Fraction Operations in Context - SHSAT Math

A construction project requires mixing concrete in the ratio $3\frac{1}{4} : 2\frac{1}{3}$ (cement to sand by weight). If 14 pounds of sand are used, how many pounds of cement are needed?
Answer: $19\frac{1}{2}$ pounds of cement are required Convert mixed numbers: $3\frac{1}{4} = \frac{13}{4}$ and $2\frac{1}{3} = \frac{7}{3}$ . Set up proportion: $\frac{13/4}{7/3} = \frac{x}{14}$ . Cross multiply: $\frac{13}{4} \times 14 = \frac{7}{3} \times x$ . This gives $\frac{13 \times 14}{4} = \frac{7x}{3}$ , so $\frac{182}{4} = \frac{7x}{3}$ , which simplifies to $\frac{91}{2} = \frac{7x}{3}$ . Solving: $x = \frac{91}{2} \times \frac{3}{7} = \frac{273}{14} = 19\frac{1}{2}$ pounds. Choice A results from computational errors in the cross multiplication. Choice B comes from rounding errors. Choice D uses incorrect ratio simplification or calculation mistakes.
A juice concentrate requires dilution with water in the ratio $1:4$ (concentrate to water). If Maria has $\frac{3}{4}$ cup of concentrate and wants to use all of it, but only has $2\frac{1}{2}$ cups of water available, how much additional water does she need to make the mixture according to the specified ratio?
Answer: $\frac{1}{2}$ cup of additional water is needed This problem tests your understanding of ratios and proportional reasoning. When you see a ratio problem, always identify what the ratio means and calculate the required amounts based on what you actually have. The ratio $1:4$ means for every 1 part concentrate, you need 4 parts water. Since Maria has $\frac{3}{4}$ cup of concentrate, she needs $\frac{3}{4} \times 4 = 3$ cups of water total to maintain the proper ratio. Maria currently has $2\frac{1}{2} = \frac{5}{2}$ cups of water available. To find how much additional water she needs: $3 - \frac{5}{2} = \frac{6}{2} - \frac{5}{2} = \frac{1}{2}$ cup. This confirms answer B is correct. Let's examine why the other answers are wrong. Answer A ( $\frac{3}{4}$ cup) likely comes from mistakenly thinking you need the same amount of additional water as concentrate. Answer C (1 cup) might result from incorrectly calculating $4 - 3 = 1$ without properly accounting for the actual amounts involved. Answer D (no additional water needed) suggests incorrectly thinking $2\frac{1}{2}$ cups is already sufficient, perhaps by confusing the ratio or miscalculating the required water amount. For ratio problems on the SHSAT, always multiply the amount you have by the ratio to find what you need, then subtract what you already have. Write out your conversions between mixed numbers and improper fractions carefully to avoid calculation errors.
A pizza is cut into equal slices. Tom eats $\frac{1}{3}$ of the pizza, Jerry eats $\frac{1}{4}$ of what remains, and Spike eats $\frac{1}{2}$ of what's left after Jerry. What fraction of the original pizza is still uneaten?
Answer: $\frac{1}{4}$ of the original pizza remains When you encounter problems where people consume fractions of what remains (rather than fractions of the original), you need to track what's left after each person eats, not add up what everyone consumed. Let's work through this step-by-step. Start with the whole pizza (1). Tom eats $\frac{1}{3}$ of the original pizza, leaving $1 - \frac{1}{3} = \frac{2}{3}$ of the pizza. Jerry eats $\frac{1}{4}$ of what remains after Tom, which is $\frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}$ of the original pizza. This leaves $\frac{2}{3} - \frac{1}{6} = \frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$ of the original pizza. Spike eats $\frac{1}{2}$ of what's left after Jerry, which is $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ of the original pizza. This leaves $\frac{1}{2} - \frac{1}{4} = \frac{1}{4}$ of the original pizza uneaten. Choice A ( $\frac{1}{6}$ ) incorrectly assumes only Tom and Jerry ate pizza. Choice B ( $\frac{3}{8}$ ) likely comes from adding fractions incorrectly or misinterpreting "what remains." Choice C ( $\frac{5}{12}$ ) represents a calculation error, possibly from incorrectly converting fractions or mixing up the sequential eating process. The key strategy here is to carefully track what remains after each step rather than trying to add up consumption amounts. Always work sequentially through "what remains" problems, updating your remaining amount after each person's turn.
A fish tank contains $8\dfrac{3}{8}$ gallons of water. After some water is siphoned out, $5\dfrac{5}{6}$ gallons remain. How many gallons were removed?
Answer: $2\dfrac{13}{24}$ gallons When you encounter word problems involving fractions being removed or taken away, you're looking at a subtraction problem. The key is to subtract the final amount from the initial amount to find what was removed. To find how many gallons were removed, you need to calculate: $8\frac{3}{8} - 5\frac{5}{6}$ First, convert both mixed numbers to improper fractions or find a common denominator. The denominators are 8 and 6, so the least common denominator is 24. Convert $8\frac{3}{8}$ to twenty-fourths: $8\frac{3}{8} = 8\frac{9}{24}$ Convert $5\frac{5}{6}$ to twenty-fourths: $5\frac{5}{6} = 5\frac{20}{24}$ Now subtract: $8\frac{9}{24} - 5\frac{20}{24}$ Since $\frac{9}{24} < \frac{20}{24}$ , you need to borrow 1 from the whole number: $7\frac{33}{24} - 5\frac{20}{24} = 2\frac{13}{24}$ This confirms answer A is correct. Looking at the wrong answers: B) $2\frac{11}{24}$ likely results from an arithmetic error in the borrowing process. C) $3\frac{1}{2}$ suggests someone subtracted the fractions incorrectly or made an error with the whole numbers. D) $2\frac{5}{12}$ indicates using 12 as the common denominator instead of 24, leading to computational mistakes. Remember: when subtracting mixed numbers, always find the LCD first, then handle borrowing carefully when the first fraction is smaller than the second.
An orchard covers $2\dfrac{1}{3}$ acres. If $\dfrac{3}{5}$ of the orchard will be replanted this year, how many acres will be replanted?
Answer: $1\dfrac{2}{5}$ acres When you see a problem asking for a fraction "of" something, you're dealing with multiplication of fractions. The word "of" signals multiplication, so you need to multiply $\frac{3}{5}$ by the total acreage. First, convert the mixed number to an improper fraction: $2\frac{1}{3} = \frac{7}{3}$ (since $2 \times 3 + 1 = 7$ ). Now multiply: $\frac{3}{5} \times \frac{7}{3} = \frac{21}{15}$ . Simplify by dividing both numerator and denominator by 3: $\frac{21}{15} = \frac{7}{5}$ . Convert back to a mixed number: $\frac{7}{5} = 1\frac{2}{5}$ acres. Choice A gives us $1\frac{2}{5}$ acres, which matches our calculation. Choice B ( $1\frac{1}{3}$ acres) likely comes from incorrectly adding the fractions $\frac{3}{5} + \frac{1}{3}$ instead of multiplying. Choice C ( $1\frac{4}{5}$ acres) might result from calculation errors when converting between mixed numbers and improper fractions. Choice D ( $2\frac{1}{5}$ acres) appears to come from mistakenly adding $\frac{3}{5}$ to the original $2\frac{1}{3}$ instead of finding $\frac{3}{5}$ of it. Remember: when you see "fraction of amount" problems, always convert mixed numbers to improper fractions before multiplying, then convert your final answer back to a mixed number if needed. Double-check by estimating— $\frac{3}{5}$ of about 2 acres should give you something between 1 and 2 acres, which $1\frac{2}{5}$ satisfies.
At a bake sale, $2\dfrac{1}{2}$ chocolate pies and $1\dfrac{3}{4}$ apple pies were sold from a total of 7 pies. How many pies remain?
Answer: $2\dfrac{3}{4}$ pies When you encounter word problems involving mixed numbers and subtraction, your goal is to find what remains after removing parts from a whole. Here, you need to subtract the total pies sold from the original amount. First, add up the pies that were sold: $2\frac{1}{2} + 1\frac{3}{4}$ . To add mixed numbers, convert them to improper fractions or find a common denominator. Using common denominators: $2\frac{1}{2} = 2\frac{2}{4}$ , so $2\frac{2}{4} + 1\frac{3}{4} = 3\frac{5}{4} = 4\frac{1}{4}$ pies sold. Now subtract from the total: $7 - 4\frac{1}{4} = 6\frac{4}{4} - 4\frac{1}{4} = 2\frac{3}{4}$ pies remain. Looking at the wrong answers: Choice B ( $3\frac{1}{2}$ ) likely comes from incorrectly adding the sold pies as $3\frac{1}{2}$ instead of $4\frac{1}{4}$ , perhaps by adding $\frac{1}{2} + \frac{3}{4} = \frac{1}{2}$ instead of $\frac{5}{4}$ . Choice C ( $4\frac{1}{4}$ ) is the total amount sold, not what remains—this suggests confusion about what the question asks. Choice D ( $\frac{11}{28}$ ) appears to result from incorrectly working with fractions, possibly multiplying instead of adding or using wrong denominators. The correct answer is A: $2\frac{3}{4}$ pies. Strategy tip: In word problems with mixed numbers, always double-check whether you're finding what was used or what remains. Convert everything to the same denominator before calculating, and verify your final answer makes logical sense given the context.
A bakery uses $\frac{2}{5}$ cup of sugar for every $\frac{3}{4}$ cup of flour in their bread recipe. If they use $2\frac{1}{4}$ cups of flour, how much sugar do they need?
Answer: $\frac{6}{5}$ cups of sugar total Sugar and flour are in a fixed ratio, so set up a proportion: $\frac{2/5}{3/4} = \frac{x}{9/4}$ , since $2\frac{1}{4} = \frac{9}{4}$ . The flour amount is $\frac{9}{4} \div \frac{3}{4} = 3$ times the recipe's flour, so the sugar is $3 \times \frac{2}{5} = \frac{6}{5}$ cups. The value $\frac{9}{10}$ comes from multiplying $\frac{2}{5} \times \frac{9}{4} = \frac{18}{20}$ and forgetting to divide by the $\frac{3}{4}$ cup of flour in the original ratio. The value $\frac{4}{5}$ only doubles the sugar instead of tripling it. The value $1\frac{2}{5}$ comes from adding $1$ to the recipe's $\frac{2}{5}$ cup instead of scaling it.
A paint shop mixes $\dfrac{3}{8}$ quart of pigment into each gallon of base paint. How many quarts of pigment are needed to tint 12 gallons?
Answer: $4\dfrac{1}{2}$ quarts This is a unit rate problem where you need to find the total amount of one ingredient when you know the rate per unit and the number of units. When you see "per gallon" or "each gallon," you're dealing with a rate that you'll multiply by the total number of gallons. Since the paint shop uses $\frac{3}{8}$ quart of pigment for each gallon of base paint, you multiply this rate by 12 gallons: $\frac{3}{8} \times 12$ . To multiply a fraction by a whole number, multiply the numerator by the whole number: $\frac{3 \times 12}{8} = \frac{36}{8}$ . Simplify by dividing both numerator and denominator by 4: $\frac{36}{8} = \frac{9}{2} = 4\frac{1}{2}$ quarts. Choice A ( $4\frac{1}{2}$ quarts) is correct based on this calculation. Choice B ( $3\frac{3}{4}$ quarts) likely results from incorrectly multiplying $\frac{3}{8} \times 10$ instead of 12, or from computational errors in the fraction arithmetic. Choice C ( $4\frac{3}{8}$ quarts) suggests adding $\frac{3}{8}$ to 4 instead of properly converting $\frac{9}{2}$ , possibly from misunderstanding mixed number conversion. Choice D ( $5\frac{1}{4}$ quarts) might come from calculation errors or incorrectly setting up the multiplication. Remember: when you see rate problems with fractions, set up the multiplication carefully and take your time with fraction arithmetic. Converting improper fractions to mixed numbers is often the final step, so double-check that conversion by ensuring your mixed number equals your improper fraction.
During two training days, a runner completed $4\dfrac{1}{4}$ miles on Monday and $3\dfrac{2}{3}$ miles on Tuesday. What is the total distance the runner covered on those two days?
Answer: $7\dfrac{11}{12}$ miles When you encounter mixed numbers in addition problems, you need to add the whole numbers and fractions separately, then combine them properly. To find the total distance, add $4\frac{1}{4} + 3\frac{2}{3}$ . Start by adding the whole numbers: $4 + 3 = 7$ . Next, add the fractions $\frac{1}{4} + \frac{2}{3}$ . Since these fractions have different denominators, find the least common denominator (LCD). The LCD of 4 and 3 is 12. Convert each fraction: $\frac{1}{4} = \frac{3}{12}$ and $\frac{2}{3} = \frac{8}{12}$ . Now add: $\frac{3}{12} + \frac{8}{12} = \frac{11}{12}$ . The total distance is $7 + \frac{11}{12} = 7\frac{11}{12}$ miles, which is choice A. Choice B ( $7\frac{5}{6}$ ) results from incorrectly using 6 as the common denominator instead of 12. Choice C (8 miles) comes from rounding or incorrectly assuming the fractions add to 1 whole. Choice D ( $8\frac{1}{12}$ ) happens when you mistakenly think $\frac{1}{4} + \frac{2}{3} = 1\frac{1}{12}$ instead of just $\frac{11}{12}$ . When adding mixed numbers, always double-check your LCD calculation and fraction conversions. A quick way to verify: since $\frac{11}{12}$ is close to 1 whole, your answer should be close to $4 + 3 + 1 = 8$ , and $7\frac{11}{12}$ fits this expectation perfectly.
A rectangular garden plot has an area of $\frac{5}{6}$ square meters. If its length is $\frac{5}{4}$ meters, and the width needs to be increased by $\frac{1}{8}$ meter for a new design, what will be the new width?
Answer: $\frac{19}{24}$ meters in total width First find the current width using Area = length × width: $\frac{5}{6} = \frac{5}{4} \times w$ , so $w = \frac{5}{6} \div \frac{5}{4} = \frac{5}{6} \times \frac{4}{5} = \frac{4}{6} = \frac{2}{3}$ meters. The new width is the current width plus the increase: $\frac{2}{3} + \frac{1}{8} = \frac{16}{24} + \frac{3}{24} = \frac{19}{24}$ meters. Choice B results from incorrectly calculating the original width as $\frac{3}{4}$ , then adding $\frac{1}{8}$ . Choice C uses wrong area calculations. Choice D represents $\frac{19}{24}$ incorrectly simplified or uses computational errors.
A storage tank originally contains $2\dfrac{5}{6}$ gallons of water. After more water is added, the tank holds $3\dfrac{3}{4}$ additional gallons. How much water is now in the tank?
Answer: $6\dfrac{7}{12}$ gallons This problem tests your ability to add mixed numbers, which requires finding a common denominator and potentially regrouping. To find the total water in the tank, you need to add the original amount ( $2\frac{5}{6}$ gallons) to the additional water ( $3\frac{3}{4}$ gallons). First, convert both fractions to have a common denominator. The least common multiple of 6 and 4 is 12, so: $2\frac{5}{6} = 2\frac{10}{12}$ and $3\frac{3}{4} = 3\frac{9}{12}$ . Now add: $2\frac{10}{12} + 3\frac{9}{12} = 5\frac{19}{12}$ . Since $\frac{19}{12}$ is an improper fraction ( $\frac{19}{12} = 1\frac{7}{12}$ ), you need to regroup: $5\frac{19}{12} = 5 + 1\frac{7}{12} = 6\frac{7}{12}$ gallons. Choice A ( $6\frac{7}{12}$ ) is correct. Choice B ( $6\frac{5}{12}$ ) represents an error in adding the fractional parts—you might get this if you incorrectly calculated $\frac{10}{12} + \frac{9}{12} = \frac{17}{12}$ instead of $\frac{19}{12}$ . Choice C ( $5\frac{11}{12}$ ) occurs if you add the fractions correctly but forget to regroup the improper fraction. Choice D ( $5\frac{7}{12}$ ) happens if you subtract instead of add, or make errors in both the whole number and fractional calculations. When adding mixed numbers, always check that your final fractional part is in lowest terms and proper form. Converting to a common denominator first prevents calculation errors.
A spool holds $4\dfrac{1}{2}$ yards of ribbon. If each party favor requires $\dfrac{3}{8}$ yard, how many complete party favors can be made from the full spool?
Answer: 12 favors When you encounter a division problem with mixed numbers and fractions, you need to find how many times one quantity fits into another. This is a classic "how many groups" division scenario. First, convert the mixed number to an improper fraction: $4\frac{1}{2} = \frac{9}{2}$ yards. Now divide the total ribbon by the amount needed per favor: $\frac{9}{2} \div \frac{3}{8}$ . To divide fractions, multiply by the reciprocal: $\frac{9}{2} \times \frac{8}{3} = \frac{72}{6} = 12$ . Since the question asks for complete party favors, 12 is your answer. Let's examine why the other choices are incorrect. Choice A (9 favors) likely comes from incorrectly converting $4\frac{1}{2}$ to the decimal 4.5, then dividing 4.5 by 0.5 instead of $\frac{3}{8}$ — a common mistake when students confuse $\frac{3}{8}$ with $\frac{1}{2}$ . Choice C (14 favors) might result from calculation errors in the fraction multiplication, perhaps getting $\frac{9 \times 8}{2 \times 3} = \frac{72}{6}$ wrong. Choice D (16 favors) could come from multiplying instead of dividing, or from other computational mistakes with the fractions. The key strategy here is methodical fraction operations: convert mixed numbers to improper fractions, then multiply by the reciprocal to divide. Always double-check that your answer makes sense — 12 favors using $\frac{3}{8}$ yard each equals $4\frac{1}{2}$ yards total, confirming our solution.
A potter starts with 7 pounds of clay. If each small vase uses $1\dfrac{3}{8}$ pounds of clay, how much clay will remain after exactly 4 vases are made?
Answer: $1\dfrac{1}{2}$ lb When you encounter word problems involving fractions and repeated operations, break the problem into clear steps: identify what you're using, calculate the total usage, then subtract from what you started with. You begin with 7 pounds of clay and need to make 4 vases, each requiring $1\frac{3}{8}$ pounds. First, calculate the total clay used: $4 \times 1\frac{3}{8}$ . Convert the mixed number to an improper fraction: $1\frac{3}{8} = \frac{11}{8}$ . So you need $4 \times \frac{11}{8} = \frac{44}{8} = 5\frac{1}{2}$ pounds total. Now subtract the clay used from your starting amount: $7 - 5\frac{1}{2} = 1\frac{1}{2}$ pounds remaining. This confirms answer choice B. Let's examine why the other answers are incorrect. Choice A ( $1\frac{1}{4}$ lb) results from a calculation error, likely miscalculating $4 \times 1\frac{3}{8}$ as $5\frac{3}{4}$ instead of $5\frac{1}{2}$ . Choice C ( $1\frac{3}{4}$ lb) comes from calculating $4 \times 1\frac{3}{8}$ as $5\frac{1}{4}$ , another computational mistake with fraction multiplication. Choice D (2 lb) suggests the error of treating $1\frac{3}{8}$ as approximately 1.25 and calculating $4 \times 1.25 = 5$ , then $7 - 5 = 2$ . When working with mixed numbers in multi-step problems, always convert to improper fractions first to avoid calculation errors. Double-check your arithmetic by converting your final answer back to verify it makes sense in context.
A rectangular piece of fabric has length $2\frac{2}{3}$ feet and width $1\frac{5}{8}$ feet. If $\frac{2}{5}$ of this fabric is used for a project, what is the area of the fabric that remains unused?
Answer: $2\frac{3}{5}$ square feet of fabric remain unused This problem tests your ability to work with mixed numbers and apply fraction operations in a multi-step area problem. When you see questions involving "remaining" amounts, always think: find the total, then subtract what's used. First, find the total area of the fabric by multiplying length × width. Convert the mixed numbers to improper fractions: $2\frac{2}{3} = \frac{8}{3}$ and $1\frac{5}{8} = \frac{13}{8}$ . The total area is $\frac{8}{3} \times \frac{13}{8} = \frac{104}{24} = \frac{13}{3}$ square feet. Since $\frac{2}{5}$ of the fabric is used, the remaining fraction is $1 - \frac{2}{5} = \frac{3}{5}$ . The unused area is $\frac{3}{5} \times \frac{13}{3} = \frac{39}{15} = \frac{13}{5} = 2\frac{3}{5}$ square feet, confirming answer C. Looking at the wrong answers: Choice A ( $2\frac{7}{15}$ ) likely comes from incorrectly adding fractions instead of finding $\frac{3}{5}$ of the total area. Choice B ( $1\frac{17}{30}$ ) suggests errors in both the area calculation and the fraction of fabric remaining. Choice D ( $2\frac{3}{8}$ ) appears to result from calculation mistakes when converting between mixed numbers and improper fractions. Strategy tip: In multi-step fraction problems, convert all mixed numbers to improper fractions early to avoid errors. Always double-check that your "remaining" fraction plus the "used" fraction equals 1 before proceeding with calculations.
A hiker has completed $4\dfrac{1}{2}$ miles of a trail that is $5\dfrac{3}{4}$ miles long. How many miles of the trail does the hiker still need to walk?
Answer: $1\dfrac{1}{4}$ miles To find the distance left, subtract the distance walked from the trail's length: $5\frac{3}{4} - 4\frac{1}{2}$ . Rewrite $4\frac{1}{2}$ as $4\frac{2}{4}$ so the fractions share a denominator. Subtract the whole numbers, $5 - 4 = 1$ , and the fractions, $\frac{3}{4} - \frac{2}{4} = \frac{1}{4}$ . The hiker has $1\frac{1}{4}$ miles left. Check by adding: $1\frac{1}{4} + 4\frac{2}{4} = 5\frac{3}{4}$ . The value $10\frac{1}{4}$ comes from adding the two distances instead of subtracting. The value $1\frac{3}{4}$ subtracts the whole numbers but keeps the $\frac{3}{4}$ without subtracting $\frac{1}{2}$ ; adding it back to $4\frac{1}{2}$ gives $6\frac{1}{4}$ , too long. The value $\frac{3}{4}$ drops the whole-number part; adding it back gives only $5\frac{1}{4}$ .
A trip consisted of $1\dfrac{3}{4}$ hours by bus followed by $2\dfrac{2}{5}$ hours by train. What was the total travel time?
Answer: $4\dfrac{3}{20}$ hours When you encounter mixed numbers in addition problems, you need to add the whole number parts and fractional parts separately, then combine them into a single mixed number. To find the total travel time, add $1\frac{3}{4} + 2\frac{2}{5}$ . First, add the whole numbers: $1 + 2 = 3$ . Next, add the fractions $\frac{3}{4} + \frac{2}{5}$ . Since these have different denominators, find the least common denominator (LCD). The LCD of 4 and 5 is 20. Convert each fraction: $\frac{3}{4} = \frac{15}{20}$ and $\frac{2}{5} = \frac{8}{20}$ . Now add: $\frac{15}{20} + \frac{8}{20} = \frac{23}{20}$ . Since $\frac{23}{20} = 1\frac{3}{20}$ , you have $3 + 1\frac{3}{20} = 4\frac{3}{20}$ hours. Choice A ( $4\frac{3}{20}$ ) is correct. Choice B ( $4\frac{1}{10}$ ) results from incorrectly adding $\frac{3}{4} + \frac{2}{5}$ as $\frac{5}{9}$ or making an error when converting to twentieths. Choice C ( $3\frac{3}{5}$ ) happens if you mistakenly convert $\frac{3}{4}$ to $\frac{3}{5}$ instead of finding the proper common denominator. Choice D ( $5\frac{1}{20}$ ) occurs if you add an extra hour somewhere in your calculation. Remember: when adding mixed numbers, always find the LCD for the fractions, and if your fractional sum exceeds 1, convert the improper fraction to a mixed number and add the whole number part to your running total.
A water tank is $\frac{2}{3}$ full. After using $\frac{1}{4}$ of the water currently in the tank, what fraction of the tank's total capacity remains?
Answer: $\frac{1}{2}$ of the tank's capacity The tank starts $\frac{2}{3}$ full. Using $\frac{1}{4}$ of the current water means using $\frac{1}{4} \times \frac{2}{3} = \frac{2}{12} = \frac{1}{6}$ of the tank's total capacity. Water remaining: $\frac{2}{3} - \frac{1}{6} = \frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$ of the tank's capacity. Choice A results from incorrectly calculating $\frac{2}{3} - \frac{1}{4} = \frac{8}{12} - \frac{3}{12} = \frac{5}{12}$ . Choice C adds instead of subtracting: $\frac{2}{3} + \frac{1}{6} = \frac{7}{12}$ . Choice D results from $\frac{2}{3} - \frac{1}{12} = \frac{8}{12} - \frac{1}{12} = \frac{7}{12}$ , then incorrectly converting.
A marinade recipe uses $\dfrac{3}{4}$ cup of oil for $2\dfrac{1}{2}$ pounds of meat. How many cups of oil are needed per pound of meat?
Answer: $\dfrac{3}{10}$ cup This is a unit rate problem where you need to find how much oil is required for each pound of meat. When you see "per pound" or "per unit," you're looking for a rate, which means dividing one quantity by another. To find cups of oil per pound of meat, divide the total oil by the total meat: $\dfrac{3}{4} \div 2\dfrac{1}{2}$ . First, convert the mixed number to an improper fraction: $2\dfrac{1}{2} = \dfrac{5}{2}$ . Now you have $\dfrac{3}{4} \div \dfrac{5}{2}$ . To divide fractions, multiply by the reciprocal: $\dfrac{3}{4} \times \dfrac{2}{5} = \dfrac{6}{20} = \dfrac{3}{10}$ . Looking at the wrong answers: Choice B ( $\dfrac{3}{8}$ ) results from incorrectly multiplying $\dfrac{3}{4} \times \dfrac{1}{2}$ instead of dividing by $2\dfrac{1}{2}$ . Choice C ( $\dfrac{2}{5}$ ) comes from flipping the division, calculating $2\dfrac{1}{2} \div \dfrac{3}{4}$ instead. Choice D ( $\dfrac{5}{6}$ ) appears when students add the fractions incorrectly or make computation errors with mixed numbers. The correct answer is A: $\dfrac{3}{10}$ cup per pound. Strategy tip: For unit rate problems, always divide the "thing you want per unit" by the "number of units." Set up your division as $\dfrac{\text{what you want}}{\text{per what}}$ , and remember that dividing by a mixed number requires converting it to an improper fraction first.
A recipe calls for $\dfrac{3}{4}$ cup of sugar for one batch of muffins. If Maya wants to make $\dfrac{5}{2}$ batches, how many cups of sugar will she need in total?
Answer: $1\dfrac{7}{8}$ cups When you encounter a word problem involving fractions and multiplication, the key is recognizing that "of" typically means multiply. Here, you need to find how much sugar is required for multiple batches. To solve this, multiply the sugar per batch by the number of batches: $\frac{3}{4} \times \frac{5}{2}$ . When multiplying fractions, multiply the numerators together and the denominators together: $\frac{3 \times 5}{4 \times 2} = \frac{15}{8}$ . To convert this improper fraction to a mixed number, divide 15 by 8: $15 ÷ 8 = 1$ remainder $7$ , so $\frac{15}{8} = 1\frac{7}{8}$ cups. Looking at the wrong answers: Choice B ( $1\frac{5}{8}$ ) represents a common error where students might incorrectly multiply $\frac{3}{4} \times \frac{5}{2}$ as $\frac{3 \times 5}{4 \times 2}$ but then confuse the numerator, getting $\frac{13}{8}$ instead of $\frac{15}{8}$ . Choice C ( $1\frac{3}{4}$ ) could result from adding instead of multiplying: $\frac{3}{4} + \frac{5}{2} = \frac{3}{4} + \frac{10}{4} = \frac{13}{4} = 1\frac{1}{4}$ (though this doesn't match exactly, it's in the ballpark of this incorrect approach). Choice D ( $1\frac{1}{2}$ ) might come from incorrectly simplifying or miscalculating the fraction arithmetic. The correct answer is A: $1\frac{7}{8}$ cups. Strategy tip: In fraction word problems, always double-check your arithmetic by converting your final mixed number back to an improper fraction to verify your multiplication was correct.
In a school fundraiser, Team A sold $\frac{3}{8}$ of the total tickets, Team B sold $\frac{1}{3}$ of the remaining tickets, and Team C sold the rest. What fraction of the total tickets did Team C sell?
Answer: $\frac{5}{12}$ of the total tickets After Team A sold $\frac{3}{8}$ , the remaining fraction is $1 - \frac{3}{8} = \frac{5}{8}$ . Team B sold $\frac{1}{3}$ of this remaining amount: $\frac{1}{3} \times \frac{5}{8} = \frac{5}{24}$ . Total sold by A and B: $\frac{3}{8} + \frac{5}{24} = \frac{9}{24} + \frac{5}{24} = \frac{14}{24} = \frac{7}{12}$ . Team C sold the rest: $1 - \frac{7}{12} = \frac{5}{12}$ . Choice B incorrectly adds all fractions: $\frac{3}{8} + \frac{1}{3} + \text{remaining}$ . Choice C represents what Team B actually sold. Choice D results from calculation errors in the subtraction.

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