Basic Math Operations - Mathematics and Statistics for the Environment - page 53

2.9 DIMENSIONAL ANALYSIS
Dimensional analysis is a problem-solving method that uses the fact that any number or expression
can be multiplied by 1 without changing its value. It is a useful technique used to check if a problem
is set up correctly. In using dimensional analysis to check a math setup, we work with the dimen-
sions (units of measure) only—not with numbers.
An example of dimensional analysis that is common to everyday life is the unit pricing found in
many hardware stores. A shopper can purchase a 1-lb box of nails for 98¢ at a local hardware store,
but a nearby warehouse store sells a 5-lb bag of the same nails for $3.50. The shopper will analyze
this problem almost without thinking about it. The solution calls for reducing the problem to the
price per pound. The pound is selected without much thought because it is the unit common to both
stores. The shopper will pay 70¢ a pound for the nails at the warehouse store but 98¢ at the local
hardware store. Implicit in the solution to this problem is knowing the unit price, which is expressed
in dollars per pound ($/lb).

Note: Unit factors may be made from any two terms that describe the same or equivalent amounts
of what we are interested in; for example, we know that 1 inch = 2.54 centimeters.

In order to use the dimensional analysis method, we must know how to perform three basic
operations.

2.9.1 b asiC o peration : d ivision oF u nits
To complete a division of units, always ensure that all units are written in the same format; it is best
to express a horizontal fraction (such as gal/ft 2 ) as a vertical fraction.

Horizontal to vertical

gal
ft

3

gal/ft o

3

lb
in.

psito

2

The same procedures are applied in the following examples.

3

ft
min

3

ft /min becomes

s
min

s/minbecomes

2.9.2 b asiC o peration : d ivide by a F raCtion
We must know how to divide by a fraction. For example,

lb
day
min
day

lb
day

day
min

becomes ×