48 Unit Conversions

Introduction

How many minutes in three hours? How many cents in $43.57? Some unit conversions are easy enough to do in our heads without mistakes. What about converting 60 mph into fps? Not so easy, I’m guessing. It’s easy to get conversions wrong-particularly if you don’t write anything down. Sloppy habits work for easy problems, but not for anything vaguely complicated. There’s just too much to track of.

You can, of course, resort to the internet: “Google, how many m/s is 60 miles per hour?” (and this technique is OK if the internet is available and you trust yourself to use it correctly).

This section shows you a mostly foolproof (paper and pencil) method for doing unit conversions. Even if you make mistakes with this method, you can should be able to catch the mistakes before you get out your calculator. Do easy conversions in your head if you want, but for anything important (or vaguely complicated) use the factor label method!

Factor Label Method

The factor label method for converting units relies on two simple ideas:

1. When you multiply anything by one, it remains the same
2. Any fraction with the same quantity in the numerator and denominator equals one.

The basic strategy is to multiply the original measurement by a string of fractions, each equal to one. Each unit conversion fraction is structured to “cancel out” units you don’t want and replace them with the units you do want. It’s easiest to show with some examples.

Basic example

How many minutes in 3 hours? This conversion is so easy, you probably wouldn’t use the method for it, but it shows how the method works.

[latex]3 \:hr = 3 \:hr \times \frac{60 \: min}{1 \: hr} =\frac{3 \: hr \times 60 \: min}{1 \: hr} =180 \:min[/latex]

The unit conversion fraction has 60 minutes on top and 1 hour on bottom. Since top an bottom are equivalent, the fraction is equal to one. I chose to put hours in the denominator to make the “hr” unit in “3 hr” cancel out. (There are units of hr in both numerator and denominator).

“Chain” example

Suppose you want to convert 10 ks into hours. You know there’s a factor of 1000 involved for “kilo” and factor(s) of 60 involved for getting from seconds to hours. But should you divide or multiply? The factor label is especially useful for doing a series of conversions, one after another.

Simply chain together a series of conversion fractions. Each fraction should cancel out units you don’t want and replace them with something “better.”

[latex]10 \:ks = 10 \:ks \times \frac{1000 \: s}{1 \: ks} \times \frac{1 \: min}{60 \: s} \times \frac{1 \: hr}{60 \: min}[/latex]

Multiply all the numbers on the top of the fraction together, multiply all the numbers on the bottom together and cancel out units that appear on both top and bottom, and you get something simple:

[latex]=\frac{10,000}{3600} \:hr=2.8 \:hr[/latex]

The method will catch mistakes! Suppose you start this way:

[latex]10 \:ks = 10 \:ks \times \frac{1 \: ks}{1000 \: s} \times \frac{60 \: s}{1 \: min} \times \frac{60 \: min}{1 \: hr}=[/latex]

You probably see the problem already- the “ks” unit in the first conversion fraction does not cancel out the “ks” in “10 ks.” If you continue, you get something pretty confusing:

[latex]=\frac{10 \times 60 \times 60}{1000} \times \: \frac{ks \times ks}{hr}??[/latex]

The nonsense unit at the end is a sure sign that something went wrong. Go back and try again. Flip one (or more) of the conversion fractions. As you practice, you’ll get a feel for spotting exactly which one(s) to flip.

Compound units example

Sometimes you have units that involve more than one unit (e.g miles per hour, square feet, etc). Simply do each conversion separately. Suppose you want to know what 70 mph (miles per hour) is in m/s and you know 1 mile = 1609 m.

I find it helpful to show that explicitly by writing 70 miles per hour as 70 miles divided by one hour:

[latex]\frac{70 \:mile}{1 \:hr} = \frac{70 \:mile}{1 \:hr} \times \frac{1609 \:m}{1 \:mile} \times \frac{1 \: hr}{60 \:min}\times \frac{1 \: min}{60 \: s}[/latex]

The first fraction changes the distance unit and the next two deal with the time unit. Finishing up:

[latex]=\frac{70 \times 1609}{60 \times 60} \frac{m}{s}=31 \frac{m}{s}[/latex]

Resources:

https://brownmath.com/bsci/convert.htm