Lesson 3: Rates

Property

A rate is a ratio that compares two quantities measured in different units. It is often expressed as a fraction where the numerator and denominator have different units.
For a quantity $a$ of one unit and a quantity $b$ of another unit, the rate is written as $\frac{a}{b}$.

Examples

• 150 miles in 3 hours can be written as the rate $\frac{150 \text{ miles}}{3 \text{ hours}}$.
• 400 words typed in 5 minutes can be written as the rate $\frac{400 \text{ words}}{5 \text{ minutes}}$.
• 6 dollars for 2 pounds of apples can be written as the rate $\frac{6 \text{ dollars}}{2 \text{ pounds}}$.

Explanation

A rate provides a way to measure how one quantity changes in relation to another. For example, speed is a rate that measures distance per unit of time. When writing a rate, it is crucial to include the units to understand what is being compared. The fraction bar in a rate means "per" or "for each".

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

Double number lines show equivalent rates by aligning corresponding values on parallel lines.
To find intermediate values:
(1) identify the unit rate;
(2) locate known values on both lines;
(3) find halfway points by averaging;
(4) use the unit rate to calculate missing values.

Examples