Compute Unit Rates With Fractions - 7th Grade Math

A snack mix uses $\tfrac{3}{4}$ cup of raisins for $\tfrac{1}{2}$ of a batch. How many cups of raisins are needed for 1 whole batch?
Answer: $\tfrac{3}{2}$ cups per batch 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.
A snack mix uses $\tfrac{3}{4}$ cup of raisins for $\tfrac{1}{2}$ of a batch. How many cups of raisins are needed for 1 whole batch?
Answer: $\tfrac{3}{2}$ cups per batch 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.
A recipe uses $\tfrac{2}{3}$ cup of sugar to make $\tfrac{1}{6}$ of a batch of cookies. How many cups of sugar are used per 1 full batch?
Answer: $4$ cups per batch 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).
A painter finishes $\tfrac{3}{5}$ of a wall in $\tfrac{1}{2}$ hour. At this constant rate, how many walls can the painter finish per hour?
Answer: $\tfrac{6}{5}$ walls per hour 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: amount per ONE unit of denominator (miles per 1 hour, cups per 1 batch). From fractional ratio: (3/5 wall)/(1/2 hour) is 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 1 hour finishes 6/5 walls (per-unit meaning). In this example, painter finishes 3/5 wall in 1/2 hour, calculate (3/5)/(1/2): invert 1/2 to 2/1, multiply (3/5)×(2/1)=6/5, units: walls per hour = 6/5. The correct complex fraction division gives the unit rate of 6/5 walls per hour. Common errors include multiplying fractions instead of dividing ((3/5)×(1/2)=3/10 wrong operation), using reciprocal of wrong fraction, arithmetic wrong (6/5=1.5 not fraction), dividing backwards ((1/2)/(3/5)=5/6 reversed), or units inverted (hours per wall). 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 fractions ((3/5)×(2/1)=6/5), (5) simplify (6/5), (6) include units (6/5 walls per hour).
A student bikes $\tfrac{3}{4}$ mile in $\tfrac{1}{2}$ hour. What is the student's speed in miles per hour?
Answer: $\tfrac{3}{2}$ miles per hour 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. In this case, the student bikes 3/4 mile in 1/2 hour, so the speed is (3/4)/(1/2) = (3/4) × (2/1) = 6/4 = 3/2 miles per hour. Common errors include multiplying fractions instead of dividing, like (3/4) × (1/2) = 3/8, using the reciprocal of the wrong fraction, arithmetic mistakes such as 6/4 = 3/3 = 1, dividing backwards as (1/2)/(3/4) = 2/3, or inverting units to hours per mile. To solve, identify the ratio (3/4 mile per 1/2 hour), write as complex fraction (3/4)/(1/2), convert to multiplication by reciprocal ÷ (1/2) = × (2/1), multiply (3/4) × (2/1) = 6/4, simplify to 3/2, and add units: 3/2 miles per hour. Understanding 'per' as division helps, like (3/4) ÷ (1/2) = 3/2, and for comparison, a rate of 3/2 mph is faster than 1 mph since 1.5 > 1.
A science club uses $\tfrac{1}{2}$ liter of solution to fill $\tfrac{1}{4}$ of a container. How many liters of solution are needed to fill 1 whole container at the same rate?
Answer: $2$ liters per container 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.
A bicyclist rides $\tfrac{2}{3}$ mile in $\tfrac{1}{4}$ hour. What is the bicyclist's speed in miles per hour (mph)?
Answer: $\tfrac{8}{3}$ mph 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).
A student reads $\tfrac{3}{4}$ of a chapter in $\tfrac{1}{3}$ hour. At this rate, how many chapters can the student read in 1 hour?
Answer: $\tfrac{9}{4}$ chapters per hour 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).
A science club grows $\tfrac{3}{2}$ pounds of tomatoes from a garden plot that is $\tfrac{1}{4}$ of an acre. What is the yield in pounds per acre?
Answer: $6$ pounds per acre 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).
A student jogs $\tfrac{3}{4}$ mile in $\tfrac{1}{2}$ hour. What is the student's speed in miles per hour (mph)?
Answer: $\tfrac{3}{2}$ mph 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).
A printer uses $\tfrac{3}{4}$ of an ink cartridge to print $\tfrac{1}{2}$ of a class set of posters. What is the unit rate in cartridges per 1 class set of posters?
Answer: $\tfrac{3}{2}$ cartridges per class set 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.
A gardener harvests $\tfrac{3}{2}$ pounds of tomatoes from $\tfrac{1}{4}$ of a garden bed. At this rate, how many pounds of tomatoes would be harvested from 1 whole garden bed?
Answer: $6$ pounds per bed 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.
A store sells $\tfrac{3}{4}$ pound of grapes for $\tfrac{1}{2}$ dollar. What is the unit price in dollars per pound?
Answer: $\tfrac{2}{3}$ dollar per pound 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.
A recipe uses $\tfrac{2}{3}$ cup of sugar to make $\tfrac{1}{6}$ of a batch of cookies. How many cups of sugar are needed for 1 whole batch?
Answer: $4$ cups per batch 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.
A runner completes $\tfrac{2}{3}$ mile in $\tfrac{1}{6}$ hour. What is the runner's speed in miles per hour?
Answer: $4$ miles per hour 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.
A painter finishes $\tfrac{3}{5}$ of a wall in $\tfrac{1}{2}$ hour. At this rate, how many walls can the painter finish per hour?
Answer: $\tfrac{6}{5}$ walls per hour 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.
A machine produces $\frac{5}{6}$ yard of fabric every $\frac{2}{9}$ hour. What is the unit rate in yards per hour?
Answer: $3\frac{3}{4}$ yards per hour at this production rate 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.
A gardener harvests $\tfrac{3}{2}$ pounds of tomatoes from $\tfrac{1}{4}$ of a garden bed. At this rate, how many pounds of tomatoes would be harvested from 1 whole garden bed?
Answer: $6$ pounds per bed 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.
A student bikes $\tfrac{3}{4}$ mile in $\tfrac{1}{2}$ hour. What is the student's speed in miles per hour?
Answer: $\tfrac{3}{2}$ miles per hour 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. In this case, the student bikes 3/4 mile in 1/2 hour, so the speed is (3/4)/(1/2) = (3/4) × (2/1) = 6/4 = 3/2 miles per hour. Common errors include multiplying fractions instead of dividing, like (3/4) × (1/2) = 3/8, using the reciprocal of the wrong fraction, arithmetic mistakes such as 6/4 = 3/3 = 1, dividing backwards as (1/2)/(3/4) = 2/3, or inverting units to hours per mile. To solve, identify the ratio (3/4 mile per 1/2 hour), write as complex fraction (3/4)/(1/2), convert to multiplication by reciprocal ÷ (1/2) = × (2/1), multiply (3/4) × (2/1) = 6/4, simplify to 3/2, and add units: 3/2 miles per hour. Understanding 'per' as division helps, like (3/4) ÷ (1/2) = 3/2, and for comparison, a rate of 3/2 mph is faster than 1 mph since 1.5 > 1.
A runner drinks $\tfrac{2}{3}$ liter of water in $\tfrac{1}{6}$ of an hour. How many liters per hour is that?
Answer: $4$ liters per hour This question tests computing unit rates from ratios of fractions by dividing complex fractions: (2/3 liter)/(1/6 hour) simplified using reciprocal (2/3)×(6/1) with units. Unit rate means the amount per one unit of the denominator, such as liters per 1 hour. From the fractional ratio: (2/3 liter)/(1/6 hour) is a complex fraction (2/3)/(1/6), simplify by dividing fractions: (2/3)÷(1/6)=(2/3)×(6/1)=12/3=4 liters per hour (multiply by reciprocal of denominator, simplify). Interpretation: 4 liters per hour means the runner drinks 4 liters in each hour (per-unit meaning). A common error is taking reciprocal of numerator instead, leading to 1/4 or 1/9, or arithmetic errors like 12/3=3. Steps: (1) identify ratio (2/3 liter per 1/6 hour), (2) write as complex fraction ((2/3)/(1/6)), (3) convert division to multiplication (÷(1/6)=×(6/1)), (4) multiply ((2/3)×(6/1)=12/3), (5) simplify to 4, (6) include units (4 liters per hour). Understanding: 'per' means division, so consumption per hour requires dividing amount by time fraction.

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