Lesson 3: Rates

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

When two rates are given, it can be difficult to determine which rate is higher or lower because they have different values.
It is not until both rates are converted to the same unit (a unit of one) that the comparison becomes easy.
This is useful for finding the better deal or determining which object is moving faster.

Examples

• Store A sells 10 pens for 2 dollars. Store B sells 12 pens for 3 dollars. Store A's rate is $\frac{2}{10} = 0.20$ dollars per pen. Store B's is $\frac{3}{12} = 0.25$ dollars per pen. Store A is cheaper.
• A train travels 210 miles in 3 hours. A bus travels 130 miles in 2 hours. The train's speed is $\frac{210}{3} = 70$ mph. The bus's speed is $\frac{130}{2} = 65$ mph. The train is faster.
• One faucet fills a 10-gallon tub in 4 minutes. Another fills a 12-gallon tub in 5 minutes. The first faucet's rate is $\frac{10}{4} = 2.5$ gal/min. The second is $\frac{12}{5} = 2.4$ gal/min. The first faucet is faster.

Explanation

To compare different deals or speeds, convert them to a common language: the unit rate. By finding the 'cost per one' or 'distance per one,' you can easily see which option is cheaper, faster, or better.