Unit Conversions

If you need to convert a number from one unit to another, use this method.  (This is one place where it’s always safest to write it out—doing it in your head makes it much more likely that you won’t notice if a conversion factor comes out backwards.)

You need to start thinking of units as if they are variables.  A value of “100 m” literally means “100 × 1 m” or “100 × m”.

This means that a conversion factor is actually an equation.  An example is the conversion:

100 cm = 1 m

This is actually an equation, so we can do algebra on it.  (Don’t panic!  It’s easy.) 

If we divide both sides by 1 m, we get the following:

Because , we can multiply anything by  and it won’t change the value.  So if we want to convert 1.35 m to cm, we simply multiply it by  (which is the same as multiplying it by 1):

Notice that meters on top cancelled out meters on the bottom, and the only unit left was cm on the top.

Many students find it easier to work with the conversion factors by placing them in a grid:

1.35 m	100 cm
1	1 m

The most important thing to remember is that only the starting number and its units go in the first column (before the double line).  If the units don’t make a fraction, put the number 1 on the bottom.  All of the conversion factors go after the double line.

That was a simple example, but this method works just as well for more complicated problems.  Suppose you had to convert 60 miles/hour to feet/second:

Here is the same problem using the grid:

60 mi.	5280 ft	1 hr.	1 min.
1 hr.	1 mi.	60 min.	60 s