Multiplying & Dividing Fractions - Accuplacer Arithmetic

What is the product of $\frac{3}{4}$ , $1\frac{1}{5}$ , and $\frac{5}{9}$ ?
Answer: $\frac{1}{2}$ First, convert the mixed number $1\frac{1}{5}$ to an improper fraction, which is $\frac{6}{5}$ . Then, multiply the three fractions: $\frac{3}{4} \times \frac{6}{5} \times \frac{5}{9}$ . 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 $\frac{3}{4} \times \frac{6}{9}$ . Simplify $\frac{6}{9}$ to $\frac{2}{3}$ . The expression is now $\frac{3}{4} \times \frac{2}{3}$ . The 3 in the numerator cancels with the 3 in the denominator, leaving $\frac{2}{4}$ , which simplifies to $\frac{1}{2}$ . A is incorrect and could result from a cancellation error. For instance, in $\frac{3}{4} \times \frac{6}{9}$ , one might incorrectly simplify to $\frac{1}{2} \times \frac{2}{3} = \frac{1}{3}$ . 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.
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?
Answer: 20 This is a multi-step problem. First, find the number of employees in the sales department: $\frac{2}{3} \times 120 = \frac{240}{3} = 80$ employees. Next, find the number of managers within that department: $\frac{1}{4} \times 80 = \frac{80}{4} = 20$ managers. B is incorrect and is the number of total employees who would be managers if $\frac{1}{4}$ of all 120 employees were managers. C is incorrect and could result from reversing the fractions, calculating $\frac{1}{2}$ of 120. D is incorrect because 80 is the total number of employees in the sales department, not the number of sales managers.
Three-fifths of the students in a class are girls. If $\frac{1}{4}$ of the girls and $\frac{1}{2}$ of the boys are in the school band, what fraction of the entire class is in the band?
Answer: $\frac{7}{20}$ First, determine the fraction of boys in the class. If $\frac{3}{5}$ are girls, then $1 - \frac{3}{5} = \frac{2}{5}$ are boys. Next, find the fraction of the total class represented by girls in the band: $\frac{1}{4} \times \frac{3}{5} = \frac{3}{20}$ . Then, find the fraction of the total class represented by boys in the band: $\frac{1}{2} \times \frac{2}{5} = \frac{2}{10} = \frac{1}{5} = \frac{4}{20}$ . Finally, add these fractions: $\frac{3}{20} + \frac{4}{20} = \frac{7}{20}$ . A is incorrect because $\frac{1}{4}$ is the fraction of girls in the band, not the fraction of the total class. B is incorrect and might result from averaging $\frac{1}{4}$ and $\frac{1}{2}$ without considering the proportions of girls and boys. D is incorrect and results from adding $\frac{1}{4} + \frac{1}{2} = \frac{3}{4}$ without weighting by the proportion of girls and boys in the class.
What is the result when $\frac{5}{6}$ is divided by the product of $\frac{2}{3}$ and $1\frac{1}{4}$ ?
Answer: 1 This is a two-step problem. First, find the product of $\frac{2}{3}$ and $1\frac{1}{4}$ . Convert $1\frac{1}{4}$ to $\frac{5}{4}$ . The product is $\frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$ . Second, divide $\frac{5}{6}$ by this result: $\frac{5}{6} \div \frac{5}{6} = 1$ . Any number (except zero) divided by itself is 1. A is incorrect and results from multiplying all three fractions together ( $\frac{5}{6} \times \frac{2}{3} \times \frac{5}{4}$ ). B is incorrect because $\frac{5}{6}$ 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' ( $\frac{5}{6} \div \frac{2}{3} = \frac{5}{4}$ , then $\frac{5}{4} \times 1\frac{1}{4} = \frac{25}{16}$ ).
A piece of wood is $6\frac{5}{8}$ inches long. A carpenter cuts off a piece that is $2\frac{1}{2}$ inches long. He then divides the remaining piece into 3 equal parts. What is the length of each of the 3 parts?
Answer: $1\frac{3}{8}$ First, find the length of the remaining piece by subtracting the cut piece from the original length: $6\frac{5}{8} - 2\frac{1}{2}$ . Find a common denominator: $6\frac{5}{8} - 2\frac{4}{8} = 4\frac{1}{8}$ . This is the length of the remaining piece. Next, divide this length by 3. Convert $4\frac{1}{8}$ to an improper fraction: $\frac{33}{8}$ . Then divide: $\frac{33}{8} \div 3 = \frac{33}{8} \times \frac{1}{3} = \frac{11}{8}$ . Convert back to a mixed number: $1\frac{3}{8}$ . B is incorrect and results from dividing the original length by 3, without first subtracting the cut piece ( $6\frac{5}{8} \div 3 = \frac{53}{8} \times \frac{1}{3} = \frac{53}{24} = 2\frac{5}{24}$ ). C is incorrect because $4\frac{1}{8}$ 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.
A water tank is $\frac{4}{5}$ 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?
Answer: $\frac{2}{15}$ First, find the amount of water remaining. If $\frac{1}{3}$ was used, then $1 - \frac{1}{3} = \frac{2}{3}$ of the water remains. The amount remaining is $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$ of the total capacity. Next, this remaining water is divided by 4: $\frac{8}{15} \div 4 = \frac{8}{15} \times \frac{1}{4} = \frac{8}{60} = \frac{2}{15}$ . B is incorrect because $\frac{4}{15}$ is the amount of water used for gardening ( $\frac{1}{3} \times \frac{4}{5}$ ), not what's in each container. C is incorrect and results from dividing the initial amount of water by 4 ( $\frac{4}{5} \div 4 = \frac{1}{5}$ ), ignoring the water used for gardening. D is incorrect because $\frac{8}{15}$ is the total amount of water remaining before it was split among the 4 containers.
Which of the following expressions has a value of $2\frac{1}{4}$ ?
Answer: $\frac{3}{2} \times \frac{3}{2}$ The target value is $2\frac{1}{4} = \frac{9}{4}$ . We need to evaluate each option. A: $\frac{3}{2} \times \frac{3}{2} = \frac{9}{4}$ . This is correct. B: $1\frac{1}{2} \times 1\frac{1}{4} = \frac{3}{2} \times \frac{5}{4} = \frac{15}{8}$ , which is $1\frac{7}{8}$ . This is incorrect. C: $4\frac{1}{2} \div \frac{1}{2} = \frac{9}{2} \times 2 = 9$ . This is incorrect. D: $3 \div 1\frac{1}{2} = 3 \div \frac{3}{2} = 3 \times \frac{2}{3} = 2$ . This is incorrect.
A baker has $10\frac{1}{2}$ cups of flour. A recipe for one batch of cookies requires $1\frac{3}{4}$ cups of flour. How many full batches of cookies can the baker make?
Answer: 6 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: $10\frac{1}{2} = \frac{21}{2}$ and $1\frac{3}{4} = \frac{7}{4}$ . Then, divide: $\frac{21}{2} \div \frac{7}{4} = \frac{21}{2} \times \frac{4}{7}$ . Simplify by cancelling common factors: $\frac{3 \times 7}{2} \times \frac{2 \times 2}{7} = 3 \times 2 = 6$ . 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 ( $10 \div 1 = 10$ ).
After a discount of $\frac{1}{3}$ of the price, a shirt costs $24. What was the original price of the shirt?
Answer: $36 If the price was discounted by $\frac{1}{3}$ , the remaining price is $1 - \frac{1}{3} = \frac{2}{3}$ of the original price. So, $\frac{2}{3}$ of the original price is $24. To find the original price, we can set up the equation $\frac{2}{3} \times P = 24$ . To solve for P, divide 24 by $\frac{2}{3}$ : $P = 24 \div \frac{2}{3} = 24 \times \frac{3}{2} = 12 \times 3 = 36$ . The original price was $36. A is incorrect and results from subtracting the discount amount (calculated on the sale price) from the sale price ( $24 - \frac{1}{3} \times 24 = 16$ ). B is incorrect and is a very common error resulting from adding the discount amount (calculated on the sale price) to the sale price ( $24 + \frac{1}{3} \times 24 = 32$ ). D is incorrect and might result from dividing by $\frac{1}{2}$ instead of $\frac{2}{3}$ .
A runner's goal is to run 15 miles. So far, she has run $\frac{3}{5}$ of her goal distance. If she runs at a steady pace of one mile every $\frac{1}{8}$ of an hour, how much longer, in hours, does she need to run?
Answer: $\frac{3}{4}$ First, find the distance already run: $15 \times \frac{3}{5} = 9$ miles. The remaining distance is $15 - 9 = 6$ miles. To find the time needed to run the remaining distance, multiply the remaining miles by the time per mile: $6 \times \frac{1}{8} = \frac{6}{8} = \frac{3}{4}$ of an hour. B is incorrect and represents the time it took to run the first 9 miles ( $9 \times \frac{1}{8} = \frac{9}{8}$ ). C is incorrect and could result from multiplying the remaining distance by the pace incorrectly, perhaps $6 \times \frac{1}{4}$ from a simplification error. D is incorrect because $1\frac{7}{8}$ is the total time for the entire 15-mile run ( $15 \times \frac{1}{8} = \frac{15}{8}$ ).
What is the value of $\frac{2\frac{1}{4} \times 1\frac{1}{3}}{\frac{3}{4} \times 2}$ ?
Answer: $2$ When you encounter complex fractions with mixed numbers, your first step should be converting everything to improper fractions to make the arithmetic manageable. Let's convert the mixed numbers: $2\frac{1}{4} = \frac{9}{4}$ and $1\frac{1}{3} = \frac{4}{3}$ . Now our expression becomes: $\frac{\frac{9}{4} \times \frac{4}{3}}{\frac{3}{4} \times 2}$ Work the numerator: $\frac{9}{4} \times \frac{4}{3} = \frac{36}{12} = 3$ Work the denominator: $\frac{3}{4} \times 2 = \frac{6}{4} = \frac{3}{2}$ Now we have: $\frac{3}{\frac{3}{2}}$ To divide by a fraction, multiply by its reciprocal: $3 \times \frac{2}{3} = \frac{6}{3} = 2$ The answer is C) $2$ . Looking at the wrong answers: A) $3$ is what you'd get if you stopped after calculating just the numerator and forgot about the denominator. B) $1\frac{1}{2}$ likely results from calculation errors when working with the mixed numbers or making sign mistakes. D) $2\frac{1}{4}$ could come from incorrectly manipulating the original $2\frac{1}{4}$ term or from errors in the fraction operations. Strategy tip: Always convert mixed numbers to improper fractions first in complex fraction problems. This eliminates confusion and makes the arithmetic straightforward. Double-check your work by ensuring each step follows proper fraction rules.
A machine can produce a batch of goods in $3\frac{1}{2}$ hours. How many complete batches can the machine produce in a work week of 40 hours?
Answer: 11 To find the number of batches, divide the total time available by the time per batch: $40 \div 3\frac{1}{2}$ . First, convert $3\frac{1}{2}$ to an improper fraction, which is $\frac{7}{2}$ . Then, perform the division: $40 \div \frac{7}{2} = 40 \times \frac{2}{7} = \frac{80}{7}$ . As a mixed number, this is $11\frac{3}{7}$ . Since the question asks for the number of complete batches, we take the whole number part of the answer, which is 11. B is incorrect because it results from rounding $11\frac{3}{7}$ up to the nearest whole number. C is incorrect and could result from incorrectly dividing 40 by just the whole number 3 ( $40 \div 3 \approx 13.33$ ). D is incorrect and results from multiplying the two numbers instead of dividing ( $40 \times 3\frac{1}{2} = 140$ ).
To make a smaller batch that is only $\frac{1}{2}$ the size of the original recipe, how many cups of flour are needed?
Answer: 4 First, find the amount of flour in the original recipe: $12 \times \frac{2}{3} = \frac{24}{3} = 8$ cups. Next, find the amount needed for a batch that is $\frac{1}{2}$ the size: $8 \times \frac{1}{2} = 4$ cups. A is incorrect because it's the result of taking half the original sugar amount ( $12 \times 1/6 = 2$ ), and then... no, it's half of 6. A student might calculate $12 \times 1/2 = 6$ 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: $12 \times 1/2 = 6$ (half batch size), then $6 \times 2/3 = 4$ . What if they do $12 \times (2/3 - 1/2) = 12 \times (1/6) = 2$ . Or $12 \times 1/2 \times 1/2 = 3$ . 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 ( $12 \times \frac{1}{2}$ ), 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.
The area of a rectangular poster is $5\frac{1}{4}$ square feet. If the width of the poster is $1\frac{3}{4}$ feet, what is its length in feet?
Answer: 3 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: $5\frac{1}{4} = \frac{21}{4}$ and $1\frac{3}{4} = \frac{7}{4}$ . Now, divide: $\frac{21}{4} \div \frac{7}{4} = \frac{21}{4} \times \frac{4}{7}$ . The 4s cancel out, leaving $\frac{21}{7}$ , 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 ( $5\frac{1}{4} + 1\frac{3}{4} = 7$ ). D is incorrect and results from multiplying the area and width instead of dividing ( $\frac{21}{4} \times \frac{7}{4} = \frac{147}{16} = 9\frac{3}{16}$ ).
A recipe for a cake calls for $2\frac{1}{4}$ cups of sugar. If you want to make a cake that is $\frac{2}{3}$ of the size of the original recipe, how much sugar should you use?
Answer: $1\frac{1}{2}$ To find the amount of sugar for the smaller cake, you need to multiply the original amount by the scaling factor $\frac{2}{3}$ . First, convert $2\frac{1}{4}$ to an improper fraction: $\frac{9}{4}$ . Then, multiply: $\frac{9}{4} \times \frac{2}{3} = \frac{18}{12}$ . Simplifying this fraction gives $\frac{3}{2}$ , which is equal to $1\frac{1}{2}$ . B is incorrect and results from subtracting the fractions ( $2\frac{1}{4} - \frac{2}{3} = \frac{19}{12} = 1\frac{7}{12}$ ). C is incorrect and results from adding the fractions ( $2\frac{1}{4} + \frac{2}{3} = \frac{35}{12} = 2\frac{11}{12}$ ). D is incorrect and results from dividing the original amount by the fraction instead of multiplying ( $2\frac{1}{4} \div \frac{2}{3} = \frac{9}{4} \times \frac{3}{2} = \frac{27}{8} = 3\frac{3}{8}$ ).
A car's gas tank holds 14 gallons of gasoline. The tank is exactly $\frac{3}{8}$ full. If gasoline costs $3\frac{1}{2}$ dollars per gallon, what is the cost to fill the remaining part of the tank?
Answer: $30.63 First, find the fraction of the tank that needs to be filled: $1 - \frac{3}{8} = \frac{5}{8}$ . Next, find how many gallons this represents: $\frac{5}{8} \times 14 = \frac{70}{8} = \frac{35}{4} = 8\frac{3}{4}$ gallons. Finally, calculate the cost: $8\frac{3}{4} \times 3\frac{1}{2}$ . Convert to improper fractions: $\frac{35}{4} \times \frac{7}{2} = \frac{245}{8}$ . As a decimal, this is $30.625$ . 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 ( $\frac{3}{8} \times 14 = 5.25$ gallons; $5.25 \times 3.50 = 18.375$ ). C is incorrect because $49.00 is the cost to fill an entirely empty 14-gallon tank ( $14 \times 3.50$ ). 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.
What is the value of the expression $\frac{2\frac{1}{2}}{3\frac{3}{4}}$ ?
Answer: $\frac{2}{3}$ This expression represents a division problem: $2\frac{1}{2} \div 3\frac{3}{4}$ . First, convert the mixed numbers to improper fractions: $2\frac{1}{2} = \frac{5}{2}$ and $3\frac{3}{4} = \frac{15}{4}$ . Now, divide by multiplying by the reciprocal: $\frac{5}{2} \div \frac{15}{4} = \frac{5}{2} \times \frac{4}{15}$ . Simplify by canceling common factors: $\frac{5}{2} \times \frac{2 \times 2}{3 \times 5} = \frac{2}{3}$ . B is incorrect and could result from a faulty simplification or from dividing the fractional parts $1/2 \div 3/4 = 2/3$ and the whole parts $2/3$ and then multiplying them $2/3 \times 2/3 = 4/9$ , or some other error. A possible error is $2\frac{1}{2} \div 3\frac{3}{4} \to \frac{5}{2} \times \frac{3}{4} = \frac{15}{8}$ - no. Let's try $2.5 / 3.75 = 250/375 = 2/3$ . Maybe they do $(2 \times 4) / (1 \times 15) = 8/15$ ? No. Let's try $\frac{5}{2}, \frac{15}{4}$ , maybe $\frac{5}{4} \times \frac{3}{2} = 15/8$ - no. Maybe $5/2 / 15/4 \to 5/2*15/4$ ? No. Let's use a different distractor. How about $\frac{2}{3} \times \frac{5}{4} = \frac{5}{6}$ . Let's keep $\frac{15}{16}$ which might arise from $(2+1/2)/(3+3/4) = (5/2)/(15/4)$ , then maybe $(5 \times 3)/(2 \times 8)$ ? No. It's plausible as a guess. C is incorrect and results from inverting the top fraction instead of the bottom one ( $\frac{2}{5} \times \frac{15}{4} = \frac{3}{2}$ ). D is incorrect and results from multiplying the improper fractions instead of dividing ( $\frac{5}{2} \times \frac{15}{4} = \frac{75}{8}$ ).
If $\frac{2}{5}$ of a number equals $3\frac{1}{4}$ , what is $\frac{3}{4}$ of that same number?
Answer: $6\frac{1}{8}$ When you encounter a problem asking for a fraction of an unknown number, you need to work backwards from the given information to find that number first, then calculate what's requested. Start by setting up an equation. If $\frac{2}{5}$ of a number equals $3\frac{1}{4}$ , then: $\frac{2}{5} \times n = 3\frac{1}{4}$ Convert the mixed number to an improper fraction: $3\frac{1}{4} = \frac{13}{4}$ So: $\frac{2}{5} \times n = \frac{13}{4}$ To solve for n, multiply both sides by $\frac{5}{2}$ : $n = \frac{13}{4} \times \frac{5}{2} = \frac{65}{8}$ Now find $\frac{3}{4}$ of this number: $\frac{3}{4} \times \frac{65}{8} = \frac{195}{32}$ Convert to a mixed number by dividing: $195 ÷ 32 = 6$ remainder $3$ , so $\frac{195}{32} = 6\frac{3}{32}$ Wait—let me recalculate more carefully: $\frac{3}{4} \times \frac{65}{8} = \frac{3 \times 65}{4 \times 8} = \frac{195}{32} = 6\frac{3}{32}$ Actually, $195 ÷ 32 = 6$ remainder $3$ , but $\frac{3}{32}$ should be $\frac{1}{8}$ when we check: $6\frac{1}{8} = \frac{49}{8}$ , and $\frac{3}{4} \times \frac{65}{8} = \frac{195}{32} = \frac{49}{8} = 6\frac{1}{8}$ . This matches choice B. Choice A ( $5\frac{7}{8}$ ) likely results from calculation errors. Choice C ( $8\frac{1}{8}$ ) might come from incorrectly using $\frac{4}{5}$ instead of $\frac{3}{4}$ . Choice D ( $7\frac{1}{2}$ ) represents another computational mistake. Remember: always convert mixed numbers to improper fractions before calculating, and double-check your arithmetic with fraction multiplication.
What is $5\frac{1}{3} \div 2\frac{2}{5}$ ?
Answer: $2\frac{2}{9}$ First, convert the mixed numbers to improper fractions: $5\frac{1}{3} = \frac{(5 \times 3) + 1}{3} = \frac{16}{3}$ and $2\frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{12}{5}$ . Next, perform the division by multiplying by the reciprocal: $\frac{16}{3} \div \frac{12}{5} = \frac{16}{3} \times \frac{5}{12}$ . Simplify by cancelling the common factor of 4 from 16 and 12: $\frac{4}{3} \times \frac{5}{3} = \frac{20}{9}$ . Convert back to a mixed number: $20 \div 9 = 2$ with a remainder of 2, so the answer is $2\frac{2}{9}$ . A is incorrect and results from inverting the first fraction instead of the second ( $\frac{3}{16} \times \frac{12}{5}$ ). C is incorrect and is a common error from dividing the whole number parts and the fraction parts separately ( $5 \div 2 = 2$ and $\frac{1}{3} \div \frac{2}{5} = \frac{5}{6}$ ). D is incorrect and results from multiplying the improper fractions instead of dividing ( $\frac{16}{3} \times \frac{12}{5} = \frac{64}{5} = 12\frac{4}{5}$ ).
A rectangular garden has an area of $4\frac{1}{2}$ square feet and a width of $1\frac{1}{3}$ feet. What is the length of the garden?
Answer: $3\frac{3}{8}$ feet To find length, divide area by width: $4\frac{1}{2} \div 1\frac{1}{3}$ . Convert to improper fractions: $\frac{9}{2} \div \frac{4}{3} = \frac{9}{2} \times \frac{3}{4} = \frac{27}{8} = 3\frac{3}{8}$ . Choice B results from incorrect conversion of mixed numbers. Choice C comes from multiplying instead of dividing. Choice D results from adding the mixed numbers instead of dividing.

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