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# How to Convert Units of Measurement

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. 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 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" 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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- Darin Franklin |
DarinFranklin@home.com