Lesson 5: Understand Rates and Unit Rates

Property

A rate compares two quantities of different units. A rate is usually written as a fraction. When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain.

Examples

• If Bob drove his car 525 miles in 9 hours, the rate is written as $\frac{525 \text{ miles}}{9 \text{ hours}}$, which simplifies to $\frac{175 \text{ miles}}{3 \text{ hours}}$.

• A price of 595 dollars for 40 hours of work is the rate $\frac{595 \text{ dollars}}{40 \text{ hours}}$.

Property

A rate is a ratio of two quantities.
The unit rate is the amount of one quantity that corresponds to 1 unit of the other quantity.
The designation of unit rate must be clear about the choice and order of the units.

For a ratio $a:b$ with $b \neq 0$, the unit rate is $\frac{a}{b}$ units of the first quantity per 1 unit of the second quantity.

Examples

• If you pay 9 dollars for 3 sandwiches, the unit rate is found by dividing: $9 \div 3 = 3$ dollars per sandwich.
• A cyclist travels 30 miles in 2 hours. The unit rate for her speed is $30 \div 2 = 15$ miles per hour.
• A team scores 45 points in 3 quarters. Their unit rate is $45 \div 3 = 15$ points per quarter.

Property

For any rate that compares two different quantities, you can calculate two different unit rates. If a rate compares quantity A to quantity B, the two unit rates are found by calculating:

Examples