01 / 21

Unit Rates
With Fractions

Master complex fractions and learn to compute unit rates involving fractional quantities: lengths, areas, speeds, and more.

7th Grade Math

What Is a Rate?

A rate compares two quantities measured in different units. It tells you how much of one thing corresponds to a certain amount of another.

Speed
Miles per hour: distance ÷ time
Price
Dollars per pound: cost ÷ weight
Density
People per square mile: population ÷ area

What Is a Unit Rate?

A unit rate is a rate where the second quantity is exactly 1. It answers: "How much of the first quantity for each one of the second?"

Rate 120 miles 2 hours
\(\div 2\)
 Unit Rate 60 miles 1 hour

Why Do Fractions Show Up?

Real-world measurements are rarely whole numbers. A painter might cover \(\frac{3}{4}\) of a wall in \(\frac{1}{2}\) hour. A recipe might use \(\frac{2}{3}\) cup of flour for \(\frac{1}{4}\) batch.

Walking
\(\frac{1}{2}\) mile in \(\frac{1}{4}\) hour
Painting
\(\frac{3}{4}\) wall in \(\frac{1}{2}\) hour

The Core Rule

To find a unit rate from a ratio of fractions, divide the numerator fraction by the denominator fraction:

\(\frac{a}{b}\)
\(\frac{c}{d}\)
\(=\)
\(\frac{a}{b}\)
\(\times\)
\(\frac{d}{c}\)
\(=\)
\(\frac{ad}{bc}\)
 Multiply by the reciprocal of the denominator

Visualizing the Flip

Think of the complex fraction as a division problem. The fraction bar means "divided by":

WRITE
\(\frac{\color{#4ade80}{\frac{1}{2}}}{\color{#fbbf24}{\frac{1}{4}}}\)
REWRITE
\(\frac{1}{2}\) \(\div\) \(\frac{1}{4}\)
FLIP + MULTIPLY
\(\frac{1}{2}\) \(\times\) \(\frac{4}{1}\)

Rate vs. Unit Rate

Rate Unit Rate
Definition Compares two quantities Second quantity equals 1
Example \(\frac{1}{2}\) mile per \(\frac{1}{4}\) hour 2 miles per 1 hour
Form \(\frac{a}{b}\) \(\frac{a}{1}\)
How to find Given in the problem Divide: \(\text{numerator} \div \text{denominator}\)

How to Compute It

1
Identify the two quantities and their units: which is the numerator, which is the denominator.
2
Write the complex fraction: place the numerator quantity on top and the denominator quantity on the bottom.
3
Rewrite as division: \(\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d}\).
4
Multiply by the reciprocal: \(\frac{a}{b} \times \frac{d}{c}\).
5
Simplify and attach the correct units (per 1 of the denominator unit).

Example 1: Walking Speed

A person walks \(\frac{1}{2}\) mile in \(\frac{1}{4}\) hour. What is the unit rate in miles per hour?

1
Complex fraction: \(\dfrac{\frac{1}{2} \text{ mi}}{\frac{1}{4} \text{ hr}}\)
2
Rewrite: \(\frac{1}{2} \div \frac{1}{4}\)
3
Reciprocal: \(\frac{1}{2} \times \frac{4}{1} = \frac{4}{2}\)
4
Simplify: \(\frac{4}{2} = \) 2 miles per hour

Example 2: Painting Rate

A painter covers \(\frac{3}{4}\) of a wall in \(\frac{2}{3}\) hour. What is the unit rate in walls per hour?

1
Complex fraction: \(\dfrac{\frac{3}{4} \text{ wall}}{\frac{2}{3} \text{ hr}}\)
2
Rewrite: \(\frac{3}{4} \div \frac{2}{3}\)
3
Reciprocal: \(\frac{3}{4} \times \frac{3}{2} = \frac{9}{8}\)
4
Simplify: \(\frac{9}{8} = 1\frac{1}{8}\) walls per hour

Example 3: Area per Hour

A lawn mower cuts \(\frac{5}{6}\) acre in \(\frac{5}{12}\) hour. Find the unit rate in acres per hour.

1
Complex fraction: \(\dfrac{\frac{5}{6}}{\frac{5}{12}}\)
2
Reciprocal: \(\frac{5}{6} \times \frac{12}{5}\)
3
Multiply: \(\frac{5 \times 12}{6 \times 5} = \frac{60}{30}\)
4
Simplify: \(\frac{60}{30} = \) 2 acres per hour

When the Answer Is a Fraction

Unit rates are not always whole numbers. A recipe uses \(\frac{2}{3}\) cup of sugar for \(\frac{3}{4}\) batch. Find cups per batch.

1
\(\dfrac{\frac{2}{3}}{\frac{3}{4}} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}\)

Choosing the Right "Per 1"

The same situation can produce two different unit rates depending on which quantity you put in the denominator.

Miles per Hour
\(\frac{1/2 \text{ mi}}{1/4 \text{ hr}} = 2\) mi/hr
Hours per Mile
\(\frac{1/4 \text{ hr}}{1/2 \text{ mi}} = \frac{1}{2}\) hr/mi

Handling Mixed Numbers

If the problem gives a mixed number, convert it to an improper fraction first.

1
Problem: A cyclist rides \(1\frac{1}{2}\) miles in \(\frac{3}{4}\) hour. Find miles per hour.
2
Convert: \(1\frac{1}{2} = \frac{3}{2}\)
3
Divide: \(\frac{3}{2} \div \frac{3}{4} = \frac{3}{2} \times \frac{4}{3} = \frac{12}{6} = \) 2 mph

Common Mistakes

✕ Flipping the Wrong Fraction
Students flip the numerator instead of the denominator. Always flip the bottom fraction.
✕ Forgetting to Convert Mixed Numbers
Attempting to divide with a mixed number like \(1\frac{1}{2}\) directly leads to errors.
✕ Swapping Numerator and Denominator
Placing the wrong quantity on top gives the reciprocal of the correct answer. Read the units carefully.

Summary at a Glance

Step What You Do Example
1. Set Up Write the complex fraction \(\dfrac{\frac{3}{4}}{\frac{2}{3}}\)
2. Rewrite Change to division \(\frac{3}{4} \div \frac{2}{3}\)
3. Reciprocal Flip the divisor \(\frac{3}{4} \times \frac{3}{2}\)
4. Multiply Straight across \(\frac{9}{8}\)
5. Simplify Reduce or convert \(1\frac{1}{8}\) per unit

Question 1 of 5

A snail travels \(\frac{1}{3}\) meter in \(\frac{1}{6}\) hour. What is the snail's speed in meters per hour?

Question 2 of 5

A machine fills \(\frac{3}{5}\) of a tank in \(\frac{2}{5}\) hour. What is the unit rate in tanks per hour?

Question 3 of 5

A recipe uses \(\frac{3}{4}\) cup of flour for \(\frac{1}{2}\) batch. How many cups of flour are needed per 1 full batch?

Question 4 of 5

A runner covers \(2\frac{1}{2}\) miles in \(\frac{5}{6}\) hour. What is the unit rate in miles per hour?

Question 5 of 5

Which expression correctly represents the unit rate for \(\frac{2}{3}\) gallon per \(\frac{4}{5}\) minute?