Dimensional Analysis Many math problems involve answers with labels, such as “3 feet” instead of just “3”, or “45 miles per hour” instead of just “45”. (These labels are also called “units” or “units of measurement”.) Whenever you need to change the labels (or units) in a problem, you can use a process called dimensional analysis, which involves using ratios and setting up proportions. Before getting into the steps involved with dimensional analysis, take a step back and recall a rule about fractions: If you need to change the numerator or denominator of a fraction, both must be multiplied (or divided) by the same number so that the overall value of the fraction is not changed. 2 7 + , both fractions need a common denominator of 12. The first 3 12 fraction can have its denominator changed to 12 by multiplying the numerator and denominator by 4: For example, to add

4 is a fraction called a conversion factor, which always has a value of 1. (Any fraction 4 with the same value in the numerator and denominator is equal to 1.) In this way, the value of the original fraction isn’t changed since anything multiplied by 1 equals itself.

This is the same way dimensional analysis works. To change the units of a numerical quantity, multiply it by a conversion factor (a fraction with a value equal to 1). example 1

12 feet is equal to how many yards? The answer to this problem will still have a value equal to 12 feet, but the label is being changed to yards.

The hardest part of dimensional analysis is to determine the conversion factor. In this example, what is the relationship between feet and yards?

So now convert this relationship to a fraction equal to 1:

3 1 nor may look like a fraction equal to 1, but it’s the units that make each 1 3 fraction have a value of 1. Neither

But which one is the conversion factor needed in this problem? Since the answer is to be in “yards”, use the conversion factor with “yards” in the numerator:

That way, the label “feet” cancels:

So 12 feet = 4 yards. example 2

The Madhany family traveled 3560 miles on a trip across the United States. Since one mile is about 1.6 kilometers, which of the following is closest to the total number of kilometers in 3560 miles? A. B. C. D.

5700 kilometers 4540 kilometers 3558 kilometers 2225 kilometers

In this problem, miles must be converted to kilometers and the conversion factor needed is given to you right in the problem.

The dimensional analysis can be set up as the original quantity multiplied by a conversion factor:

Next, cancel like units from each numerator and denominator:

Multiply 3560 × 1.6

So 3560 miles equals approximately 5700 kilometers. The answer is A. example 3

The largest natural lake in Massachusetts is Assawompsett Pond which has an area of 2,656 acres. What is the approximate area in square miles? A. B. C. D.

4 sq. mi. 7 sq. mi. 17 sq. mi. 40 sq. mi.

In this problem, acres must be converted to square miles, but the conversion factor needed is not given in the problem. It would need to be provided on a reference sheet or from some other source in order to answer this question. (In this case, you need to know that 1 square mile equals 640 acres.) The dimensional analysis can now be set up the same way as the previous example:

Divide 2656 ÷ 640

So 2,656 acres is approximately 4 square miles. The answer is A. example 4

If a car is traveling 30 miles per hour, how many feet per minute is the car moving? This problem changes a rate, which has two units of measurement, distance and time. Dimensional analysis can be used to convert units of measure within a rate because a rate can be expressed as a fraction. We’ll just need two conversion factors instead of one. The conversion factors for changing miles to feet and hours to minutes are:

Writing this all out so we can cancel like units:

A car traveling 30 miles per hour is also moving 2640 feet per minute.

Name ______________________________ Use dimensional analysis to solve. 1) During lunch, a local fast-food restaurant sold 48 quarter-pound hamburgers. How many kilograms of hamburgers were sold?

2) Bill and Sheila’s living room is 18 feet long and 12 feet wide. How many square yards is the area of their living room?

3) A mining company removes 30 tons of rock from a quarry every day. If the company works around the clock, (24 hours per day), how many pounds of rock are removed from the quarry each minute?

4) 1 cubic meter is how many cubic millimeters? Express your answer in scientific notation.

Use dimensional analysis to solve. 5) Deanna’s little sister wants to go on a carnival ride, but only children taller than 120 centimeters may go on it. Deanna’s sister is 4 feet tall. Can she go on the ride?

6) Light travels about 300,000 kilometers per second. The Sun is approximately 150,000,000 kilometers away from the Earth. How many minutes does it take light from the Sun to reach Earth?

7) Boston and Springfield are 90 miles apart in Massachusetts. How many kilometers apart are the two cities?

8) The distance around the track at the high school is 440 yards. How many meters long is the track?