MULTIPLICATION
The fact that multiplication by a fraction does not
increase the value of the product may confuse those who remember the definition
of multiplication presented earlier for whole numbers. It
was stated that 4(5) means 5 is taken as an
addend
4 times. How is it then that .1/2(4) is 2, a number
less than 4? Obviously our Idea of multiplication
must be broadened.
Consider the following products:
Notice that as the multiplier decreases, the product
decreases, until, when the multiplier is a
fraction, the product is less than 4 and continues
to decrease as the fraction decreases. The
fraction introduces the "part of" idea: 1/2(4)
means 1/2 of 4; 1/4(4) means 1/4 of 4.
The definition of multiplication stated for
whole numbers may be extended to include fractions.
Since 4(5) means that 5 is to be used 4 times
as an addend, we can say that with fractions the numerator of the multiplier
tells how many times the numerator of the
multiplicand is to be used as an addend. By
the same reasoning, the denominator of the multiplier tells how
many times the denominator of the multiplicand is to be used as an addend. The
following examples illustrate the use of this idea:
1. The fraction 1/12 is multiplied by the whole number
4 as follows:
This example shows that 4 (1/12) is the same as 4(1)/12.
Another way of thinking about the multiplication of l/12 by 4 is as follows:
2. The fraction 2/3 is multiplied by l/2 as follows:
member that we can simplify division by showing both dividend and divisor as
the indicated From these examples a general
rule is developed: To find the product of two
or more fractions multiply their numerators
together and write the result as the
numerator of the product; multiply their
denominators and write the result as the
denominator of the product; reduce the answer
to lowest terms.
In using this rule with whole numbers, write each
whole number as a fraction with 1 as the denominator.
For example, multiply 4 times 1/12 as
follows:
In using this rule with mixed numbers, rewrite all mixed numbers as improper
fractions before applying the rule, as follows:
A second method of multiplying mixed numbers
makes use of the distributive law. This
law states that a multiplier applied to a twopart
expression is distributed over both parts. For
example, to multiply 6 1/3 by 4 we may rewrite
6 1/3 as 6 + l/3. Then the problem can be written
as 4(6 + 1/3) and the multiplication proceeds as
follows:
4(6 + 1/3) = 24 + 4/3
= 25 + l/3
= 25 1/3
Cancellation
Computation can be considerably reduced by dividing
out (CANCELING) factors common to both the
numerator and the denominator. We recognize a
fraction as an indicated division. Thinking
of 6/9 as an indicated division, we remember that we can simplify division by
showing both dividend and divisor as the indicated products
of their factors and then dividing like factors,
or canceling. Thus,
Dividing the factor 3 in the numerator by 3 in the
denominator gives the following simplified result:
This method is most advantageous when done before
any other computation. Consider the example,
The product in factored form is
Rather than doing the multiplying and then reducing
the result 6/30 it is simpler to cancel like
factors first, as follows:
Likewise,
Here we mentally factor 6 to the form 3 x 2, and
4 to the form 2 x 2. Cancellation is a valuable
tool in shortening operations with fractions.
The general rule may be applied to mixed numbers
by simply changing them to improper fractions.
Thus,
Practice problems. Determine the following products,
using the general rule and canceling where
possible:
Answers:
The following problem illustrates the multiplication of fractions in a
practical situation.
EXAMPLE: Find the distance between the center lines of the first and fifth
rivets connecting the two metal plates shown
in figure 47 (A). SOLUTION: The distance between two adjacent rivets,
centerline to centerline, is 4 1/2 times the
diameter of one of them.
Thus,
There are 4 such spaces between the first and fifth
rivets. Therefore, the total distance, D, is
found as follows:
Figure 47.Application of multiplication of fractions in
determining rivet spacing.
The distance is 11 a inches Practice problem. Find the distance between the
centers of the two rivets shown in figure 47 (B).
Answer:
