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Unit Rates with Fractional Quantities

Grade 7·21 problems·Common Core Math - Grade 7·standard·7-rp-a-1

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Recall / Warm-Up

A car travels 180 miles in 3 hours. What is the unit rate in miles per hour?

$60$

$540$

$\frac{1}{60}$

$183$

What is $\frac{3}{4} \div \frac{1}{2}$ ?

$\frac{3}{8}$

$\frac{3}{2}$

$\frac{2}{3}$

$\frac{5}{4}$

Rewrite $\frac{2}{3} \div \frac{4}{5}$ as multiplication by the reciprocal, then compute the result. Express as a fraction in simplest form.

Fluency Practice

Simplify the complex fraction $\dfrac{1/2}{1/4}$ . Express as a whole number or fraction in simplest form.

A cyclist rides $\frac{3}{4}$ mile in $\frac{1}{2}$ hour. Compute the unit rate in miles per hour. Express as a fraction in simplest form.

Compute $\frac{4}{5} \div \frac{2}{3}$ . Express as a fraction in simplest form.

Compute $\frac{5}{6} \div \frac{5}{3}$ . Express as a fraction in simplest form.

Which of the following correctly evaluates $\frac{3}{8} \div \frac{3}{4}$ ?

$\frac{9}{32}$

$\frac{1}{2}$

$2$

$\frac{3}{2}$

Varied Practice

A recipe uses $\frac{2}{5}$ cup of oil for every $\frac{3}{4}$ cup of flour. Which expression correctly sets up the unit rate in cups of oil per cup of flour?

$\dfrac{2/5}{3/4}$

$\dfrac{3/4}{2/5}$

$\dfrac{2}{5} \times \dfrac{3}{4}$

$\dfrac{2}{5} + \dfrac{3}{4}$

A printer uses $\frac{2}{3}$ pint of ink to print $\frac{1}{2}$ page. The complex fraction for pints of ink per page is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ / ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . The simplified unit rate is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ pints per page.

numerator:

denominator:

unit rate:

A scale model is built so that $\frac{1}{3}$ foot of the model represents $\frac{5}{6}$ foot of the actual building. What is the scale in actual feet per model foot?

$\frac{5}{2}$

$\frac{5}{18}$

$\frac{2}{5}$

$\frac{1}{3}$

A floor tile covers $\frac{3}{4}$ square foot and costs $\frac{5}{6}$ dollar. What is the cost in dollars per square foot? Express as a fraction in simplest form.

Word Problems

A runner covers $\frac{5}{8}$ mile in $\frac{1}{4}$ hour.

What is the runner's speed in miles per hour? Express as a fraction in simplest form.

A bulk store sells $\frac{3}{8}$ pound of almonds for $\frac{3}{4}$ dollar.

What is the price per pound in dollars? Express as a whole number or fraction in simplest form.

A printing press uses $\frac{3}{5}$ pint of ink to print $\frac{2}{3}$ of a newspaper page.

What is the unit rate in pints of ink per full page? Express as a fraction in simplest form.

What is the unit rate in pages printed per pint of ink? Express as a fraction in simplest form.

A map uses $\frac{1}{4}$ inch to represent $\frac{1}{3}$ mile.

What is the map's scale in inches per mile? Express as a fraction in simplest form.

Error Analysis

$Two-column contrast card. Left (yellow): wrong setup puts hours in numerator, giving 2/3 mph with a red X. Right (teal): correct setup puts miles in numerator, giving 3/2 mph with a checkmark.$

Maya found the unit rate in miles per hour when a person walks $\frac{3}{4}$ mile in $\frac{1}{2}$ hour:

Step 1: Unit rate = $\frac{1}{2} \div \frac{3}{4}$
Step 2: $= \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$ miles per hour

What error did Maya make?

Maya divided hours by miles — since the unit rate is miles per hour, miles ( $\frac{3}{4}$ ) should be in the numerator and hours ( $\frac{1}{2}$ ) in the denominator

Maya used the wrong reciprocal — she should have multiplied by $\frac{3}{4}$ instead of $\frac{4}{3}$

Maya's arithmetic is wrong — $\frac{1}{2} \times \frac{4}{3}$ does not equal $\frac{2}{3}$

Maya should have added the fractions: $\frac{1}{2} + \frac{3}{4} = \frac{5}{4}$ miles per hour

$Two-column contrast card. Left (yellow): student multiplies by 1/2 (no flip), getting 3/8, with a red X. Right (teal): correct step multiplies by 2/1 (reciprocal of 1/2), getting 3/2, with a checkmark.$

Carlos computed $\frac{3}{4} \div \frac{1}{2}$ :

Step 1: Rewrite as multiplication: $\frac{3}{4} \times \frac{1}{2}$
Step 2: $= \frac{3}{8}$

What error did Carlos make?

Carlos multiplied by $\frac{1}{2}$ instead of flipping it — he should have multiplied by $\frac{2}{1}$ (the reciprocal of $\frac{1}{2}$ )

Carlos set up the division in the wrong direction — he should have computed $\frac{1}{2} \div \frac{3}{4}$

Carlos should have found a common denominator of 4 before dividing

Carlos made an arithmetic error — $\frac{3}{4} \times \frac{1}{2}$ does not equal $\frac{3}{8}$

Challenge

Runner A covers $\frac{5}{8}$ mile in $\frac{1}{2}$ hour. Runner B covers $\frac{3}{4}$ mile in $\frac{2}{3}$ hour. Compute both speeds in miles per hour, then find the difference. By how many miles per hour is the faster runner ahead? Express as a fraction in simplest form.

Pipe 1 fills $\frac{2}{3}$ gallon per $\frac{1}{4}$ minute. Pipe 2 fills $\frac{1}{2}$ gallon per $\frac{1}{6}$ minute. Compute the unit rate in gallons per minute for each pipe and explain which pipe fills a tank faster.

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