Page 22 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 2] MATHEMATICAL METHODS IN CHEMISTRY 11

How many dimes are there in 9.40 dollars? We know that

10 dimes = 1 dollar or 1 dime = 0.1 dollar
We may divide both sides of the first of these equations by 10 dimes or by 1 dollar, yielding

10 dimes 1 dollar 10 dimes 1 dollar
= or =
10 dimes 10 dimes 1 dollar 1 dollar
Since the numerator and denominator (top and bottom) of the fraction on the left side of the first equation are
the same, the ratio is equal to 1. The ratio 1 dollar/10 dimes is therefore equal to 1. By analogous argument, the
first ratio of the equation to the right is also equal to 1. That being the case, we can multiply any quantity by
either ratio without changing the value of that quantity, because multiplying by 1 does not change the value of
anything. We call each ratio a factor; the units are the labels.
We can use the equation 1 dime = 0.1 dollar to arrive at the following equivalent equations:
1 dime 0.1 dollar 1 dime 0.1 dollar
= =
1 dime 1 dime 0.1 dollar 0.1 dollar
Students often like to use the equations above to avoid using decimal fractions, but these might be more useful
later (Sec. 2.3).
To use the factor-label method, start with the quantity given (not a rate or ratio). Multiply that quantity by a
factor, or more than one factor, until an answer with the desired units is obtained.
Back to the problem:
10 dimes 1 dime

9.40 dollars = or 9.40 dollars =
1 dollar 0.1 dollar
The dollar in the denominator cancels the dollars in the quantity given (the unit, not the number). It does not
matter if the units are singular (dollar) or plural (dollars). We multiply by the number in the numerator of the
ratio and divide by the number in the denominator. That gives us
10 dimes 1 dime

9.40 dollars = 94 dimes or 9.40 dollars = 94 dimes
1dollar 0.1dollar
EXAMPLE 2.1. How many dollars are there in 220 dimes?

1 dollar
Ans. 220 dimes = 22.00 dollars
10 dimes
In this case, the unit dimes canceled. Suppose we had multiplied by the original ratio:
2
10 dimes 2200 dimes
220 dimes =
1 dollar dollar
Indeed, this expression has the same value, but the units are unfamiliar and the answer is useless.
More than one factor might be required in a single problem. The steps can be done one at a time, but it is
more efficient to do them all at once.

EXAMPLE 2.2. Calculate the number of seconds in 2.50 h.
Ans. We can first calculate the number of minutes in 2.50 h.
60 min

2.50 h = 150 min
1h
Then we can change the minutes to seconds:
60 s

150 min = 9000 s
1 min