Simplify fractions.
Every fraction can be written in infinitely many equivalent forms. For example, some
equivalent forms of are:
...Of all of the equivalent forms in which we can write a fraction, we often need to
determine the one that is in simplest form.
Simplest Form of a FractionA fraction is in simplest form,or lowest terms,when the numerator and
denominator have no common factors other than 1.2
3
4
6
6
9
8
12
10
15
12
18
14
21
16
24
18
27
20
30
10
1553.1 An Introduction to Fractions 213EXAMPLE (^5) Write 4 as an equivalent fraction with a denominator of 6.
StrategyWe will express 4 as the fraction and build an equivalent fraction by
multiplying it by
WHYSince we need to multiply the denominator of by 6 to obtain a
denominator of 6, it follows that should be the form of 1 that is used to build an
equivalent fraction for
Solution
Write 4 as a fraction:
1
Build an equivalent fraction by multiplying by a form of 1:.
Multiply the numerators.
Multiply the denominators.
24
6
4 6
1 6
6
6 ^14
14
1
6
6
4 4 ^41.
4
1
4
1.6
64
16
6.4
1Self Check 5
Write 10 as an equivalent
fraction with a denominator
of 3.
Now TryProblem 57Self Check 6
Are the following fractions in
simplest form?
a.b.Now TryProblem 616
20
4
21
EXAMPLE 6
Are the following fractions in simplest form? a. b.StrategyWe will determine whether the numerator and denominator have any
common factors other than 1.
WHYIf the numerator and denominator have no common factors other than 1,
the fraction is in simplest form.
Solution
a.The factors of the numerator, 12, are: 1 ,2, 3 ,4,6,12
The factors of the denominator, 27, are: 1 , 3 ,9,27
Since the numerator and denominator have a common factor of 3, the fraction
is notin simplest form.
b.The factors of the numerator, 5, are: 1 ,5
The factors of the denominator, 8, are: 1 ,2,4,8
Since the only common factor of the numerator and denominator is 1, the fraction
is in simplest form.