## Unit Conversions

This chapter explains a method for converting any units of measurement, including derived units such as square meters or miles per hour, given appropriate conversion factors. The list of common conversion factors is given. This chapter also provides real-world examples of when such conversions might need to be made. Read the chapter and work through the examples.

#### Rational Expressions - Dimensional Analysis

##### Objective: Use dimensional analysis to perform single unit, dual unit, square unit, and cubed unit conversions.

One application of rational expressions deals with converting units. When we convert units of measure we can do so by multiplying several fractions together in a process known as dimensional analysis. The trick will be to decide what fractions to multiply. When multiplying, if we multiply by 1, the value of the expression does not change. One written as a fraction can look like many different things as long as the numerator and denominator are identical in value. Notice the numerator and denominator are not identical in appearance, but rather identical in value. Below are several fractions, each equal to one where numerator and denominator are identical in value.

The last few fractions that include units are called conversion factors. We can make a conversion factor out of any two measurements that represent the same distance. For example, . We could then make a conversion factor because both values are the same, the fraction is still equal to one. Similarly we could make a conversion factor . The trick for conversions will be to use the correct fractions.

The idea behind dimensional analysis is we will multiply by a fraction in such a way that the units we don't want will divide out of the problem. We found out when multiplying rational expressions that if a variable appears in the numerator and denominator we can divide it out of the expression. It is the same with units. Consider the following conversion.

#### Example 1.

In the previous example, we had to use the conversion factor so the miles would divide out. If we had used we would not have been able to divide out the miles. This is why when doing dimensional analysis it is very important to use units in the set-up of the problem, so we know how to correctly set up the conversion factor.

#### Example 2.

If pound = ounces, how many pounds ounces?

The same process can be used to convert problems with several units in them. Consider the following example.

#### Example 3.

A student averaged 45 miles per hour on a trip. What was the student's speed in feet per second?

If the units are two-dimensional (such as square inches - in ) or three-dimensional (such as cubic feet - ) we will need to put the same exponent on the conversion factor. So if we are converting square inches in to square , the conversion factor would be squared, . Similarly if the units are cubed, we will cube the convesion factor.

#### Example 4.

When calculating area or volume, be sure to use the units and multiply them as well.

#### Example 5.

A room is 10 ft by 12 ft. How many square yards are in the room?

To focus on the process of conversions, a conversion sheet has been included at the end of this lesson which includes several conversion factors for length, volume, mass, and time in both English and Metric units.

The process of dimensional analysis can be used to convert other types of units as well. If we can identify relationships that represent the same value we can make them into a conversion factor.

#### Example 6.

A child is prescribed a dosage of of a certain drug and is allowed to refill his prescription twice. If there are 60 tablets in a prescription, and each tablet has , how many doses are in the ?

**World View Note: **Only three countries in the world still use the English system commercially: Liberia (Western Africa), Myanmar (between India and Vietnam), and the USA.

Source: Tyler Wallace, http://www.wallace.ccfaculty.org/book/7.8%20Dimentional%20Analysis.pdf

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