The conversion of numbers from one system of units to another often
puzzles students, but if it is treated as just another problem in arithmetic
using arithmetic's rules the problems disappear.

For example: Knowing that there are 2.54 cm in 1.0 inch, how many cm
are there in 15 inches?

Of course the answer is simple

15 in = 15.0 X 2.54 = 38.1 cm

But what actually was being done here? The complete solution is as follows:

Notice that the units were treated just like arithmetic quantities and
the "in" were canceled.

Let's do one that is not quite as obvious! How many (mm)^{2}
are there in 4.0 (in)^{2}? The solution is:

Notice that the quantities 2.54 cm/in and 10 mm/cm were used but, because
the units had to be squared, then the numbers that accompanied them had
also to be squared.

Another example: Convert 30 mi/hr into m/s. This is a common conversion. It is necessary to know that there are 1.6 km in 1 mile and 3600 s in 1 hr (60 X 60).

Notice that the only tricky part here is the time conversion 3600 s/hr
which is the wrong way up for our conversion but of course it is equally
true that there are (1/3600) hr/s.

So long as the relevant relations between the quantities are assembled
in advance, then any conversion can be performed using these strict arithmetic
rules.

Try this more complicated conversion:

An old (and ridiculous) unit of thermal conductivity sometimes still encountered in building materials is

Btu.hr^{-1}.in.F^{-1}.ft^{-2}

where 1 Btu = 1054.8 J

1 in = 2.54 cm

1 ft^{2} = 0.0929 m^{2}

9 F = 5 C

What is 1 (Btu.hr^{-1}.in.F^{-1}.ft^{-2}) in
proper units (W.m^{-1}.C^{-1})?