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 ).

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.

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.

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.

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.

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).

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).

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.

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).

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).