Page 209 - Basic College Mathematics with Early Integers

186 C HAPTE R 3 I FRACTIONS

a factor of 2.That is, 2 is a common factor of 4 and 6.The process of writing a fraction
in simplest form is called simplifying the fraction.
4 2
To simplify and write it as , let’s first study a few properties. Recall from
6 3
Section 3.1 that any nonzero number n divided by itself is 1.

Any nonzero number n divided by itself is 1.
5 17 24 n
= 1, = 1, = 1, or, in general, = 1
5 17 24 n

a c
Also, in general, if and are fractions (with b and d not 0), the following is true.
b d

#
a c a * c
= #
#
b d b d
These properties allow us to do the following:
#
4 = 2 2 = 2 # 2 = 1 # 2 = 2 When 1 is multiplied by a number, the result is
#
6 2 3 2 3 3 3 the same number.

This is 1
12
PRACTICE 4 Example 4 Write in simplest form:
30 20
Write in simplest form:
45 Solution: Notice that 12 and 20 have a common factor of 4.
#
12 4 3 4 3 3 3
= = # = 1 # =
#
20 4 5 4 5 5 5
3
Since 3 and 5 have no common factors (other than 1), is in simplest form.
5
Work Practice 4
If you have trouble finding common factors, write the prime factorization of the
numerator and the denominator.
42
PRACTICE 5 Example 5 Write in simplest form:
39 66
Write in simplest form:
51 Solution: Let’s write the prime factorizations of 42 and 66.
# #
42 2 3 7 2 3 7 7 7
# #
= = # # = 1 1 =
# #
66 2 3 11 2 3 11 11 11
Work Practice 5
In the example above, you may have saved time by noticing that 42 and 66 have a
common factor of 6.
#
42 6 7 6 7 7 7
= = # = 1 # =
#
66 6 11 6 11 11 11

Writing the prime factorizations of the numerator and the denominator is helpful Copyright 2012 Pearson Education, Inc.
in finding any common factors.
Answers
2 13
4. 5. *Note: We will study this concept further in the next section.
3 17