Middle School Math Quiz: Compute Unit Rates With Fractions
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Compute Unit Rates With FractionsQuestion 1 of 20

A bag of apples costs \\tfrac{3}{4}forfor\tfrac{1}{2}$ pound. What is the unit price in dollars per pound?

\\tfrac{3}{2}$ per pound
\\tfrac{2}{3}$ per pound
\\tfrac{1}{2}$ per pound
\\tfrac{3}{8}$ per pound
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Middle School Math Quiz

Middle School Math Quiz: Compute Unit Rates With Fractions

Practice Compute Unit Rates With Fractions in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Compute Unit Rates With Fractions, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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.

All questions

Question 1

A bag of apples costs \\tfrac{3}{4}forfor\tfrac{1}{2}$ pound. What is the unit price in dollars per pound?

  1. \\tfrac{3}{2}$ per pound (correct answer)
  2. \\tfrac{2}{3}$ per pound
  3. \\tfrac{1}{2}$ per pound
  4. \\tfrac{3}{8}$ per pound
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: ( 34\frac{3}{4} dollar ) / ( 12\frac{1}{2} pound ) simplified using reciprocal ( 34\frac{3}{4} ) × ( 21\frac{2}{1} ) with units. Unit rate: amount per ONE unit of denominator (miles per 1 hour, cups per 1 batch). From fractional ratio: ( 34\frac{3}{4} dollar ) / ( 12\frac{1}{2} pound ) is complex fraction ( 34\frac{3}{4} ) / ( 12\frac{1}{2} ), simplify by dividing fractions: ( 34\frac{3}{4} ) ÷ ( 12\frac{1}{2} ) = ( 34\frac{3}{4} ) × ( 21\frac{2}{1} ) = 64\frac{6}{4} = 32\frac{3}{2} dollars per pound (multiply by reciprocal of denominator, simplify). Interpretation: 32\frac{3}{2} dollars per pound means for each 1 pound costs 32\frac{3}{2} dollars (per-unit meaning). In this example, bag costs 34\frac{3}{4} dollar for 12\frac{1}{2} pound, calculate ( 34\frac{3}{4} ) / ( 12\frac{1}{2} ): invert 12\frac{1}{2} to 21\frac{2}{1}, multiply ( 34\frac{3}{4} ) × ( 21\frac{2}{1} ) = 64\frac{6}{4}, simplify to 32\frac{3}{2}, units: dollars per pound = 32\frac{3}{2}. The correct complex fraction division gives the unit rate of 32\frac{3}{2} dollars per pound. Common errors include multiplying fractions instead of dividing ( ( 34\frac{3}{4} ) × ( 12\frac{1}{2} ) = 38\frac{3}{8} wrong operation ), using reciprocal of wrong fraction, arithmetic wrong ( 64\frac{6}{4} = 1.2 not fraction ), dividing backwards ( ( 12\frac{1}{2} ) / ( 34\frac{3}{4} ) = 23\frac{2}{3} reversed ), or units inverted (pounds per dollar). Steps: (1) identify ratio ( 34\frac{3}{4} dollar per 12\frac{1}{2} pound ), (2) write as complex fraction ( ( 34\frac{3}{4} ) / ( 12\frac{1}{2} ) ), (3) convert division to multiplication ( ÷ ( 12\frac{1}{2} ) = × ( 21\frac{2}{1} ) ), (4) multiply fractions ( ( 34\frac{3}{4} ) × ( 21\frac{2}{1} ) = 64\frac{6}{4} ), (5) simplify ( 64\frac{6}{4} = 32\frac{3}{2} ), (6) include units ( 32\frac{3}{2} dollars per pound ).

Question 2

A science club uses 12\tfrac{1}{2} liter of solution to fill 14\tfrac{1}{4} of a container. How many liters of solution are needed to fill 1 whole container at the same rate?

  1. 34\tfrac{3}{4} liter per container
  2. 12\tfrac{1}{2} liter per container
  3. 22 liters per container (correct answer)
  4. 18\tfrac{1}{8} liter per container
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (a/b)/(c/d) simplified using reciprocal (a/b)×(d/c) with units. Unit rate means the amount per one unit of the denominator, such as miles per 1 hour or cups per 1 batch; for example, from a fractional ratio like (1/2 mile)/(1/4 hour), form the complex fraction (1/2)/(1/4) and simplify by dividing fractions: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2 miles per hour, meaning the traveler covers 2 miles in each hour. For instance, if someone walks 1/2 mile in 1/4 hour, calculate (1/2)/(1/4) by inverting 1/4 to 4/1 and multiplying (1/2) × (4/1) = 4/2 = 2 miles per hour; similarly, a recipe using 2/3 cup per 1/3 batch gives (2/3)/(1/3) = (2/3) × (3/1) = 6/3 = 2 cups per batch. Here, 1/2 liter fills 1/4 container, so liters per container is (1/2)/(1/4) = (1/2) × (4/1) = 4/2 = 2 liters per container. Errors include multiplying (1/2) × (1/4) = 1/8, incorrect reciprocal, arithmetic like 4/2 = 1, backwards (1/4)/(1/2) = 1/2, or units as containers per liter. Solve: identify (1/2 liter per 1/4 container), write (1/2)/(1/4), ÷ (1/4) = × 4, (1/2) × 4 = 2, simplify, units: 2 liters per container. Division for 'per', reciprocal method key, compare to 1 liter per container to determine more needed.

Question 3

A store sells 34\tfrac{3}{4} pound of grapes for 12\tfrac{1}{2} dollar. What is the unit price in dollars per pound?​​

  1. 23\tfrac{2}{3} dollars per pound
  2. 23\tfrac{2}{3} dollar per pound (correct answer)
  3. 38\tfrac{3}{8} dollar per pound
  4. 14\tfrac{1}{4} dollar per pound
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (a/b)/(c/d) simplified using reciprocal (a/b)×(d/c) with units. Unit rate means the amount per one unit of the denominator, such as miles per 1 hour or cups per 1 batch; for example, from a fractional ratio like (1/2 mile)/(1/4 hour), form the complex fraction (1/2)/(1/4) and simplify by dividing fractions: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2 miles per hour, meaning the traveler covers 2 miles in each hour. For instance, if someone walks 1/2 mile in 1/4 hour, calculate (1/2)/(1/4) by inverting 1/4 to 4/1 and multiplying (1/2) × (4/1) = 4/2 = 2 miles per hour; similarly, a recipe using 2/3 cup per 1/3 batch gives (2/3)/(1/3) = (2/3) × (3/1) = 6/3 = 2 cups per batch. For this store, 1/2 dollar for 3/4 pound means dollars per pound is (1/2)/(3/4) = (1/2) × (4/3) = 4/6 = 2/3 dollar per pound. Errors might include multiplying (1/2) × (3/4) = 3/8, wrong reciprocal, arithmetic like 4/6 = 2/2 = 1, backwards division (3/4)/(1/2) = 3/2, or units as pounds per dollar. Solve by identifying ratio (1/2 dollar per 3/4 pound), writing (1/2)/(3/4), converting ÷ (3/4) = × (4/3), multiplying (1/2) × (4/3) = 4/6, simplifying to 2/3, adding units: 2/3 dollar per pound. Remember 'per' as division, use reciprocal for simplification, and compare to another rate like 1/2 dollar per pound to see which is cheaper.

Question 4

A student buys 12\tfrac{1}{2} pound of trail mix for 34\tfrac{3}{4} dollar. What is the unit price in dollars per pound?​​

  1. 38\tfrac{3}{8} dollar per pound
  2. 12\tfrac{1}{2} dollar per pound
  3. 32\tfrac{3}{2} dollars per pound (correct answer)
  4. 23\tfrac{2}{3} dollar per pound
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (a/b)/(c/d) simplified using reciprocal (a/b)×(d/c) with units. Unit rate means the amount per one unit of the denominator, such as miles per 1 hour or cups per 1 batch; for example, from a fractional ratio like (1/2 mile)/(1/4 hour), form the complex fraction (1/2)/(1/4) and simplify by dividing fractions: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2 miles per hour, meaning the traveler covers 2 miles in each hour. For instance, if someone walks 1/2 mile in 1/4 hour, calculate (1/2)/(1/4) by inverting 1/4 to 4/1 and multiplying (1/2) × (4/1) = 4/2 = 2 miles per hour; similarly, a recipe using 2/3 cup per 1/3 batch gives (2/3)/(1/3) = (2/3) × (3/1) = 6/3 = 2 cups per batch. Student buys 3/4 dollar for 1/2 pound, so dollars per pound is (3/4)/(1/2) = (3/4) × (2/1) = 6/4 = 3/2 dollars per pound. Errors: (3/4) × (1/2) = 3/8, wrong reciprocal, 6/4 = 1.25 not 1.5, backwards (1/2)/(3/4) = 2/3, or pounds per dollar. Solve: identify (3/4 dollar per 1/2 pound), (3/4)/(1/2), ÷ (1/2) = × 2, (3/4) × 2 = 3/2, simplify, units: 3/2 dollars per pound. Division for 'per', reciprocal key, compare to 1 dollar per pound to assess value.

Question 5

A science club grows 32\tfrac{3}{2} pounds of tomatoes from a garden plot that is 14\tfrac{1}{4} of an acre. What is the yield in pounds per acre?

  1. 16\tfrac{1}{6} pound per acre
  2. 38\tfrac{3}{8} pound per acre
  3. 66 pounds per acre (correct answer)
  4. 52\tfrac{5}{2} pounds per acre
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (3/2 pound)/(1/4 acre) simplified using reciprocal (3/2)×(4/1) with units. Unit rate: amount per ONE unit of denominator (miles per 1 hour, cups per 1 batch). From fractional ratio: (3/2 pound)/(1/4 acre) is complex fraction (3/2)/(1/4), simplify by dividing fractions: (3/2)÷(1/4)=(3/2)×(4/1)=12/2=6 pounds per acre (multiply by reciprocal of denominator, simplify). Interpretation: 6 pounds per acre means from each 1 acre grows 6 pounds (per-unit meaning). In this example, grows 3/2 pounds from 1/4 acre, calculate (3/2)/(1/4): invert 1/4 to 4/1, multiply (3/2)×(4/1)=12/2, simplify to 6, units: pounds per acre = 6. The correct complex fraction division gives the unit rate of 6 pounds per acre. Common errors include multiplying fractions instead of dividing ((3/2)×(1/4)=3/8 wrong operation), using reciprocal of wrong fraction, arithmetic wrong (12/2=5), dividing backwards ((1/4)/(3/2)=1/6 reversed), or units inverted (acres per pound). Steps: (1) identify ratio (3/2 pound per 1/4 acre), (2) write as complex fraction ((3/2)/(1/4)), (3) convert division to multiplication (÷(1/4)=×(4/1)), (4) multiply fractions ((3/2)×(4/1)=12/2), (5) simplify (12/2=6), (6) include units (6 pounds per acre).

Question 6

A student buys 12\tfrac{1}{2} pound of trail mix for 34\tfrac{3}{4} dollar. What is the unit price in dollars per pound?

  1. 23\tfrac{2}{3} dollar per pound
  2. 38\tfrac{3}{8} dollar per pound
  3. 32\tfrac{3}{2} dollars per pound (correct answer)
  4. 12\tfrac{1}{2} dollar per pound
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (a/b)/(c/d) simplified using reciprocal (a/b)×(d/c) with units. Unit rate means the amount per one unit of the denominator, such as miles per 1 hour or cups per 1 batch; for example, from a fractional ratio like (1/2 mile)/(1/4 hour), form the complex fraction (1/2)/(1/4) and simplify by dividing fractions: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2 miles per hour, meaning the traveler covers 2 miles in each hour. For instance, if someone walks 1/2 mile in 1/4 hour, calculate (1/2)/(1/4) by inverting 1/4 to 4/1 and multiplying (1/2) × (4/1) = 4/2 = 2 miles per hour; similarly, a recipe using 2/3 cup per 1/3 batch gives (2/3)/(1/3) = (2/3) × (3/1) = 6/3 = 2 cups per batch. Student buys 3/4 dollar for 1/2 pound, so dollars per pound is (3/4)/(1/2) = (3/4) × (2/1) = 6/4 = 3/2 dollars per pound. Errors: (3/4) × (1/2) = 3/8, wrong reciprocal, 6/4 = 1.25 not 1.5, backwards (1/2)/(3/4) = 2/3, or pounds per dollar. Solve: identify (3/4 dollar per 1/2 pound), (3/4)/(1/2), ÷ (1/2) = × 2, (3/4) × 2 = 3/2, simplify, units: 3/2 dollars per pound. Division for 'per', reciprocal key, compare to 1 dollar per pound to assess value.

Question 7

A recipe uses 23\tfrac{2}{3} cup of flour for 13\tfrac{1}{3} of a batch. How many cups of flour are needed per 1 full batch?

  1. 29\tfrac{2}{9} cup per batch
  2. 22 cups per batch (correct answer)
  3. 12\tfrac{1}{2} cup per batch
  4. 32\tfrac{3}{2} cups per batch
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (2/3 cup)/(1/3 batch) simplified using reciprocal (2/3)×(3/1) with units. Unit rate means the amount per one unit of the denominator, such as cups per 1 batch. From the fractional ratio: (2/3 cup)/(1/3 batch) is a complex fraction (2/3)/(1/3), simplify by dividing fractions: (2/3)÷(1/3)=(2/3)×(3/1)=6/3=2 cups per batch (multiply by reciprocal of denominator, simplify). Interpretation: 2 cups per batch means for each full batch, 2 cups of flour are needed (per-unit meaning). A common error is multiplying fractions instead of dividing, like (2/3)×(1/3)=2/9, which is wrong, or using the reciprocal of the wrong fraction, leading to 1/2 or other errors. Steps: (1) identify ratio (2/3 cup per 1/3 batch), (2) write as complex fraction ((2/3)/(1/3)), (3) convert division to multiplication (÷(1/3)=×(3/1)), (4) multiply ((2/3)×(3/1)=6/3), (5) simplify to 2, (6) include units (2 cups per batch). Understanding: 'per' means division, so (2/3) per (1/3) = (2/3)÷(1/3), and this scales up the partial amount to a full batch correctly.

Question 8

A store charges 34\tfrac{3}{4} dollar for 12\tfrac{1}{2} pound of grapes. What is the cost per pound?

  1. 32\tfrac{3}{2} dollars per pound (correct answer)
  2. 38\tfrac{3}{8} dollar per pound
  3. 12\tfrac{1}{2} dollar per pound
  4. 23\tfrac{2}{3} dollar per pound
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (3/4 dollar)/(1/2 pound) simplified using reciprocal (3/4)×(2/1) with units. Unit rate means the amount per one unit of the denominator, such as dollars per 1 pound. From the fractional ratio: (3/4 dollar)/(1/2 pound) is a complex fraction (3/4)/(1/2), simplify by dividing fractions: (3/4)÷(1/2)=(3/4)×(2/1)=6/4=3/2 dollars per pound (multiply by reciprocal of denominator, simplify). Interpretation: 3/2 dollars per pound means for each pound, the cost is 1.5 dollars (per-unit meaning). A common error is dividing backwards, like (1/2)/(3/4)=2/3, which inverts the units to pounds per dollar, or arithmetic mistakes like 6/4=3/4 instead of 3/2. Steps: (1) identify ratio (3/4 dollar per 1/2 pound), (2) write as complex fraction ((3/4)/(1/2)), (3) convert division to multiplication (÷(1/2)=×(2/1)), (4) multiply ((3/4)×(2/1)=6/4), (5) simplify to 3/2, (6) include units (3/2 dollars per pound). Understanding: 'per' means division, so cost per pound requires dividing cost by weight, ensuring the unit rate is correctly oriented.

Question 9

A container holds 34\tfrac{3}{4} liter of juice. If that amount fills 13\tfrac{1}{3} of the container, what is the container's full capacity in liters?

  1. 94\tfrac{9}{4} liters (correct answer)
  2. 14\tfrac{1}{4} liter
  3. 11 ⁣2\tfrac{1}{1\!2} liters
  4. 19\tfrac{1}{9} liter
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (3/4 liter)/(1/3 full) simplified using reciprocal (3/4)×(3/1) with units. Unit rate: amount per ONE unit of denominator (miles per 1 hour, cups per 1 batch). From fractional ratio: (3/4 liter)/(1/3 full) is complex fraction (3/4)/(1/3), simplify by dividing fractions: (3/4)÷(1/3)=(3/4)×(3/1)=9/4 liters per full (multiply by reciprocal of denominator, simplify). Interpretation: 9/4 liters means the full capacity is 9/4 liters (per-unit meaning). In this example, 3/4 liter fills 1/3 of container, calculate (3/4)/(1/3): invert 1/3 to 3/1, multiply (3/4)×(3/1)=9/4, units: liters = 9/4 liters full capacity. The correct complex fraction division gives the unit rate of 9/4 liters. Common errors include multiplying fractions instead of dividing ((3/4)×(1/3)=1/4 wrong operation), using reciprocal of wrong fraction, arithmetic wrong (9/4=2.5 not fraction), dividing backwards ((1/3)/(3/4)=4/9 not matching), or units missing. Steps: (1) identify ratio (3/4 liter per 1/3 full), (2) write as complex fraction ((3/4)/(1/3)), (3) convert division to multiplication (÷(1/3)=×(3/1)), (4) multiply fractions ((3/4)×(3/1)=9/4), (5) simplify (9/4), (6) include units (9/4 liters).

Question 10

A recipe uses 23\tfrac{2}{3} cup of sugar to make 16\tfrac{1}{6} of a batch of cookies. How many cups of sugar are used per 1 full batch?

  1. 14\tfrac{1}{4} cup per batch
  2. 23\tfrac{2}{3} cup per batch
  3. 19\tfrac{1}{9} cup per batch
  4. 44 cups per batch (correct answer)
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (2/3 cup)/(1/6 batch) simplified using reciprocal (2/3)×(6/1) with units. Unit rate: amount per ONE unit of denominator (miles per 1 hour, cups per 1 batch). From fractional ratio: (2/3 cup)/(1/6 batch) is complex fraction (2/3)/(1/6), simplify by dividing fractions: (2/3)÷(1/6)=(2/3)×(6/1)=12/3=4 cups per batch (multiply by reciprocal of denominator, simplify). Interpretation: 4 cups per batch means for each 1 batch uses 4 cups (per-unit meaning). In this example, recipe uses 2/3 cup for 1/6 batch: (2/3)/(1/6)=(2/3)×(6/1)=12/3=4 cups per batch. The correct complex fraction division gives the unit rate of 4 cups per batch. Common errors include multiplying fractions instead of dividing ((2/3)×(1/6)=2/18=1/9 wrong operation), using reciprocal of wrong fraction, arithmetic wrong (12/3=3), dividing backwards ((1/6)/(2/3)=1/4 reversed), or units inverted (batches per cup). Steps: (1) identify ratio (2/3 cup per 1/6 batch), (2) write as complex fraction ((2/3)/(1/6)), (3) convert division to multiplication (÷(1/6)=×(6/1)), (4) multiply fractions ((2/3)×(6/1)=12/3), (5) simplify (12/3=4), (6) include units (4 cups per batch).

Question 11

A student jogs 34\tfrac{3}{4} mile in 12\tfrac{1}{2} hour. What is the student's speed in miles per hour (mph)?

  1. 38\tfrac{3}{8} mph
  2. 16\tfrac{1}{6} mph
  3. 23\tfrac{2}{3} mph
  4. 32\tfrac{3}{2} mph (correct answer)
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (3/4 mile)/(1/2 hour) simplified using reciprocal (3/4)×(2/1) with units. Unit rate: amount per ONE unit of denominator (miles per 1 hour, cups per 1 batch). From fractional ratio: (3/4 mile)/(1/2 hour) is complex fraction (3/4)/(1/2), simplify by dividing fractions: (3/4)÷(1/2)=(3/4)×(2/1)=6/4=3/2 miles per hour (multiply by reciprocal of denominator, simplify). Interpretation: 3/2 mph means in each 1 hour jogs 3/2 miles (per-unit meaning). In this example, student jogs 3/4 mile in 1/2 hour, calculate (3/4)/(1/2): invert 1/2 to 2/1, multiply (3/4)×(2/1)=6/4, simplify to 3/2, units: miles per hour = 3/2 mph. The correct complex fraction division gives the unit rate of 3/2 mph. Common errors include multiplying fractions instead of dividing ((3/4)×(1/2)=3/8 wrong operation), using reciprocal of wrong fraction, arithmetic wrong (6/4=3/3=1), dividing backwards ((1/2)/(3/4)=2/3 reversed), or units inverted (hours per mile not mph). Steps: (1) identify ratio (3/4 mile per 1/2 hour), (2) write as complex fraction ((3/4)/(1/2)), (3) convert division to multiplication (÷(1/2)=×(2/1)), (4) multiply fractions ((3/4)×(2/1)=6/4), (5) simplify (6/4=3/2), (6) include units (3/2 miles per hour).

Question 12

A gardener harvests 32\tfrac{3}{2} pounds of tomatoes from 14\tfrac{1}{4} of a garden bed. At this rate, how many pounds of tomatoes would be harvested from 1 whole garden bed?​​

  1. 56\tfrac{5}{6} pound per bed
  2. 16\tfrac{1}{6} pound per bed
  3. 66 pounds per bed (correct answer)
  4. 38\tfrac{3}{8} pound per bed
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (a/b)/(c/d) simplified using reciprocal (a/b)×(d/c) with units. Unit rate means the amount per one unit of the denominator, such as miles per 1 hour or cups per 1 batch; for example, from a fractional ratio like (1/2 mile)/(1/4 hour), form the complex fraction (1/2)/(1/4) and simplify by dividing fractions: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2 miles per hour, meaning the traveler covers 2 miles in each hour. For instance, if someone walks 1/2 mile in 1/4 hour, calculate (1/2)/(1/4) by inverting 1/4 to 4/1 and multiplying (1/2) × (4/1) = 4/2 = 2 miles per hour; similarly, a recipe using 2/3 cup per 1/3 batch gives (2/3)/(1/3) = (2/3) × (3/1) = 6/3 = 2 cups per batch. Gardener harvests 3/2 pounds from 1/4 bed, so pounds per bed is (3/2)/(1/4) = (3/2) × (4/1) = 12/2 = 6 pounds per bed. Mistakes: (3/2) × (1/4) = 3/8, wrong reciprocal, 12/2 = 5, backwards (1/4)/(3/2) = 1/6, or beds per pound. Steps: identify (3/2 pounds per 1/4 bed), (3/2)/(1/4), ÷ (1/4) = × 4, (3/2) × 4 = 6, simplify, units: 6 pounds per bed. 'Per' as division, reciprocal method, compare to 5 pounds per bed to see higher yield.

Question 13

A gardener harvests 32\tfrac{3}{2} pounds of tomatoes from 14\tfrac{1}{4} of a garden bed. At this rate, how many pounds of tomatoes would be harvested from 1 whole garden bed?

  1. 56\tfrac{5}{6} pound per bed
  2. 66 pounds per bed (correct answer)
  3. 16\tfrac{1}{6} pound per bed
  4. 38\tfrac{3}{8} pound per bed
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (a/b)/(c/d) simplified using reciprocal (a/b)×(d/c) with units. Unit rate means the amount per one unit of the denominator, such as miles per 1 hour or cups per 1 batch; for example, from a fractional ratio like (1/2 mile)/(1/4 hour), form the complex fraction (1/2)/(1/4) and simplify by dividing fractions: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2 miles per hour, meaning the traveler covers 2 miles in each hour. For instance, if someone walks 1/2 mile in 1/4 hour, calculate (1/2)/(1/4) by inverting 1/4 to 4/1 and multiplying (1/2) × (4/1) = 4/2 = 2 miles per hour; similarly, a recipe using 2/3 cup per 1/3 batch gives (2/3)/(1/3) = (2/3) × (3/1) = 6/3 = 2 cups per batch. Gardener harvests 3/2 pounds from 1/4 bed, so pounds per bed is (3/2)/(1/4) = (3/2) × (4/1) = 12/2 = 6 pounds per bed. Mistakes: (3/2) × (1/4) = 3/8, wrong reciprocal, 12/2 = 5, backwards (1/4)/(3/2) = 1/6, or beds per pound. Steps: identify (3/2 pounds per 1/4 bed), (3/2)/(1/4), ÷ (1/4) = × 4, (3/2) × 4 = 6, simplify, units: 6 pounds per bed. 'Per' as division, reciprocal method, compare to 5 pounds per bed to see higher yield.

Question 14

A snack mix uses 34\tfrac{3}{4} cup of raisins for 12\tfrac{1}{2} of a batch. How many cups of raisins are needed for 1 whole batch?

  1. 38\tfrac{3}{8} cup per batch
  2. 32\tfrac{3}{2} cups per batch (correct answer)
  3. 23\tfrac{2}{3} cup per batch
  4. 14\tfrac{1}{4} cup per batch
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (a/b)/(c/d) simplified using reciprocal (a/b)×(d/c) with units. Unit rate means the amount per one unit of the denominator, such as miles per 1 hour or cups per 1 batch; for example, from a fractional ratio like (1/2 mile)/(1/4 hour), form the complex fraction (1/2)/(1/4) and simplify by dividing fractions: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2 miles per hour, meaning the traveler covers 2 miles in each hour. For instance, if someone walks 1/2 mile in 1/4 hour, calculate (1/2)/(1/4) by inverting 1/4 to 4/1 and multiplying (1/2) × (4/1) = 4/2 = 2 miles per hour; similarly, a recipe using 2/3 cup per 1/3 batch gives (2/3)/(1/3) = (2/3) × (3/1) = 6/3 = 2 cups per batch. The mix uses 3/4 cup for 1/2 batch, so cups per batch is (3/4)/(1/2) = (3/4) × (2/1) = 6/4 = 3/2 cups per batch. Common errors: (3/4) × (1/2) = 3/8, wrong reciprocal, 6/4 = 3/3 = 1, backwards (1/2)/(3/4) = 2/3, or batches per cup. Steps: identify (3/4 cup per 1/2 batch), (3/4)/(1/2), ÷ (1/2) = × 2, (3/4) × 2 = 3/2, simplify, units: 3/2 cups per batch. 'Per' means division, use reciprocal, compare to 1 cup per batch to see more raisins needed.

Question 15

A machine produces 56\frac{5}{6} yard of fabric every 29\frac{2}{9} hour. What is the unit rate in yards per hour?

  1. 527\frac{5}{27} yards per hour at this production rate
  2. 3343\frac{3}{4} yards per hour at this production rate (correct answer)
  3. 1245\frac{12}{45} yards per hour at this production rate
  4. 154\frac{15}{4} yards per hour at this production rate
Explanation: To find yards per hour, compute 5/62/9=56×92=4512=154=334\frac{5/6}{2/9} = \frac{5}{6} \times \frac{9}{2} = \frac{45}{12} = \frac{15}{4} = 3\frac{3}{4} yards per hour. Choice A results from multiplying 56×29\frac{5}{6} \times \frac{2}{9}. Choice C shows an unreduced fraction from incorrect computation. Choice D shows the improper fraction form of the correct answer.

Question 16

A conveyor belt moves 710\frac{7}{10} meter of material in 215\frac{2}{15} minute. The same belt needs to move 8 meters of material. How long will this take?

  1. It will take 16105\frac{16}{105} minute to move the material completely
  2. It will take 11211\frac{1}{21} minutes to move the material completely
  3. It will take 5145\frac{1}{4} minutes to move the material completely
  4. It will take 3221\frac{32}{21} minutes to move the material completely (correct answer)
Explanation: First find the unit rate: 7/102/15=710×152=10520=214\frac{7/10}{2/15} = \frac{7}{10} \times \frac{15}{2} = \frac{105}{20} = \frac{21}{4} meters per minute. To move 8 meters: 821/4=8×421=3221\frac{8}{21/4} = 8 \times \frac{4}{21} = \frac{32}{21} minutes. Choice A results from multiplying the original fractions. Choice B converts 3221\frac{32}{21} incorrectly to mixed number form. Choice C represents a calculation error in the division step.

Question 17

A store sells 34\tfrac{3}{4} pound of grapes for 12\tfrac{1}{2} dollar. What is the unit price in dollars per pound?

  1. 23\tfrac{2}{3} dollar per pound (correct answer)
  2. 38\tfrac{3}{8} dollar per pound
  3. 32\tfrac{3}{2} dollars per pound
  4. 14\tfrac{1}{4} dollar per pound
Explanation: Unit rate is dollars divided by pounds: (1/2)/(3/4) = (1/2)x(4/3) = 4/6 = 2/3 dollar per pound, matching choice A. Choice B multiplies the fractions instead of dividing them. Choice C divides in the reverse order, computing pounds per dollar instead of dollars per pound. Choice D comes from subtracting the fractions instead of dividing them.

Question 18

A runner completes 23\tfrac{2}{3} mile in 16\tfrac{1}{6} hour. What is the runner's speed in miles per hour?​​

  1. 14\tfrac{1}{4} miles per hour
  2. 19\tfrac{1}{9} miles per hour
  3. 29\tfrac{2}{9} miles per hour
  4. 44 miles per hour (correct answer)
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (a/b)/(c/d) simplified using reciprocal (a/b)×(d/c) with units. Unit rate means the amount per one unit of the denominator, such as miles per 1 hour or cups per 1 batch; for example, from a fractional ratio like (1/2 mile)/(1/4 hour), form the complex fraction (1/2)/(1/4) and simplify by dividing fractions: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2 miles per hour, meaning the traveler covers 2 miles in each hour. For instance, if someone walks 1/2 mile in 1/4 hour, calculate (1/2)/(1/4) by inverting 1/4 to 4/1 and multiplying (1/2) × (4/1) = 4/2 = 2 miles per hour; similarly, a recipe using 2/3 cup per 1/3 batch gives (2/3)/(1/3) = (2/3) × (3/1) = 6/3 = 2 cups per batch. The runner completes 2/3 mile in 1/6 hour, so speed is (2/3)/(1/6) = (2/3) × (6/1) = 12/3 = 4 miles per hour. Mistakes: multiplying (2/3) × (1/6) = 2/18 = 1/9, wrong reciprocal, arithmetic 12/3 = 3, backwards (1/6)/(2/3) = 1/4, or hours per mile. Steps: identify (2/3 mile per 1/6 hour), write (2/3)/(1/6), ÷ (1/6) = × 6, (2/3) × 6 = 4, simplify, units: 4 miles per hour. 'Per' as division, reciprocal for complex fractions, compare 4 mph to 3 mph to see faster.

Question 19

A recipe calls for 34\frac{3}{4} cup of flour for every 16\frac{1}{6} cup of sugar. Maria wants to know how many cups of flour she needs per cup of sugar. What is the unit rate of flour to sugar?

  1. 18\frac{1}{8} cup of flour per cup of sugar
  2. 29\frac{2}{9} cup of flour per cup of sugar
  3. 4124\frac{1}{2} cups of flour per cup of sugar (correct answer)
  4. 92\frac{9}{2} cups of flour per cup of sugar
Explanation: To find the unit rate, divide 34\frac{3}{4} by 16\frac{1}{6}: 3/41/6=34×61=184=92=412\frac{3/4}{1/6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = \frac{9}{2} = 4\frac{1}{2}. Choice A results from multiplying the fractions instead of dividing. Choice B comes from incorrectly computing 16÷34\frac{1}{6} \div \frac{3}{4}. Choice D is the improper fraction form but wasn't converted to mixed number form as expected.

Question 20

A student reads 34\tfrac{3}{4} of a chapter in 13\tfrac{1}{3} hour. At this rate, how many chapters can the student read in 1 hour?

  1. 14\tfrac{1}{4} hour per chapter
  2. 14\tfrac{1}{4} chapters per hour
  3. 19\tfrac{1}{9} chapters per hour
  4. 94\tfrac{9}{4} chapters per hour (correct answer)
Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (a/b)/(c/d) simplified using reciprocal (a/b)×(d/c) with units. Unit rate means the amount per one unit of the denominator, such as miles per 1 hour or cups per 1 batch; for example, from a fractional ratio like (1/2 mile)/(1/4 hour), form the complex fraction (1/2)/(1/4) and simplify by dividing fractions: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2 miles per hour, meaning the traveler covers 2 miles in each hour. For instance, if someone walks 1/2 mile in 1/4 hour, calculate (1/2)/(1/4) by inverting 1/4 to 4/1 and multiplying (1/2) × (4/1) = 4/2 = 2 miles per hour; similarly, a recipe using 2/3 cup per 1/3 batch gives (2/3)/(1/3) = (2/3) × (3/1) = 6/3 = 2 cups per batch. Student reads 3/4 chapter in 1/3 hour, so chapters per hour is (3/4)/(1/3) = (3/4) × (3/1) = 9/4 chapters per hour. Errors: (3/4) × (1/3) = 3/12 = 1/4, wrong reciprocal, 9/4 = 2/4 = 1/2, backwards (1/3)/(3/4) = 4/9, or hours per chapter. Solve: identify (3/4 chapter per 1/3 hour), (3/4)/(1/3), ÷ (1/3) = × 3, (3/4) × 3 = 9/4, simplify, units: 9/4 chapters per hour. Division for 'per', reciprocal method, compare to 2 chapters per hour (9/4 = 2.25 > 2).