7th Grade Math Quiz: Compute Unit Rates With Fractions

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 7th Grade 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.

Question 1

1. $\frac{1}{8}$ cup of flour per cup of sugar
2. $\frac{2}{9}$ cup of flour per cup of sugar
3. $4\frac{1}{2}$ cups of flour per cup of sugar (correct answer)
4. $\frac{9}{2}$ cups of flour per cup of sugar

Explanation: To find the unit rate, divide $\frac{3}{4}$ by $\frac{1}{6}$: $\frac{3/4}{1/6} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = \frac{9}{2} = 4\frac{1}{2}$. Choice A results from multiplying the fractions instead of dividing. Choice B comes from incorrectly computing $\frac{1}{6} \div \frac{3}{4}$. Choice D is the improper fraction form but wasn't converted to mixed number form as expected.

Question 2

1. $\frac{5}{27}$ yards per hour at this production rate
2. $3\frac{3}{4}$ yards per hour at this production rate (correct answer)
3. $\frac{12}{45}$ yards per hour at this production rate
4. $\frac{15}{4}$ yards per hour at this production rate

Explanation: To find yards per hour, compute $\frac{5/6}{2/9} = \frac{5}{6} \times \frac{9}{2} = \frac{45}{12} = \frac{15}{4} = 3\frac{3}{4}$ yards per hour. Choice A results from multiplying $\frac{5}{6} \times \frac{2}{9}$. Choice C shows an unreduced fraction from incorrect computation. Choice D shows the improper fraction form of the correct answer.

Question 3

1. It will take $\frac{16}{105}$ minute to move the material completely
2. It will take $1\frac{1}{21}$ minutes to move the material completely
3. It will take $5\frac{1}{4}$ minutes to move the material completely
4. It will take $\frac{32}{21}$ minutes to move the material completely (correct answer)

Explanation: First find the unit rate: $\frac{7/10}{2/15} = \frac{7}{10} \times \frac{15}{2} = \frac{105}{20} = \frac{21}{4}$ meters per minute. To move 8 meters: $\frac{8}{21/4} = 8 \times \frac{4}{21} = \frac{32}{21}$ minutes. Choice A results from multiplying the original fractions. Choice B converts $\frac{32}{21}$ incorrectly to mixed number form. Choice C represents a calculation error in the division step.

Question 4

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

Question 5

1. $\tfrac{3}{4}$ liter per container
2. $\tfrac{1}{2}$ liter per container
3. $2$ liters per container (correct answer)
4. $\tfrac{1}{8}$ 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 6

1. $\tfrac{3}{10}$ wall per hour
2. $\tfrac{6}{5}$ walls per hour (correct answer)
3. $\tfrac{5}{6}$ wall per hour
4. $\tfrac{1}{5}$ wall per hour

Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (3/5 wall)/(1/2 hour) simplified using reciprocal (3/5)×(2/1) with units. Unit rate means the amount per one unit of the denominator, such as walls per 1 hour. From the fractional ratio: (3/5 wall)/(1/2 hour) is a complex fraction (3/5)/(1/2), simplify by dividing fractions: (3/5)÷(1/2)=(3/5)×(2/1)=6/5 walls per hour (multiply by reciprocal of denominator, simplify). Interpretation: 6/5 walls per hour means in each hour, the painter finishes 1.2 walls (per-unit meaning). A common error is multiplying instead of dividing, like (3/5)×(1/2)=3/10, or taking reciprocal incorrectly leading to 5/6. Steps: (1) identify ratio (3/5 wall per 1/2 hour), (2) write as complex fraction ((3/5)/(1/2)), (3) convert division to multiplication (÷(1/2)=×(2/1)), (4) multiply ((3/5)×(2/1)=6/5), (5) simplify (already 6/5), (6) include units (6/5 walls per hour). Understanding: 'per' means division, so walls per hour is walls divided by hours, scaling the partial work to a full hour.

Question 7

1. $\tfrac{2}{3}$ dollars per pound
2. $\tfrac{2}{3}$ dollar per pound (correct answer)
3. $\tfrac{3}{8}$ dollar per pound
4. $\tfrac{1}{4}$ 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 8

1. $\tfrac{3}{8}$ dollar per pound
2. $\tfrac{1}{2}$ dollar per pound
3. $\tfrac{3}{2}$ dollars per pound (correct answer)
4. $\tfrac{2}{3}$ 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 9

1. $\tfrac{3}{2}$ mph
2. $\tfrac{3}{8}$ mph
3. $\tfrac{2}{12}$ mph
4. $\tfrac{8}{3}$ mph (correct answer)

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

Question 10

1. $\tfrac{1}{6}$ pound per acre
2. $\tfrac{3}{8}$ pound per acre
3. $6$ pounds per acre (correct answer)
4. $\tfrac{5}{2}$ 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 11

1. $\tfrac{2}{9}$ miles per hour
2. $\tfrac{1}{4}$ miles per hour
3. $\tfrac{1}{9}$ miles per hour
4. $4$ 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 12

1. $\tfrac{1}{4}$ hour per chapter
2. $\tfrac{1}{4}$ chapters per hour
3. $\tfrac{1}{9}$ chapters per hour
4. $\tfrac{9}{4}$ 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).

Question 13

1. $\tfrac{1}{4}$ cup per batch
2. $\tfrac{2}{3}$ cup per batch
3. $\tfrac{1}{9}$ cup per batch
4. $4$ 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 14

1. $\tfrac{3}{8}$ pound per bed
2. $\tfrac{2}{3}$ pound per bed
3. $6$ pounds per bed (correct answer)
4. $\tfrac{5}{4}$ pounds per bed

Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (3/2 pounds)/(1/4 bed) simplified using reciprocal (3/2)×(4/1) with units. Unit rate means the amount per one unit of the denominator, such as pounds per 1 bed. From the fractional ratio: (3/2 pounds)/(1/4 bed) is a complex fraction (3/2)/(1/4), simplify by dividing fractions: (3/2)÷(1/4)=(3/2)×(4/1)=12/2=6 pounds per bed (multiply by reciprocal of denominator, simplify). Interpretation: 6 pounds per bed means each full garden bed produces 6 pounds of tomatoes (per-unit meaning). A common error is multiplying fractions wrongly, like (3/2)×(1/4)=3/8, or dividing backwards to get 2/3. Steps: (1) identify ratio (3/2 pounds per 1/4 bed), (2) write as complex fraction ((3/2)/(1/4)), (3) convert division to multiplication (÷(1/4)=×(4/1)), (4) multiply ((3/2)×(4/1)=12/2), (5) simplify to 6, (6) include units (6 pounds per bed). Understanding: 'per' means division, so production per bed scales the partial yield to a full bed accurately.

Question 15

1. $\tfrac{2}{3}$ dollar per pound
2. $\tfrac{3}{8}$ dollar per pound
3. $\tfrac{3}{2}$ dollars per pound (correct answer)
4. $\tfrac{1}{2}$ 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 16

1. $\tfrac{5}{6}$ cup per batch
2. $\tfrac{1}{9}$ cup per batch
3. $\tfrac{1}{4}$ cup per batch
4. $4$ cups per batch (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. Here, the recipe uses 2/3 cup for 1/6 batch, so cups per batch is (2/3)/(1/6) = (2/3) × (6/1) = 12/3 = 4 cups per batch. Common errors include multiplying instead of dividing like (2/3) × (1/6) = 2/18 = 1/9, reciprocal of the wrong fraction, arithmetic errors such as 12/3 = 3, dividing backwards as (1/6)/(2/3) = 1/4, or units as batches per cup. Steps: identify ratio (2/3 cup per 1/6 batch), write (2/3)/(1/6), convert ÷ (1/6) = × 6, multiply (2/3) × 6 = 4, simplify to 4, add units: 4 cups per batch. 'Per' means division, so always use the reciprocal method for complex fractions, and compare rates like 4 cups vs 2 cups to see which recipe uses more sugar.

Question 17

1. $\tfrac{1}{2}$ chapter per hour
2. $\tfrac{1}{8}$ chapter per hour
3. $2$ chapters per hour (correct answer)
4. $\tfrac{1}{2}$ hour per chapter

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

Question 18

1. $\tfrac{2}{9}$ cup per batch
2. $2$ cups per batch (correct answer)
3. $\tfrac{1}{2}$ cup per batch
4. $\tfrac{3}{2}$ 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 19

1. $\tfrac{3}{2}$ dollars per pound (correct answer)
2. $\tfrac{3}{8}$ dollar per pound
3. $\tfrac{1}{2}$ dollar per pound
4. $\tfrac{2}{3}$ 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 20

1. $\tfrac{2}{3}$ cartridge per class set
2. $\tfrac{5}{4}$ cartridges per class set
3. $\tfrac{3}{2}$ cartridges per class set (correct answer)
4. $\tfrac{3}{8}$ cartridge per class set

Explanation: This question tests computing unit rates from ratios of fractions by dividing complex fractions: (3/4 cartridge)/(1/2 set) simplified using reciprocal (3/4)×(2/1) with units. Unit rate means the amount per one unit of the denominator, such as cartridges per 1 class set. From the fractional ratio: (3/4 cartridge)/(1/2 set) is a complex fraction (3/4)/(1/2), simplify by dividing fractions: (3/4)÷(1/2)=(3/4)×(2/1)=6/4=3/2 cartridges per set (multiply by reciprocal of denominator, simplify). Interpretation: 3/2 cartridges per set means 1.5 cartridges are used for each full class set (per-unit meaning). A common error is dividing backwards to get 2/3, or multiplying to 3/8. Steps: (1) identify ratio (3/4 cartridge per 1/2 set), (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 cartridges per class set). Understanding: 'per' means division, so cartridges per set scales the partial usage to a full set.