Fraction Word Problems - ISEE Lower Level Quantitative Reasoning

A carpenter has a board that is 12 feet long. He needs to cut it into shelves that are each $2\frac{1}{4}$ feet long. How many full shelves can he cut from the board?
Answer: 5 shelves 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.
A baker starts with a 10-pound bag of flour. He uses $\frac{1}{4}$ of the bag for bread. He then uses $\frac{1}{3}$ of the remaining flour for muffins. How many pounds of flour are left in the bag?
Answer: 5 pounds 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.
In a bag of fruit, $\frac{2}{3}$ of the fruits are apples. The rest are oranges and pears. There are 6 oranges, which is $\frac{1}{2}$ of the number of non-apple fruits. How many total fruits are in the bag?
Answer: 36 fruits 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 $12 \div \frac{1}{3} = 12 \times 3 = 36$ .
At a concert, $\frac{1}{5}$ of the attendees are children. Of the adults, the number of women is three times the number of men. If there are 120 men at the concert, how many people are at the concert in total?
Answer: 600 people 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.
A library had 60 visitors one morning. Of these visitors, $\frac{2}{3}$ were adults. Later, $\frac{1}{4}$ of the adults who were there in the morning left. How many adults were still in the library?
Answer: 30 adults 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.
Mr. Chen owns a 10-acre plot of land. He divides the entire plot into smaller lots that each measure $\frac{2}{5}$ of an acre. How many smaller lots can he create?
Answer: 25 lots 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.
A large truck is carrying a load of bricks that weighs $\frac{3}{4}$ of a ton. A smaller truck is carrying a load that is $\frac{2}{3}$ as heavy as the large truck's load. What is the weight, in tons, of the smaller truck's load?
Answer: $\frac{1}{2}$ of a ton 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.
At a certain school, $\frac{3}{5}$ of the students are girls. Of the girls, $\frac{1}{4}$ have brown hair. What fraction of the students at the school are girls who do NOT have brown hair?
Answer: $\frac{9}{20}$ of the students 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.
Maria spent $\frac{1}{3}$ of her money on a book and $\frac{1}{2}$ of her remaining money on a snack. If she has $6.00 left, how much money did she start with?
Answer: $18.00 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, $\text{Start} = \$12.00 \div \frac{2}{3} = \$12.00 \times \frac{3}{2} = \$18.00$ .
A spool contains a ribbon that is $5\frac{1}{2}$ yards long. If the ribbon is cut into shorter pieces that are each $\frac{1}{4}$ of a yard long, how many full pieces can be cut?
Answer: 22 pieces 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.
A rectangular garden is $4\frac{1}{2}$ meters long and $2\frac{2}{3}$ meters wide. If $\frac{1}{3}$ of the garden's area is planted with tomatoes, what is the area of the section with tomatoes?
Answer: 4 square meters 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.
A large water jug holds 6 liters. A small cup holds $\frac{3}{8}$ of a liter. How many full cups can be filled from the jug?
Answer: 16 cups 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.
After selling $\frac{2}{5}$ of his comic books, Leo has 18 comic books left. How many comic books did he have originally?
Answer: 30 comic books 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.
Jamal had $48. He spent $\frac{1}{3}$ of his money on a video game. He then spent $\frac{1}{4}$ of the money that was left on a snack. How much money does Jamal have now?
Answer: $24 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: $\$32 - \$8 = \$24$ .
A recipe for soup requires $\frac{2}{3}$ cup of carrots and a recipe for stew requires $\frac{3}{4}$ cup of carrots. If you make half of the soup recipe and a full stew recipe, how many total cups of carrots do you need?
Answer: 1 $\frac{1}{12}$ cups 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.
A fuel tank is $\frac{3}{4}$ full and contains 120 gallons of fuel. What is the volume of fuel in the tank when it is $\frac{1}{2}$ full?
Answer: 80 gallons 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.
A painter used $\frac{1}{3}$ of a can of paint for a wall. He then used $\frac{1}{2}$ of the remaining paint for a door. What fraction of the original can of paint is left?
Answer: $\frac{1}{3}$ of the can 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.
A container holds 2 gallons of juice. Sara pours out $\frac{1}{4}$ of the juice. Then she and her two friends share the remaining juice equally. How many gallons of juice does each person get?
Answer: $\frac{1}{2}$ gallon 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.
A recipe that makes 1 dozen cookies requires $\frac{3}{4}$ cup of flour. If Sarah wants to make 16 cookies, how much flour will she need?
Answer: 1 cup 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.

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