Dimensional Analysis

Conversions Made Easy

Objectives

• Understand the need to convert units.• After this lesson, you will be able to:

– Convert between units within the metric system.– Convert between historical units.– Convert between U.S. Customary units.– Convert from one system to another.

Why convert?

• Most of the world works within the metric system– The U.S.A. is among only three countries that do

not make extensive use of the metric system.

• When working with numbers from a system that is not familiar to you, being able to place them into a system you are familiar with will help you understand their significance.

Understanding Fractions is Key!

• Often times, the relationship between two systems is expressed as a fraction we call a conversion factor.

• These take the form of an expression consisting of two equal values express in different units.

Examples of Conversion Factors

• 1 mile is equal to 5280 feet, so a the possible conversion factors that we can make from this relationship are:

Take Note!

• Since the values in the denominator and the numerator represent the same thing, the conversion factor is a fraction that equals 1 (even though it does not look like it at first).

• What happens any time we multiply a number by 1?– We end up with a value that is equal to the value

of the original number!

Using Conversion Factors

• If we were given the task of finding how many miles there are between us and the ground when we are flying in an airplane, we would need two things:– Our height in feet.– The conversion factor that relates feet to miles.

The Problem

• When flying across the country, a Boeing 767 cruises at an altitude of 33,000 feet. If there are 5,280 feet in a mile, how many miles above the ground is a 767 flying?

The Importance of Units

• The key to properly setting up and solving this problem lies with the units.

• The units tell us how to set up the conversion factor.

• The goal when setting up the conversion factor is to get the units of the measurement given to us in the problem to cancel out, leaving only the units that we desire our answer to be expressed in.

The Set-up

• We begin by writing our given value (do not forget the units!)33,000 feet

• We then draw a “crossbar” which gives us a place to place the conversion factor.

33,000 feet

Using the Correct Conversion Factor

• Now that we have our cross bar in place, we are ready to fill up the upper and lower spaces on the right hand side of the cross bar with the information given to us in our conversion factor.

• We can either place 5,280 ft over 1 mile, or 1 mile over 5,280 ft…only one way is correct!

• Because our given value is in feet, we have to set our conversion factor up so that feet cancel.

• The only way to get this to happen is to place feet in the denominator.

The Solution

•Since the units “feet” are both in the denominator and the numerator, they cancel out.•This leaves us with “miles” as our only remaining unit, which just happens to be the unit we are looking for!

33,000 feet 1 mile5,280 feet

=

Solving

• To solve the previous expression, we multiply all the numbers across the top and divide by the numbers on the bottom.

• To prevent errors, make sure that you hit “enter” when you are finished multiplying and are ready to divide

• 33,000 x 1 “enter” ÷ 5,280 = 6.25 miles (don’t forget the units!!!)

Things To Consider

• Since all conversion factors are represented by different numbers whose values are the same, we are never changing the value of our given number when we use conversion factors.

• In order to ensure the proper use of conversion factors, place the value with the units that we desire in our answer on the top of the cross bar, and the given units on the bottom.