Jse of Dimensions
29
Calculation is to divide the total number of units of one property by the total
number of units of another property to which it corresponds. Thus, if we are
told that 4.2 gallons weigh 10.5 pounds, we express the relation by the ratio
10.5 Ib _ 25 _lb_
4.2 gal
' gal
which is read as "2.5 pounds per gallon." A few examples will help in understanding the use of units.
PROBLEM:
If apples cost 30 cents per dozen, how many can be bought for 50 cents?
SOLUTION:
Set up an equation that will eliminate "cents" and give "apples" in its place, as
follows.
(a) number of apples =
12
50 x 12 apples
-
~
= 2
°
apples
or (b) number of apples = (50 cents) LQ^^\ (l2
50 x 1 x 12 apples
= - — — — — = 20 apples
In (a) we divided 50 cents by 30 cents/doz, while in (b) we multiplied 50 cents by 1
doz/30 cents, the reciprocal of 30 cents/doz. These are equivalent operations and
you should feel comfortable with either method. In each case, "cents" and "doz"
cancel out, leaving only "apples," as we had hoped.
A somewhat more sophisticated method for setting up the problem is to use
the negative exponent, - 1, for units that appear in the denominator of a set of
units. Thus the term "per dozen" may be written as doz"
1
. The preceding
problem could have been set up in the following form:
number of apples = I —
;——) (12 apples dee"
1
) = 20 apples
\ 30 cents dezrv
PROBLEM:
Find the number of feet in 1.5 mi (one mile is 5280 ft).