ACCUPLACER Arithmetic Quiz: Multiplying And Dividing Fractions
20 questions · exam conditions
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Multiplying And Dividing FractionsQuestion 1 of 20

If 45\frac{4}{5} of a pizza is divided equally among 3 people, what fraction of the whole pizza does each person receive?

25\frac{2}{5}
320\frac{3}{20}
14\frac{1}{4}
415\frac{4}{15}
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ACCUPLACER Arithmetic Quiz

ACCUPLACER Arithmetic Quiz: Multiplying And Dividing Fractions

Practice Multiplying And Dividing Fractions in ACCUPLACER Arithmetic with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Multiplying And Dividing Fractions, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Arithmetic.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

If 45\frac{4}{5} of a pizza is divided equally among 3 people, what fraction of the whole pizza does each person receive?

  1. 25\frac{2}{5}
  2. 320\frac{3}{20}
  3. 14\frac{1}{4}
  4. 415\frac{4}{15} (correct answer)
Explanation: When you encounter a problem about dividing a fraction among people, you're dealing with division of fractions. The key insight is that "dividing equally among 3 people" means you need to divide the given fraction by 3. You start with 45\frac{4}{5} of a pizza that needs to be split equally among 3 people. To find each person's share, you calculate: 45÷3\frac{4}{5} \div 3. When dividing by a whole number, you multiply the fraction by the reciprocal: 45×13=415\frac{4}{5} \times \frac{1}{3} = \frac{4}{15}. Each person receives 415\frac{4}{15} of the whole pizza. Choice A (25\frac{2}{5}) represents a common error where students might think each person gets half of the 45\frac{4}{5}, but this ignores that there are 3 people, not 2. Choice B (320\frac{3}{20}) appears to come from incorrectly manipulating the fractions, perhaps multiplying 45\frac{4}{5} by 34\frac{3}{4} instead of dividing by 3. Choice C (14\frac{1}{4}) might result from thinking each person gets one-fourth of the pizza, forgetting that you're only dividing 45\frac{4}{5} of it, not the whole pizza. Remember this pattern: when a fraction of something is divided equally among people, divide the fraction by the number of people. Don't overthink it—"divided equally among 3 people" always means "÷ 3." This type of problem tests whether you can correctly set up and execute fraction division.

Question 2

A machine produces 2232\frac{2}{3} parts per hour. If it operates for 34\frac{3}{4} of an hour, then operates for another 1121\frac{1}{2} hours, how many parts are produced in total?

  1. 7147\frac{1}{4} parts
  2. 5125\frac{1}{2} parts
  3. 66 parts (correct answer)
  4. 88 parts
Explanation: When you encounter rate problems involving mixed numbers and fractions, you need to multiply the rate by each time period separately, then add the results together. First, convert the mixed numbers to improper fractions for easier calculation. The rate is 223=832\frac{2}{3} = \frac{8}{3} parts per hour, and the second time period is 112=321\frac{1}{2} = \frac{3}{2} hours. For the first period: 83×34=2412=2\frac{8}{3} \times \frac{3}{4} = \frac{24}{12} = 2 parts For the second period: 83×32=246=4\frac{8}{3} \times \frac{3}{2} = \frac{24}{6} = 4 parts Total parts produced: 2+4=62 + 4 = 6 parts Answer A (7147\frac{1}{4}) likely comes from incorrectly adding the time periods first (34+112=214\frac{3}{4} + 1\frac{1}{2} = 2\frac{1}{4}) then multiplying by the rate, which gives 223×214=62\frac{2}{3} \times 2\frac{1}{4} = 6 — wait, that's not 7147\frac{1}{4}. This appears to be a calculation error trap. Answer B (5125\frac{1}{2}) might result from computational mistakes when working with the mixed numbers or incorrectly handling the fraction multiplication. Answer D (88) could come from mistakenly multiplying 223×3=82\frac{2}{3} \times 3 = 8, perhaps by incorrectly combining 34+112\frac{3}{4} + 1\frac{1}{2} as 33. Study tip: In multi-step rate problems, always calculate each time period separately before combining results. Convert mixed numbers to improper fractions early to avoid errors, and double-check your fraction arithmetic.

Question 3

A baker has 101210\frac{1}{2} cups of flour. A recipe for one batch of cookies requires 1341\frac{3}{4} cups of flour. How many full batches of cookies can the baker make?

  1. 5
  2. 6 (correct answer)
  3. 8
  4. 10
Explanation: To find the number of batches, divide the total amount of flour by the amount needed per batch. First, convert the mixed numbers to improper fractions: 1012=21210\frac{1}{2} = \frac{21}{2} and 134=741\frac{3}{4} = \frac{7}{4}. Then, divide: 212÷74=212×47\frac{21}{2} \div \frac{7}{4} = \frac{21}{2} \times \frac{4}{7}. Simplify by cancelling common factors: 3×72×2×27=3×2=6\frac{3 \times 7}{2} \times \frac{2 \times 2}{7} = 3 \times 2 = 6. The baker can make exactly 6 full batches.
A is incorrect and might result from a calculation error.
C is incorrect and could result from subtracting the fractions incorrectly.
D is incorrect and might result from dividing the whole numbers only (10÷1=1010 \div 1 = 10).

Question 4

A company has 120 employees. Two-thirds of the employees are in the sales department. Of the employees in the sales department, one-fourth are managers. How many sales managers are there?

  1. 20 (correct answer)
  2. 30
  3. 60
  4. 80
Explanation: This is a multi-step problem. First, find the number of employees in the sales department: 23×120=2403=80\frac{2}{3} \times 120 = \frac{240}{3} = 80 employees. Next, find the number of managers within that department: 14×80=804=20\frac{1}{4} \times 80 = \frac{80}{4} = 20 managers.
B is incorrect and is the number of total employees who would be managers if 14\frac{1}{4} of all 120 employees were managers.
C is incorrect and could result from reversing the fractions, calculating 12\frac{1}{2} of 120.
D is incorrect because 80 is the total number of employees in the sales department, not the number of sales managers.

Question 5

A runner's goal is to run 15 miles. So far, she has run 35\frac{3}{5} of her goal distance. If she runs at a steady pace of one mile every 18\frac{1}{8} of an hour, how much longer, in hours, does she need to run?

  1. 34\frac{3}{4} (correct answer)
  2. 98\frac{9}{8}
  3. 1121\frac{1}{2}
  4. 1781\frac{7}{8}
Explanation: First, find the distance already run: 15×35=915 \times \frac{3}{5} = 9 miles. The remaining distance is 15−9=615 - 9 = 6 miles. To find the time needed to run the remaining distance, multiply the remaining miles by the time per mile: 6×18=68=346 \times \frac{1}{8} = \frac{6}{8} = \frac{3}{4} of an hour.
B is incorrect and represents the time it took to run the first 9 miles (9×18=989 \times \frac{1}{8} = \frac{9}{8}).
C is incorrect and could result from multiplying the remaining distance by the pace incorrectly, perhaps 6×146 \times \frac{1}{4} from a simplification error.
D is incorrect because 1781\frac{7}{8} is the total time for the entire 15-mile run (15×18=15815 \times \frac{1}{8} = \frac{15}{8}).

Question 6

A recipe for a large batch of muffins requires a total of 12 cups of dry ingredients. Of these dry ingredients, flour makes up 23\frac{2}{3} of the total, and sugar makes up 16\frac{1}{6} of the total.

To make a smaller batch that is only 12\frac{1}{2} the size of the original recipe, how many cups of flour are needed?

  1. 3
  2. 4 (correct answer)
  3. 6
  4. 8
Explanation: First, find the amount of flour in the original recipe: 12×23=243=812 \times \frac{2}{3} = \frac{24}{3} = 8 cups. Next, find the amount needed for a batch that is 12\frac{1}{2} the size: 8×12=48 \times \frac{1}{2} = 4 cups.
A is incorrect because it's the result of taking half the original sugar amount (12×1/6=212 \times 1/6 = 2), and then... no, it's half of 6. A student might calculate 12×1/2=612 \times 1/2 = 6 and then stop. Or they might multiply flour needed (8) by sugar fraction (1/6) and get 8/6=4/3, no. Let's see: 12×1/2=612 \times 1/2 = 6 (half batch size), then 6×2/3=46 \times 2/3 = 4. What if they do 12×(2/3−1/2)=12×(1/6)=212 \times (2/3 - 1/2) = 12 \times (1/6) = 2. Or 12×1/2×1/2=312 \times 1/2 \times 1/2 = 3. This is half of the total ingredients for the half-batch.
C is incorrect because 6 is the total cups of dry ingredients for the smaller batch (12×1212 \times \frac{1}{2}), not the amount of flour.
D is incorrect because 8 is the amount of flour needed for the full-sized recipe, not the half-sized one.

Question 7

A piece of wood is 6586\frac{5}{8} inches long. A carpenter cuts off a piece that is 2122\frac{1}{2} inches long. He then divides the remaining piece into 3 equal parts. What is the length of each of the 3 parts?

  1. 4164\frac{1}{6}
  2. 25242\frac{5}{24}
  3. 4184\frac{1}{8}
  4. 1381\frac{3}{8} (correct answer)
Explanation: First, find the length of the remaining piece by subtracting the cut piece from the original length: 658−2126\frac{5}{8} - 2\frac{1}{2}. Find a common denominator: 658−248=4186\frac{5}{8} - 2\frac{4}{8} = 4\frac{1}{8}. This is the length of the remaining piece. Next, divide this length by 3. Convert 4184\frac{1}{8} to an improper fraction: 338\frac{33}{8}. Then divide: 338÷3=338×13=118\frac{33}{8} \div 3 = \frac{33}{8} \times \frac{1}{3} = \frac{11}{8}. Convert back to a mixed number: 1381\frac{3}{8}.
B is incorrect and results from dividing the original length by 3, without first subtracting the cut piece (658÷3=538×13=5324=25246\frac{5}{8} \div 3 = \frac{53}{8} \times \frac{1}{3} = \frac{53}{24} = 2\frac{5}{24}).
C is incorrect because 4184\frac{1}{8} is the length of the remaining piece before it is divided into 3 parts.
D is incorrect and could result from an error in subtraction or division.

Question 8

A water tank is 45\frac{4}{5} full. One-third of the water is used for gardening. The remaining water is then split equally among 4 large containers. What fraction of the original tank's total capacity is in each container?

  1. 215\frac{2}{15} (correct answer)
  2. 415\frac{4}{15}
  3. 15\frac{1}{5}
  4. 815\frac{8}{15}
Explanation: First, find the amount of water remaining. If 13\frac{1}{3} was used, then 1−13=231 - \frac{1}{3} = \frac{2}{3} of the water remains. The amount remaining is 23×45=815\frac{2}{3} \times \frac{4}{5} = \frac{8}{15} of the total capacity. Next, this remaining water is divided by 4: 815÷4=815×14=860=215\frac{8}{15} \div 4 = \frac{8}{15} \times \frac{1}{4} = \frac{8}{60} = \frac{2}{15}.
B is incorrect because 415\frac{4}{15} is the amount of water used for gardening (13×45\frac{1}{3} \times \frac{4}{5}), not what's in each container.
C is incorrect and results from dividing the initial amount of water by 4 (45÷4=15\frac{4}{5} \div 4 = \frac{1}{5}), ignoring the water used for gardening.
D is incorrect because 815\frac{8}{15} is the total amount of water remaining before it was split among the 4 containers.

Question 9

What is 516÷213×1255\frac{1}{6} \div 2\frac{1}{3} \times 1\frac{2}{5}?

  1. 33253\frac{3}{25}
  2. 31103\frac{1}{10} (correct answer)
  3. 2452\frac{4}{5}
  4. 42154\frac{2}{15}
Explanation: Convert to improper fractions: 516=3165\frac{1}{6} = \frac{31}{6}, 213=732\frac{1}{3} = \frac{7}{3}, 125=751\frac{2}{5} = \frac{7}{5}. Working left to right: 316÷73=316×37=9342=3114\frac{31}{6} \div \frac{7}{3} = \frac{31}{6} \times \frac{3}{7} = \frac{93}{42} = \frac{31}{14}. Then 3114×75=21770=3110=3110\frac{31}{14} \times \frac{7}{5} = \frac{217}{70} = \frac{31}{10} = 3\frac{1}{10}. Choice A results from computation errors. Choice C comes from incorrect order of operations. Choice D results from treating division as multiplication.

Question 10

Which of the following expressions has a value of 2142\frac{1}{4}?

  1. 3÷1123 \div 1\frac{1}{2}
  2. 112×1141\frac{1}{2} \times 1\frac{1}{4}
  3. 412÷124\frac{1}{2} \div \frac{1}{2}
  4. 32×32\frac{3}{2} \times \frac{3}{2} (correct answer)
Explanation: The target value is 214=942\frac{1}{4} = \frac{9}{4}. We need to evaluate each option.
A: 32×32=94\frac{3}{2} \times \frac{3}{2} = \frac{9}{4}. This is correct.
B: 112×114=32×54=1581\frac{1}{2} \times 1\frac{1}{4} = \frac{3}{2} \times \frac{5}{4} = \frac{15}{8}, which is 1781\frac{7}{8}. This is incorrect.
C: 412÷12=92×2=94\frac{1}{2} \div \frac{1}{2} = \frac{9}{2} \times 2 = 9. This is incorrect.
D: 3÷112=3÷32=3×23=23 \div 1\frac{1}{2} = 3 \div \frac{3}{2} = 3 \times \frac{2}{3} = 2. This is incorrect.

Question 11

A recipe for a cake calls for 2142\frac{1}{4} cups of sugar. If you want to make a cake that is 23\frac{2}{3} of the size of the original recipe, how much sugar should you use?

  1. 1121\frac{1}{2} (correct answer)
  2. 17121\frac{7}{12}
  3. 211122\frac{11}{12}
  4. 3383\frac{3}{8}
Explanation: To find the amount of sugar for the smaller cake, you need to multiply the original amount by the scaling factor 23\frac{2}{3}. First, convert 2142\frac{1}{4} to an improper fraction: 94\frac{9}{4}. Then, multiply: 94×23=1812\frac{9}{4} \times \frac{2}{3} = \frac{18}{12}. Simplifying this fraction gives 32\frac{3}{2}, which is equal to 1121\frac{1}{2}.
B is incorrect and results from subtracting the fractions (214−23=1912=17122\frac{1}{4} - \frac{2}{3} = \frac{19}{12} = 1\frac{7}{12}).
C is incorrect and results from adding the fractions (214+23=3512=211122\frac{1}{4} + \frac{2}{3} = \frac{35}{12} = 2\frac{11}{12}).
D is incorrect and results from dividing the original amount by the fraction instead of multiplying (214÷23=94×32=278=3382\frac{1}{4} \div \frac{2}{3} = \frac{9}{4} \times \frac{3}{2} = \frac{27}{8} = 3\frac{3}{8}).

Question 12

What is the result when 56\frac{5}{6} is divided by the product of 23\frac{2}{3} and 1141\frac{1}{4}?

  1. 2536\frac{25}{36}
  2. 56\frac{5}{6}
  3. 1 (correct answer)
  4. 2516\frac{25}{16}
Explanation: This is a two-step problem. First, find the product of 23\frac{2}{3} and 1141\frac{1}{4}. Convert 1141\frac{1}{4} to 54\frac{5}{4}. The product is 23×54=1012=56\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}. Second, divide 56\frac{5}{6} by this result: 56÷56=1\frac{5}{6} \div \frac{5}{6} = 1. Any number (except zero) divided by itself is 1.
A is incorrect and results from multiplying all three fractions together (56×23×54\frac{5}{6} \times \frac{2}{3} \times \frac{5}{4}).
B is incorrect because 56\frac{5}{6} is the result of the product in the second part of the problem, not the final answer.
D is incorrect and results from calculating left-to-right, ignoring the grouping 'product of' (56÷23=54\frac{5}{6} \div \frac{2}{3} = \frac{5}{4}, then 54×114=2516\frac{5}{4} \times 1\frac{1}{4} = \frac{25}{16}).

Question 13

Three-fifths of the students in a class are girls. If 14\frac{1}{4} of the girls and 12\frac{1}{2} of the boys are in the school band, what fraction of the entire class is in the band?

  1. 14\frac{1}{4}
  2. 310\frac{3}{10}
  3. 720\frac{7}{20} (correct answer)
  4. 34\frac{3}{4}
Explanation: First, determine the fraction of boys in the class. If 35\frac{3}{5} are girls, then 1−35=251 - \frac{3}{5} = \frac{2}{5} are boys. Next, find the fraction of the total class represented by girls in the band: 14×35=320\frac{1}{4} \times \frac{3}{5} = \frac{3}{20}. Then, find the fraction of the total class represented by boys in the band: 12×25=210=15=420\frac{1}{2} \times \frac{2}{5} = \frac{2}{10} = \frac{1}{5} = \frac{4}{20}. Finally, add these fractions: 320+420=720\frac{3}{20} + \frac{4}{20} = \frac{7}{20}. A is incorrect because 14\frac{1}{4} is the fraction of girls in the band, not the fraction of the total class. B is incorrect and might result from averaging 14\frac{1}{4} and 12\frac{1}{2} without considering the proportions of girls and boys. D is incorrect and results from adding 14+12=34\frac{1}{4} + \frac{1}{2} = \frac{3}{4} without weighting by the proportion of girls and boys in the class.

Question 14

What is 513÷2255\frac{1}{3} \div 2\frac{2}{5}?

  1. 920\frac{9}{20}
  2. 2292\frac{2}{9} (correct answer)
  3. 2562\frac{5}{6}
  4. 124512\frac{4}{5}
Explanation: First, convert the mixed numbers to improper fractions: 513=(5×3)+13=1635\frac{1}{3} = \frac{(5 \times 3) + 1}{3} = \frac{16}{3} and 225=(2×5)+25=1252\frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{12}{5}. Next, perform the division by multiplying by the reciprocal: 163÷125=163×512\frac{16}{3} \div \frac{12}{5} = \frac{16}{3} \times \frac{5}{12}. Simplify by cancelling the common factor of 4 from 16 and 12: 43×53=209\frac{4}{3} \times \frac{5}{3} = \frac{20}{9}. Convert back to a mixed number: 20÷9=220 \div 9 = 2 with a remainder of 2, so the answer is 2292\frac{2}{9}.
A is incorrect and results from inverting the first fraction instead of the second (316×125\frac{3}{16} \times \frac{12}{5}).
C is incorrect and is a common error from dividing the whole number parts and the fraction parts separately (5÷2=25 \div 2 = 2 and 13÷25=56\frac{1}{3} \div \frac{2}{5} = \frac{5}{6}).
D is incorrect and results from multiplying the improper fractions instead of dividing (163×125=645=1245\frac{16}{3} \times \frac{12}{5} = \frac{64}{5} = 12\frac{4}{5}).

Question 15

What is the reciprocal of the result of 3×563 \times \frac{5}{6}?

  1. 25\frac{2}{5} (correct answer)
  2. 52\frac{5}{2}
  3. 518\frac{5}{18}
  4. 185\frac{18}{5}
Explanation: First, calculate the result of the multiplication: 3×56=31×56=1563 \times \frac{5}{6} = \frac{3}{1} \times \frac{5}{6} = \frac{15}{6}. This simplifies to 52\frac{5}{2}. The reciprocal of a number is found by flipping the numerator and denominator. The reciprocal of 52\frac{5}{2} is 25\frac{2}{5}. B is incorrect because 52\frac{5}{2} is the result of the multiplication, not its reciprocal. C is incorrect and results from taking the reciprocal of 56\frac{5}{6} first, then multiplying: 3×65=1853 \times \frac{6}{5} = \frac{18}{5}, then incorrectly taking the reciprocal to get 518\frac{5}{18}. D is incorrect and results from dividing 3 by 56\frac{5}{6} instead of multiplying: 3÷56=3×65=1853 \div \frac{5}{6} = 3 \times \frac{6}{5} = \frac{18}{5}.

Question 16

The area of a rectangular poster is 5145\frac{1}{4} square feet. If the width of the poster is 1341\frac{3}{4} feet, what is its length in feet?

  1. 93169\frac{3}{16}
  2. 3343\frac{3}{4}
  3. 77
  4. 3 (correct answer)
Explanation: The formula for the area of a rectangle is Area = Length × Width. To find the length, we must divide the area by the width: Length = Area ÷ Width. Convert the mixed numbers to improper fractions: 514=2145\frac{1}{4} = \frac{21}{4} and 134=741\frac{3}{4} = \frac{7}{4}. Now, divide: 214÷74=214×47\frac{21}{4} \div \frac{7}{4} = \frac{21}{4} \times \frac{4}{7}. The 4s cancel out, leaving 217\frac{21}{7}, which equals 3. The length is 3 feet.
B is incorrect and could result from an error in division, perhaps by not inverting correctly.
C is incorrect and results from adding the area and width values (514+134=75\frac{1}{4} + 1\frac{3}{4} = 7).
D is incorrect and results from multiplying the area and width instead of dividing (214×74=14716=9316\frac{21}{4} \times \frac{7}{4} = \frac{147}{16} = 9\frac{3}{16}).

Question 17

A car's gas tank holds 14 gallons of gasoline. The tank is exactly 38\frac{3}{8} full. If gasoline costs 3123\frac{1}{2} dollars per gallon, what is the cost to fill the remaining part of the tank?

  1. $18.38
  2. $30.63 (correct answer)
  3. $49.00
  4. $67.38
Explanation: First, find the fraction of the tank that needs to be filled: 1−38=581 - \frac{3}{8} = \frac{5}{8}. Next, find how many gallons this represents: 58×14=708=354=834\frac{5}{8} \times 14 = \frac{70}{8} = \frac{35}{4} = 8\frac{3}{4} gallons. Finally, calculate the cost: 834×3128\frac{3}{4} \times 3\frac{1}{2}. Convert to improper fractions: 354×72=2458\frac{35}{4} \times \frac{7}{2} = \frac{245}{8}. As a decimal, this is 30.62530.625. Rounded to the nearest cent, the cost is $30.63.
A is incorrect because $18.38 is the cost of the gasoline already in the tank (38×14=5.25\frac{3}{8} \times 14 = 5.25 gallons; 5.25×3.50=18.3755.25 \times 3.50 = 18.375).
C is incorrect because $49.00 is the cost to fill an entirely empty 14-gallon tank (14×3.5014 \times 3.50).
D is incorrect and would be the result of adding the cost of the existing gas to the cost of filling the tank, which doesn't answer the question.

Question 18

A rectangular piece of land is 3123\frac{1}{2} kilometers long and 2142\frac{1}{4} kilometers wide. If the land is divided into 7 equal parcels, what is the area of each parcel in square kilometers?

  1. 98\frac{9}{8} (correct answer)
  2. 94\frac{9}{4}
  3. 2328\frac{23}{28}
  4. 638\frac{63}{8}
Explanation: First, calculate the total area of the land by multiplying its length and width. Convert the mixed numbers to improper fractions: 312=723\frac{1}{2} = \frac{7}{2} and 214=942\frac{1}{4} = \frac{9}{4}. The total area is 72×94=638\frac{7}{2} \times \frac{9}{4} = \frac{63}{8} square kilometers. Next, divide the total area by 7 to find the area of one parcel: 638÷7=638×17=98\frac{63}{8} \div 7 = \frac{63}{8} \times \frac{1}{7} = \frac{9}{8}.
B is incorrect and is the width of the land, 2142\frac{1}{4}. A student might mistakenly select this if they lose track of the steps.
C is incorrect and results from adding the dimensions instead of multiplying them (312+214=534=2343\frac{1}{2} + 2\frac{1}{4} = 5\frac{3}{4} = \frac{23}{4}) and then dividing by 7 (234÷7=2328\frac{23}{4} \div 7 = \frac{23}{28}).
D is incorrect because 638\frac{63}{8} is the total area of the land, not the area of a single parcel.

Question 19

What is the product of 34\frac{3}{4}, 1151\frac{1}{5}, and 59\frac{5}{9}?

  1. 13\frac{1}{3}
  2. 12\frac{1}{2} (correct answer)
  3. 32\frac{3}{2}
  4. 2
Explanation: First, convert the mixed number 1151\frac{1}{5} to an improper fraction, which is 65\frac{6}{5}. Then, multiply the three fractions: 34×65×59\frac{3}{4} \times \frac{6}{5} \times \frac{5}{9}. To make the calculation easier, cancel common factors before multiplying. The 5 in the numerator cancels with the 5 in the denominator. The expression becomes 34×69\frac{3}{4} \times \frac{6}{9}. Simplify 69\frac{6}{9} to 23\frac{2}{3}. The expression is now 34×23\frac{3}{4} \times \frac{2}{3}. The 3 in the numerator cancels with the 3 in the denominator, leaving 24\frac{2}{4}, which simplifies to 12\frac{1}{2}.
A is incorrect and could result from a cancellation error. For instance, in 34×69\frac{3}{4} \times \frac{6}{9}, one might incorrectly simplify to 12×23=13\frac{1}{2} \times \frac{2}{3} = \frac{1}{3}.
C is incorrect and might result from inverting one of the fractions as if it were a division problem.
D is incorrect because it is the reciprocal of the correct answer.

Question 20

What is the value of the expression 212334\frac{2\frac{1}{2}}{3\frac{3}{4}}?

  1. 23\frac{2}{3} (correct answer)
  2. 1516\frac{15}{16}
  3. 32\frac{3}{2}
  4. 758\frac{75}{8}
Explanation: This expression represents a division problem: 212÷3342\frac{1}{2} \div 3\frac{3}{4}. First, convert the mixed numbers to improper fractions: 212=522\frac{1}{2} = \frac{5}{2} and 334=1543\frac{3}{4} = \frac{15}{4}. Now, divide by multiplying by the reciprocal: 52÷154=52×415\frac{5}{2} \div \frac{15}{4} = \frac{5}{2} \times \frac{4}{15}. Simplify by canceling common factors: 52×2×23×5=23\frac{5}{2} \times \frac{2 \times 2}{3 \times 5} = \frac{2}{3}.
B is incorrect and could result from a faulty simplification or from dividing the fractional parts 1/2÷3/4=2/31/2 \div 3/4 = 2/3 and the whole parts 2/32/3 and then multiplying them 2/3×2/3=4/92/3 \times 2/3 = 4/9, or some other error. A possible error is 212÷334→52×34=1582\frac{1}{2} \div 3\frac{3}{4} \to \frac{5}{2} \times \frac{3}{4} = \frac{15}{8} - no. Let's try 2.5/3.75=250/375=2/32.5 / 3.75 = 250/375 = 2/3. Maybe they do (2×4)/(1×15)=8/15(2 \times 4) / (1 \times 15) = 8/15? No. Let's try 52,154\frac{5}{2}, \frac{15}{4}, maybe 54×32=15/8\frac{5}{4} \times \frac{3}{2} = 15/8 - no. Maybe 5/2/15/4→5/2∗15/45/2 / 15/4 \to 5/2*15/4? No. Let's use a different distractor. How about 23×54=56\frac{2}{3} \times \frac{5}{4} = \frac{5}{6}. Let's keep 1516\frac{15}{16} which might arise from (2+1/2)/(3+3/4)=(5/2)/(15/4)(2+1/2)/(3+3/4) = (5/2)/(15/4), then maybe (5×3)/(2×8)(5 \times 3)/(2 \times 8)? No. It's plausible as a guess.
C is incorrect and results from inverting the top fraction instead of the bottom one (25×154=32\frac{2}{5} \times \frac{15}{4} = \frac{3}{2}).
D is incorrect and results from multiplying the improper fractions instead of dividing (52×154=758\frac{5}{2} \times \frac{15}{4} = \frac{75}{8}).