ISEE Middle Level Mathematics Achievement Quiz: Fraction Multiplication Division

What this quiz covers

This quiz focuses on Fraction Multiplication Division, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Mathematics Achievement.

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.

Question 1

1. 3 feet
2. 12 feet (correct answer)
3. 15 feet
4. 27 feet

Explanation: First, calculate the length Liam painted: (72 \times \frac{3}{8} = 9 \times 3 = 27) feet. Next, find the length of the remaining portion: (72 - 27 = 45) feet. Then, calculate the length Chloe painted, which is (\frac{1}{3}) of the remaining part: (45 \times \frac{1}{3} = 15) feet. Finally, find the difference between the lengths they painted: (27 - 15 = 12) feet. Liam painted 12 more feet than Chloe.

Question 2

1. (3\frac{5}{6}) square feet
2. (5\frac{1}{4}) square feet (correct answer)
3. (7\frac{7}{8}) square feet
4. (11\frac{13}{16}) square feet

Explanation: First, calculate the total area of the painting by multiplying its length and width. Convert mixed numbers to improper fractions: (3\frac{1}{2} = \frac{7}{2}) and (2\frac{1}{4} = \frac{9}{4}). Area = (\frac{7}{2} \times \frac{9}{4} = \frac{63}{8}) square feet. Next, find (\frac{2}{3}) of this area: (\frac{63}{8} \times \frac{2}{3} = \frac{126}{24}). Simplify by dividing the numerator and denominator by their greatest common divisor, 6: (\frac{21}{4}). Convert this to a mixed number: (5\frac{1}{4}) square feet.

Question 3

1. 27 gallons
2. 30 gallons
3. 75 gallons (correct answer)
4. 90 gallons

Explanation: Let C be the total capacity. The problem states that (\frac{3}{5}) of the total capacity is 45 gallons, so (\frac{3}{5}C = 45). To find C, divide 45 by (\frac{3}{5}): (C = 45 \div \frac{3}{5} = 45 \times \frac{5}{3} = \frac{225}{3} = 75). The total capacity is 75 gallons.

Question 4

1. $176
2. $192 (correct answer)
3. $224
4. $240

Explanation: First, calculate the sale price. A discount of (\frac{1}{4}) means the price is (1 - \frac{1}{4} = \frac{3}{4}) of the original. Sale price = ($320 \times \frac{3}{4} =$240). The coupon gives an additional (\frac{1}{5}) off this sale price, so the customer pays (1 - \frac{1}{5} = \frac{4}{5}) of the sale price. Final price = ($240 \times \frac{4}{5} =$192).

Question 5

1. 5 km (correct answer)
2. (4\frac{4}{5}) km
3. (7\frac{1}{2}) km
4. (31\frac{1}{4}) km

Explanation: To find the distance traveled, multiply the total length of the trail by the fraction completed. First, convert the mixed number to an improper fraction: (12\frac{1}{2} = \frac{25}{2}). Now multiply: (\frac{25}{2} \times \frac{2}{5}). You can simplify by canceling the 2s and dividing 25 by 5: (\frac{25}{2} \times \frac{2}{5} = \frac{5}{1} \times \frac{1}{1} = 5). The hiker has traveled 5 kilometers.

Question 6

1. 10
2. 17
3. 18 (correct answer)
4. 20

Explanation: To find the number of batches, divide the total amount of sugar by the amount needed per batch. First, convert the mixed number to an improper fraction: (13\frac{1}{2} = \frac{27}{2}). Now, divide: (\frac{27}{2} \div \frac{3}{4} = \frac{27}{2} \times \frac{4}{3}). Simplify by cross-cancellation: (27 \div 3 = 9) and (4 \div 2 = 2). The calculation becomes (\frac{9}{1} \times \frac{2}{1} = 18). The baker can make exactly 18 full batches.

Question 7

1. (\frac{7}{8}) laps
2. 12 laps
3. 16 laps
4. 14 laps (correct answer)

Explanation: This is a division problem involving mixed numbers and fractions. When you need to find how many times one quantity fits into another, you divide the total by the individual amount. To find the number of laps, you need to divide the total distance by the length of one lap: $3\frac{1}{2} \div \frac{1}{4}$ First, convert the mixed number to an improper fraction: $3\frac{1}{2} = \frac{7}{2}$ Now divide: $\frac{7}{2} \div \frac{1}{4}$ Remember that dividing by a fraction is the same as multiplying by its reciprocal: $\frac{7}{2} \times \frac{4}{1} = \frac{28}{2} = 14$ The runner completed 14 laps, making D correct. Looking at the wrong answers: A) $\frac{7}{8}$ laps results from incorrectly multiplying $3\frac{1}{2} \times \frac{1}{4}$ instead of dividing—this gives you a fraction of a lap, which doesn't make sense when someone ran multiple miles. B) 12 laps comes from converting $3\frac{1}{2}$ incorrectly to $\frac{6}{2}$ and then calculating $\frac{6}{2} \times 4 = 12$. C) 16 laps results from converting $3\frac{1}{2}$ to 4 (rounding up) and then multiplying by 4. When solving "how many times" problems, always ask yourself whether your answer makes logical sense. If someone ran $3\frac{1}{2}$ miles on a $\frac{1}{4}$-mile track, they must have completed more than 3 laps but fewer than 20, which helps you verify that 14 is reasonable.

Question 8

1. 36
2. 45
3. 60 (correct answer)
4. 72

Explanation: First, determine the fraction of the entire band that plays the trumpet. This is (\frac{3}{5} \times \frac{1}{3} = \frac{3}{15} = \frac{1}{5}). So, (\frac{1}{5}) of the total band members are trumpet players. If T is the total number of members, then (\frac{1}{5}T = 12). To find T, multiply 12 by 5: (12 \times 5 = 60). There are 60 members in the band.

Question 9

1. 86
2. 88 (correct answer)
3. 88.5
4. 90

Explanation: First, find the average of the three test scores: ((80 + 90 + 100) \div 3 = 270 \div 3 = 90). This average counts for (\frac{2}{3}) of the final grade, so its contribution is (90 \times \frac{2}{3} = 60) points. The final exam score is 84, and it counts for (\frac{1}{3}) of the grade, so its contribution is (84 \times \frac{1}{3} = 28) points. The final grade is the sum of these two parts: (60 + 28 = 88).

Question 10

1. 9 ounces
2. 16 ounces
3. 24 ounces
4. 36 ounces (correct answer)

Explanation: This is a two-step problem working backwards. Let L be the weight of the leftover pizza. David ate (\frac{2}{3}) of L, which weighed 6 ounces. So, (\frac{2}{3}L = 6). To find L, calculate (6 \div \frac{2}{3} = 6 \times \frac{3}{2} = 9) ounces. So, the leftover pizza weighed 9 ounces. This leftover amount was (\frac{1}{4}) of the original pizza's weight, P. So, (\frac{1}{4}P = 9). To find P, calculate (9 \div \frac{1}{4} = 9 \times 4 = 36) ounces.

Question 11

1. (\frac{1}{4}) yard (correct answer)
2. (\frac{1}{3}) yard
3. (\frac{1}{2}) yard
4. (\frac{3}{4}) yard

Explanation: First, find how many pieces can be cut by dividing the total length by the length of one piece: (16 \div \frac{3}{4} = 16 \times \frac{4}{3} = \frac{64}{3} = 21\frac{1}{3}). This means 21 full pieces can be cut. The fractional part, (\frac{1}{3}), represents (\frac{1}{3}) of a piece, not (\frac{1}{3}) of a yard. To find the leftover length, calculate the total length used for the 21 pieces: (21 \times \frac{3}{4} = \frac{63}{4} = 15\frac{3}{4}) yards. Subtract the used length from the original length: (16 - 15\frac{3}{4} = \frac{1}{4}) yard.

Question 12

1. 1
2. 13
3. 16 (correct answer)
4. 18

Explanation: First, convert the total length of the wood into inches. Since 1 foot = 12 inches, 6 feet is (6 \times 12 = 72) inches. Next, convert the length of the smaller pieces to an improper fraction: (4\frac{1}{2} = \frac{9}{2}) inches. To find the number of pieces, divide the total length by the length of one piece: (72 \div \frac{9}{2} = 72 \times \frac{2}{9}). Simplify: (\frac{72}{9} \times 2 = 8 \times 2 = 16). The carpenter can cut 16 full pieces.

Question 13

1. (\frac{1}{15})
2. (\frac{2}{15}) (correct answer)
3. (4\frac{4}{5})
4. (7\frac{1}{2})

Explanation: To find the amount of juice in each glass, divide the total amount of juice by the number of glasses. This is calculated as (\frac{4}{5} \div 6). To divide by a whole number, multiply by its reciprocal: (\frac{4}{5} \times \frac{1}{6} = \frac{4}{30}). This fraction can be simplified by dividing both the numerator and the denominator by 2, which gives (\frac{2}{15}). Each glass contains (\frac{2}{15}) of a liter.

Question 14

1. (1\frac{1}{2}) cups
2. (1\frac{7}{8}) cups (correct answer)
3. (3\frac{1}{4}) cups
4. (\frac{3}{10}) cup

Explanation: To find the total amount of flour needed, multiply the amount per dozen by the number of dozens. First, convert the mixed number to an improper fraction: (2\frac{1}{2} = \frac{5}{2}). Then, multiply: (\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}). Convert the result back to a mixed number: (\frac{15}{8} = 1\frac{7}{8}) cups.

Question 15

1. (39\frac{3}{8}) miles per hour
2. (69\frac{1}{3}) miles per hour
3. 70 miles per hour (correct answer)
4. 75 miles per hour

Explanation: Average speed is calculated by dividing the total distance by the total time. First, convert the distance to an improper fraction: (52\frac{1}{2} = \frac{105}{2}) miles. Now, divide the distance by the time: (\frac{105}{2} \div \frac{3}{4} = \frac{105}{2} \times \frac{4}{3}). Simplify before multiplying: (\frac{105 \div 3}{2 \div 2} \times \frac{4 \div 2}{3 \div 3} = \frac{35}{1} \times \frac{2}{1} = 70). The car's average speed is 70 miles per hour.

Question 16

1. $18,000 (correct answer)
2. $24,000
3. $36,000
4. $48,000

Explanation: This problem requires multiplying the total budget by a series of fractions. The amount for social media is (240,000 \times \frac{1}{5} \times \frac{1}{2} \times \frac{3}{4}). First step: Marketing budget = (240,000 \times \frac{1}{5} = 48,000\). Second step: Online advertising = \(48,000 \times \frac{1}{2} = 24,000\). Third step: Social media = \(24,000 \times \frac{3}{4} = $18,000).

Question 17

1. (\frac{1}{2}) gallon
2. (1\frac{1}{4}) gallons (correct answer)
3. (1\frac{11}{20}) gallons
4. (1\frac{3}{4}) gallons

Explanation: First, find the amount of paint used for the wall. Convert the starting amount to an improper fraction: (2\frac{1}{2} = \frac{5}{2}). Paint for wall = (\frac{5}{2} \times \frac{1}{5} = \frac{1}{2}) gallon. The amount remaining after painting the wall is (\frac{5}{2} - \frac{1}{2} = \frac{4}{2} = 2) gallons. Then, he used (\frac{3}{4}) of a gallon for a door. The final amount left is (2 - \frac{3}{4} = 1\frac{1}{4}) gallons.

Question 18

1. 20 (correct answer)
2. 30
3. 60
4. 80

Explanation: First, find the number of fiction books: (120 \times \frac{1}{3} = 40). The number of remaining books is (120 - 40 = 80). Of these 80 books, (\frac{3}{4}) are non-fiction: (80 \times \frac{3}{4} = 60). The rest of the 80 books are reference books, so subtract the non-fiction books from the remainder: (80 - 60 = 20). There are 20 reference books.

Question 19

1. 18 acres
2. 24 acres
3. 54 acres
4. 36 acres (correct answer)

Explanation: This problem involves working with fractions in multiple steps, which is common on fraction word problems. You need to carefully track what fraction refers to what quantity at each stage. Start by finding how many acres are planted with corn: $\frac{3}{5} \times 90 = \frac{3 \times 90}{5} = \frac{270}{5} = 54$ acres of corn initially. Next, the farmer sells $\frac{1}{3}$ of his cornfield to a neighbor. This means he sells $\frac{1}{3}$ of the 54 acres of corn: $\frac{1}{3} \times 54 = 18$ acres sold. Therefore, the farmer has left: $54 - 18 = 36$ acres of corn remaining. Looking at the wrong answers: Choice A (18 acres) represents the amount of cornfield sold, not what remains. Choice B (24 acres) might result from incorrectly calculating $\frac{2}{3}$ of 36 instead of $\frac{2}{3}$ of 54. Choice C (54 acres) is the original amount of corn planted, ignoring the sale entirely. The key strategy for multi-step fraction problems is to work through them systematically: identify what fraction applies to which quantity, calculate each step completely before moving to the next, and always check that your final answer makes sense in context. Here, 36 acres should be less than the original 54 acres but more than half of it, which it is.

Question 20

1. (\frac{2}{3}) pounds
2. (1\frac{3}{7}) pounds
3. (1\frac{1}{2}) pounds (correct answer)
4. (73\frac{1}{2}) pounds

Explanation: To find the amount for each section, divide the total weight by the number of sections. First, convert the mixed number to an improper fraction: (10\frac{1}{2} = \frac{21}{2}). Then, divide by 7: (\frac{21}{2} \div 7 = \frac{21}{2} \times \frac{1}{7} = \frac{21}{14}). Simplify the fraction by dividing the numerator and denominator by 7: (\frac{3}{2}), which is equal to (1\frac{1}{2}) pounds.