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Converting Unit Rates

Author: Sophia

what's covered
In this lesson, you will learn how to convert a unit rate in a given scenario. Specifically, this lesson will cover:

Table of Contents

1. Converting Units

When converting from one unit to another, we multiply a quantity by a conversion factor. A conversion factor is a fraction equivalent to 1, so multiplying it by a quantity doesn't change its actual measure. However, the fraction certainly doesn't look like it's equal to one (at least not until you inspect it). The expressions in the numerator and denominator are equal quantities but measured in different units. During unit conversion, units are also canceled out when they appear in both the numerator and the denominator.

EXAMPLE

Convert 1.5 feet into inches, using the fact that 1 foot is equivalent to 12 inches.

fraction numerator 1.5 space feet over denominator 1 end fraction times fraction numerator 12 space inches over denominator 1 space foot end fraction Create conversion factor with feet and inches so that units of feet cancel, leaving inches
fraction numerator 1.5 space up diagonal strike feet over denominator 1 end fraction times fraction numerator 12 space inches over denominator 1 space up diagonal strike foot end fraction Multiply numbers in numerator and denominator
fraction numerator 18 space inches over denominator 1 end fraction Simplify
18 space inches Our Solution


2. Unit Rates

A unit rate is a ratio between two quantities with different units. Speed is a unit rate because it is the ratio between distance traveled and the time taken to travel that distance, like 20 miles per hour, or fraction numerator 20 space miles over denominator hour end fraction. There are two different types of units that make up this ratio, distance and time.

Another characteristic of a unit rate is that a unit rate has a denominator of 1. Let's think about speed again: we wouldn't normally say that a car travels 120 miles per 2 hours. Instead, we simplify the ratio to 1 hour, and say 60 miles per hour or fraction numerator 60 space miles over denominator 1 space hour end fraction.

Here are some more examples of unit rates:

  • dollars per hour, such as fraction numerator $ 17.50 over denominator 1 space hour end fraction
  • words per minutes, such as fraction numerator 200 space words over denominator 1 space minute end fraction
  • calories per serving, such as fraction numerator 250 space calories over denominator 1 space serving end fraction
  • steps per day, such as fraction numerator 3 comma 000 space steps over denominator 1 space day end fraction

3. Converting Unit Rates

When converting unit rates, we need to make use of multiple conversion factors: as many conversion factors as needed to convert from one quantity to another for every unit involved in our unit rate. In the example below, we will work more with miles per hour. First, we'll need to convert miles to feet, but also hours to seconds. Let's first list our conversion factors:

  • 1 mile = 5280 feet
  • 1 foot = 0.0002 miles
  • 1 hour = 3600 seconds
  • 1 second = 0.00028 hours
hint
When using a conversion factor to create a fraction equal to 1, think about what unit you wish to convert into. In most cases, this will be the numerator of the fraction, which places the unit you wish to cancel in the denominator of that fraction. This is because the unit you wish to cancel will likely be in the numerator of another fraction. Units are canceled when they appear in the numerator and denominator of the combined fraction.

hint
Keep in mind that although we have listed some very helpful conversion factors, we may not need to use all of them. For example, as we will see when we are converting from miles per hour into feet per second, we will use the conversion factor that 1 space mile equals 5280 space feet, instead of 1 space foot equals 0.0002 space miles. This all depends on which unit you want to convert into and which unit you wish to cancel.

EXAMPLE

Convert 60 miles per hour to feet per second.

fraction numerator 60 space mi over denominator 1 space hour end fraction times fraction numerator 5280 space ft over denominator 1 space mi end fraction Start with fraction numerator 60 space mi over denominator 1 space hour end fraction and use the conversion factor fraction numerator 5280 space ft over denominator 1 space mi end fraction to cancel miles
fraction numerator 60 space up diagonal strike mi over denominator 1 space hour end fraction times fraction numerator 5280 over denominator 1 space up diagonal strike mi end fraction times fraction numerator 1 space hour over denominator 3600 space sec end fraction Use a second conversion factor fraction numerator 1 space hour over denominator 3600 space sec end fraction to cancel hours
fraction numerator 60 space up diagonal strike mi over denominator 1 space up diagonal strike hour end fraction times fraction numerator 5280 space ft over denominator 1 space up diagonal strike mi end fraction times fraction numerator 1 space up diagonal strike hour over denominator 3600 space sec end fraction Multiply numbers in numerator and denominator, noting which units were canceled and which units remained
fraction numerator 60 times 5280 times 1 space ft over denominator 1 times 1 times 3600 space sec end fraction Evaluate
fraction numerator 316800 space feet over denominator 3600 space sec end fraction Divide 316800 by 3600 for a denominator of 1
fraction numerator 88 space feet over denominator 1 space sec end fraction Our Solution

60 miles per hour is equivalent to 88 feet per second.

summary
When converting units, we multiply a quantity by a conversion factor. Unit rates are ratios comparing two quantities with different units. Conversion factors, which can be used to convert between different units, can also then be used when converting unit rates.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License