**Converting one unit to another changes nothing: it
just expresses a distance, a volume, a mass (etc) in different
units.** In effect, you are multiplying by one. Using it
is the easiest way to find out how many miles someone from Canada
means when they say that the hockey rink is about 42 km away.
Here’s what you do:

1 mi
42 ~~km~~ × ———————— = 26 mi This works because 1 mi = 1.61 km
1.61 ~~km~~

**Multiplying by any conversion factor is just like
multiplying by one. You change the number but not the
quantity.**

The number of miles is directly proportional to the number of
kilometers. The algebraic equation that leads to the math above
is:

*x* mi 1 mi
——————— = ——————— which is:
42 km 1.61 km

1 mi
*x* mi = ——————— × 42 ~~km~~ so *x* mi = 26 mi
1.61 ~~km~~
Notice especially the way the units cancel each other out so
that you are left with the units you want and you cancel the units you have.

These proportions are called **unit factors**
because they are equal to the number one. Remember, multiplying
by the number one does not change the numerical value! The unit
factor is used to convert one unit to another unit as in the
examples on this page. Pick the unit factor you use carefully as
the units must cancel out. If the unit to be cancelled is not
part of a fraction or is in the numerator then the unit factor
must be written with that unit in the denominator. Here are more
examples:

How many seconds are in one year?
365 ~~day~~ 24 ~~hr~~ 60 ~~min~~ 60 s
1 ~~yr~~ × ——————— × ——————— × ——————— × ——————— = 31,536,000 s
1 ~~yr~~ 1 ~~day~~ 1 ~~hr~~ 1 ~~min~~
Notice that the unit you want to cancel goes on the opposite side of the
conversion factor from the unit you want to replace it: we want the unit
‘yr’ to cancel to be replaced by ‘day’.
Express 25 miles per hour as meters per second
~~mi~~ 1 ~~km~~ 1000 m 1 ~~hr~~ 1 ~~min~~
25 ——— × ——————— × ——————— × ——————— × ——————— = 11 m/s
~~hr~~ 0.621 ~~mi~~ 1 ~~km~~ 60 ~~min~~ 60 s

##### Key Point

The math for all of this is really quite simple: multiply by
the top, divide by the bottom. Do this for each conversion
factor, cancelling units as you go. Do not record intermediate
values from your calculator: do all calculations as one long
series.

### Unit Factor Basic Skills

When it comes to knowing how to use the Unit Factor Process
(also known as dimensional analisys) there are two basic skills.
First, building a unit factor. Second, setting up and carrying out
the calculation.

#### Building a Unit Factor

One quantity (like how much milk is left in the jug) can be
expressed using different units: saying 1 quart is left is the same
as saying 2 pints are left or saying 4 cups are left or even 950 mL
are left. A unit equality gives the relationship between two units.
It can be used to make two unit factors.

For example:

1 lb = 0.454 kg

can be written as either of the following:

1 lb 0.454 kg
——————— ———————
0.454 kg 1 lb

This works because of some simple algebra:

1 lb 0.454 kg 1 lb
1 lb = 0.454 kg ----------- = ----------- ----------- = 1
0.454 kg 0.454 kg 0.454 kg
**divide both sides by 0.454 kg** **0.454 kg/0.454 kg = 1!**

The math above gives the unit factor on the left, but what about
the one on the right? Can you figure out the necessary math to get
that one?

#### Setting up Calculations

Which of the unit factors do you choose to solve a problem? It
depends.

**If you have a quantity expressed in kg then
choose the unit factor on the left.**

1 lb
5.0 kg × ——————— = 11 lb
0.454 kg

0.454 kg
11 lb × ——————— = 5.0 kg
1 lb

**If you have a quantity expressed in lb
then choose the unit factor on the right.**

Why do you choose the unit factors as described in the examples
above? You choose them that way so that you cancel out the units
you are changing *from* and introduce the units you are
changing *to*.

Here is a bigger example.

Convert 2 years to milliseconds (ms).

2 yr = 2 yr/1

2 yr 365 day 24 hr 60 min 60 s 1,000 ms
------ x -------- x -------- x -------- x -------- x ---------- = 6.3072 × 10^{10} ms
1 1 yr 1 day 1 hr 1 min 1s
Enter this into your calculator as 2 × 365 × 24 × 60 × 60 × 1,000 = 6.3072 × 10^{10}

If it helps you to keep track of what you are doing then make a
small chart like the one at left. The important thing to note in
this example is that the unit to be cancelled out is always on the
opposite side of the fraction bar.

Here is an example in which you must cancel units both in the
numerator and the denominator:

Convert 2.88 × 10^{4} km/hr to mi/s

28,800 km 1 hr 1 min 0.621 mi
------------ x -------- x -------- x -------- = 4.97 mi/s
1 hr 60 min 60 s 1 km

Type into your calculator: 2.88e4 ÷ 60 ÷ 60
× 0.621 =