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.
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
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 quarts
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"
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 ----------
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,
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.
55 miles
1609.34 meters
88513.7 meters
-------------- x
--------------
= -----------
hour 1 mile hour
Then, convert hours to seconds:
--------------
x
----------
x
----------
hour 60 minutes 60 seconds
88513.7 meters
=
------- ------
60 x 60 second
= 24.59 meters/second