Skill: Interpret a rate as a comparison of quantities

A rate compares two quantities by division. The units tell you how the quantities are being compared.

For example, suppose a cyclist travels $(42)$ miles in $3$ hours. What is the cyclist’s rate?

Step 1: Identify the two quantities.

• Distance: $(42)$ miles
• Time: $3$ hours

Step 2: Divide the distance by the time.

$$\text{Rate}=\frac{\text{distance}}{\text{time}} =\frac{42\text{ miles}}{3\text{ hours}} =14\text{ miles per hour}$$

Step 3: Interpret the units.

The rate $(14)$ miles per hour means the cyclist travels 14 miles for every 1 hour.

This is a multiplicative comparison: for each hour, multiply $1$ hour by $(14)$ miles per hour to get $(14)$ miles.

It is not an additive difference. Subtracting $(42-3)$ would compare unlike quantities in a way that does not describe the cyclist’s travel. Dividing gives a meaningful rate with units: $(14)$ miles per hour.