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Finding Unit Rates

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Finding Unit Rates

In the last example, we calculated that Bob was driving at a rate of $$\frac{\text{175 miles}}{\text{3 hours}}.$$ This tells us that every three hours, Bob will travel $$175$$ miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of $$1$$ unit is referred to as a unit rate.

Definition: Unit Rate

A unit rate is a rate with denominator of $$1$$ unit.

Unit rates are very common in our lives. For example, when we say that we are driving at a speed of $$68$$ miles per hour we mean that we travel $$68$$ miles in $$1$$ hour. We would write this rate as $$68$$ miles/hour (read $$68$$ miles per hour). The common abbreviation for this is $$68$$ mph. Note that when no number is written before a unit, it is assumed to be $$1.$$

So $$68$$ miles/hour really means $$\text{68 miles/1 hour.}$$

Two rates we often use when driving can be written in different forms, as shown:

$$68$$ miles in $$1$$ hour$$\frac{\text{68 miles}}{\text{1 hour}}$$$$68$$ miles/hour$$68$$ mph$$\text{68 miles per hour}$$
$$36$$ miles to $$1$$ gallon$$\frac{\text{36 miles}}{\text{1 gallon}}$$$$36$$ miles/gallon$$36$$ mpg$$\text{36 miles per gallon}$$

Another example of unit rate that you may already know about is hourly pay rate. It is usually expressed as the amount of money earned for one hour of work. For example, if you are paid $$\text{\12.50}$$ for each hour you work, you could write that your hourly (unit) pay rate is $$\text{\12.50/hour}$$ (read $$\text{\12.50}$$ per hour.)

To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of $$1.$$

Example

Anita was paid $$\text{\384}$$ last week for working $$\text{32 hours}.$$ What is Anita’s hourly pay rate?

Solution

 Start with a rate of dollars to hours. Then divide. $$\text{\384 last week for 32 hours}$$ Write as a rate. $$\frac{\384}{\text{32 hours}}$$ Divide the numerator by the denominator. $$\frac{\12}{\text{1 hour}}$$ Rewrite as a rate. $$\12/\text{hour}$$

Anita’s hourly pay rate is $$\text{\12}$$ per hour.

Example

Sven drives his car $$455$$ miles, using $$14$$ gallons of gasoline. How many miles per gallon does his car get?

Solution

 $$\text{455 miles to 14 gallons of gas}$$ Write as a rate. $$\frac{\text{455 miles}}{\text{14 gallons}}$$ Divide 455 by 14 to get the unit rate. $$\frac{\text{32.5 miles}}{\text{1 gallon}}$$
Sven’s car gets $$32.5$$ miles/gallon, or $$32.5$$ mpg.