ISEE Lower Level Quantitative Reasoning Quiz: Fraction Word Problems

What this quiz covers

This quiz focuses on Fraction Word Problems, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Lower Level Quantitative Reasoning.

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. (\frac{1}{6}) of the can
2. (\frac{1}{4}) of the can
3. (\frac{1}{3}) of the can (correct answer)
4. (\frac{5}{6}) of the can

Explanation: First, find the fraction of paint remaining after painting the wall: (1 - \frac{1}{3} = \frac{2}{3}). Next, find the amount of paint used for the door, which is (\frac{1}{2}) of the remaining (\frac{2}{3}): (\frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}). The amount of paint left is the amount that remained after painting the wall minus the amount used for the door: (\frac{2}{3} - \frac{1}{3} = \frac{1}{3}). So, (\frac{1}{3}) of the original can is left.

Question 2

1. 27 comic books
2. 30 comic books (correct answer)
3. 36 comic books
4. 45 comic books

Explanation: If Leo sold (\frac{2}{5}) of his comic books, the fraction he has left is (1 - \frac{2}{5} = \frac{3}{5}). The problem states that this remaining fraction is equal to 18 comic books. So, (\frac{3}{5}) of the original total is 18. To find the original total, we can set up the equation (\frac{3}{5} \times \text{Total} = 18). Solving for the total gives (\text{Total} = 18 \div \frac{3}{5} = 18 \times \frac{5}{3} = 6 \times 5 = 30). Leo originally had 30 comic books.

Question 3

1. 18 fruits
2. 24 fruits
3. 30 fruits
4. 36 fruits (correct answer)

Explanation: First, find the total number of non-apple fruits. Since 6 oranges make up (\frac{1}{2}) of the non-apple fruits, the total number of non-apple fruits is (6 \times 2 = 12). Next, determine what fraction of the total fruits are non-apples. If (\frac{2}{3}) are apples, then (1 - \frac{2}{3} = \frac{1}{3}) of the fruits are non-apples. We know that this (\frac{1}{3}) is equal to 12 fruits. To find the total number of fruits, solve (\frac{1}{3} \times \text{Total} = 12). The total is (12 \div \frac{1}{3} = 12 \times 3 = 36).

Question 4

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

Explanation: First, calculate the amount of juice poured out: (\frac{1}{4} \times 2 = \frac{1}{2}) gallon. Next, find the amount of juice remaining: (2 - \frac{1}{2} = 1\frac{1}{2}) gallons. Sara and her two friends make a total of 3 people. They share the remaining juice equally, so divide the remaining amount by 3: (1\frac{1}{2} \div 3 = \frac{3}{2} \div 3 = \frac{3}{2} \times \frac{1}{3} = \frac{3}{6} = \frac{1}{2}) gallon. Each person gets (\frac{1}{2}) gallon of juice.

Question 5

1. (\frac{3}{20}) of the students
2. (\frac{2}{5}) of the students
3. (\frac{9}{20}) of the students (correct answer)
4. (\frac{3}{4}) of the students

Explanation: First, determine the fraction of girls who do not have brown hair. If (\frac{1}{4}) of the girls have brown hair, then (1 - \frac{1}{4} = \frac{3}{4}) of the girls do not. Next, find what fraction this group represents of the entire school. Since girls make up (\frac{3}{5}) of the school, multiply this by the fraction of girls without brown hair: (\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}). Thus, (\frac{9}{20}) of the students are girls without brown hair.

Question 6

1. 20 pieces
2. 21 pieces
3. 22 pieces (correct answer)
4. 23 pieces

Explanation: To find the number of pieces, divide the total length of the ribbon by the length of each piece. First, convert the mixed number to an improper fraction: (5\frac{1}{2} = \frac{11}{2}). Now, divide (\frac{11}{2}) by (\frac{1}{4}): (\frac{11}{2} \div \frac{1}{4} = \frac{11}{2} \times \frac{4}{1} = \frac{44}{2} = 22). Therefore, 22 full pieces can be cut.

Question 7

1. 4 shelves
2. 5 shelves (correct answer)
3. 6 shelves
4. 27 shelves

Explanation: To find the number of shelves, divide the total length of the board by the length of one shelf. First, convert the mixed number to an improper fraction: (2\frac{1}{4} = \frac{9}{4}). Now, divide 12 by (\frac{9}{4}): (12 \div \frac{9}{4} = 12 \times \frac{4}{9} = \frac{48}{9}). To find how many full shelves can be cut, convert the improper fraction to a mixed number: (\frac{48}{9} = 5\frac{3}{9} = 5\frac{1}{3}). This means he can cut 5 full shelves and will have a piece of wood left over.

Question 8

1. 2 (\frac{1}{4}) cups
2. 24 cups
3. 18 cups
4. 16 cups (correct answer)

Explanation: When you see a question asking "how many full cups can be filled," you're dealing with a division problem where you need to find how many smaller units fit into a larger unit. To solve this, divide the total amount of water by the capacity of each cup: $6 \div \frac{3}{8}$. When dividing by a fraction, multiply by its reciprocal instead: $6 \times \frac{8}{3}$. This gives you $\frac{6 \times 8}{3} = \frac{48}{3} = 16$ full cups. Let's examine why the other answers are incorrect. Choice A (2¼ cups) likely comes from incorrectly multiplying $6 \times \frac{3}{8} = \frac{18}{8} = 2\frac{1}{4}$, but this tells you how much water would be in 6 cups, not how many cups you can fill. Choice B (24 cups) might result from confusing the fraction and thinking each cup holds $\frac{1}{4}$ liter, leading to $6 \div \frac{1}{4} = 24$. Choice C (18 cups) could come from multiplying 6 by the numerator 3, ignoring the denominator entirely. Remember: when a problem asks "how many of X fit into Y," you're dividing Y by X. The key trap here is remembering to flip and multiply when dividing by fractions, and being careful not to multiply when you should divide. Always check if your answer makes sense—16 cups holding $\frac{3}{8}$ liter each should give you close to 6 liters total.

Question 9

1. 10 adults
2. 15 adults
3. 30 adults (correct answer)
4. 40 adults

Explanation: First, find the initial number of adults: (\frac{2}{3} \times 60 = 40) adults. Next, find the number of adults who left: (\frac{1}{4} \times 40 = 10) adults. Finally, subtract the number of adults who left from the initial number of adults to find how many remain: (40 - 10 = 30) adults.

Question 10

1. 4 lots
2. 20 lots
3. 25 lots (correct answer)
4. 50 lots

Explanation: To find the number of smaller lots, divide the total area of the land by the area of each lot. The calculation is (10 \div \frac{2}{5}). To divide by a fraction, multiply by its reciprocal: (10 \times \frac{5}{2} = \frac{50}{2} = 25). He can create 25 lots.

Question 11

1. $20
2. $24 (correct answer)
3. $32
4. $36

Explanation: First, calculate the cost of the video game: (\frac{1}{3} \times $48 =$16). Next, find the amount of money remaining after buying the game: ($48 -$16 = 32\). Then, calculate the cost of the snack, which is \(\frac{1}{4}\) of the remaining money: \(\frac{1}{4} \times 32 = 8\). Finally, subtract the cost of the snack from the amount that was remaining to find the final amount: \(32 - $8 =$24).

Question 12

1. 1(\frac{1}{12}) cups (correct answer)
2. 1 cup
3. (\frac{5}{7}) cup
4. 1(\frac{5}{12}) cups

Explanation: When you encounter fraction word problems involving recipes, focus on identifying what portions of each recipe you're making, then carefully add the amounts together. First, determine how many carrots you need for each recipe. The soup recipe calls for $\frac{2}{3}$ cup, but you're only making half the recipe, so you need $\frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3}$ cup. The stew recipe requires $\frac{3}{4}$ cup, and you're making the full recipe, so you need $\frac{3}{4}$ cup. Now add these amounts: $\frac{1}{3} + \frac{3}{4}$. To add fractions, find a common denominator. The least common multiple of 3 and 4 is 12. Convert both fractions: $\frac{1}{3} = \frac{4}{12}$ and $\frac{3}{4} = \frac{9}{12}$. Adding gives you $\frac{4}{12} + \frac{9}{12} = \frac{13}{12} = 1\frac{1}{12}$ cups. Looking at the wrong answers: Choice B (1 cup) results from incorrectly adding $\frac{1}{3} + \frac{3}{4}$ without finding a proper common denominator. Choice C ($\frac{5}{7}$ cup) comes from adding the numerators and denominators incorrectly ($\frac{1+3}{3+4}$). Choice D ($1\frac{5}{12}$ cups) happens if you forget to take half of the soup recipe and use the full $\frac{2}{3}$ cup instead. Remember to read recipe problems carefully—always check whether you're making full or partial recipes before calculating ingredient amounts. Convert to common denominators methodically to avoid arithmetic errors.

Question 13

1. (\frac{1}{12}) of a ton
2. 1(\frac{5}{12}) tons
3. 1(\frac{1}{8}) tons
4. (\frac{1}{2}) of a ton (correct answer)

Explanation: When you see a problem involving fractions of quantities, you need to multiply fractions to find "a fraction of" something. Here, you're looking for what $\frac{2}{3}$ of $\frac{3}{4}$ ton equals. To find $\frac{2}{3}$ of $\frac{3}{4}$ ton, multiply the fractions: $\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$. The smaller truck's load weighs $\frac{1}{2}$ ton. Answer choice A ($\frac{1}{12}$ ton) represents a common error where students subtract the fractions instead of multiplying: $\frac{3}{4} - \frac{2}{3} = \frac{9}{12} - \frac{8}{12} = \frac{1}{12}$. This makes no sense logically since the smaller truck should carry a substantial portion of the larger truck's load. Answer choice B ($1\frac{5}{12}$ tons) results from adding the fractions: $\frac{3}{4} + \frac{2}{3} = \frac{9}{12} + \frac{8}{12} = \frac{17}{12} = 1\frac{5}{12}$. This also doesn't match the problem's requirement. Answer choice C ($1\frac{1}{8}$ tons) might come from incorrectly calculating $\frac{2}{3} \times \frac{3}{4}$ and getting a mixed number, but this exceeds the original load weight, which is impossible. Remember: when you see "is $\frac{a}{b}$ as heavy as" or similar language, you're multiplying fractions. The key phrase "as heavy as" signals multiplication, not addition or subtraction.

Question 14

1. 60 gallons
2. 160 gallons
3. 90 gallons
4. 80 gallons (correct answer)

Explanation: When you encounter fraction word problems involving capacity, your goal is to find the total capacity first, then calculate what any other fraction would contain. Since the tank is $\frac{3}{4}$ full and contains 120 gallons, you can set up the equation: $\frac{3}{4} \times \text{total capacity} = 120$. To find the total capacity, divide 120 by $\frac{3}{4}$, which is the same as multiplying by $\frac{4}{3}$: $120 \times \frac{4}{3} = 160$ gallons total capacity. Now that you know the tank holds 160 gallons when completely full, you can find how much it contains when $\frac{1}{2}$ full: $\frac{1}{2} \times 160 = 80$ gallons. Looking at the wrong answers: Choice A (60 gallons) represents a common error where students incorrectly think that half of $\frac{3}{4}$ full would be half of 120 gallons. Choice B (160 gallons) is the total tank capacity, not the amount when half full. Choice C (90 gallons) might result from incorrectly calculating $\frac{3}{4}$ of 120 gallons, then adding or subtracting incorrectly. The correct answer is D (80 gallons). Strategy tip: In fraction capacity problems, always work backwards to find the total capacity first. Once you have that key piece of information, you can calculate any other fractional amount. Don't try to work directly from one fraction to another—you'll likely make calculation errors.

Question 15

1. 2 (\frac{1}{2}) pounds
2. 4 (\frac{1}{6}) pounds
3. 5 pounds (correct answer)
4. 7 (\frac{1}{2}) pounds

Explanation: First, calculate the flour used for bread: (\frac{1}{4} \times 10 = \frac{10}{4} = 2.5) pounds. Then, find the amount of flour remaining: (10 - 2.5 = 7.5) pounds. Next, calculate the flour used for muffins, which is (\frac{1}{3}) of the remaining amount: (\frac{1}{3} \times 7.5 = 2.5) pounds. Finally, subtract the flour used for muffins from the amount that was remaining: (7.5 - 2.5 = 5) pounds. There are 5 pounds of flour left.

Question 16

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

Explanation: First, determine what fraction of the recipe Sarah is making. A dozen is 12 cookies. Sarah is making 16 cookies, so she is making (\frac{16}{12}) of the recipe. This fraction simplifies to (\frac{4}{3}). Next, multiply the amount of flour by this fraction: (\frac{3}{4} \times \frac{4}{3} = \frac{12}{12} = 1). Sarah will need 1 cup of flour.

Question 17

1. 600 people (correct answer)
2. 500 people
3. 480 people
4. 720 people

Explanation: This problem tests your ability to work with fractions and proportional relationships by building up from given information to find a total. Start with what you know: there are 120 men at the concert. Since the number of women is three times the number of men, there are $3 \times 120 = 360$ women. This means there are $120 + 360 = 480$ adults total. Now use the fraction information. If $\frac{1}{5}$ of attendees are children, then $\frac{4}{5}$ must be adults. Since you found there are 480 adults, you can set up the equation: $\frac{4}{5} \times \text{total} = 480$. Solving for the total: $\text{total} = 480 \times \frac{5}{4} = 600$ people. Looking at the wrong answers: Choice B (500) likely comes from incorrectly assuming adults make up $\frac{4}{5}$ of 500, but $\frac{4}{5} \times 500 = 400 \neq 480$. Choice C (480) is the trap of giving just the number of adults instead of the total attendees. Choice D (720) might result from incorrectly thinking children are $\frac{1}{4}$ of the total instead of $\frac{1}{5}$, or from calculation errors with the ratios. The correct answer is A (600 people). Strategy tip: In multi-step fraction problems, always work backwards from concrete numbers to check your answer. Here, verify: $\frac{1}{5} \times 600 = 120$ children, and $600 - 120 = 480$ adults, which matches what you calculated.

Question 18

1. $18.00 (correct answer)
2. $24.00
3. $30.00
4. $36.00

Explanation: Work backwards from the final amount. The 6.00 she has left is the amount after spending \(\frac{1}{2}\) of her money on a snack, which means 6.00 is the other (\frac{1}{2}). So, before buying the snack, she had ($6.00 \times 2 =$12.00). This 12.00 was the amount remaining after she spent \(\frac{1}{3}\) of her original money on a book. This means the 12.00 represents (1 - \frac{1}{3} = \frac{2}{3}) of her starting money. To find the starting amount, solve the equation (\frac{2}{3} \times \text{Start} = 12.00\). So, \(\text{Start} = 12.00 \div \frac{2}{3} = $12.00 \times \frac{3}{2} =$18.00).

Question 19

1. 4 square meters (correct answer)
2. 8 square meters
3. 12 square meters
4. 16 square meters

Explanation: First, calculate the total area of the garden by multiplying its length and width. Convert the mixed numbers to improper fractions: (4\frac{1}{2} = \frac{9}{2}) and (2\frac{2}{3} = \frac{8}{3}). The area is (\frac{9}{2} \times \frac{8}{3} = \frac{72}{6} = 12) square meters. Next, find the area planted with tomatoes, which is (\frac{1}{3}) of the total area: (\frac{1}{3} \times 12 = 4). The area with tomatoes is 4 square meters.