Skill: Interpret rates as comparisons of quantities

A rate compares two quantities with different units. To find a unit rate, divide so that one of the quantities is $1$. The word “per” tells you to divide.

Example:
Store A sells 8 notebooks for 12$. Store B sells 5 notebooks for$$8$. Which store has the lower cost per notebook?

1. Find Store A’s cost per notebook:

$$\frac{\$12}{8\text{ notebooks}}=\$1.50\text{ per notebook}$$

This means each notebook costs 1.50$ at Store A.

2. Find Store B’s cost per notebook:

$$\frac{\$8}{5\text{ notebooks}}=\$1.60\text{ per notebook}$$

This means each notebook costs 1.60$ at Store B.

3. Compare the unit rates:

$$\$1.50\text{ per notebook}<\$1.60\text{ per notebook}$$

Therefore, Store A has the lower rate, so it is the better buy.

The rate is a multiplicative comparison: the number of notebooks is multiplied by the cost per notebook to find the total cost. For example, at Store A,

$$8\cdot \$1.50=\$12.$$