Mixed-Number Problems - ISEE Upper Level Mathematics Achievement

1. Tom's morning jog consists of three segments: $2\frac{1}{6}$ miles uphill, $1\frac{3}{4}$ miles on flat ground, and $1\frac{5}{12}$ miles downhill. What is the total distance of his jog?
   Answer: $5\frac{1}{3}$ miles 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.
2. A farmer harvests $12\frac{3}{4}$ bushels of corn from one field and $8\frac{5}{6}$ bushels from another field. If he sells $15\frac{1}{3}$ bushels, how many bushels does he have left?
   Answer: $6\frac{1}{4}$ bushels 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.
3. A chef prepares soup using $2\frac{1}{2}$ pounds of vegetables per batch. If she makes $1\frac{3}{5}$ batches, how many pounds of vegetables does she use?
   Answer: $4$ pounds 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.
4. A seamstress has $8\frac{1}{3}$ yards of fabric. She uses $2\frac{3}{4}$ yards for a dress and $1\frac{5}{6}$ yards for a skirt. How much fabric remains?
   Answer: $3\frac{3}{4}$ yards 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.
5. Emma drinks $1\frac{3}{8}$ liters of water in the morning and $2\frac{5}{6}$ liters in the afternoon. Her daily goal is $5$ liters. How much more water does she need to drink to reach her goal?
   Answer: $\frac{19}{24}$ liters 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.
6. Jason runs $3\frac{2}{5}$ miles on Monday and $2\frac{3}{4}$ miles on Tuesday. If he wants to run a total of $10$ miles over three days, how many miles must he run on Wednesday?
   Answer: $3\frac{17}{20}$ miles 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.
7. Kevin runs $4\frac{3}{5}$ miles each day for $2\frac{1}{4}$ days. What is the total distance he runs?
   Answer: $10\frac{7}{20}$ miles 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.
8. A student reads $1\frac{2}{3}$ hours on Monday, $2\frac{1}{4}$ hours on Tuesday, and $\frac{5}{6}$ hour on Wednesday. What is the total reading time for the three days?
   Answer: $4\frac{3}{4}$ hours 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.
9. A recipe calls for $2\frac{3}{4}$ cups of flour. Maria has already added $1\frac{5}{8}$ cups of flour to her mixing bowl. How much more flour does she need to add?
   Answer: $1\frac{1}{8}$ cups Subtract what Maria has added from what the recipe needs, using a common denominator. Write $2\frac{3}{4}$ in eighths: $2\frac{3}{4} = 2\frac{6}{8}$. Then subtract: $2\frac{6}{8} - 1\frac{5}{8} = 1\frac{1}{8}$ cups. Check by adding back: $1\frac{5}{8} + 1\frac{1}{8} = 2\frac{6}{8} = 2\frac{3}{4}$. An answer of $4\frac{3}{8}$ cups adds the two amounts instead of subtracting. An answer of $\frac{7}{8}$ cups comes from subtracting the fractions backward, getting $-\frac{1}{8}$, and then taking that from the 1 whole cup. An answer of $1\frac{3}{8}$ cups is $\frac{1}{4}$ cup too much, as the check by adding back shows. Maria needs $1\frac{1}{8}$ more cups.
10. A baker uses $2\frac{1}{4}$ pounds of sugar to make one batch of cookies. If she wants to make $3\frac{1}{2}$ batches, how many pounds of sugar will she need in total?
    Answer: $7\frac{7}{8}$ pounds This question tests your ability to multiply mixed numbers, a key skill for real-world applications involving fractional quantities. To find the total sugar needed, you must multiply the amount per batch by the number of batches: $2\frac{1}{4} \times 3\frac{1}{2}$. First, convert both mixed numbers to improper fractions. For $2\frac{1}{4}$: multiply the whole number by the denominator and add the numerator: $(2 \times 4) + 1 = 9$, so $2\frac{1}{4} = \frac{9}{4}$. For $3\frac{1}{2}$: $(3 \times 2) + 1 = 7$, so $3\frac{1}{2} = \frac{7}{2}$. Now multiply: $\frac{9}{4} \times \frac{7}{2} = \frac{63}{8}$. Convert back to a mixed number by dividing: $63 \div 8 = 7$ remainder $7$, giving you $7\frac{7}{8}$ pounds. Choice A ($5\frac{3}{4}$) results from incorrectly adding the mixed numbers instead of multiplying them. Choice B ($6\frac{7}{8}$) comes from calculation errors, possibly multiplying $2 \times 3\frac{1}{2}$ instead of using the full $2\frac{1}{4}$. Choice D ($8\frac{1}{8}$) suggests computational mistakes in the fraction multiplication or conversion process. When multiplying mixed numbers, always convert to improper fractions first—this prevents errors that occur when trying to work with the mixed form directly. Double-check by estimating: $2\frac{1}{4} \times 3\frac{1}{2}$ should be close to $2 \times 4 = 8$, confirming that $7\frac{7}{8}$ is reasonable.
11. A water tank holds $15\frac{2}{3}$ gallons when full. If $9\frac{5}{8}$ gallons have been used, how much water remains in the tank?
    Answer: $6\frac{1}{24}$ gallons This problem tests your ability to subtract mixed numbers, which requires careful handling of both whole numbers and fractions with different denominators. To find how much water remains, you need to subtract the amount used from the total capacity: $15\frac{2}{3} - 9\frac{5}{8}$. First, convert both mixed numbers to improper fractions or find a common denominator for the fractional parts. The LCD of 3 and 8 is 24. Convert the fractions: $\frac{2}{3} = \frac{16}{24}$ and $\frac{5}{8} = \frac{15}{24}$ Now you have: $15\frac{16}{24} - 9\frac{15}{24} = 6\frac{1}{24}$ Since $\frac{16}{24} > \frac{15}{24}$, you can subtract directly: the whole numbers give you $15 - 9 = 6$, and the fractions give you $\frac{16}{24} - \frac{15}{24} = \frac{1}{24}$. Choice A gives $6\frac{1}{24}$, which is correct. Choice B shows $6\frac{1}{12}$, which would result from incorrectly using 12 as the common denominator instead of 24. Choice C gives $5\frac{23}{24}$, which suggests an error in borrowing when subtracting fractions—treating this as if you needed to borrow from the whole number when you actually didn't. Choice D shows $25\frac{7}{24}$, which appears to come from adding the numbers instead of subtracting. When subtracting mixed numbers, always find the LCD first and check whether you need to borrow from the whole number before performing the subtraction.
12. A carpenter needs to cut a board that is $8\frac{1}{3}$ feet long into pieces that are each $1\frac{2}{9}$ feet long. How many complete pieces can be cut from the board?
    Answer: $6$ pieces To find how many complete pieces fit, divide the board length by the piece length and keep only the whole number. Write both lengths as improper fractions: $8\frac{1}{3} = \frac{25}{3}$ and $1\frac{2}{9} = \frac{11}{9}$. $\frac{25}{3} \div \frac{11}{9} = \frac{25}{3} \times \frac{9}{11} = \frac{225}{33} = \frac{75}{11} = 6\frac{9}{11}$ That is 6 complete pieces with part of a seventh left over. Check: 6 pieces use $6 \times \frac{11}{9} = \frac{66}{9} = 7\frac{1}{3}$ feet, which fits. Seven pieces would need $7 \times \frac{11}{9} = \frac{77}{9} = 8\frac{5}{9}$ feet, which is longer than the board. So 7 pieces comes from rounding up instead of down, and 8 or 9 pieces are even further over. The carpenter can cut 6 complete pieces.
13. A swimming pool is being filled at a rate of $2\frac{2}{5}$ gallons per minute. How many gallons will be added to the pool in $1\frac{1}{4}$ hours?
    Answer: $180$ gallons 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.
14. Lisa is making a quilt that requires $6\frac{3}{8}$ yards of fabric. She has already purchased $2\frac{1}{4}$ yards of blue fabric and $1\frac{5}{6}$ yards of green fabric. How much more fabric does she need?
    Answer: $2\frac{7}{24}$ yards 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.
15. A recipe for punch calls for $3\frac{1}{3}$ cups of fruit juice and $1\frac{3}{4}$ cups of soda water. If Maria wants to make $\frac{1}{2}$ of the recipe, how many total cups of liquid will she need?
    Answer: $2\frac{13}{24}$ cups 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 Now add these amounts. To add fractions, find a common denominator. The LCD of 3 and 8 is 24: $\frac{5}{3} = \frac{40}{24}$ and $\frac{7}{8} = \frac{21}{24}$ Total: $\frac{40}{24} + \frac{21}{24} = \frac{61}{24} = 2\frac{13}{24}$ cups This confirms answer B is correct. Answer A ($2\frac{17}{24}$) likely comes from calculation errors in finding the common denominator. Answer C ($5\frac{1}{12}$) appears to be the full recipe total rather than half. Answer D ($10\frac{1}{6}$) seems to result from adding the original amounts incorrectly, then doubling instead of halving. Remember: always convert mixed numbers to improper fractions before multiplying, and double-check that you're applying the correct operation (half means multiply by $\frac{1}{2}$, not divide by 2).
16. A recipe for trail mix calls for $1\frac{1}{2}$ cups of nuts for every $\frac{3}{4}$ cup of dried fruit. If Sarah wants to use $3$ cups of dried fruit, how many cups of nuts will she need?
    Answer: $6$ cups 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.
17. A fabric store sells ribbon by the yard. Mrs. Johnson buys $4\frac{2}{3}$ yards of red ribbon and $3\frac{1}{4}$ yards of blue ribbon. She uses $2\frac{5}{12}$ yards for a project. How much ribbon does she have left?
    Answer: $5\frac{1}{2}$ yards 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.
18. A delivery truck travels $3\frac{1}{4}$ miles to its first stop, then $2\frac{2}{3}$ miles to its second stop, and finally $1\frac{5}{12}$ miles back to the depot. What is the total distance traveled?
    Answer: $7\frac{1}{3}$ miles 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.
19. A machinist needs to drill holes that are $\frac{3}{16}$ inch in diameter. If the drill bit creates holes that are $\frac{1}{32}$ inch larger than specified, what is the actual diameter of each hole?
    Answer: $\frac{7}{32}$ inch 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.
20. A construction worker pours concrete in three sections: $3\frac{2}{5}$ cubic yards in the morning, $2\frac{3}{10}$ cubic yards at lunch, and $1\frac{7}{10}$ cubic yards in the afternoon. What is the total amount of concrete poured?
    Answer: $7\frac{2}{5}$ cubic yards Add mixed numbers by giving the fractions a common denominator, then adding the whole numbers and fractions separately. Tenths work for all three: $3\frac{2}{5} = 3\frac{4}{10}$, so the amounts are $3\frac{4}{10}$, $2\frac{3}{10}$, and $1\frac{7}{10}$. Whole numbers: $3 + 2 + 1 = 6$. Fractions: $\frac{4}{10} + \frac{3}{10} + \frac{7}{10} = \frac{14}{10} = 1\frac{4}{10}$. Total: $6 + 1\frac{4}{10} = 7\frac{4}{10} = 7\frac{2}{5}$ cubic yards. An answer of $7\frac{3}{10}$ comes from converting $\frac{2}{5}$ to $\frac{3}{10}$ instead of $\frac{4}{10}$. An answer of $6\frac{4}{5}$ is too small, since the fractions alone add to more than 1. An answer of $6\frac{12}{25}$ comes from mishandling the denominators instead of converting to tenths. The total is $7\frac{2}{5}$ cubic yards.

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