by Donald Proctor

We will resort to a bit of silliness to make a point about what conversion factors should be and how they should be used. Imagine that I have a 3-ft-long rattlesnake and want to know if it is long enough to make a belt to hold up my pants. Since I don’t know my waist size in feet, please tell me what to do to get the snake length in inches.

Did I hear you say that I should multiply by 12? Let’s do it.

          Snake length × 12 = 3 ft × 12. So, 12 snake lengths = 36 ft.

Now I know that these snakes are 36 ft long per dozen, but I still don’t know the length of this one snake expressed in inches. Any more ideas?

Only multiply the right side by 12 and leave the left side alone, you say? Wait just a second. I have two things, each equal to the other: a snake length (A) and 3 ft (B). You want me to make one of those things 12 times as large but leave the other unchanged, and then pretend that they are still equal? Is that even legal?

Let’s drop the snake for the moment and consider the single most important rule of all of algebra. If two things that are equal are treated equally, they should still be equal! If not treated equally, they ain’t equal no more! (The rule is correct even if the grammar ain’t!)

Let’s test the rule:  1 ft = 12 in. Two things equal to each other. Now let’s divide both of these equal things by the same thing, namely by 1 ft.

          (1 ft/1 ft) = (12 in./1 ft), which is the same as 1 = (12 in./ft)

This seems to indicate that the value of 1 (with no units) is exactly equal to 12 in./ft. Logically then, I could multiply the snake length times 1 and the 3 ft by 12 in./ft to get the following: Snake length × 1 = 3 ft × (12 in./ft) = 36 in.

This leads us to an important rule: Every conversion factor that can logically and legitimately be used must be a factor exactly equal to 1.0 (no units of measure). Examples of such conversion factors are: 12 in./ft,  1 ft/12 in., 2000 lb/ton, 24 h/d, etc.

A proper conversion factor does not change the absolute value of anything. It only changes the units of measure of that value. For example, 3 days and 72 hours are the same amount of time, just expressed in different units.

Also note that any two (or more) conversion factors multiplied together simply make another conversion factor. Some examples are listed below:

                      12 in./ft × 12 in./ft = 144 in.2/ft²
              60 sec/min × 60 min/h × 24 h/d = 86,400 sec/d
  86,400 sec/d × 365d/yr × 100 yr/century = 3,153,600,000 sec/century (ignoring leap years)

Any fraction consisting of a numerator (upper term) exactly equal to the denominator (bottom term) is equal to 1.0 and is therefore a conversion factor. For example, 30.48 cm/12 in. = 1.0 is a conversion factor that could be used to find how many centimeters there are in 65 in.

            65 in. × (30.48 cm/12 in.) = 165.1 cm (Note that inches cancel out.)

If you will get in the habit of always including the units of measure on all quantitative values in your solutions and always using proper conversion factors, you will probably find that it will allow you to avoid about half of the mistakes you might otherwise make. The most common mistake is assuming that the units are what you think they should be when you should have used a conversion factor but forgot to do so. If the units don’t come out correctly, the numbers are probably wrong, too.