LIKE AND UNLIKE FRACTIONS
We have shown that like fractions are added by
simply adding the numerators and keeping the denominator.
Thus,
or
Similarly we can subtract like fractions by subtracting
the numerators.
The following examples will show that like fractions
may be divided by dividing the numerator of the dividend by the numerator of
the divisor.
SOLUTION: We may state the problem as a question:
‘How many times does 1/8 appear in 3/8 or
how many times may 1/8 be taken from 3/8?"
We see that l/8 can be subtracted from 3/8 three
times. Therefore,
3/8 ÷ 1/8 = 3
When the denominators of fractions are unequal, the fractions are said to be
unlike. Addition, subtraction, or division cannot be performed directly on
unlike fractions. The proper application of
the fundamental rule, however,
can change their form so that they become
like fractions; then all the rules for like
fractions apply.
LOWEST COMMON DENOMINATOR
To change unlike fractions to like fractions, it
is necessary to find a COMMON DENOMINATOR and it is usually advantageous to find
the LOWEST COMMON DENOMINATOR (L C D).
This is nothing more than the least common multiple
of the denominators.
Least Common Multiple
If a number is a multiple of two or more different
numbers, it is called a COMMON MULTIPLE.
Thus, 24 is a common multiple of 6 and 2.
There are many common multiples of these
numbers. The numbers 36, 48, and 54, to name
a few, are also common multiples of 6 and 2.
The smallest of the common multiples of a set
of numbers is called the LEAST COMMON MULTIPLE.
It is abbreviated LCM. The least common
multiple of 6 and 2 is 6. To find the least
common multiple of a set of numbers, first
separate each of the numbers into prime factors.
Suppose that we wish to find the LCM of 14, 24,
and 30. Separating these numbers into prime
factors we have
14 =2 x 7
24 = 2^{3} x 3
30 = 2 x 3 x 5
The LCM will contain each of the various prime factors
shown. Each prime factor is used the greatest
number of times that it occurs in any one of
the numbers. Notice that 3, 5, and 7 each occur
only once in any one number. On the other
hand, 2 occurs three times in one number.
We get the following result:
LCM = 2^{3} x 3 x 5 x 7
=
840
Thus, 840 is the least common multiple of 14, 24,
and 30.
Greatest Common Divisor
The largest number that can be divided into each
of two or more given numbers without a remainder
is called the GREATEST COMMON DIVISOR of the
given numbers. It is abbreviated GCD. It is
also sometimes called the HIGHEST COMMON
FACTOR.
In finding the GCD of a set of numbers, separate the numbers into prime
factors just as for LCM. The GCD is the
product of only those factors that appear in
all of the numbers. Notice in the example of
the previous section that 2 is the greatest
common divisor of 14, 24, and 30. Find the
GCD of 650,900, and 700. The procedure is as follows:
Notice that 2 and 52 are factors of each number. The greatest common divisor
is 2 x 25 = 50.
USING THE LCD
Consider the example
The numbers 2 and 3 are both prime; so the LCD
is 6.
Therefore
and
Thus, the addition of $ and i is performed as follows:
In the example
10 is the LCD.
Therefore,
Practice problems. Change the fractions in each
of the following groups to like fractions with
least common denominators:
Answers:
