Before you read

Answer first — see what you already know.

During a cross-country road trip, a family drives 195 miles on 6.5 gallons of gas. What is the car's fuel efficiency in miles per gallon?

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Answer:

Before you read

Answer first — see what you already know.

An industrial copier produces 32 pages in 4 minutes. At this rate, how many pages can it produce in 15 minutes?

Check your answer

Answer:

Theory

Understanding Rates

A rate compares two quantities with different units. Examples: 60 miles per hour, $3.50 per gallon, 200 words per minute.

A unit rate has a denominator of 1:
$\text{Unit rate} = \frac{\text{Total quantity}}{\text{Total units}}$

Unit rate = Distance ÷ Time

Example: If you drive 150 miles in 3 hours, your unit rate (speed) is $\frac{150}{3} = 50$ miles per hour.

Example 1

A factory produces 840 widgets in 12 hours. What is the production rate in widgets per hour?

Unit rate $= \frac{840}{12} = 70$ widgets per hour

70 widgets per hour

Example 2

Store A sells 5 apples for $4.00. Store B sells 8 apples for $6.00. Which store has the better deal?

Store A: $\frac{\$4.00}{5} = \$0.80$ per apple

Store B: $\frac{\$6.00}{8} = \$0.75$ per apple

Store B is cheaper per apple.

Store B ($0.75 per apple)

Theory

Using Rates to Solve Problems

The key formula is:
$\text{Quantity} = \text{Rate} \times \text{Time}$

This rearranges to:
- $\text{Rate} = \frac{\text{Quantity}}{\text{Time}}$
- $\text{Time} = \frac{\text{Quantity}}{\text{Rate}}$

For combined rates (two workers doing the same job), add the rates:
$\text{Combined rate} = \text{Rate}_1 + \text{Rate}_2$

Example 1

Pipe A fills a pool in 6 hours. Pipe B fills it in 4 hours. How long does it take both pipes together?

Rate A $= \frac{1}{6}$ pool/hour. Rate B $= \frac{1}{4}$ pool/hour.

Combined rate $= \frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}$ pool/hour

Time $= \frac{1}{5/12} = \frac{12}{5} = 2.4$ hours

\frac{12}{5}

hours (2 hours 24 minutes)

2 more sections including examples, practice problems, and step-by-step solutions.

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