Unit Conversion Using

Dimensional Analysis

Objective N-Q.1Use units as a way to understand problems and to guide the solution of multi-step problems

Why do you need to know how to convert units?

We use measurements every day of our lives. When you go to the store, items are sold by weight, volume, length, number of items, etc. It is important to know how these units of measurement compare to each other and how to readily convert between various units.

Let’s explore what is involved when we need to make a conversion between different units. We need to start with conversion factors.

What is a conversion factor?

A conversion factor is a ratio written in fraction form that expresses the same quantity (an equality) using two different units. Every conversion factor can be written two ways, which are reciprocals of each other.

For example:

Same Quantity 2 Conversion Factors1 foot = 12 inches

7.48 gal = 1 ft3 or

Write each of the given equal quantities as 2 reciprocal conversion factors. 2 conversion factors

60 seconds =1 minute

1 mile = 5280 feet

1.06 quarts = 1 liter

*All conversion factors must be labeled correctly

What is dimensional analysis?

Dimensional analysis (DA) is a systematic mathematical method using conversion factors to move from one unit of measurement to another.

Here is a flow chart showing DA basics.

Choose the conversion

factor with the given unit in the denominator

Start with the given quantity

Calculate answer by

multiplying and dividing

Example of a one-step conversion using DA

Convert 153 cm to inches (1 inch = 2.54 cm)

1. Write down the given quantity (153 cm)2. Write the equality as two conversion

factors

or

3. Choose the conversion factor that has the given unit in the denominator and the desired unit in the numerator.

Step 1 Step 3 Answer

153cm 1inch2.54cm

=60.2 inches

*Notice the unit of centimeters is cancelled

Now its your turn!

Convert 13.5 feet to inches (1 foot = 12 inches)

1. Write down the given quantity 2. Write the equality as two conversion

factors

3. Choose the conversion factor that has the given unit in the denominator and the desired unit in the numerator.

Step 1 Step 3 Answer

=¿

Using DA in a multi-step conversion

Follow the same steps as before, but use as many conversion factors as necessary to obtain the desired unit.

Convert 3.5 miles to inches

Step 1 Step 3 Answer

Example of a multi-step conversion using DA

3.5miles 5280 feet1mile

12 inches1 foot

=221760 inches

Now it’s your turn!

On a recent trip to Canada it took 93.6 liters of gasoline to completely fill the empty gas tank in your truck. What is the capacity of the gas tank in gallons?Use DA to set up the conversions and solve! (1 gallon = 4 quarts) (1 liter = 1.06 quarts)

One last note about DA!

All of the conversion factors we have used thus far have been based on equalities with different units. 1 inch = 2.54 cm 1 foot = 12 inches

However, a conversion factor can be written from any type of relation. Consider the problem below:

Suppose you can make 17 pasties in 45 minutes. How many pasties can you make in 3.5 hours?

Notice that 1 hour = 60 min (an equality), however,17 pasties ≠ 45 min (not an equality), but rather 17 pasties ~ 45 min. Conversion factors can still be written

One last note about DA!

All of the conversion factors we have used thus far have been based on equalities with different units. 1 inch = 2.54 cm 1 foot = 12 inches

However, a conversion factor can be written from any type of relation. Consider the problem below:

Suppose you can make 17 pasties in 45 minutes. How many pasties can you make in 3.5 hours?