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\(\frac{1}{2}\)
1

Unit Rate
with Fractions

Master solving unit-rate problems involving fractions and decimals, and learn to interpret your results with correct units.

Pre-Algebra

What Is a Rate?

A rate compares two quantities measured in different units. For example, "60 miles in 2 hours" is a rate because it relates distance (miles) to time (hours).

Speed
60 miles per 2 hours
Price
$4.50 for 3 pounds
Work
15 pages in 4 minutes

What Is a Unit Rate?

A unit rate is a rate where the second quantity is exactly 1. It tells you how much of the first quantity corresponds to one unit of the second.

Rate
\(\displaystyle \frac{60 \text{ miles}}{2 \text{ hours}}\)
(two units of time) ÷ 2 Unit Rate
\(\displaystyle \frac{30 \text{ miles}}{1 \text{ hour}}\)
(one unit of time)

Why Fractions Make It Interesting

In many real-world problems the quantities themselves are fractions. For example: "A painter uses \(\frac{3}{4}\) gallon of paint to cover \(\frac{1}{2}\) wall." You still need to find how much paint covers one whole wall.

This lesson will teach you the exact process to handle these problems confidently, every time.

The Core Rule

Unit Rate Formula
\(\displaystyle \text{Unit Rate} = \frac{\text{Quantity A}}{\text{Quantity B}}\)
Divide so Quantity B becomes 1

Visualizing Divide by a Fraction

Step 1: Write the division
\(\displaystyle \frac{3/4}{1/2}\)
Step 2: Multiply by reciprocal
\(\displaystyle \frac{3}{4} \times \frac{2}{1}\)
Step 3: Simplify
\(\displaystyle \frac{6}{4} = \frac{3}{2} = 1\tfrac{1}{2}\)

Rate vs. Unit Rate

Rate Unit Rate
Definition Compares two quantities in different units Rate with denominator = 1
Example \(\frac{3}{4}\) gal per \(\frac{1}{2}\) wall \(1\frac{1}{2}\) gal per 1 wall
How to find State the relationship directly Divide numerator by denominator
Best for Describing a specific situation Comparing and scaling

How to Solve Unit Rate Problems

1
Identify the two quantities and their units. Decide which unit should be "per 1."
2
Set up the division: place the quantity you want on top, the "per" quantity on the bottom.
3
Divide the fractions: multiply the numerator fraction by the reciprocal of the denominator fraction.
4
Simplify and convert to a mixed number or decimal if needed.
5
Attach the units: write your answer as "[value] [unit A] per 1 [unit B]."

Example 1: Walking Speed

Mia walks \(\frac{3}{4}\) mile in \(\frac{1}{4}\) hour. What is her speed in miles per hour?

Set up
\(\displaystyle \frac{3/4 \text{ mi}}{1/4 \text{ hr}}\)
Multiply by reciprocal
\(\displaystyle \frac{3}{4} \times \frac{4}{1} = \frac{12}{4}\)
\(= 3\) miles per hour

Example 2: Recipe Scaling

A recipe uses \(\frac{2}{3}\) cup of sugar for every \(\frac{3}{5}\) batch. How many cups of sugar per one whole batch?

1
Set up: \(\displaystyle \frac{2/3 \text{ cup}}{3/5 \text{ batch}}\)
2
Multiply by reciprocal: \(\displaystyle \frac{2}{3} \times \frac{5}{3} = \frac{10}{9}\)
3
Simplify: \(\displaystyle \frac{10}{9} = 1\frac{1}{9}\) cups per batch

Example 3: Comparing Prices

Store A sells \(\frac{3}{8}\) lb of cheese for \(\$\frac{5}{4}\). Store B sells \(\frac{1}{2}\) lb for \(\$\frac{7}{6}\). Which store has the lower price per pound?

Store A
\(\frac{5/4}{3/8} = \frac{5}{4} \times \frac{8}{3} = \frac{40}{12} = \frac{10}{3} \approx\) $3.33/lb
Store B
\(\frac{7/6}{1/2} = \frac{7}{6} \times \frac{2}{1} = \frac{14}{6} = \frac{7}{3} \approx\) $2.33/lb

Handling Decimals

Sometimes the problem mixes decimals with fractions. The same rule applies: divide the two quantities.

1
Convert decimals to fractions if needed: \(0.75 = \frac{3}{4}\), \(0.5 = \frac{1}{2}\).
2
Divide as usual: multiply by the reciprocal.
3
Or divide directly: \(0.75 \div 0.5 = 1.5\). Both methods work.

Interpreting Units Correctly

The order of division determines the units of your answer. Pay close attention to which quantity goes on top.

Miles ÷ Hours
= miles per hour
(speed)
Hours ÷ Miles
= hours per mile
(pace)

Example 4: Mixed Numbers

A car travels \(2\frac{1}{2}\) miles on \(\frac{1}{8}\) gallon of gas. What is the fuel efficiency in miles per gallon?

1
Convert mixed number: \(2\frac{1}{2} = \frac{5}{2}\)
2
Set up: \(\displaystyle \frac{5/2 \text{ mi}}{1/8 \text{ gal}}\)
3
Multiply by reciprocal: \(\displaystyle \frac{5}{2} \times \frac{8}{1} = \frac{40}{2} = 20\)

Common Mistakes to Avoid

✕ Flipping the Wrong Fraction
Flip only the denominator (bottom fraction). The numerator stays the same.
✕ Forgetting to Convert Mixed Numbers
Always rewrite \(2\frac{1}{3}\) as \(\frac{7}{3}\) before you multiply or divide.
✕ Swapping the Order
Dividing \(\frac{\text{hours}}{\text{miles}}\) instead of \(\frac{\text{miles}}{\text{hours}}\) gives the wrong unit.

Summary: Quick Reference

Scenario Setup Result
Whole numbers \(60 \div 2\) 30 mph
Fraction ÷ fraction \(\frac{3/4}{1/2} = \frac{3}{4} \times \frac{2}{1}\) \(\frac{3}{2}\) = 1.5
Mixed number ÷ fraction \(\frac{5/2}{1/8} = \frac{5}{2} \times \frac{8}{1}\) 20
Decimal ÷ decimal \(0.75 \div 0.5\) 1.5

Question 1 of 5

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

Question 2 of 5

A recipe calls for \(\frac{5}{6}\) cup of flour for every \(\frac{2}{3}\) dozen cookies. How many cups of flour are needed per dozen?

Question 3 of 5

A runner goes \(3\frac{1}{2}\) miles in \(\frac{7}{10}\) hour. What is the speed in miles per hour?

Question 4 of 5

Store X sells \(\frac{3}{4}\) lb of turkey for \(\$4.50\). Store Y sells \(\frac{2}{5}\) lb for \(\$2.20\). Which store has the lower unit price?

Question 5 of 5

When finding a unit rate from \(\displaystyle \frac{a/b}{c/d}\), which operation do you perform?