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.

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.

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.

First find gallons per complete trip: $$\frac{3/4}{5/8} = \frac{3}{4} \times \frac{8}{5} = \frac{24}{20} = \frac{6}{5} = 1\frac{1}{5}$$ gallons for the whole trip. Choice A results from calculating $$\frac{5/8}{3/4}$$. Choice C comes from multiplying $$\frac{3}{4} \times \frac{5}{8}$$. Choice D shows the improper fraction form of the correct answer.

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).