Dimensional Analysis and Conversions for Firefighter Exam Study Guide

What Is Dimensional Analysis?

Dimensional analysis is a technique for solving problems that involve units or conversions that is taught in many engineering schools. It is a very useful technique in some areas of the emergency services, especially in EMS, where drug and fluid administration rates need to be calculated.

Dimensional analysis is based on multiplication of improper fractions. The key to using it is to keep the units with the numbers. The units cancel out in much the same way as fractions do when multiplying. Writing out your work using dimensional analysis makes it much easier to keep track of what you are doing, and to double check your work. If your units do not properly cancel out to give you the unit that you want for the answer, your answer is wrong.

How Is It Used?

Dimensional analysis is most useful in doing conversions of numbers from one unit to another, such as in converting a value in miles per hour to feet per second. Most people probably do not know that to convert miles per hour to feet per second, you multiply by 1.4666667.However, you can perform this conversion using more commonly known information.

To use dimensional analysis, you need to examine the information that you are given in a problem and convert it into an improper fraction. Say you want to convert 45 miles per hour to feet per second. You know, or need to know, the following information:

• What is the speed in mph (45 miles / hour)?
• How any minutes are in an hour (60 minutes / hour)?
• How many seconds are in a minute (60 seconds / minute)?
• How many feet are in a mile (5,280 feet / mile)?

You can write these values with either unit on top or bottom. It depends on what you are starting with and what you want to end up with. Miles per hour and feet per second both have a distance unit on top and a time unit on bottom. Therefore, you should set the start of the problem this way:

You then need to write the other conversions as fractions so that the units cancel out:

Remember, any whole number that you use as a conversion can be written as a fraction either over 1 or 1 over itself. You need to make sure the units cancel out. Once you cancel the units you do the math in the same way that you do multiplication of fractions. You multiply the top numbers together, in this case 45 × 5,280 × 1 × 1 = 237,600, and the bottom numbers together, 1 × 1 × 60 × 60 = 3,600, and divide the top number by the bottom number. The result can be written over 1, or can be written in a more common form as 66 feet / second (fps).

Here are a few problems to try in which dimensional analysis would be useful:

91. If your truck is pumping water at 700 gallons per minute, how many gallons of water are pumped in 2 hours?

Answer: 84,000 gallons

92. Convert 250 feet per minute to miles per hour.

Answer: 2.84 miles per hour

93. If a grass fire burns a total of 320 acres of farmland, how many square miles of farmland were burned? (Hint : 1 acre = 43,560 square feet; a square mile is the same as a square that is 5,280 feet on each side)

Answer: square miles