Converting Units

1. Introduction to Converting Units

Unit conversion is a simple process to overlook, but doing so can have dramatic effects on the results of any project we may be working on. For example, suppose someone told us that a machine needs to be able to handle a 50,000 N load. If we were in a country that uses the metric system, that may make perfect sense to us. If we are in the United States, where English units are used, that may not make any sense at all. The need then arises to convert 50,000 N into pounds (lbs), so that we can more easily understand the quantities we are working with. Unit conversion allows us to express 50,000 N as roughly 11,240 lbs, or 5.62 tons. While the numbers and units are different, the actual quantity they represent is the same.

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Notice that each of these values has different units, but they represent the same amount of force. That is why unit conversion is important. It allows us to represent quantities in terms of measurements we understand or need to work with.

Let’s next look at how to do some unit conversion.

2. Simple Unit Conversion

To convert units, we first need to understand something called a conversion factor, which is basically a fraction equal to 1 that relates two different units.

step by step

1. List the value you are given.
2. Determine the conversion factor.
3. Multiply the conversion factor by the given value, making sure that the units we start with cancel, and the unit we are looking for will be left.
4. Simplify as needed.

EXAMPLE

Suppose we have 160 cups of water, and we want to determine how many gallons of water this is. Before we make any calculations, we might recall that there are 16 cups in one gallon of water. This knowledge will help us determine what our conversion factor will be.

So, how do we begin converting?

For the problem involving cups to water, here is what this would look like:

	Conversion factor: 1 gallon = 16 cups Place 16 cups in the denominator so that the “cups” cancel.
	Multiply across numerators and denominators.
	The units of cups cancel.
	Our solution

try it

Try doing the following calculations on your own and then check the solutions.

Convert 7200 seconds into hours, using the fact that 1 hour = 3600 seconds.

	Conversion factor: 1 hour = 3600 seconds
	Multiply across numerators and denominators.
	The units of seconds cancel.
	Our solution

Convert 2 miles into feet, using the fact that 1 mile = 5280 feet.

	Conversion factor: 5280 feet = 1 mile
	Multiply the numerators and denominators.
	Our solution

3. Multi-Step Unit Conversion

In the above example of converting seconds to hours, suppose we did not know a conversion factor between hours and seconds. Do you think we can make the conversion?

Of course! In cases such as this, we may wish to use multiple conversion factors to help us make a conversion. For example, we may know that there are 60 seconds in 1 minute, and 60 minutes in 1 hour.

Therefore, we can do the following:

EXAMPLE

	Conversion factors: 1 min = 60 sec; 1 hr = 60 min
	Multiply the numerators and denominators.
	The units of seconds and minutes cancel, leaving hours.
	Our solution

try it

Try your hand at converting 3 meters into inches. Note that there are 2.54 centimeters in 1 inch, and 100 centimeters in 1 meter.

Convert 3 meters into inches.

To solve this problem, we can implement the conversion factors 2.54 centimeter = 1 inch and 100 centimeter = 1 meter, and string the conversion factors together.

	Conversion factors: 1 m = 100 cm; 1 in = 2.54 cm
	Multiply the numerators and denominators.
	The units of meters and centimeters cancel, leaving inches.
	Our solution

4. Converting Units of Area and Volume

Sometimes when converting units, we may need to convert between squared units (area) or cubed units (volume). In these instances, we follow the same process as before, but we have to be careful with our conversion factors. Let's look at some examples.

EXAMPLE

Suppose we are told that we need to convert 200 square feet into square inches. How would we go about doing this calculation?

Before we begin the conversion, we need to determine if we are using the correct conversion factor. Here, we may be tempted to use 12 inches in 1 foot, making the conversion factor (12 inches / 1 foot). However, we would be incorrect when making this calculation. In truth, there are 12 inches • 12 inches, or 144 square inches in 1 square foot.

Here is a picture to help visualize this.



Each side is 1 foot, or 12 inches. This means that each block is

This also means that the entire figure has area

Therefore, the conversion between square feet and square inches is

Going through the calculations, we would do the following:

	Conversion factor: 1 sq ft = 144 sq in
	Multiply the numerators and denominators.
	The units of square feet cancel, leaving square inches.
	Our solution

Let’s now look at a real-life situation in which we need to use conversions involving volume.

IN CONTEXT

A home cook has made a pot of soup for a sick friend. He’s let the soup cool enough to store but now must decide which storage container to use. The pot is in the shape of a cylinder, so he takes quick measurements: The diameter is 10 inches, and the depth of the soup is about 5 inches. His storage containers are also cylindrical but have measurements in quarts. How can he figure out how much liquid is in the pot? Since the volume will be measured in cubic inches, we need a conversion factor between cubic inches and quarts. After doing some research, it is found that 1 quart is 57.75 cubic inches.

To solve this problem, here is what we need to do:

• Find the volume of soup in the pot.
• Convert the result to quarts.

A) Find the volume of the soup.

	Here is the formula for the volume of a cylinder with a circular base.
	Since the diameter of the base is 10, the radius is 5.
	Simplify the result.
	Using 3.14 for π, approximate the volume.

To ensure that the soup fits in the container, we’ll round this to 400 in³, just because this is a nice number. Note that rounding up helps us to play it safe. We would never round down since our goal is to make sure the container is large enough.

B) Now, apply the conversion factor to determine the number of quarts.

	Place the cubic inches in the denominator so that these units cancel out and we are left with quarts.
	Simplify (cubic inches drop).
	Approximate to the nearest tenth.

Conclusion: Any container larger than 6.926 quarts would suffice. In reality, 7 quarts might be a tight squeeze, so anything larger will be optimal.

Note: This is almost 2 gallons of soup! 

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The key idea here is that we must match squared units with squared units when choosing the conversion factor.

EXAMPLE

Suppose you want to convert 5 cubic feet to liters. How would you make this conversion, given that there are approximately 30.48 centimeters in 1 foot and 0.001 liters in 1 cubic centimeter?

Here, we have to first determine a conversion factor between cubic feet and liters. Right now, we do not have a conversion factor relating the two; however, we do know how many centimeters are in 1 foot. Therefore, if we take the cube of 30.48 centimeters, we would know how many cubic centimeters there are in 1 cubic foot. With this knowledge, we can use the other conversion factor we are given to convert between cubic centimeters to liters.

Using this conversion factor, we can now begin making our conversion.

	Here are the conversion factors for to and to L.
	Multiply the numerators and denominators.
	The units of and cancel, leaving L.
	Our solution (rounded to the hundredths place)

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When working with volume conversion, you must make sure your conversion factor is converting between cubic units.