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Converting Unit Rates

Converting Unit Rates

Author: Sophia Tutorial
Description:

Convert a unit rate in a given scenario.

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what's covered
  1. Converting Units
  2. Unit Rates
  3. Converting Unit Rates

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. To illustrate this idea, let's convert 1.5 feet into inches, using the fact that 1 foot is equivalent to 12 inches.

fraction numerator 1.5 space f e e t over denominator 1 end fraction times fraction numerator 12 space i n c h e s over denominator 1 space f o o t end fraction

Create fraction multiplication
fraction numerator open parentheses 1.5 times 12 close parentheses open parentheses f e e t times i n c h e s close parentheses over denominator open parentheses 1 times 1 close parentheses open parentheses f o o t close parentheses end fraction

Multiply numbers, multiply units
fraction numerator 18 space i n c h e s over denominator 1 end fraction

Units of feet cancel
18 space i n c h e s
Our Solution


2. Unit Rates

A unit rate is a ratio between two quantities with different units. For examples, speed is a unit rate, because it is the ratio between distance traveled and the time taken to travel that distance. 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.


3. Converting Unit Rates

When converting unit rates, we need to make use of multiple conversion factors: as many conversion factors needed to convert from one quantity to another for every unit involved in our unit rate. Let's work more with miles per hour, and convert 60 miles per hour into feet per second. We'll need to convert miles to kilometers, but also hours to seconds. Let's first list our conversion factors:

1 space m i l e space equals space 5280 space f e e t
1 space f o o t space equals space 0.0002 space m i l e s
1 space h o u r space equals space 3600 space s e c o n d s
1 space s e c o n d space equals space 0.00028 space h o u r s

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 mile = 5280 feet, instead of 1 foot = 0.0002 miles. This all depends on which unit you want to convert into and which unit you wish to cancel.

fraction numerator 60 space m i over denominator 1 space h o u r end fraction times fraction numerator 5280 space f t over denominator 1 space m i end fraction

This conversion factor cancels miles
fraction numerator 60 space up diagonal strike m i end strike over denominator 1 space h o u r end fraction times fraction numerator 5280 over denominator 1 space up diagonal strike m i end strike end fraction times fraction numerator 1 space h o u r over denominator 3600 space s e c end fraction

The second conversion factor cancels hours
fraction numerator 60 times 5280 times 1 space f t over denominator 1 times 1 times 3600 space s e c end fraction

Multiply all fractions, noting canceled units
fraction numerator 316800 space f e e t over denominator 3600 space s e c end fraction

Divide 316800 by 3600, for a denominator of 1
88 space f e e t space p e r space s e c o n d
Our Solution

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.