One-half of the pants are black. One-fifth of the pants are tan. How maThere are 50 pairs of pants. One-half of the pants are black. One-fifth of the pants are tan. How many pairs of pants are not black or tan.
First, determine what fraction of pants are black and tan:
1/2 + 1/5
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F2&frac2=1%2F5&pl=Add']fraction addition calculator[/URL], we get 7/10.
So the rest of the pants are 1 - 7/10.
1 can be written as 10/10.
So we have 10/10 - 7/10 = 3/10
3/10 * 50 = 150/10 = [B]15[/B]
There are two numbers. The sum of 4 times the first number and 3 times the second number is 24. TheThere are two numbers. The sum of 4 times the first number and 3 times the second number is 24. The difference between 2 times the first number and 3 times the second number is 24. Find the two numbers.
Let the first number be x and the second number be y. We have 2 equations:
[LIST=1]
[*]4x + 3y = 24
[*]2x - 3y = 24
[/LIST]
Without doing anything else, we can add the 2 equations together to eliminate the y term:
(4x + 2x) + (3y - 3y) = (24 + 24)
6x = 48
Divide each side by 6:
[B]x = 8
[/B]
Substitute this into equation (1)
4(8) + 3y = 24
32 + 3y = 24
[URL='https://www.mathcelebrity.com/1unk.php?num=32%2B3y%3D24&pl=Solve']Type 32 + 3y = 24 into our search engine[/URL] and we get [B]y = 2.6667[/B].
There is a bag filled with 5 blue, 6 red and 2 green marbles. A marble is taken at random from the bThere is a bag filled with 5 blue, 6 red and 2 green marbles. A marble is taken at random from the bag, the colour is noted and then it is replaced. Another marble is taken at random. What is the probability of getting exactly 1 blue?
Find the total number of marbles in the bag:
Total marbles = 5 blue + 6 red + 2 green
Total marbles = 13
The problem asks for exactly one blue in 2 draws [I]with replacement[/I]. Which means you could draw as follows:
Blue, Not Blue
Not Blue, Blue
The probability of drawing a blue is 5/13, since we replace the marbles in the bag each time.
The probability of not drawing a blue is (6 + 2)/13 = 8/13
And since each of the 2 draws are independent of each other, we multiply the probability of each draw:
Blue, Not Blue = 5/13 * 8/13 =40/169
Not Blue, Blue = 8/13 * 5/13 = 40/169
We add both probabilities since they both count under our scenario:
40/169 + 40/169 = 80/169
Checking our [URL='https://www.mathcelebrity.com/fraction.php?frac1=80%2F169&frac2=3%2F8&pl=Simplify']fraction simplification calculator[/URL], we see you cannot simplify this fraction anymore.
So our probability stated in terms of a fraction is 80/169
[URL='https://www.mathcelebrity.com/perc.php?num=80&den=169&pcheck=1&num1=16&pct1=80&pct2=70&den1=80&idpct1=10&hltype=1&idpct2=90&pct=82&decimal=+65.236&astart=12&aend=20&wp1=20&wp2=30&pl=Calculate']Stated in terms of a decimal[/URL], it's 0.4734
There were 150 students at a dance. There were 16 more boys than girls. How many boys were there?Set up two equations:
(1) b = g + 16
(2) b + g = 150
Substitute equation (1) into (2)
(g + 16) + g = 150
Combine like terms
2g + 16 = 150
Subtract 16 from each side
2g = 134
Divide each side by 2 to isolate g
g = 67
Substitute this into equation (1)
b = 67 + 16
[B]b = 83[/B]
There were 175 tickets sold for the upcoming event in the gym. If students tickets cost $5 and adultThere were 175 tickets sold for the upcoming event in the gym. If students tickets cost $5 and adult tickets are $8, tell me how many tickets were sold if gate receipts totaled $1028?
Let s be the number of student tickets and a be the number of adult tickets. We are given:
a + s = 175
8a + 5s = 1028
There are 3 ways to solve this, all of which give us:
[B]a = 51
s = 124
[/B]
[URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+s+%3D+175&term2=8a+%2B+5s+%3D+1028&pl=Substitution']Substitution Method[/URL]
[URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+s+%3D+175&term2=8a+%2B+5s+%3D+1028&pl=Elimination']Elimination Method[/URL]
[URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+s+%3D+175&term2=8a+%2B+5s+%3D+1028&pl=Cramers+Method']Cramers Method[/URL]
Think of a number. Double the number. Subtract 6 from the result and divide the answer by 2. The quoThink of a number. Double the number. Subtract 6 from the result and divide the answer by 2. The quotient will be 20. What is the number
Let's call our number n.
Double the number means we multiply n by 2:
2n
Subtract 6 from the result means we subtract 6 from 2n:
2n - 6
Divide the answer by 2:
(2n - 6)/2
We can simplify this as n - 3
The quotient will be 20. This means the simplified term above is set equal to 20:
[B]n - 3 = 20 [/B] <-- This is our algebraic expression
If you want to take it a step further, and solve for n in the algebraic expression above, we [URL='https://www.mathcelebrity.com/1unk.php?num=n-3%3D20&pl=Solve']type this expression into our calculator[/URL], and get:
n = 23
Tickets for a concert were priced at $8 for students and $10 for nonstudents. There were 1340 ticketTickets for a concert were priced at $8 for students and $10 for nonstudents. There were 1340 tickets sold for a total of $12,200. How many student tickets were sold?
Let s be the number of student tickets and n be the number of nonstudent tickets:
[LIST=1]
[*]n + s = 1340
[*]10n + 8s = 12200
[/LIST]
Use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=n+%2B+s+%3D+1340&term2=10n+%2B+8s+%3D+12200&pl=Cramers+Method']simultaneous equation calculator[/URL]:
n = 740
[B]s = 600[/B]
Tina's mom made brownies. When tinas friend came over they ate 1/3 of the brownies. Her sister ate 2Tina's mom made brownies. When tinas friend came over they ate 1/3 of the brownies. Her sister ate 2 and her dad ate 4. If there are 26 brownies left. How many did her mom make
Let b denote the number of brownies Tina's mom made. We're given:
b - 1/3b - 2 - 4 = 26
Combining like terms, we have:
2b/3 - 6 = 26
Add 6 to each side, we get:
2b/3 = 32
To solve this equation for b, we [URL='https://www.mathcelebrity.com/prop.php?num1=2b&num2=32&den1=3&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']type it in our math engine[/URL] and we get:
b = [B]48[/B]
To make an international telephone call, you need the code for the country you are calling. The codeTo make an international telephone call, you need the code for the country you are calling. The code for country A, country B, and C are three consecutive integers whose sum is 90. Find the code for each country.
If they are three consecutive integers, then we have:
[LIST=1]
[*]B = A + 1
[*]C = B + 1, which means C = A + 2
[*]A + B + C = 90
[/LIST]
Substitute (1) and (2) into (3)
A + (A + 1) + (A + 2) = 90
Combine like terms
3A + 3 = 90
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=3a%2B3%3D90&pl=Solve']equation calculator[/URL], we get:
[B]A = 29[/B]
Which means:
[LIST]
[*]B = A + 1
[*]B = 29 + 1
[*][B]B = 30[/B]
[*]C = A + 2
[*]C = 29 + 2
[*][B]C = 31[/B]
[/LIST]
So we have [B](A, B, C) = (29, 30, 31)[/B]
Tom is 2 years older than Sue and Bill is twice as old as Tom. If you add all their ages and subtraTom is 2 years older than Sue and Bill is twice as old as Tom. If you add all their ages and subtract 2, the sum is 20. How old is Bill?
Let t be Tom's age., s be Sue's age, and b be Bill's age. We have the following equations:
[LIST=1]
[*]t = s + 2
[*]b = 2t
[*]s + t + b - 2 = 20
[/LIST]
Get (2) in terms of s
(2) b = 2(s + 2) <-- using (1), substitute for t
So we have (3) rewritten with substitution as:
s + (s + 2) + 2(s + 2) - 2 = 20
s + (s + 2) + 2s + 4 - 2 = 20
Group like terms:
(s + s + 2s) + (2 + 4 - 2) = 20
4s + 4 = 20
Run this through our [URL='https://www.mathcelebrity.com/1unk.php?num=4s%2B4%3D20&pl=Solve']equation calculator [/URL]to get s = 4
Above, we had b = 2(s + 2)
Substituting s = 4, we get:
2(4 + 2) = 2(6) = [B]12
Bill is 12 years old[/B]
Tom is the deli manager at a grocery store. He needs to schedule employees to staff the deli departmTom is the deli manager at a grocery store. He needs to schedule employees to staff the deli department at least 260 person-hours per week. Tom has one part-time employeewho works 20 hours per week. Each full-time employee works 40 hours per week. Write an inequality to determine n, the number of full-time employees Tom must schedule, so that his employees will work at least 260 person-hours per week.
Set up the inequality:
[LIST]
[*]Add the part-timer's hours of 20
[*]Full time hours is 40 times n employees
[*]At least means greater than or equal to, so we use the >= sign
[/LIST]
[B]40n + 20 >= 260[/B]
TrapezoidsFree Trapezoids Calculator - This calculator determines the following items for a trapezoid based on given inputs:
* Area of trapezoid
* Perimeter of a Trapezoid
Triangle InequalityFree Triangle Inequality Calculator - This calculator displays 2 scenarios
1) Enter 3 sides of a triangle, and it will determine if the side lengths satisfy the properties of the triangle inequality and form a triangle
2) Enter 2 sides of a triangle, and this will determine an acceptable range for the length of the 3rd side of a triangle so that the 3rd side respects the Triangle Inequality.
Triangular NumberFree Triangular Number Calculator - This calculator determines the nth triangular number. Generates composite numbers.
Trig MeasurementFree Trig Measurement Calculator - Given an angle θ, this calculates the following measurements:
Sin(θ) = Sine
Cos(θ) = Cosine
Tan(θ) = Tangent
Csc(θ) = Cosecant
Sec(θ) = Secant
Cot(θ) = Cotangent
Arcsin(x) = θ = Arcsine
Arccos(x) = θ = Arccosine
Arctan(x) =θ = Arctangent
Also converts between Degrees and Radians and Gradians
Coterminal Angles as well as determine if it is acute, obtuse, or right angle. For acute angles, a cofunction will be determined. Also shows the trigonometry function unit circle
True False EquationsFree True False Equations Calculator - Determines if a set of addition and subtraction of numbers on each side of an equation are equivalent.
Also known as true or false equations
Trump stamps sold at $1.25 and Obama stamps sold at $2 . How many of each stamp was sold if 700 stamTrump stamps sold at $1.25 and Obama stamps sold at $2 . How many of each stamp was sold if 700 stamps were sold making $1250
Let o be the number of Obama stamps. Let t be the number of Trump stamps. We have two equations:
[LIST=1]
[*]o + t = 700
[*]2o + 1.25t = 1250
[/LIST]
Using our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=o%2Bt%3D700&term2=2o%2B1.25t%3D1250&pl=Cramers+Method']simultaneous equations calculator[/URL], we get:
[B]o = 500
t = 200[/B]
Twice a first number decreased by a second number is 16. The first number increased by 3 times the sTwice a first number decreased by a second number is 16. The first number increased by 3 times the second number is 1. Find the numbers.
Let the first number be x and the second number be y. We're given:
[LIST=1]
[*]2x - y = 16
[*]x + 3y = 1
[/LIST]
Using our simultaneous equations calculator, you can solve this 3 ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2x+-+y+%3D+16&term2=x+%2B+3y+%3D+1&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2x+-+y+%3D+16&term2=x+%2B+3y+%3D+1&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=2x+-+y+%3D+16&term2=x+%2B+3y+%3D+1&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter what method we use, we get the same answers:
[B]x = 7
y = -2
(x, y) = (7, -2)
[/B]
Let's check our work in equation 1:
2(7) - -2 ? 16
14 + 2 ? 16
16 = 16 <-- Check
Let's check our work in equation 2:
7 + 3(-2) ? 1
7 - 6 ? 1
1 = 1 <-- Check
Two dice are rolled. Determine the probability of the following. Rolling an even number or a numberTwo dice are rolled. Determine the probability of the following. Rolling an even number or a number greater than 6
We want P(X = Even) or P(X>6)
With 2 dice, our die totals are:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Evens are: 2, 4, 6, 8, 10, 12
> 6 = 7, 8, 9, 10, 11, 12
When the problem states [I]or[/I] it means either of the sets. When we take the union of both sets, we get:
2,4,6,7,8,9,10,11,12
This is 9 possible entries out of 12:
9/12
We can simplify this by dividing top and bottom by 3:
P(X = Even) or P(X>6) = [B]3/4 or 0.75[/B]
Two mechanics worked on a car. the first mechanic worked for 5 hours snd the second mechanic workedTwo mechanics worked on a car. the first mechanic worked for 5 hours snd the second mechanic worked for 15 hours. Together they charged a total of $2375. What was the rate charged per hour by each mechanic if the sum of the two rates was $235 per hour?
Setup equations where x is the rate of the first mechanic and y is the rate of the second mechanic:
[LIST]
[*]5x + 15y = 2375
[*]x + y = 235
[/LIST]
Using Cramers method with our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=5x+%2B+15y+%3D+2375&term2=x+%2B+y+%3D+235&pl=Cramers+Method']simultaneous equations calculator[/URL], we get:
[LIST]
[*][B]x = 115[/B]
[*][B]y = 120[/B]
[/LIST]
Two numbers have a sum of 20. Determine the lowest possible sum of their squares.Two numbers have a sum of 20. Determine the lowest possible sum of their squares.
If sum of two numbers is 20, let one number be x. Then the other number would be 20 - x.
The sum of their squares is:
x^2+(20 - x)^2
Expand this and we get:
x^2 + 400 - 40x + x^2
Combine like terms:
2x^2 - 40x + 400
Rewrite this:
2(x^2 - 20x + 100 - 100) + 400
2(x - 10)^2 - 200 + 400
2(x−10)^2 + 200
The sum of squares of two numbers is sum of two positive numbers, one of which is a constant of 200.
The other number, 2(x - 10)^2, can change according to the value of x. The least value could be 0, when x=10
Therefore, the minimum value of sum of squares of two numbers is 0 + 200 = 200 when x = 10.
If x = 10, then the other number is 20 - 10 = 10.
Two numbers have a sum of 20. If one number is p, express the other in terms of p.Two numbers have a sum of 20. If one number is p, express the other in terms of p.
If the sum is 20 and one number is p, then let the other number be q.
We have: p + q = 20
We want q, so we subtract p from each side:
[B]q = 20 - p[/B]
Two numbers have a sum of 59. If one number is q, express the other number on terms of qTwo numbers have a sum of 59. If one number is q, express the other number on terms of q
The other number is [B]59 - q[/B].
Add them together, you get q + (59 - q) = 59.
two numbers have an average of 2100 and one number is $425 more than the other number. What are thetwo numbers have an average of 2100 and one number is $425 more than the other number. What are the numbers
Let the first number be x and the second number be y. We're given two equations:
[LIST=1]
[*](x + y)/2 = 2100 (Average)
[*]y = x + 425
[/LIST]
Rearrange equation (1) by cross multiplying
x + y = 2 * 2100
x + y = 4200
So we have our revised set of equations:
[LIST=1]
[*]x + y = 4200
[*]y = x + 425
[/LIST]
Substituting equation (2) into equation (1) for y, we get:
x + (x + 425) = 4200
Combining like terms, we get:
2x + 425 = 4200
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B425%3D4200&pl=Solve']equation solver[/URL], we get:
x = [B]1887.5[/B]
Which means using equation (2), we get
y = 1887.5 + 425
y = [B]2312.5[/B]
Two numbers have the sum of 40 if one number is P express the other in terms of PTwo numbers have the sum of 40 if one number is P express the other in terms of P
We write this as P + (40 - P) = 40
So the other number is [B]40 - P[/B]
Two numbers that total 44 and have a difference of 6Two numbers that total 44 and have a difference of 6.
Let the two numbers be x and y. We're given the following equations:
[LIST=1]
[*]x + y = 44 <-- Total means a sum
[*]x - y = 6
[/LIST]
Add the two equations together:
(x + x) + (y - y) = 44 + 6
Cancelling the y terms, we have:
2x = 50
[URL='https://www.mathcelebrity.com/1unk.php?num=2x%3D50&pl=Solve']Typing this equation into the search engine[/URL], we get:
[B]x = 25
[/B]
Rearranging equation (2) above, we get:
y = x - 6
Substituting x = 25 into this, we get:
y = 25 - 6
[B]y = 19[/B]
Two numbers total 12, and their differences is 20. Find the two numbers.Two numbers total 12, and their differences is 20. Find the two numbers.
Let the first number be x. Let the second number be y. We're given two equations:
[LIST=1]
[*]x + y = 12
[*]x - y = 20
[/LIST]
Since we have y coefficients of (-1 and 1) that cancel, we add the two equations together:
(x + x) + (y - y) = 12 + 20
The y terms cancel, so we have:
2x = 32
[URL='https://www.mathcelebrity.com/1unk.php?num=2x%3D32&pl=Solve']Type this equation into our search engine[/URL] and we get:
x = [B]16[/B]
Substitute this value of x = 16 back into equation 1:
16 + y = 12
[URL='https://www.mathcelebrity.com/1unk.php?num=16%2By%3D12&pl=Solve']Typing this equation into our search engine[/URL], we get:
y = [B]-4
[/B]
Now, let's check our work for both equations:
[LIST=1]
[*]16 - 4 = 12
[*]16 - -4 --> 16 + 4 = 20
[/LIST]
So these both check out.
(x, y) = ([B]16, -4)[/B]
Two numbers total 83 and have a difference of 17 find the two numbersLet the numbers be x and y. Set up our givens:
[LIST=1]
[*]x + y = 83
[*]x - y = 17
[/LIST]
[U]Add equation (1) to equation (2)[/U]
x + x + y - y = 83 + 17
[U]The y-terms cancel out:[/U]
2x = 100
[U]Divide each side by 2:[/U]
2x/2= 100/2
x = [B]50[/B]
[U]
Plug x = 50 into equation (1)[/U]
50 + y = 83
[U]Subtract 50 from each side:[/U]
50 - 50 + y = 83 - 50
[U]Cancel the 50 on the left side:[/U]
y = [B]33
[/B]
So our two numbers (x, y) = (33, 50)
[MEDIA=youtube]jajO043ChUM[/MEDIA]
Tyrone re sells 3 pairs of Yeezys and a pair of Nikes for 250$. Nucci re sells a pair of Yeezys andTyrone re sells 3 pairs of Yeezys and a pair of Nikes for 250$. Nucci re sells a pair of Yeezys and Nikes for 150$ How much does a pair of Yeezys cost?
Let y be the cost of Yeezy's and n be the cost of Nike's. We're given two equations:
[LIST=1]
[*]3y + n = 250
[*]y + n = 150
[/LIST]
We have a system of equations, and we can solve it using one of three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=3y+%2B+n+%3D+250&term2=y+%2B+n+%3D+150&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=3y+%2B+n+%3D+250&term2=y+%2B+n+%3D+150&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=3y+%2B+n+%3D+250&term2=y+%2B+n+%3D+150&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter what method we choose, we get:
[LIST]
[*][B]n = 100[/B]
[*][B]y = 50[/B]
[/LIST]
Unit CircleFree Unit Circle Calculator - Determines if coordinates for a unit circle are valid, or calculates a variable for unit circle coordinates
Unit FractionFree Unit Fraction Calculator - Determines the unit fraction for a fraction.
Unknown NumberFree Unknown Number Calculator - Determines the unknown number needed to make an equation true.
Use the definite integral to find the area between the x-axis and the function f(x)= x^2-x-12 over tUse the definite integral to find the area between the x-axis and the function f(x)= x^2-x-12 over the interval [ -5, 10].
Using our [URL='http://www.mathcelebrity.com/dfii.php?term1=x%5E2-x-12&fpt=0&ptarget1=0&ptarget2=0&itarget=-5%2C10&starget=0%2C1&nsimp=8&pl=Integral']integral calculator[/URL], we get:
[B]157.5[/B]
Use the information below to determine the weight of 500 gallons of water. a) There are 1.057 quartUse the information below to determine the weight of 500 gallons of water.
a) There are 1.057 quarts in a liter and 4 quarts in a gallon
b) A cubic decimeter of water is a liter of water
c) A cubic decimeter of water weighs one kilogram
d) There are 2.2 pounds in a kilogram
[LIST]
[*]500 gallons = 2000 quarts
[*]2000 quarts / 1.057 quarts in a liter = 1892.15 liters
[*]1892.15 liters weight 1892.15 kilograms
[*]1892.15 kilograms x 2.2 pounds = [B]4163 pounds[/B]
[/LIST]
Van needs to enter a formula into a spreadsheet to show the outputs of an arithmetic sequence that sVan needs to enter a formula into a spreadsheet to show the outputs of an arithmetic sequence that starts with 13 and continues to add seven to each output. For now, van needs to know what the 15th output will be. Complete the steps needed to determine the 15th term in sequence.
Given a first term a1 of 13 and a change amount of 7, expand the series
The explicit formula for an [I]arithmetic series[/I] is an = a1 + (n - 1)d
d represents the common difference between each term, an - an - 1
Looking at all the terms, we see the common difference is 7, and we have a1 = 13
Therefore, our explicit formula is an = 13 + 7(n - 1)
If n = 15, then we plug it into our explicit formula above:
an = 13 + 7(n - 1)
a(15) = 15 + 7(15 - 1)
a(15) = 15 + 7 * 14
a(15) = 15 + 98
a(15) = [B]113[/B]
VectorsFree Vectors Calculator - Given 2 vectors A and B, this calculates:
* Length (magnitude) of A = ||A||
* Length (magnitude) of B = ||B||
* Sum of A and B = A + B (addition)
* Difference of A and B = A - B (subtraction)
* Dot Product of vectors A and B = A x B
A ÷ B (division)
* Distance between A and B = AB
* Angle between A and B = θ
* Unit Vector U of A.
* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).
* Cauchy-Schwarz Inequality
* The orthogonal projection of A on to B, projBA and and the vector component of A orthogonal to B → A - projBA
Also calculates the horizontal component and vertical component of a 2-D vector.
Victoria is 4 years older than her neighbor. The sum of their ages is no more than 14 years.Victoria is 4 years older than her neighbor. The sum of their ages is no more than 14 years.
Let Victoria's age be v. And her neighbor's age be n. We're given:
[LIST=1]
[*]v = n + 4
[*]v + n <=14 <-- no more than means less than or equal to
[/LIST]
Substitute Equation (1) into Inequality (2):
(n + 4) + n <= 14
Combine like terms:
2n + 4 <= 14
[URL='https://www.mathcelebrity.com/1unk.php?num=2n%2B4%3C%3D14&pl=Solve']Typing this inequality into our search engine[/URL], we get:
n <= 5
Substituting this into inequality (2):
v + 5 <= 14
[URL='https://www.mathcelebrity.com/1unk.php?num=v%2B5%3C%3D14&pl=Solve']Typing this inequality into our search engine[/URL], we get:
[B]v <= 9[/B]
VolatilityFree Volatility Calculator - Given a set of stock prices, this determines expected rates of return and volatility
Walking Distance (Pedometer)Free Walking Distance (Pedometer) Calculator - Given a number of steps and a distance per stride in feet, this calculator will determine how far you walk in other linear measurements.
What can we conclude if the coefficient of determination is 0.94?What can we conclude if the coefficient of determination is 0.94?
[LIST]
[*]Strength of relationship is 0.94
[*]Direction of relationship is positive
[*]94% of total variation of one variable(y) is explained by variation in the other variable(x).
[*]All of the above are correct
[/LIST]
[B]94% of total variation of one variable(y) is explained by variation in the other variable(x)[/B]. The coefficient of determination explains ratio of explained variation to the total variation.
What does y=f(x) meanWhat does y=f(x) mean
It means y = a function of the variable x.
x is the independent variable and y is the dependent variable.
f(x) means a function in terms of x
What is the 1000th term in the series 0, 7, 14, 21, … ?Map this out as a function with term number n and value
1, 0
2, 7
3, 14
4, 21
The values jump by 7, but they do so as the n - 1 term.
We have the series formula S(n) = 7(n - 1)
The problem asks for S(1000)
S(1000) = 7(1000 - 1)
S(1000) = 7(999)
S(1000) = [B]6,993[/B]
[MEDIA=youtube]ZF10Ec29XKo[/MEDIA]
What is the 7th number in the following pattern: 3.2, 4.4, 5.6, 6.8, ...What is the 7th number in the following pattern: 3.2, 4.4, 5.6, 6.8, ...
This is an arithmetic sequence with an increase amount of 1.2. Each term S(n) is found by adding 1.2 to the prior term.
S(1) = 3.2
S(2) = 3.2 + 1.2 = 4.4
S(3) = 4.4 + 1.2 = 5.6
S(4) = 5.6 + 1.2 = 6.8
S(5) = 6.8 + 1.2 = 8.0
S(6) = 8.0 + 1.2 = 9.2
S(7) = 9.2 + 1.2 = [B]10.4[/B]
What is the missing number? 0, 1, 1, 2, 3, 5, 8, __, 21, 34We can't add a common number of multiply a common number to solve this.
But what happens if we start at the 3rd term and add the 2 prior terms?
[LIST]
[*]0 + 1= 1
[*]1 + 1 = 2
[*]1 + 2 = 3
[*]3 + 5 = 8
[*]5 + 8 = [B]13[/B]
[/LIST]
[B]This is known as the Fibonacci Sequence.
[MEDIA=youtube]9SZ1TJGwyLw[/MEDIA][/B]
What is the ratio of the area of a circle to the area of a square drawn around that circle? ExpressWhat is the ratio of the area of a circle to the area of a square drawn around that circle? Express your answer in terms of pi.
Area of a circle = pir^2
area of a square = (2r)^2 = 4r^2
Ratio = pir^2/4r^2
Ratio = [B]pi/4[/B]
When an alligator is born it is about 8 inches long each year they grow 12 inches determine the ageWhen an alligator is born it is about 8 inches long each year they grow 12 inches determine the age and years of 116 inch alligator?
Calculate inches to grow to get to 116
116 - 8 = 108
Now figure out how many years it takes growing at 12 inches per year, using y as years
12y = 108
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=12y%3D108&pl=Solve']equation calculator[/URL], we get:
[B]y = 9[/B]
When Esteban left for college, his parents decided to give him an allowance of $100 every 4 weeks. TWhen Esteban left for college, his parents decided to give him an allowance of $100 every 4 weeks. They told Esteban that he could decide how he wanted raises to his allowance determined.
Choice #1 - A raise of $10 every 4 weeks
Choice #2 - A raise of $1.50 each week
What choice should Esteban pick?
Choice 1:
[LIST]
[*]First 4 weeks = $100
[*]Weeks 5 - 8 = $110
[*]Weeks -9-12 = $120
[*]Total = $330
[/LIST]
Choice 2:
[LIST]
[*]1st week = $25
[*]2nd week = $26.50
[*]3rd week = $28
[*]4th week - $29.50
[*]5th week = $31.00
[*]6th week = $32.50
[*]7th week = $34.00
[*]8th week = $35.50
[*]9th week = $37.00
[*]10th week = $38.50
[*]11th week = $40.00
[*]12th week = $41.50
[*]Total = [B]$399[/B]
[/LIST]
[B]Choice 2 is the better option[/B]
Which is a better buy 42 bows for $7.14 or 120 bows for $25.20Let's compare in terms of what 1 dollar gets you:
Use our [URL='http://www.mathcelebrity.com/betterbuy.php?p1=42&p2=120&q1=7.14&q2=25.20&pl=Better+Buy']better buy calculator[/URL] to show 120 bows for $25.20 is a better buy.
Which of the following is equivalent to 3(2x + 1)(4x + 1)?Which of the following is equivalent to 3(2x + 1)(4x + 1)?
[LIST]
[*]A) 45x
[*]B) 24x^2 + 3
[*]C) 24x^2 + 18x + 3
[*]D) 18x^2 + 6
[/LIST]
First, [URL='https://www.mathcelebrity.com/binomult.php?term1=2x%2B1&term2=4x%2B1&pl=Expand+Product+of+2+Binomials+using+FOIL']multiply the binomials[/URL]:
We get 8x^2 + 6x + 1
Now multiply this polynomial by 3:
3(8x^2 + 6x + 1) = [B]24x^2 + 18x + 3, answer C[/B]
Wind Chill FactorFree Wind Chill Factor Calculator - This calculator determines the wind chill factor given a temperature in F° and a wind speed in miles per hour (mph). Simply enter your temperature and wind speed and press the button
Write a model that utilizes all three explanatory variables with no interaction or quadratic terms.Write a model that utilizes all three explanatory variables with no interaction or quadratic terms. Choose the correct answer below.
A. y i = B0 + B1x1 + B2x2 + B3x3 + e i
B. y i = B0 + B1x1 + B2x2 + B3x3x2 + e i
C. y i = B1x1 + B2x2 + B3x3 + ei
D. None of the above equations satisfy all of the conditions
[B]A. y i = B0 + B1x1 + B2x2 + B3x3 + e i[/B]
X minus 5 plus x equals 79X minus 5 plus x equals 79
x minus 5
x - 5
plus x
x - 5 + x
equals 79
x - 5 + x = 79
Group like terms:
(x + x) - 5 = 79
[B]2x - 5 = 79[/B]
x/3 - g = a for xx/3 - g = a for x
Add g to each side so we can isolate the x term:
x/3 - g + g = a + g
Cancel the g terms on the left side and we get:
x/3 = a + g
Multiply each side by 3 to isolate x:
3(x/3) = 3(a + g)
Cancelling the 3's on the left side, we get:
x = [B]3(a + g)[/B]
x/y + 9 = n for xx/y + 9 = n for x
Subtract 9 from each side to isolate the x term:
x/y + 9 - 9 = n - 9
Cancel the 9's on the left side and we get:
x/y = n - 9
Because we have a fraction on the left side, we can cross multiply the denominator y by n - 9
[B]x =[/B] [B]y(n - 9)[/B]
x/y + 9 = n for yx/y + 9 = n for y
First, subtract 9 from each side to isolate the y term:
x/y + 9 - 9 = n - 9
Cancel the 9's on the left side, and we get:
x/y = n - 9
Cross multiply:
x = y(n - 9)
Divide each side by (n - 9):
x/(n - 9) = y(n - 9)/(n - 9)
Cancel the (n - 9) on the right side, and we get:
y = [B]x/(n - 9)[/B]
You and a friend collect acorns from a field. After g minutes u have collected(10 + 2g) acorns and yYou and a friend collect acorns from a field. After g minutes u have collected(10 + 2g) acorns and your friend has collected (5g - 2) acorns. How many total acorns have you and your friend collected
Add both acorn collections together:
(10 + 2g) + (5g - 2)
Group like terms:
(5 + 2)g + 10 - 2
[B]7g + 8[/B]
You and your friend are playing a number-guessing game. You ask your friend to think of a positive nYou and your friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add three. If your friend's final answer is 53, what was the original number chosen?
Let n be our original number.
Square the number means we raise n to the power of 2:
n^2
Multiply the result by 2:
2n^2
And then add three:
2n^2 + 3
If the friend's final answer is 53, this means we set 2n^2 + 3 equal to 53:
2n^2 + 3 = 53
To solve for n, we subtract 3 from each side, to isolate the n term:
2n^2 + 3 - 3 = 53 - 3
Cancel the 3's on the left side, and we get:
2n^2 = 50
Divide each side of the equation by 2:
2n^2/2 = 50/2
Cancel the 2's, we get:
n^2 = 25
Take the square root of 25
n = +-sqrt(25)
n = +-5
We are told the number is positive, so we discard the negative square root and get:
n = [B]5[/B]
You are buying boxes of cookies at a bakery. Each box of cookies costs $4. In the equation below, cYou are buying boxes of cookies at a bakery. Each box of cookies costs $4. In the equation below, c represents the number of boxes of cookies you buy, and d represents the amount the cookies will cost you (in dollars). The relationship between these two variables can be expressed by the following equation: d=4c. Identify the dependent and independent variables.
[B]The variable d is dependent, and c is independent since the value of d is determined by c.[/B]
You earned $141 last week babysitting and cleaning. You earned $5 per hour babysitting and $7 per hoYou earned $141 last week babysitting and cleaning. You earned $5 per hour babysitting and $7 per hour cleaning. You worked 9 more hours babysitting than cleaning. How many hours did you work last week?
Let b be the hours of babysitting and c be the hours of cleaning. We're given two equations:
[LIST=1]
[*]b = c + 9
[*]5b + 7c = 141
[/LIST]
Substitute equation (1) into (2):
5(c + 9) + 7c = 141
Multiply through:
5c + 45 + 7c = 141
Combine like terms:
12c + 45 = 141
[URL='https://www.mathcelebrity.com/1unk.php?num=12c%2B45%3D141&pl=Solve']Typing this equation into our search engine[/URL], we get:
c = 8
Now substitute this value of c back into Equation (1):
b = 8 + 9
b = 17
The total hours worked (t) is:
t = b + c
t = 17 + 8
t = [B]25[/B]
You have $535 in your wallet and want to buy pizzas that cost $3 each. How much money will you haveYou have $535 in your wallet and want to buy pizzas that cost $3 each. How much money will you have left after buying 178 pizzas?
Calculate total cost of pizzas:
Cost = 178 * 3
Cost = 534
Determine money leftover
Money leftover = Money in Wallet - Cost of pizzas
Money leftover = 535 - 534
Money leftover = [B]1[/B]
You have a total of 42 math and science problems for homework. You have 10 more math problems than sYou have a total of 42 math and science problems for homework. You have 10 more math problems than science problems. How many problems do you have in each subject?
Let m be the math problems and s be the science problems. We have two equations:
(1) m + s = 42
(2) m = s + 10
Substitute (2) into (1)
(s + 10) + s = 42
Combine like terms
2s + 10 = 42
Subtract 10 from each side
2s = 32
Divide each side by 2
[B]s = 16[/B]
So that means m = 16 + 10 --> [B]m = 26
(m, s) = (26, 16)[/B]
You purchase a car for $23,000. The car depreciates at a rate of 15% per year. Determine the valueYou purchase a car for $23,000. The car depreciates at a rate of 15% per year. Determine the value of the car after 7 years. Round your answer to the nearest cent.
Set up the Depreciation equation:
D(t) = 23,000/(1.15)^t
We want D(7)
D(7) = 23,000/(1.15)^7
D(7) = 23,000/2.66002
D(7) = [B]8,646.55[/B]
Z Score LookupFree Z Score Lookup Calculator - Given a Z-score probability statement from the list below, this will determine the probability using the normal distribution z-table.
* P(z < a)
* P(z <= a)
* P(z > a)
* P(z >= a)
* P(a < z < b)
Calculates z score probability
z=m-x+y, for xz=m-x+y, for x
This is a literal equation. Let's add subtract (m + y) from each side:
z - (m + y) = m - x + y - (m + y)
The m + y terms cancel on the right side, so we have:
z - m - y = -x
Multiply each side by -1 to isolate x:
-1(z - m - y) = -(-x)
x = [B]m + y - z[/B]
Zero-Coupon Bond PriceFree Zero-Coupon Bond Price Calculator - This calculator calculates the price of a zero-coupon bond given a face value, yield rate, and term.
zy-dm=ky/t for yzy-dm=ky/t for y
Isolate terms with y to solve this literal equation.
Subtract zy from each side:
zy - dm - zy = ky/t - zy
Cancel the zy terms on the left side, we get:
-dm = ky/t - zy
Factor out y:
y(k/t - z) = -dm
Divide each side by (k/t - z)
y = -dm/(k/t - z)
(k/t - z) can be rewritten as (k - tz)/t
We multiply -dm by the reciprocal of this quotient to get our simplified literal equation:
y = [B]-dmt/(k - tz)[/B]
