How to Convert Units of Measure

Two quantities that represent equivalent physical amounts can be written as a ratio that equals 1. For example, we know that "penny" and "nickel" are two different units where 1 nickel is the same amount of money as 5 pennies. It follows that

5 pennies = 1 nickel

therefore,

5 pennies                   1 nickel
--------- = 1     and       --------- = 1
1 nickel                    5 pennies

These ratios are called unit factors because they are equivalent to 1 (unity). Any number multiplied by 1 will be equal to itself. Therefore, we can multiply any quantity by a unit factor to change the unit of measure without changing the physical quantity.

Example 1

Q: If we have 23 nickels, what is the equivalent amount of money in pennies?

A: Multiply 23 nickels by a unit factor that will change the unit of measure from nickels to pennies.

             5 pennies
23 nickels x ---------  =  23 x 5 pennies  =  115 pennies
             1 nickel

We chose to use the unit factor that had "nickels" in the denominator because the "nickels" labels would cancel and leave us with "pennies".

Now consider the case where we do not have a single unit factor that will do the job for us. In this case, we will have to chain-link several unit factors together to obtain the desired result:

Example 2

Q: A bottle holds 10 fluid ounces of water. Using the following equivalencies, determine how many cubic centimeters of water the glass contains.

16 fl oz = 1 pint
2 pints = 1 quart
1 liter = 2.1134 pints
1 ml = 1 cubic cm
1 liter = 1000 ml

A: Start with the known quantity, 10 fl oz, and multiply it by a unit factor that will cancel the "fl oz" label:

           1 pint
10 fl oz x --------
           16 fl oz

Continue to multiply by unit factors until the desired label is obtained:

           1 pint     1 quart    1 liter         1000 ml   1 cubic cm
10 fl oz x -------- x -------- x ------------- x ------- x ----------
           16 fl oz   2 pints    1.0567 quarts   1 liter   1 ml

          10 x 1000
        = ---------------- cubic cm
          16 x 2 x 1.0567

        = 295.73 cubic cm

Of course, if we had known in the first place that 1 fl oz = 29.5737 cubic cm, we could have used a single unit factor,

           29.5737 cubic cm
10 fl oz x ---------------- = 295.737 cubic cm
           1 fl oz

Complex dimensional labels, such as miles/hour, can be converted in the same manner. Simply convert one unit at a time.

Example 3

Q: Convert the speed of 55 miles/hour into meters/second, given that

1 mile = 1609.34 meters
1 hour = 60 minutes
1 minute = 60 seconds.

A: First, convert miles to meters:

55 miles   1609.34 meters   88513.7 meters
-------- x -------------- = --------------
   hour    1 mile                   hour

Then, convert hours to seconds:

88513.7 meters   1 hour       1 minute
-------------- x ---------- x ----------  
        hour     60 minutes   60 seconds

          88513.7 meters
        = ------- ------
          60 x 60 second

        = 24.59 meters/second

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