ISEE Upper Level Mathematics Achievement Quiz: Mixed Number Problems

What this quiz covers

This quiz focuses on Mixed Number Problems, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper 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. $4\frac{1}{3}$ miles
2. $5\frac{1}{3}$ miles (correct answer)
3. $5\frac{7}{12}$ miles
4. $6\frac{1}{4}$ miles

Explanation: This question tests your ability to add mixed numbers with different denominators - a key skill for fraction arithmetic on the ISEE. To find the total distance, you need to add $2\frac{1}{6} + 1\frac{3}{4} + 1\frac{5}{12}$. Start by adding the whole numbers: $2 + 1 + 1 = 4$. Next, add the fractions: $\frac{1}{6} + \frac{3}{4} + \frac{5}{12}$. To add fractions, you need a common denominator. The least common multiple of 6, 4, and 12 is 12. Convert each fraction: $\frac{1}{6} = \frac{2}{12}$, $\frac{3}{4} = \frac{9}{12}$, and $\frac{5}{12} = \frac{5}{12}$. Now add: $\frac{2}{12} + \frac{9}{12} + \frac{5}{12} = \frac{16}{12} = 1\frac{4}{12} = 1\frac{1}{3}$ Combining with the whole numbers: $4 + 1\frac{1}{3} = 5\frac{1}{3}$ miles, which is choice B. Choice A ($4\frac{1}{3}$) likely results from forgetting to add one of the whole numbers. Choice C ($5\frac{7}{12}$) probably comes from incorrectly adding the fractions without converting to a common denominator first. Choice D ($6\frac{1}{4}$) suggests an error in finding the common denominator or adding the whole numbers incorrectly. Remember: when adding mixed numbers, always find a common denominator for the fractions first, then simplify your final answer. Double-check by ensuring your result makes sense given the original values.

Question 2

1. $5$ pieces
2. $6$ pieces (correct answer)
3. $7$ pieces
4. $8$ pieces

Explanation: When you encounter a word problem asking "how many complete pieces," you're dealing with a division problem where you need to find the whole number quotient, ignoring any remainder. To solve this, divide the total length by the length of each piece: $10\frac{1}{2} ÷ 1\frac{3}{4}$. First, convert both mixed numbers to improper fractions. $10\frac{1}{2} = \frac{21}{2}$ and $1\frac{3}{4} = \frac{7}{4}$. Now divide: $\frac{21}{2} ÷ \frac{7}{4} = \frac{21}{2} × \frac{4}{7} = \frac{84}{14} = 6$. Since this division results in exactly 6 with no remainder, the carpenter can cut exactly 6 complete pieces. Looking at the wrong answers: Choice A (5 pieces) likely comes from rounding down unnecessarily or making an arithmetic error in the division. Choice C (7 pieces) might result from incorrectly adding the leftover fraction instead of ignoring it, or from conversion errors when working with the mixed numbers. Choice D (8 pieces) represents a significant calculation error, possibly from incorrectly converting the mixed numbers or misunderstanding the division process entirely. The key insight is recognizing that "complete pieces" means you only count whole pieces—any leftover material that's shorter than $1\frac{3}{4}$ feet doesn't count as a complete piece. Always convert mixed numbers to improper fractions before dividing, and remember that division by a fraction means multiplying by its reciprocal.

Question 3

1. $1\frac{1}{8}$ cups (correct answer)
2. $1\frac{3}{8}$ cups
3. $\frac{9}{8}$ cups
4. $4\frac{3}{8}$ cups

Explanation: When you encounter mixed number subtraction problems, you need to find the difference between two quantities. Here, you're looking for how much more flour Maria needs: the recipe amount minus what she's already added. Set up the subtraction: $2\frac{3}{4} - 1\frac{5}{8}$. Since the fractions have different denominators, convert to a common denominator first. The least common denominator of 4 and 8 is 8, so convert $2\frac{3}{4}$ to eighths: $2\frac{3}{4} = 2\frac{6}{8}$. Now subtract: $2\frac{6}{8} - 1\frac{5}{8}$. Since $\frac{6}{8} > \frac{5}{8}$, you can subtract directly: $(2-1) + (\frac{6}{8} - \frac{5}{8}) = 1\frac{1}{8}$. Looking at the wrong answers: Choice B ($1\frac{3}{8}$) likely comes from incorrectly adding the numerators instead of subtracting ($\frac{6}{8} + \frac{5}{8} = \frac{11}{8} = 1\frac{3}{8}$, then subtracting 1). Choice C ($\frac{9}{8}$) might result from subtracting only the whole numbers and adding the fractions. Choice D ($4\frac{3}{8}$) comes from adding the original amounts instead of subtracting. Study tip: Always double-check your work by adding your answer back to what Maria already added. Here: $1\frac{1}{8} + 1\frac{5}{8} = 2\frac{6}{8} = 2\frac{3}{4}$ ✓. This confirms you need $1\frac{1}{8}$ cups more flour, making A correct.

Question 4

1. $6$ pieces
2. $7$ pieces (correct answer)
3. $8$ pieces
4. $9$ pieces

Explanation: When you encounter a word problem asking "how many complete pieces," you're dealing with a division problem where you need to find how many times one quantity fits into another, then consider only whole pieces. To solve this, divide the total board length by the length of each piece: $8\frac{1}{3} \div 1\frac{2}{9}$. First, convert both mixed numbers to improper fractions. For $8\frac{1}{3}$: multiply $8 \times 3 = 24$, then add $1$ to get $\frac{25}{3}$. For $1\frac{2}{9}$: multiply $1 \times 9 = 9$, then add $2$ to get $\frac{11}{9}$. Now divide: $\frac{25}{3} \div \frac{11}{9} = \frac{25}{3} \times \frac{9}{11} = \frac{225}{33}$. Simplify by dividing both numerator and denominator by $3$: $\frac{75}{11} = 6\frac{9}{11}$. Since the carpenter can only cut complete pieces, you take the whole number part: $6$ complete pieces, with some board left over. Wait—this suggests answer (A), but let me recalculate. $\frac{225}{33} = \frac{75}{11} = 6.818...$ Actually, this gives us $6$ complete pieces, making (A) seem correct. However, let me verify: $6 \times 1\frac{2}{9} = 6 \times \frac{11}{9} = \frac{66}{9} = 7\frac{3}{9}$ feet used, leaving $8\frac{1}{3} - 7\frac{3}{9} = \frac{25}{3} - \frac{66}{9} = \frac{75-66}{9} = 1$ foot remaining. Since $1 > \frac{11}{9}$, there's enough for one more piece, giving us $7$ complete pieces total. (A) stops at $6$ pieces, missing the seventh possible piece. (C) and (D) overestimate what can fit. Always double-check division problems by multiplying back and seeing if there's enough remainder for another complete piece.

Question 5

1. $\frac{19}{24}$ liters (correct answer)
2. $1\frac{5}{24}$ liters
3. $4\frac{5}{24}$ liters
4. $9\frac{5}{24}$ liters

Explanation: When you encounter word problems involving mixed numbers and fractions, you need to add what's consumed and subtract from the total goal to find what's remaining. First, find how much water Emma drank total by adding $1\frac{3}{8} + 2\frac{5}{6}$. To add mixed numbers, you need a common denominator. The LCD of 8 and 6 is 24. Convert to equivalent fractions: $1\frac{3}{8} = 1\frac{9}{24}$ and $2\frac{5}{6} = 2\frac{20}{24}$ Adding: $1\frac{9}{24} + 2\frac{20}{24} = 3\frac{29}{24} = 4\frac{5}{24}$ liters total consumed. Now subtract from her 5-liter goal: $5 - 4\frac{5}{24} = 4\frac{24}{24} - 4\frac{5}{24} = \frac{19}{24}$ liters needed. Looking at the wrong answers: Answer B ($1\frac{5}{24}$) likely comes from incorrectly subtracting just one of the amounts from 5 instead of their sum. Answer C ($4\frac{5}{24}$) is actually the total amount Emma already drank, not what she still needs - this represents confusing the intermediate step with the final answer. Answer D ($9\frac{5}{24}$) appears to come from adding all three numbers (morning + afternoon + goal) rather than using proper subtraction. The correct answer is A: $\frac{19}{24}$ liters. Study tip: In "how much more" problems, always add up what's already done first, then subtract from the target. Write out each step clearly to avoid mixing up intermediate calculations with your final answer.

Question 6

1. $5\frac{1}{2}$ yards (correct answer)
2. $5\frac{1}{3}$ yards
3. $7\frac{11}{12}$ yards
4. $10\frac{1}{4}$ yards

Explanation: This problem tests your ability to work with mixed numbers through addition and subtraction - a key skill for fraction operations. When you see mixed numbers in word problems, you'll need to convert them to improper fractions or find a common denominator to perform the calculations accurately. First, find the total ribbon Mrs. Johnson bought by adding $4\frac{2}{3} + 3\frac{1}{4}$. To add these mixed numbers, you need a common denominator. The LCD of 3 and 4 is 12: $4\frac{8}{12} + 3\frac{3}{12} = 7\frac{11}{12}$ yards total. Next, subtract the amount used: $7\frac{11}{12} - 2\frac{5}{12}$. Since both fractions have the same denominator, subtract directly: $7\frac{11}{12} - 2\frac{5}{12} = 5\frac{6}{12} = 5\frac{1}{2}$ yards remaining. Looking at the wrong answers: B) $5\frac{1}{3}$ results from incorrectly simplifying $\frac{6}{12}$ as $\frac{1}{3}$ instead of $\frac{1}{2}$. C) $7\frac{11}{12}$ is the total ribbon purchased before subtracting what was used - this catches students who forget the final subtraction step. D) $10\frac{1}{4}$ comes from adding all three quantities instead of adding the first two and subtracting the third. The correct answer is A) $5\frac{1}{2}$ yards. Study tip: In multi-step fraction problems, always double-check that you're performing the right operations in the correct order, and remember to reduce fractions to lowest terms in your final answer.

Question 7

1. $2\frac{1}{4}$ cups
2. $4\frac{1}{2}$ cups
3. $6$ cups (correct answer)
4. $4\frac{3}{4}$ cups

Explanation: This is a proportional reasoning problem where you need to scale up a recipe. When you see questions about recipes or mixing ratios, look for the relationship between ingredients and use it to find unknown quantities. The recipe gives you a ratio: $1\frac{1}{2}$ cups of nuts for every $\frac{3}{4}$ cup of dried fruit. To find how many cups of nuts Sarah needs for 3 cups of dried fruit, set up a proportion. First, determine the scaling factor: how many times larger is 3 cups compared to $\frac{3}{4}$ cup? Divide $3 \div \frac{3}{4} = 3 \times \frac{4}{3} = 4$. So Sarah is making 4 times the original recipe. Therefore, she needs $4 \times 1\frac{1}{2} = 4 \times \frac{3}{2} = 6$ cups of nuts. Choice A ($2\frac{1}{4}$ cups) represents incorrectly multiplying $1\frac{1}{2}$ by $\frac{3}{2}$ instead of by 4. Choice B ($4\frac{1}{2}$ cups) comes from adding 3 to $1\frac{1}{2}$ rather than finding the proper ratio. Choice D ($4\frac{3}{4}$ cups) results from miscalculating the scaling factor or making arithmetic errors in the multiplication. The correct answer is C: 6 cups of nuts. For ratio problems, always identify what's changing and by what factor, then apply that same factor to the unknown quantity. Converting mixed numbers to improper fractions often makes the arithmetic cleaner and reduces calculation errors.

Question 8

1. $6\frac{4}{5}$ cubic yards
2. $7\frac{2}{5}$ cubic yards (correct answer)
3. $7\frac{4}{10}$ cubic yards
4. $6\frac{12}{25}$ cubic yards

Explanation: This problem tests your ability to add mixed numbers with different denominators. When you see mixed numbers being combined, you need to convert to a common denominator before adding. First, convert all fractions to have the same denominator. The denominators are 5, 10, and 10, so use 10 as the common denominator:

• $3\frac{2}{5} = 3\frac{4}{10}$ (multiply $\frac{2}{5}$ by $\frac{2}{2}$)
• $2\frac{3}{10} = 2\frac{3}{10}$ (already has denominator 10)
• $1\frac{7}{10} = 1\frac{7}{10}$ (already has denominator 10)

Question 9

1. $2\frac{17}{24}$ cups
2. $3$ cups (correct answer)
3. $3\frac{19}{24}$ cups
4. $2\frac{23}{24}$ cups

Explanation: This is a multiplication problem involving mixed numbers. When you see "per serving" combined with a number of servings, you need to multiply the amount per serving by the total number of servings. To solve this, multiply $1\frac{1}{8}$ cups per serving by $2\frac{2}{3}$ servings. First, convert both mixed numbers to improper fractions: $1\frac{1}{8} = \frac{9}{8}$ and $2\frac{2}{3} = \frac{8}{3}$. Now multiply: $\frac{9}{8} \times \frac{8}{3} = \frac{9 \times 8}{8 \times 3} = \frac{72}{24}$ Notice that the 8s cancel out, giving you $\frac{9}{3} = 3$. So you need exactly 3 cups of milk. Looking at the wrong answers: Choice A ($2\frac{17}{24}$) likely comes from adding the mixed numbers instead of multiplying them - a common mistake when students see fractions and panic. Choice C ($3\frac{19}{24}$) might result from calculation errors when working with the improper fractions, possibly from not simplifying $\frac{72}{24}$ correctly. Choice D ($2\frac{23}{24}$) could come from similar computational mistakes or from incorrectly converting the mixed numbers to improper fractions initially. The key strategy here is recognizing the word "per" as a multiplication signal. When you see "X amount per unit" and need to find the total for multiple units, always multiply. Also, look for opportunities to cancel common factors early in fraction multiplication - it makes the arithmetic much cleaner.

Question 10

1. $3\frac{3}{4}$ yards (correct answer)
2. $4\frac{7}{12}$ yards
3. $3\frac{1}{4}$ yards
4. $12\frac{11}{12}$ yards

Explanation: This problem tests your ability to subtract mixed numbers, which requires finding a common denominator and sometimes borrowing when the subtraction becomes challenging. Start with the original amount and subtract what was used: $8\frac{1}{3} - 2\frac{3}{4} - 1\frac{5}{6}$. To work with these fractions, you need a common denominator. The least common multiple of 3, 4, and 6 is 12. Convert each mixed number: $8\frac{1}{3} = 8\frac{4}{12}$, $2\frac{3}{4} = 2\frac{9}{12}$, and $1\frac{5}{6} = 1\frac{10}{12}$. Now subtract: $8\frac{4}{12} - 2\frac{9}{12} - 1\frac{10}{12}$. Since $\frac{4}{12}$ is smaller than $\frac{9}{12} + \frac{10}{12} = \frac{19}{12}$, you need to borrow from the whole number. Convert $8\frac{4}{12}$ to $7\frac{16}{12}$. Calculate: $7\frac{16}{12} - 2\frac{9}{12} - 1\frac{10}{12} = (7-2-1) + (\frac{16}{12} - \frac{9}{12} - \frac{10}{12}) = 4 - \frac{3}{12} = 4 - \frac{1}{4} = 3\frac{3}{4}$. Choice A) $3\frac{3}{4}$ is correct. Choice B) $4\frac{7}{12}$ likely results from calculation errors with the common denominator. Choice C) $3\frac{1}{4}$ comes from incorrectly handling the borrowing process. Choice D) $12\frac{11}{12}$ suggests adding instead of subtracting the fabric used. When working with mixed number subtraction, always find the common denominator first, then handle borrowing carefully when the fraction part of the minuend is smaller than what you're subtracting.

Question 11

1. $6\frac{1}{3}$ miles
2. $7\frac{1}{3}$ miles (correct answer)
3. $6\frac{1}{4}$ miles
4. $8\frac{1}{12}$ miles

Explanation: When you encounter a problem asking for total distance, you need to add all the individual distances together. Since these are mixed numbers with different denominators, you'll need to find a common denominator to add the fractions properly. First, convert each mixed number to an improper fraction or keep them as mixed numbers and add the whole parts and fractional parts separately. Let's use the second approach: $3\frac{1}{4} + 2\frac{2}{3} + 1\frac{5}{12}$. Adding the whole numbers: $3 + 2 + 1 = 6$. For the fractions $\frac{1}{4} + \frac{2}{3} + \frac{5}{12}$, find the least common denominator. The LCD of 4, 3, and 12 is 12. Convert each fraction: $\frac{1}{4} = \frac{3}{12}$, $\frac{2}{3} = \frac{8}{12}$, and $\frac{5}{12}$ stays the same. Now add: $\frac{3}{12} + \frac{8}{12} + \frac{5}{12} = \frac{16}{12} = 1\frac{4}{12} = 1\frac{1}{3}$. Total distance: $6 + 1\frac{1}{3} = 7\frac{1}{3}$ miles, which is choice B. Choice A ($6\frac{1}{3}$) likely results from forgetting to carry over when the fractional sum exceeds 1. Choice C ($6\frac{1}{4}$) suggests adding fractions incorrectly without finding a common denominator. Choice D ($8\frac{1}{12}$) appears to come from adding whole numbers incorrectly or miscalculating the fractional sum. When adding mixed numbers, always double-check your common denominator work and remember to carry over any improper fractions to the whole number part.

Question 12

1. $2\frac{17}{24}$ cups
2. $2\frac{13}{24}$ cups (correct answer)
3. $5\frac{1}{12}$ cups
4. $10\frac{1}{6}$ cups

Explanation: When you encounter a recipe problem asking for a fraction of the original amounts, you need to multiply each ingredient by that fraction, then find the total. First, let's find how much of each liquid Maria needs for half the recipe. The original recipe calls for $3\frac{1}{3}$ cups of fruit juice and $1\frac{3}{4}$ cups of soda water. Convert to improper fractions: $3\frac{1}{3} = \frac{10}{3}$ and $1\frac{3}{4} = \frac{7}{4}$ For half the recipe, multiply each by $\frac{1}{2}$:

• Fruit juice: $\frac{10}{3} \times \frac{1}{2} = \frac{10}{6} = \frac{5}{3}$ cups
• Soda water: $\frac{7}{4} \times \frac{1}{2} = \frac{7}{8}$ cups

Question 13

1. $6\frac{1}{4}$ bushels (correct answer)
2. $5\frac{11}{12}$ bushels
3. $21\frac{7}{12}$ bushels
4. $36\frac{11}{12}$ bushels

Explanation: When you encounter word problems involving mixed numbers, you need to carefully track the operations: what's being added together versus what's being subtracted. First, find the total bushels harvested by adding the two fields: $12\frac{3}{4} + 8\frac{5}{6}$. To add mixed numbers, convert to improper fractions or add whole numbers and fractions separately. Using a common denominator of 12: $12\frac{9}{12} + 8\frac{10}{12} = 20\frac{19}{12} = 21\frac{7}{12}$ bushels total. Next, subtract what he sold: $21\frac{7}{12} - 15\frac{1}{3}$. Converting $\frac{1}{3}$ to twelfths: $21\frac{7}{12} - 15\frac{4}{12} = 6\frac{3}{12} = 6\frac{1}{4}$ bushels remaining. Looking at the wrong answers: Answer B ($5\frac{11}{12}$) likely comes from calculation errors in the addition or subtraction steps. Answer C ($21\frac{7}{12}$) represents the total harvested before selling anything—this is a classic trap where students stop after the first step without completing the problem. Answer D ($36\frac{11}{12}$) suggests adding all three quantities instead of recognizing that the sold amount should be subtracted. The correct answer is A: $6\frac{1}{4}$ bushels. Strategy tip: In multi-step word problems, identify what each number represents before calculating. Ask yourself: "Am I adding to the total or taking away from it?" Also, check that your final answer makes logical sense—the remaining amount should be less than what was originally harvested.

Question 14

1. $3$ gallons
2. $180$ gallons (correct answer)
3. $3\frac{3}{4}$ gallons
4. $75$ gallons

Explanation: This is a rate problem that requires you to multiply a rate by time, but you need to be careful with units and mixed numbers. First, convert the mixed numbers to improper fractions or decimals to make calculations easier. The rate is $2\frac{2}{5} = \frac{12}{5} = 2.4$ gallons per minute. The time is $1\frac{1}{4} = \frac{5}{4} = 1.25$ hours. Since the rate is given per minute but the time is in hours, you must convert units. There are 60 minutes in an hour, so $1.25$ hours equals $1.25 \times 60 = 75$ minutes. Now multiply: rate × time = $2.4 \times 75 = 180$ gallons. Looking at the wrong answers: Choice A ($3$ gallons) likely comes from multiplying the whole number parts only: $2 \times 1 = 2$, then rounding up slightly. Choice C ($3\frac{3}{4}$ gallons) results from multiplying the mixed numbers without converting units: $2\frac{2}{5} \times 1\frac{1}{4} = 3\frac{3}{20}$, which is close to $3\frac{3}{4}$. Choice D ($75$ gallons) comes from forgetting to multiply by the rate entirely—this is just the time conversion from hours to minutes. The correct answer is B: $180$ gallons. Remember: in rate problems, always check your units carefully. When rate and time have different units (per minute vs. hours), convert everything to the same time unit before multiplying. Setting up the calculation as "rate × time = total amount" helps ensure you don't miss any steps.

Question 15

1. $3\frac{17}{20}$ miles (correct answer)
2. $4\frac{3}{20}$ miles
3. $6\frac{3}{20}$ miles
4. $16\frac{3}{20}$ miles

Explanation: This problem tests your ability to work with mixed numbers and solve multi-step word problems involving addition and subtraction. When you see a question asking for a missing part of a total, think about setting up an equation where the known parts plus the unknown part equal the whole. First, you need to find how many miles Jason has already run by adding Monday's and Tuesday's distances. To add $3\frac{2}{5}$ and $2\frac{3}{4}$, convert to a common denominator. The LCD of 5 and 4 is 20: $3\frac{2}{5} = 3\frac{8}{20}$ and $2\frac{3}{4} = 2\frac{15}{20}$. Adding these gives $5\frac{23}{20} = 6\frac{3}{20}$ miles total for Monday and Tuesday. Since Jason wants to run 10 miles total over three days, subtract what he's already run from his goal: $10 - 6\frac{3}{20} = 9\frac{20}{20} - 6\frac{3}{20} = 3\frac{17}{20}$ miles needed on Wednesday. Choice A ($3\frac{17}{20}$) is correct. Choice B ($4\frac{3}{20}$) likely results from calculation errors when adding the mixed numbers or finding common denominators. Choice C ($6\frac{3}{20}$) is the total Jason ran on Monday and Tuesday combined—this represents missing the subtraction step entirely. Choice D ($16\frac{3}{20}$) comes from adding all three numbers instead of subtracting, showing a fundamental misunderstanding of the problem setup. Remember: when working with mixed numbers, always convert to common denominators first, and carefully track whether you need to add or subtract at each step.

Question 16

1. $2\frac{7}{24}$ yards (correct answer)
2. $2\frac{5}{24}$ yards
3. $4\frac{5}{24}$ yards
4. $10\frac{13}{24}$ yards

Explanation: When you encounter word problems involving mixed numbers, you need to add and subtract fractions with different denominators. The key is finding a common denominator and keeping track of what's being asked. First, find how much fabric Lisa already has by adding $2\frac{1}{4} + 1\frac{5}{6}$. Convert to improper fractions: $\frac{9}{4} + \frac{11}{6}$. The LCD of 4 and 6 is 12, so: $\frac{27}{12} + \frac{22}{12} = \frac{49}{12} = 4\frac{1}{12}$ yards. Next, subtract what she has from what she needs: $6\frac{3}{8} - 4\frac{1}{12}$. Convert to improper fractions: $\frac{51}{8} - \frac{49}{12}$. The LCD of 8 and 12 is 24, so: $\frac{153}{24} - \frac{98}{24} = \frac{55}{24} = 2\frac{7}{24}$ yards. Looking at the wrong answers: Choice B ($2\frac{5}{24}$) likely results from an arithmetic error when finding the common denominator or subtracting fractions. Choice C ($4\frac{5}{24}$) suggests you might have added the fabric she needs to what she already has, rather than subtracting. Choice D ($10\frac{13}{24}$) appears to come from adding all three quantities together instead of performing the correct operations. The correct answer is A: $2\frac{7}{24}$ yards. Study tip: In multi-step fraction problems, work systematically—first combine what you have, then compare to what you need. Always double-check that your final answer makes logical sense in the context of the problem.

Question 17

1. $3\frac{1}{10}$ pounds
2. $4$ pounds (correct answer)
3. $4\frac{1}{10}$ pounds
4. $2\frac{9}{10}$ pounds

Explanation: When you encounter word problems involving fractions, you need to identify the operation required. Here, the chef uses a certain amount of vegetables "per batch" and makes multiple batches, so you're finding the total by multiplying: amount per batch × number of batches. First, convert both mixed numbers to improper fractions to make multiplication easier. For $2\frac{1}{2}$: multiply the whole number by the denominator and add the numerator: $(2 \times 2) + 1 = 5$, so $2\frac{1}{2} = \frac{5}{2}$. For $1\frac{3}{5}$: $(1 \times 5) + 3 = 8$, so $1\frac{3}{5} = \frac{8}{5}$. Now multiply: $\frac{5}{2} \times \frac{8}{5} = \frac{5 \times 8}{2 \times 5} = \frac{40}{10} = 4$. The chef uses exactly 4 pounds of vegetables. Looking at the wrong answers: Choice A ($3\frac{1}{10}$) results from incorrectly adding the fractions instead of multiplying them. Choice C ($4\frac{1}{10}$) might come from calculation errors during the multiplication process or mishandling the conversion between mixed and improper fractions. Choice D ($2\frac{9}{10}$) likely stems from subtracting instead of multiplying, or from errors in converting the mixed numbers. The correct answer is B. Study tip: When you see "per" in a word problem (per batch, per hour, per mile), it usually signals multiplication. Always convert mixed numbers to improper fractions before multiplying—it prevents messy calculations and reduces errors.

Question 18

1. $6\frac{17}{20}$ miles
2. $8\frac{13}{20}$ miles
3. $10\frac{7}{20}$ miles (correct answer)
4. $18\frac{3}{20}$ miles

Explanation: This question tests your ability to multiply mixed numbers, a key skill for solving real-world problems involving fractional quantities over time. To find the total distance Kevin runs, you need to multiply his daily distance by the number of days: $4\frac{3}{5} \times 2\frac{1}{4}$. First, convert both mixed numbers to improper fractions. For $4\frac{3}{5}$: multiply the whole number by the denominator and add the numerator: $\frac{(4 \times 5) + 3}{5} = \frac{23}{5}$. For $2\frac{1}{4}$: $\frac{(2 \times 4) + 1}{4} = \frac{9}{4}$. Now multiply: $\frac{23}{5} \times \frac{9}{4} = \frac{23 \times 9}{5 \times 4} = \frac{207}{20}$. Convert back to a mixed number by dividing: $207 \div 20 = 10$ with remainder $7$, so $\frac{207}{20} = 10\frac{7}{20}$. This matches choice C. Choice A ($6\frac{17}{20}$) likely comes from adding the mixed numbers instead of multiplying them. Choice B ($8\frac{13}{20}$) might result from calculation errors when converting to improper fractions or during multiplication. Choice D ($18\frac{3}{20}$) appears to come from multiplying the whole numbers and fractions separately, then combining incorrectly. Remember: when working with mixed numbers in multiplication, always convert to improper fractions first. This eliminates the complexity of dealing with whole and fractional parts separately and reduces calculation errors.

Question 19

1. $\frac{5}{32}$ inch
2. $\frac{7}{32}$ inch (correct answer)
3. $\frac{4}{16}$ inch
4. $\frac{2}{16}$ inch

Explanation: When you encounter word problems involving fractions, the key is to identify what operation you need to perform and ensure all fractions have a common denominator before calculating. The machinist needs holes that are $\frac{3}{16}$ inch in diameter, but the drill bit creates holes $\frac{1}{32}$ inch larger than specified. To find the actual diameter, you need to add these two measurements: $\frac{3}{16} + \frac{1}{32}$. Before adding fractions, convert them to a common denominator. Since 32 is a multiple of 16, convert $\frac{3}{16}$ to thirty-seconds: $\frac{3}{16} = \frac{6}{32}$. Now you can add: $\frac{6}{32} + \frac{1}{32} = \frac{7}{32}$ inch. Choice A ($\frac{5}{32}$) represents a common error where students subtract instead of add, calculating $\frac{6}{32} - \frac{1}{32}$. Choice C ($\frac{4}{16}$) suggests incorrectly adding the numerators without finding a common denominator: $3 + 1 = 4$, keeping the denominator as 16. Choice D ($\frac{2}{16}$) might result from various calculation errors, possibly confusing which measurements to use or making arithmetic mistakes. The correct answer is B: $\frac{7}{32}$ inch. For fraction word problems on the ISEE, always identify whether you're adding or subtracting, convert to common denominators first, and double-check that your answer makes logical sense in context. Here, the actual hole should indeed be larger than the specified size, confirming our addition approach was correct.

Question 20

1. $4\frac{1}{4}$ hours
2. $4\frac{3}{4}$ hours (correct answer)
3. $5\frac{1}{12}$ hours
4. $3\frac{3}{4}$ hours

Explanation: When you encounter mixed number addition problems, you need to add fractions with different denominators, which requires finding a common denominator. To find the total reading time, add $1\frac{2}{3} + 2\frac{1}{4} + \frac{5}{6}$. First, convert to improper fractions: $\frac{5}{3} + \frac{9}{4} + \frac{5}{6}$. The least common denominator of 3, 4, and 6 is 12. Converting each fraction: $\frac{5}{3} = \frac{20}{12}$, $\frac{9}{4} = \frac{27}{12}$, and $\frac{5}{6} = \frac{10}{12}$. Adding: $\frac{20}{12} + \frac{27}{12} + \frac{10}{12} = \frac{57}{12}$ Converting back to a mixed number: $57 ÷ 12 = 4$ remainder $9$, so $\frac{57}{12} = 4\frac{9}{12} = 4\frac{3}{4}$ hours. Answer choice A ($4\frac{1}{4}$) likely results from incorrectly adding the fractional parts without finding a common denominator, perhaps just adding numerators. Choice C ($5\frac{1}{12}$) suggests an error in converting mixed numbers to improper fractions or miscalculating the whole number part when converting back. Choice D ($3\frac{3}{4}$) indicates an arithmetic error, possibly miscounting the whole number portions or making a calculation mistake with the fractions. Remember: when adding mixed numbers, either convert everything to improper fractions first (as shown) or add the whole numbers separately from the fractional parts. Always find the least common denominator before adding fractions, and double-check your arithmetic when converting between mixed and improper fractions.