1/2a-10b=c solve for a Multiply each side of the equation by 2: 2/2a - 2(10)b = 2c Simplify: a - 20b = 2c Add 20b to each side: a - 20b + 20b = 2c + 20b Cancel the 20b on the left side: [B]a = 2c + 20b [/B] You can also factor out a 2 on the left side for another version of this answer: [B]a = 2(c + 10b)[/B]

This is a symbol for a triple factorial. We have n!!! = n * (n - 3) * (n - 6) * ... * 1 Our subtraction of 3 never goes below one. 12!!! = [B]12 * 9 * 6 * 3 [MEDIA=youtube]xm2D7WxVjk8[/MEDIA] [/B]

Do 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]

21 apples and 49 pears. What is the largest number of baskets to make and still use up all the fruit We use our [URL='https://www.mathcelebrity.com/gcflcm.php?num1=21&num2=49&num3=&pl=GCF+and+LCM']greatest common factor calculator for GCF(21, 49)[/URL] to get: GCF(21, 49) = 7 This means with [B]7 baskets[/B]: [LIST] [*]We divide 21 apples by 7 to get 3 apples per basket [*]We divide 49 pears by 7 to get 7 pears per basket [/LIST]

24 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]

3 is a factor of 18 true or false We go to our math engine and type in [URL='https://www.mathcelebrity.com/factoriz.php?num=18&pl=Show+Factorization']factors of 18[/URL] and we see that 3 is a factor of 18, so our answer is [B]TRUE.[/B]

32 girls and 52 boys were on an overseas learning trip . They were divided into as many groups as possible where the number of groups of girls and the number of groups of boys is the same .how many boys and how many girls were in each group We want a number such that our total members divided by this number equals our group size. We take the greatest common factor (32,52) = 4 Therefore, we have: [LIST] [*][B]32/4 = 8 girls in each group[/B] [*][B]52/4 = 13 boys in each group[/B] [/LIST]

5 books and 5 bags cost $175. What is the cost of 2 books and 2 bags Let the cost of each book be b and the cost of each bag be c. We're given 5b + 5c = 175 We can factor this as: 5(b + c) = 175 Divide each side of the equation by 5, we get: (b + c) = 35 The problem asks for 2b + 2c Factor out 2: 2(b + c) we know from above that (b + c) = 35, so we substitute: 2(35) [B]70[/B]

A adalah faktor dari 36 tetapi bukan kelipatan dari 4. Tentukan nilai a sebesar yang mungkin! [URL='https://www.mathcelebrity.com/factoriz.php?num=36&pl=Show+Factorization']Kami memasukkan faktor 36 ke dalam enjin carian kami dan kami [/URL]mendapat: {1, 2, 3, 4, 6, 9, 12, 18, 36} Menyaring nombor yang bukan faktor 4, kami mendapat: [B]A = {3, 6, 9, 18}[/B]

A bag contains 3 black, 4 red, 3 yellow, and 2 green marbles. What is the probability of drawing a black and then a red marble out of the bag without replacing the black marble before drawing the red marble? The phrase [U][B]without replacement[/B][/U] is a huge clue on this problem. Take each draw and calculate the probability. Draw 1: P(Drawing a red) P(Drawing a red) = Total Red marbles n the jar / Total marbles in the jar P(Drawing a red) = 4/12 4/12 simplifies to 1/3 using a common factor of 4: P(Drawing a red) = 1/3 Draw 2: P(Drawing a black) P(Drawing a black) = Total Black marbles in the jar / Total marbles in the jar [I]We drew one red marble already. Without replacement means we do not put it back. Therefore, we have 12 - 1 = 11 marbles left in the jar.[/I] P(Drawing a black) = 3/11 The question asks, what is the the following probability: P(Drawing a Red, Drawing a Black) Because each draw is [I][U]independent[/U], [/I]we multiply each draw probability together: P(Drawing a Red, Black) = P(Drawing a Red) * P(Drawing a Black) P(Drawing a Red, Black) = 1/3 * 3/11 P(Drawing a Red, Black) = [B]1/11[/B]

A box is filled with 5 blue cards,2 red cards, and 5 yellow cards. A card is chosen at random from the box. What is the probability that it is a blue or a yellow card? Write your answer as a fraction in simplest form. We want P(B) + P(Y) P(B) = 5/12 P(Y) = 5/12 P(B) + P(Y) = 5/12 + 5/12 = 10/12 Reduce this fraction using 2 as our common factor: [B]5/6[/B]

A computer randomly generates a whole number from 1 to 25. Find the probability that the computer generates a multiple of 5 [URL='https://www.mathcelebrity.com/factoriz.php?num=25&pl=Show+Factorization']Multiples of 5[/URL]: {1, 5, 25} So we have the probability of a random number multiple of 5 is [B]3/25[/B]

A desk drawer contains 10 blue pencils, 7 red pencils, and 8 green pencils. Without looking, you draw out a pencil and then draw out a second pencil without returning the first pencil. What is the probability that the first pencil and the second pencil are both green? We are drawing without replacement. Take each draw probability: [LIST=1] [*]First draw, we have a total of 10 + 7 + 8 = 25 pencils to choose from. P(Green) = 8/25 [*]Next draw, we only have 24 total pencils, and 7 green pencils since we do not replace. Therefore, we have P(Green)= 7/24 [/LIST] Since both events are independent, we have: P(Green) * P(Green) = 8/25 * 7/24 P(Green) * P(Green) = 56/600 Using our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=56&num2=600&num3=&pl=GCF']GCF Calculator[/URL], we see the greatest common factor of 56 and 600 is 8. So we divide top and bottom of the fraction by 8. [B]P(Green) * P(Green) = 7/75[/B]

A factory put cakes into boxes of 6 how many boxes can they fill with 3285 3285 cakes / 6 cakes per box = [B]547.5 boxes[/B]

A farmer was 1/3 of his land to grow corn, a quarter of his land to grow lettuce, and 12.5% of his land to grow green beans. He uses the remaining 7 acres to grow wheat.How many total acres does the farmer own? Convert all land portions to fractions or decimals. We will do fractions: [LIST] [*]1/3 for corn [*][I]A quarter[/I] means 1/4 for lettuce [*]12.5% is 12.5/100 or 1/8 for green beans [/LIST] Now add all these up: 1/3 + 1/4 + 1/8 We need a common factor for 3, 4, and 8. Using our [URL='https://www.mathcelebrity.com/gcflcm.php?num1=3&num2=4&num3=8&pl=LCM']LCM Calculator[/URL], we get 24. 1/3 = 8/24 1/4 = 6/24 18 = 3/24 Add them all up: (8 + 6 + 3)/24 17/24 This means 17/24 of the land is used for everything but wheat. Wheat occupies (24-17)/24 = 7/24 of the land. We'll use a for the number of acres on the farm. 7a/24 = 7 [B]a = 24[/B]

A is the set of factors of 12 Type in [URL='https://www.mathcelebrity.com/factoriz.php?num=12&pl=Show+Factorization']factor 12[/URL] into our math engine and we get: A = {[B]1, 2, 3, 4, 6, 12[/B]}

A mother duck lines her 13 ducklings up behind her. In how many ways can the ducklings line up? In position one, we can have any of the 13 ducks. In position two, we can have 12 ducks, since one has to occupy position one. We subtract 1 each time until we fill up all 13 positions. We have: 13 * 12 * 11 * ... * 2 * 1 Or, 13!. [URL='https://www.mathcelebrity.com/factorial.php?num=13!&pl=Calculate+factorial']Typing 13! into our search engine[/URL], we get [B]6,227,020,800[/B] ways the ducklings can line up behind the mother duck.

A natural number greater than 1 has only itself and 1 as factors is called a [B]prime number.[/B]

A pair of numbers has an HCF (Highest Common Factor) of 3, and an LCM (Lowest Common Multiple) of 45 . If one of the numbers in the pair is 15 , what is the other number? [LIST=1] [*]Prime Factorization for 15 is 3 * 5 [*]Prime Factorization for 9 is 3 * 3 [*]LCM of (9, 15) = 35 [/LIST] [URL='https://www.mathcelebrity.com/gcflcm.php?num1=9&num2=15&num3=&pl=GCF+and+LCM']Check out this link here to see the details[/URL]

A pretzel factory has daily fixed costs of $1100. In addition, it costs 70 cents to produce each bag of pretzels. A bag of pretzels sells for $1.80. [U]Build the cost function C(b) where b is the number of bags of pretzels:[/U] C(b) = Cost per bag * b + Fixed Costs C(b) = 0.70b + 1100 [U]Build the revenue function R(b) where b is the number of bags of pretzels:[/U] R(b) = Sale price * b R(b) = 1.80b [U]Build the revenue function P(b) where b is the number of bags of pretzels:[/U] P(b) = Revenue - Cost P(b) = R(b) - C(b) P(b) = 1.80b - (0.70b + 1100) P(b) = 1.80b = 0.70b - 1100 P(b) = 1.10b - 1100

A square of an integer is the integer. Find the integer. Let the integer be n. The square means we raise n to the power of 2, so we have: n^2 = n Subtract n from each side: n^2 - n = n - n n^2 - n = 0 Factoring this, we get: n(n - 1) = 0 So n is either [B]0 or 1[/B].

A student and the marine biologist are together at t = 0. The student ascends more slowly than the marine biologist. Write an equation of a function that could represent the student's ascent. Please keep in mind the slope for the marine biologist is 12. Slope means rise over run. In this case, rise is the ascent distance and run is the time. 12 = 12/1, so for each second of time, the marine biologist ascends 12 units of distance If the student ascends slower, than the total distance gets reduced by an unknown factor, let's call it c. So we have the student's ascent function as: [B]y(t) = 12t - c[/B]

A tourist in Ireland wants to visit six different cities. How many different routes are possible? [URL='https://www.mathcelebrity.com/factorial.php?num=6!&pl=Calculate+factorial']We want 6![/URL] which is [B]720[/B]

A toy company makes "Teddy Bears". The company spends $1500 for factory expenses plus $8 per bear. The company sells each bear for $12.00 each. How many bears must this company sell in order to break even? [U]Set up the cost function C(b) where b is the number of bears:[/U] C(b) = Cost per bear * b + factory expenses C(b) = 8b + 1500 [U]Set up the revenue function R(b) where b is the number of bears:[/U] R(b) = Sale Price per bear * b R(b) = 12b [U]Break-even is where cost equals revenue, so we set C(b) equal to R(b) and solve for b:[/U] C(b) = R(b) 8b + 1500 = 12b To solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=8b%2B1500%3D12b&pl=Solve']type this equation into our search engine[/URL] and we get: b = [B]375[/B]

A toy factory makes 5,000 teddy bears per day. The supervisor randomly selects 10 teddy bears from all 5,000 teddy bears and uses this sample to estimate the mean weight of teddy bears and the sample standard deviation. How many degrees of freedom are there in the estimate of the standard deviation? DF = n - 1 DF = 10 - 1 [B]DF = 9[/B]

A vendor sells h hot dogs and s sodas. If a hot dog costs twice as much as a soda, and if the vendor takes in a total of d dollars, how many cents does a soda cost? Let the cost of the soda be p. So the cost of a hot dog is 2p. The total cost of hot dogs: 2hp The total cost of sodas: ps The total cost of both equals d. So we set the total cost of hots dogs plus sodas equal to d: 2hp + ps = d We want to know the cost of a soda (p). So we have a literal equation. We factor out p from the left side: p(2h + s) = d Divide each side of the equation by (2h + s) p(2h + s)/(2h + s) = d/(2h + s) Cancel the (2h + s) on the left side, we get: p = [B]d/(2h + s[/B])

Alex says all factors of 16 are even why is she wrong. [URL='https://www.mathcelebrity.com/factoriz.php?num=16&pl=Show+Factorization']Type in factor 16[/URL] into our search engine. We get the following factor of 16: 1, 2, 4, 8, 16 [B]All of these are even [I]except[/I] 1, which is odd. This is why Alex is wrong.[/B]

Alice is a machinist in a shirt factory. For the first 180 shirts she is paid $2.20 and then $2.90 per garment thereafter. What are her gross wages for a week in which she produces 240 shirts? Calculate commission on the first 180 shirts (Commission 1): Commission 1 = Shirts (up to 180) * $2.20 Commission 1 = 180 * $2.20 Commission 1 = $396 Calculate commission on the rest of the shirts about 180 (Commission 2): Commission 2 = Shirts Above 180 * $2.90 Commission 2 = (240 - 180) * $2.90 Commission 2 = 60 * $2.90 Commission 2 = $174 Calculate Total Commission: Total Commission = Commission 1 + Commission 2 Total Commission = $396 + $174 Total Commission = [B]$570[/B]

Ana has 24 carrots, 12 cucumbers, and 36 radishes. She wants to make identical vegetable baskets using all of the vegetables. What is the greatest number of baskets she can make The key to solving this problem is asking what is the common factor between the 3 numbers. We want the greatest common factor or GCF [URL='https://www.mathcelebrity.com/gcflcm.php?num1=12&num2=24&num3=36&pl=GCF']GCF(12, 24, 36) [/URL]= [B]12[/B] We divide up our 12 baskets into carrots, cucumbers, and radishes. Each basket of the 12 baskets has the following: [LIST=1] [*]12 cucumbers / GCF of 12 = [B]1 cucumber per basket[/B] [*]24 carrots / GCF of 12 = [B]2 carrots per basket[/B] [*]36 radishes / GCF of 12 = [B]3 radishes per basket[/B] [/LIST] [B][MEDIA=youtube]D1KTOP0h2P4[/MEDIA][/B]

At a light bulb factory 4 out of every 25 light bulbs are defective. How many light bulbs would you excpect to be defective out of 350 light bulbs Set up a proportion of light bulbs to defects where d is the number of defects per 350 light bulbs: 4/25 = b/350 [URL='https://www.mathcelebrity.com/prop.php?num1=4&num2=b&den1=25&den2=350&propsign=%3D&pl=Calculate+missing+proportion+value']Type this proportion into our search engine[/URL], and we get: b = [B]56[/B]

We are solving for x: Subtract b from each side: ax = cx - d - b Subtract cx from each side: ax - cx = -d - b Factor out x from the left side: x(a - c) = -d - b Divide each side by (a - c) x = (-d - b)/(a - c)

ax - 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]

Besides 8 and 1, what is one factor of 8. Using our [URL='http://www.mathcelebrity.com/factoriz.php?num=8&pl=Show+Factorization']factor calculator[/URL], or entering the shortcut [B]Factor 8[/B], we get the following factors: 1, 2, 4, 8 Excluding 1 and 8, we have [B]2, 4[/B]

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Bob has 4/5 of a pound of fudge. He wants to share it with Sue. How much of a pound of fudge will both of them get? If Bob shares the fudge with Sue, we assume they split equal parts. This means: We take 4/5 total and divide into 2 for 2 people: 4/5/2 This is the same as 4/5 * 1/2 4/10 This fraction is not simplified. Factor of 4 = {1, [U]2[/U], 4} Factors of 10 = {1, [U]2[/U], 5, 10} In both of these lists, we see the greatest common factor is 2. So we divide top and bottom of 4/10 by 2: 4/2 / 10 / 2 [B]2/5 Bob gets 2/5 of a pound of fudge and Sue gets [B]2/5 of a pound of fudge[/B][/B]

c increased by a factor of 20 This means we multiply c by 20: [B]20c[/B]

Claire bought 180 candies and 140 pens for goody bags for her birthday. What is the largest number of goody bags that Claire can make so that each goody bag has the same number of candies and the same number of pens? (All candies and pens should be used.) We want the greatest common factor of 180 and 140. When we [URL='https://www.mathcelebrity.com/gcflcm.php?num1=140&num2=180&num3=&pl=GCF+and+LCM']run GCF(180,140) in our calculator[/URL], we get 20. We divide our total candies and total pens by our GCF. So each bag has the following: Candies: 180/20 = [B]9 candies[/B] Pens: 140/20 = [B]7 pens[/B]

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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]

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Factors of 36 between 2 and 12 We type in [I][URL='https://www.mathcelebrity.com/factoriz.php?num=36&pl=Show+Factorization']factors of 36[/URL][/I] into our search engine and we get: {1, 2, 3, 4, 6, 9, 12, 18, 36} The problem asks for factors of 36 between 2 and 12: Between does not mean inclusive, so we have anything greater than 2 and less than 12: [B]{3, 4, 6, 9}[/B]

Find r in P(7, r) Recall the permutations formula: 7! / (7-r!) = 840. We [URL='https://www.mathcelebrity.com/factorial.php?num=7!&pl=Calculate+factorial']run 7! through our search engine[/URL] and we get: [URL='https://www.mathcelebrity.com/factorial.php?num=7!&pl=Calculate+factorial']7![/URL] = 5040 5040 / (7 - r)! = 840 Cross multiply, and we get: 5040/840 = 7 - r! 6 = (7 - r)! Since 6 = 3*2*! = 3!, we have; 3! = (7 - r)! 3 = 7 - r To solve for r, we [URL='https://www.mathcelebrity.com/1unk.php?num=3%3D7-r&pl=Solve']type this equation into our search engine[/URL] and we get: r = [B]4[/B]

Find the greatest common factor without a calculator Check out this table stacking method and the product of factors. [MEDIA=youtube]3Zjo0XRD6fw[/MEDIA]

Find the greatest number which divides 845 and 1250 This is the greatest common factor. We [URL='https://www.mathcelebrity.com/gcflcm.php?num1=845&num2=1250&num3=&pl=GCF+and+LCM']type GCF(845,1250) into our search engine [/URL]and we get: [B]5[/B]

Consider the first 8 calculations of 7 to an exponent: [LIST] [*]7^1 = 7 [*]7^2 = 49 [*]7^3 = 343 [*]7^4 = 2,401 [*]7^5 = 16,807 [*]7^6 = 117,649 [*]7^7 = 823,543 [*]7^8 = 5,764,801 [/LIST] Take a look at the last digit of the first 8 calculations: 7, 9, 3, 1, 7, 9, 3, 1 The 7, 9, 3, 1 repeats through infinity. So every factor of 4, the cycle of 7, 9, 3, 1 restarts. Counting backwards from 2013, we know that 2012 is the largest number divisible by 4: 7^2013 = 7^2012 * 7^1 The cycle starts over after 2012. Which means the last digit of 7^2013 = [B]7 [MEDIA=youtube]Z157jj8R7Yc[/MEDIA][/B]

Using our [URL='https://www.mathcelebrity.com/factoriz.php?num=72&pl=Show+Factorization']factor calculator[/URL], we get: 6 & 12 [MEDIA=youtube]91JpmUfNBb4[/MEDIA]

George and William both ran a one mile race. William won the race with a time of 4 minutes and 30 seconds. If George was 480 feet behind William when the race finished, how long did it take George to run the entire mile? (George continued to run at the same pace.) When the race was done, George completed: 5280 feet in a mile - 480 feet = 4800 feet set up a proportion of distance traveled to time where n is the time needed to run the mile 4800/4.5 = 5280/n Using our [URL='https://www.mathcelebrity.com/proportion-calculator.php?num1=4800&num2=5280&den1=4.5&den2=n&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator,[/URL] we get: n = 4.95 5280/4800 = 1.1 Setup another proportion with the 1.1 factor of distance to time: 4800 * 1.1/4.5 * 1.1 = 5280/4.95 4.95 = 4 minutes and .95*60 seconds 4.95 = [B]4 minutes and 57 seconds[/B]

Free Greatest Common Factor and Least Common Multiple Calculator - Given 2 or 3 numbers, the calculator determines the following:
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gy=-g/v+w for g Multiply each side of the equation by v to eliminate fractions: gvy = -g + vw Add g to each side: gvy + g = -g + g + vw Cancel the g's on the right side and we geT: gvy + g = vw Factor out g on the left side: g(vy + 1) = vw Divide each side of the equation by (vy + 1): g(vy + 1)/(vy + 1) = vw/(vy + 1) Cancel the (vy + 1) on the left side and we geT: g = [B]vw/(vy + 1)[/B]

Harry got 42 out of 49 correct in his test. What fraction of the marks did he get correct? The fraction correct is: 42/49 Both the numerator and denominator [URL='http://www.mathcelebrity.com/gcflcm.php?num1=42&num2=49&num3=&pl=GCF']have a common factor[/URL] of 7 Reducing top and bottom by 7, we get: [B]6/7[/B]

How many different ways could you arrange 5 books on a shelf? [URL='https://www.mathcelebrity.com/factorial.php?num=5!&pl=Calculate+factorial']Using permutations, you can type in 5![/URL] and we get: [B]120 different ways.[/B]

How many factors does 36 have We can type in [URL='https://www.mathcelebrity.com/factoriz.php?num=36&pl=Show+Factorization']factor 36[/URL] into our math engine and we get: {1, 2, 3, 4, 6, 9, 12, 18, 36} 1 * 36 2 * 18 3 * 12 4 * 9 6 * 6 This set contains [B]9 factors[/B].

How many ways can 5 people be seated in 5 seats? We have the permutation 5!. Because the first seat can have 5 different people. The next seat has 5 - 1 = 4 people since one person is in the first seat The next seat can have 5 - 2 = 3 people since we have two people in the first two seats The next seat can have 5 - 3 = 2 people since we have three people in the first three seats The next seat can have 5 - 4 = 1 people since we have four people in the first four seats [URL='https://www.mathcelebrity.com/factorial.php?num=5!&pl=Calculate+factorial']Type in 5! into our search engine[/URL], and we get 120.

Identify a pair of factors of -35 that has a sum of -2. If we [URL='https://www.mathcelebrity.com/factoriz.php?num=-35&pl=Show+Factorization']type in [I]factor -35[/I] into our search engine[/URL], we see 4 factor pairs. When we add up the factors for each pair, we see [B]7, -5[/B] added together gives us 2.

If 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 a+b=2 and a2-b2=-4, what is the value of a-b? a^2 - b^2 = -4 Factor this: (a + b)(a - b) = -4 We know from above, (a +b) = 2, so substitute: 2(a - b) = -4 Divide each side by 2 [B](a - b) = -2[/B]

If 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 a+b=16, then what is 3a+3b= Factor 3a + 3b: 3(a + b) Since we know a+b = 16, we have: 3(16) = [B]48[/B]

If f(x) = 3x + 1 and g(x) = x^2 + 2x, find x when f(g(x)) = 10 [U]Evaluate f(g(x))[/U] f(g(x)) = 3(x^2 + 2x) + 1 f(g(x)) = 3x^2 + 6x + 1 [U]When f(g(x)) = 10, we have[/U] 10 = 3x^2 + 6x + 1 [U]Subtract 10 from each side:[/U] 3x^2 + 6x - 9 = 0 Divide each side of the equation by 3 x^2 + 2x - 3 = 0 Factor, we have: (x + 3)(x - 1) = 0 So x is either [B]1 or -3[/B]

If you can buy 1⁄3 of a box of chocolates for 6 dollars, how much can you purchase for 4 dollars? Write your answer as a fraction of a box. Set up a proportion of dollars to boxes where b is the number of boxes for $4: 6/1/3 = 4/b Cross multiply: 6b = 4/3 Multiply each side by 1/6 to isolate b: b = 4/18 [URL='https://www.mathcelebrity.com/gcflcm.php?num1=4&num2=18&num3=&pl=GCF+and+LCM']Type in GCF(4,18) into the search engine[/URL]. We get a greatest common factor of 2. Divide 4 and 18 in the fraction by 2. We get the reduced fraction of: [B]b = 2/9[/B]

In 2016, National Textile installed a new textile machine in one of its factories at a cost of $300,000. The machine is depreciated linearly over 10 years with a scrap value of $10,000. (a) Find an expression for the textile machines book value in the t th year of use (0 ≤ t ≤ 10) We have a straight line depreciation. Book Value is shown on the [URL='http://www.mathcelebrity.com/depsl.php?d=&a=300000&s=10000&n=10&t=3&bv=&pl=Calculate']straight line depreciation calculator[/URL].

In a factory that manufactures tires, a machine responsible for molding the tire has a failure rate of 0.2%. If 1,000 tires are produced in a day, of which 6 are faulty, what is the difference between the experimental probability and the theoretical probability? Theoretical probability = Failure Rate * Tires Theoretical probability = 0.002 * 1000 Theoretical probability = 2 The experimental probability was given as 6, so the difference is: 6 - 2 = [B]4[/B]

In Trina's desk drawer, there are 15 paper clips and 18 rubber bands. In Kirk's office supply tray, there are 13 paper clips and 16 rubber bands. Who has a higher ratio of paper clips to rubber bands? Trina: 15/18 Kirk: 13/16 We want common denominators to compare, so we get a greatest common factor (GCF) for 16 and 18. [URL='https://www.mathcelebrity.com/gcflcm.php?num1=16&num2=18&num3=&pl=GCF+and+LCM']Running this through our search engine[/URL], we get GCF(16, 18) = 144 For Trina, 144/18 = 8 For Kirk, 144/16 = 9 We multiply Trina's fraction, top and bottom by 8: 15 * 8 / 18 * 8 120/144 We multiply Trina's fraction, top and bottom by 8: 13 * 8 / 16 * 8 104/144 [B]Trina[/B] has more in her numerator, so her ratio of paper clips to rubber bands is greater.

If we factor your expression, we get: y > x(10% + 1) y> 1.1x Since 10% is 0.1 I read it as y > 110% of x

Jack is making snack bags. He has 18 baby carrots and 42 pretzels. He wants to divide them equally among the bags. What is the greatest number of snack bags he can make? Find the [URL='http://www.mathcelebrity.com/gcflcm.php?num1=18&num2=42&num3=&pl=GCF']Greatest Common Factor[/URL] of (18, 42) = 6 6 bags for 18 carrots = 3 carrots per bag 6 bags for 42 pretzels = 7 pretzels per bag [B]6 bags is the answer[/B]

log5 = 0.699, log2 = 0.301. Use these values to evaluate log40. One of the logarithmic identities is: log(ab) = log(a) + log(b). Using the numbers 2 and 5, we somehow need to get to 40. [URL='http://www.mathcelebrity.com/factoriz.php?num=40&pl=Show+Factorization']List factors of 40[/URL]. On the link above, take a look at the bottom where it says prime factorization. We have: 40 = 2 x 2 x 2 x 5 Using our logarithmic identity, we have: log40 = log(2 x 2 x 2 x 5) Rewriting this using our identity, we have: log40 = log2 + log2 + log2 + log5 log40 = 0.301 + 0.301 + 0.301 + 0.699 log40 = [B]1.602 [MEDIA=youtube]qyG_Jkf9VDc[/MEDIA][/B]

Make 19 using only four four's 4!-4-4/4 To prove our work, we have: [URL='https://www.mathcelebrity.com/factorial.php?num=4!&pl=Calculate+factorial']4![/URL] = 24 24 - 4 = 20 Since 4/4 = 1, we have: 20 - 1 = 19

Mrs. Taylor is making identical costumes for the dancers in the dance club. She uses 126 pink ribbons and 108 yellow ribbons. a) What is the maximum possible number of costumes she can make? b) How many pink and how many yellow ribbons are on each costume? a), we want the greatest common factor (GCF) of 108 and 126. [URL='https://www.mathcelebrity.com/gcflcm.php?num1=108&num2=126&num3=&pl=GCF+and+LCM']Using our GCF calculator[/URL] we get: [B]a) 18 costumes [/B] b) Pink Ribbons per costume = Total Pink Ribbons / GCF in question a Pink Ribbons per costume = 126/18 Pink Ribbons per costume = [B]7[/B] [B][/B] Yellow Ribbons per costume = Total Yellow Ribbons / GCF in question a Yellow Ribbons per costume = 108/18 Yellow Ribbons per costume = [B]6[/B]

Free Multifactorials Calculator - Calculates the multifactorial n!^(m)

n is a factor of 16 List out factor of 16: 1 * 16 2 * 8 4 * 4 From the list above, we take the 5 [U]unique[/U] factors and build a set for n: n = [B]{1, 2, 4, 8, 16}[/B]

natural numbers that are factors of 16 Natural numbers are positive integers starting at 1. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} Of these, [URL='https://www.mathcelebrity.com/factoriz.php?num=16&pl=Show+Factorization']the only factors of 16[/URL] are: {[B]1, 2, 4, 8, 16}[/B]

n^2+n = odd Factor n^2+n: n(n + 1) We have one of two scenarios: [LIST=1] [*]If n is odd, then n + 1 is even. The product of an odd and even number is an even number [*]If n is even, then n + 1 is odd. The product of an even and odd number is an even number [/LIST]

n^2-n = even Factor n^2-n: n(n - 1) We have one of two scenarios: [LIST=1] [*]If n is odd, then n - 1 is even. The product of an odd and even number is an even number [*]If n is even, then n - 1 is odd. The product of an even and odd number is an even number [/LIST]

On your first draw, what is the probability of drawing a red card, without looking, from a shuffled deck containing 6 red cards, 6 blue cards, and 8 black cards? P(Red) = Total Red / Total Cards P(Red) = 6 red/(6 red + 6 blue + 8 black) P(Red) = 6/20 This fraction can be simplified. The [URL='https://www.mathcelebrity.com/gcflcm.php?num1=6&num2=20&num3=&pl=GCF+and+LCM']greatest common factor of 6 and 20[/URL] is 2. So we divide top and bottom of our probability by 2: P(Red) = 6/2 / 20 / 2 P(Red) = [B]3/10[/B]

P is the natural numbers that are factors of 25 we type in [I][URL='https://www.mathcelebrity.com/factoriz.php?num=25&pl=Show+Factorization']factor 25[/URL][/I] into our math engine and we get: {1, 5, 25} Since [U]all[/U] of these are natural numbers, our answer is: [B]{1, 5, 25}[/B]

p/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]

p/q=f/q-f for f To solve this literal equation for f, let's factor out f on the right side: p/q=f(1/q-1) Divide each side by (1/q - 1) p/(q(1/q - 1)) = f(1/q-1)/(1/q - 1) Cancelling the (1/q - 1) on the right side, we get: f = p/(1/q - 1) Rewriting this since (1/q -1) = (1 - q)/q since q/q = 1 we have: f = [B]pq/(1 - q)[/B]

Pete ate 1/2 a pizza, Carol ate 1/3, and Joe ate 1/4. How much pizza was eaten? Total pizza eaten is: 1/2 + 1/3 + 1/4 We need a common denominator. List out factors: 2: 1, 2, 4, 6, 8, 10, 12 3: 1, 3, 6, 9, 12 4: 1, 4, 8, 12 12 is our least common multiple: 1/2 = 6/12 1/3 = 4/12 1/4 = 3/12 So our addition becomes: (6 + 4 + 3)/12 = 13/12 of a pizza

Prove 0! = 1 Let n be a whole number, where n! represents the product of n and all integers below it through 1. The factorial formula for n is: n! = n · (n - 1) * (n - 2) * ... * 3 * 2 * 1 Written in partially expanded form, n! is: n! = n * (n - 1)! [U]Substitute n = 1 into this expression:[/U] n! = n * (n - 1)! 1! = 1 * (1 - 1)! 1! = 1 * (0)! For the expression to be true, 0! [U]must[/U] equal 1. Otherwise, 1! <> 1 which contradicts the equation above

[URL='https://www.mathcelebrity.com/proofs.php?num=prove0%21%3D1&pl=Prove']Prove 0! = 1[/URL] Let n be a whole number, where n! represents: The product of n and all integers below it through 1. The factorial formula for n is n! = n · (n - 1) · (n - 2) · ... · 3 · 2 · 1 Written in partially expanded form, n! is: n! = n · (n - 1)! [SIZE=5][B]Substitute n = 1 into this expression:[/B][/SIZE] n! = n · (n - 1)! 1! = 1 · (1 - 1)! 1! = 1 · (0)! For the expression to be true, 0! [U]must[/U] equal 1. Otherwise, 1! ≠ 1 which contradicts the equation above [MEDIA=youtube]wDgRgfj1cIs[/MEDIA]

Take 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]

Take 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]

Take two arbitrary integers, x and y We can express the odd integer x as 2a + 1 for some integer a We can express the odd integer y as 2b + 1 for some integer b x + y = 2a + 1 + 2b + 1 x + y = 2a + 2b + 2 Factor out a 2: x + y = 2(a + b + 1) Since 2 times any integer even or odd is always even, then [B]x + y by definition is even[/B]. [MEDIA=youtube]9A-qe4yZXYw[/MEDIA]

Let us take an integer x which is both even [I]and[/I] odd. [LIST] [*]As an even integer, we write x in the form 2m for some integer m [*]As an odd integer, we write x in the form 2n + 1 for some integer n [/LIST] Since both the even and odd integers are the same number, we set them equal to each other 2m = 2n + 1 Subtract 2n from each side: 2m - 2n = 1 Factor out a 2 on the left side: 2(m - n) = 1 By definition of divisibility, this means that 2 divides 1. But we know that the only two numbers which divide 1 are 1 and -1. Therefore, our original assumption that x was both even and odd must be false. [MEDIA=youtube]SMM9ubEVcLE[/MEDIA]

Free 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
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Sarah rolls 2 fair dice and adds the results from each. Work out the probability of getting a total that is a factor of 6. Factors of 6 are {6, 12} [URL='https://www.mathcelebrity.com/2dice.php?gl=1&pl=6&opdice=1&rolist=+&dby=&ndby=&montect=+']P(Roll a 6)[/URL] = 5/36 [URL='https://www.mathcelebrity.com/2dice.php?gl=1&pl=12&opdice=1&rolist=+&dby=&ndby=&montect=+']P(Roll a 12)[/URL] = 1/36 P(Roll a 6 or Roll a 12) = P(Roll a 6) + P(Roll a 12) P(Roll a 6 or Roll a 12) = 5/36 + 1/36 P(Roll a 6 or Roll a 12) = 6/36 Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=6%2F36&frac2=3%2F8&pl=Simplify']fraction simplifier[/URL], we see that: P(Roll a 6 or Roll a 12) = [B]1/6[/B]

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]

Soda cans are sold in a local store for 50 cents each. The factory has $900 in fixed costs plus 25 cents of additional expense for each soda can made. Assuming all soda cans manufactured can be sold, find the break-even point. Calculate the revenue function R(c) where s is the number of sodas sold: R(s) = Sale Price * number of units sold R(s) = 50s Calculate the cost function C(s) where s is the number of sodas sold: C(s) = Variable Cost * s + Fixed Cost C(s) = 0.25s + 900 Our break-even point is found by setting R(s) = C(s): 0.25s + 900 = 50s We [URL='https://www.mathcelebrity.com/1unk.php?num=0.25s%2B900%3D50s&pl=Solve']type this equation into our search engine[/URL] and we get: s = [B]18.09[/B]

Squaring a number equals 5 times that number. The phrase [I]a number[/I] means an arbitrary variable, let's call it x. Squaring this number: x^2 5 times this number means we multiply by 5: 5x The phrase [I]equals[/I] means we set both expressions equal to each other: [B]x^2 = 5x [/B] <-- This is our algebraic expression If you want to solve for x, then we subtract 5x from each side: x^2 - 5x = 5x - 5x Cancel the 5x on the right side, leaving us with 0: x^2 - 5x = 0 Factor out x: x(x - 5) So we get x = 0 or [B]x = 5[/B]

Free 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

The coach writes the batting order on a piece of paper. How many different ways could the list be written? We have 9 people in a line up. The total lineups are shown by: 9 * 8 * 7 * ... * 2 * 1 Or, 9!. [URL='https://www.mathcelebrity.com/factorial.php?num=9!&pl=Calculate+factorial']Typing 9! in our search engine[/URL] and we get [B]362,880[/B]

The doubling time of a population of flies is 8 hours. a) By what factor does a population increase in 24 hours? b) By what factor does the population increase in 2 weeks? a) If the population doubles every 8 hours, then it doubles 3 times in 24 hours, since 24/8 = 3. So 2 * 3 = 6. The increase factor is [B]6[/B] b) Since a) is one full day, we now need to know how much it doubles in 14 days, which is 2 weeks. We take our factor of 6 for each day, and multiply it by 14: 14 * 6 = [B]84[/B]

The earth makes a full rotation in 24 hours. Jupiter takes approximately 58 days, 15 h, and 30 min to complete a full rotation. How many hours does it take Jupiter to complete a full rotation? Show your work using the correct conversion factors. Convert 58 days, 15 h, and 30 min to hours. [LIST=1] [*]Type [URL='https://www.mathcelebrity.com/timecon.php?quant=58&pl=Calculate&type=day']58 days[/URL] into the search engine to get 1,392 hours. [*]Add 15 hours to get 1,392 + 15 = 2,007 hours [*]Now convert the 30 min to hours. [URL='https://www.mathcelebrity.com/timecon.php?quant=30&pl=Calculate&type=minute']Type 30 minutes into the search engine[/URL] to get 0.5 hours [*]Add up (1), (2), and (3) to get 1,392 + 15 + 0.5 = [B]2007.5[/B] hours for a full rotation. [/LIST]

The next number in the series 38 36 30 28 22 is Notice the change of factors. Subtract 2, Subtract 6, Subtract 2, Subtract 6. So the next number should subtract 2. 22 - 2 = [B]20 [MEDIA=youtube]x7SHk_6-aok[/MEDIA][/B]

The Palafoxes make $3,840 a month. They spend $1,600 for rent. What fraction of their income goes to rent? Rent Payment Fraction = Rent Payment / Total Income Rent Payment Fraction = 1600 / 3840 Our greatest common factor of 1600 and 3840 is 320. So if we divide 1600 and 3840 by 320, we get: Rent Payment Fraction = [B]5/12 [MEDIA=youtube]DsXk6AKT18M[/MEDIA][/B]

The 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]

There are 72 boys and 90 girls on the Math team For the next math competition Mr Johnson would like to arrange all of the students in equal rows with only girls or boys in each row with only girls or boys in each row. What is the greatest number of students that can be put in each row? To find the maximum number (n) of boys or girls in each row, we want the GCF (Greatest Common Factor) of 72 and 90. [URL='https://www.mathcelebrity.com/gcflcm.php?num1=72&num2=90&num3=&pl=GCF+and+LCM']Using our GCF calculator for GCF(72,90)[/URL], we get 18. [LIST] [*]72 boys divided by 18 = [B]4 rows of boys[/B] [*]90 girls divided by 18 = [B]5 rows of girls[/B] [/LIST]

What is the average of 7 consecutive numbers if the smallest number is called n? [LIST] [*]First number = n [*]Second number = n + 1 [*]Third number = n + 2 [*]Fourth number = n + 3 [*]Fifth number = n + 4 [*]Sixth number = n + 5 [*]Seventh number = n + 6 [/LIST] Average = Sum of all numbers / Total numbers Average = (n + n + 1 + n + 2 + n + 3 + n + 4 + n + 5 + n + 6)/7 Average = 7n + 21/7 Factor out a 7 from the top: 7(n + 3)/7 Cancel the 7's: [B]n + 3[/B]

What pair of factors of -28 has a sum of -3? We type in [I]factor -28[/I] into our search engine. Scrolling down the list of factor sums, we see: -7 + 4 = -3 So our answer is [B](-7, 4)[/B]

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Write in set builder form {all possible numbers formed by any two of the digits 1 2 5} With 3 numbers, we got [URL='https://www.mathcelebrity.com/factorial.php?num=3!&pl=Calculate+factorial']3! = 6[/URL] possible numbers formed by the two digits [LIST=1] [*]12 [*]15 [*]21 [*]25 [*]51 [*]52 [/LIST] In set builder notation, we write this as: {x : x ∈ {12, 15, 21, 25, 51, 52}) x such that x is a element of the set {12, 15, 21, 25, 51, 52}

wy - ma = ay/n for y Subtract ay/n from each side: wy - ma - ay/n = ay/n - ay/n wy - ma - ay/n = 0 Now add ma to each side: wy - ay/n = ma Factor out y: y(w - a/n) = ma Divide each side by (w - a/n) y = [B]ma/(w - a/n)[/B]

You and 5 friends go to a concert. how many different ways can you sit in the assigned seats? With 6 possible seats, the [URL='https://www.mathcelebrity.com/factorial.php?num=6!&pl=Calculate+factorial']number of unique arrangements is[/URL]: 6! = 6 x 5 x 4 x 3 x 2 x 1 = [B]720[/B]

You are making identical gift bags using 24 candles and 36 bottles of lotion. What is the greatest number of gift bags you can make with no items left over? We take the greatest common factor [URL='https://www.mathcelebrity.com/gcflcm.php?num1=24&num2=36&num3=&pl=GCF+and+LCM']GCF (24, 36) = 12[/URL] So we have a ratio of 24/12 = 2 candles and 36/12 = 3 bottles of lotion per bag giving us [B]12 bags[/B].

your classmate asserted that x^2 - 4x - 12 and 12 - 4x - x^2 has the same factors is your classmate correct Factor x^2 - 4x - 12 using binomials: (x + 2)(x - 6) Therefore, factors are x = -2, x = 6 Factor 12 - 4x - x^2 -(x - 6)(x + 2) Therefore, factors are x = -2, x = -6 Because they don't have two matching factors, your classmate is [B]incorrect.[/B]

Your salary after a 5% increase if your salary before the increase was s. If we start with s, and get a 5% increase, we will have s + 0.05s. Factor our s: [B]s(1.05) or 1.05s[/B]

z = (x + y)/mx; Solve for x Cross multiply: zmx = x + y Subtract x from each side zmx - x = y Factor out x x(zm - 1) = y Divide each side by zm - 1 x = y/(zm - 1) [MEDIA=youtube]ksxCS3YlCwY[/MEDIA]

z/w=x+z/x+y for z This is a literal equation. Let's isolate z on one side. Subtract z/x from each side. z/w - z/x = x + y Factor our z on the left side: z(1/w - 1/x) = x + y Divide each side by (1/w - 1/x) z = x + y/(1/w - 1/x) To remove reciprocals in the denominator, we rewrite 1/w - 1/x with a common denominator xw (x - w)/xw Then multiply x + y by the reciprocal z = [B](x + y)xw/(x - w)[/B]

zy-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]