Page 285 - Basic College Mathematics with Early Integers

Definitions and Concepts Examples

Section 3.2 Factors and Simplest Form

A prime number is a natural number that has exactly two 2, 3, 5, 7, 11, 13, 17, Á
different factors, 1 and itself.
A composite number is any natural number other than 1 4, 6, 8, 9, 10, 12, 14, 15, 16, Á
that is not prime.
The prime factorization of a number is the factorization Write the prime factorization of 60.
in which all the factors are prime numbers. #
60 = 6 10
# # #
3 5
= 2 3 2 5 or 2 2 # #
Fractions that represent the same portion of a whole are
called equivalent fractions.

3 = 12
4 16
2
A fraction is in simplest form or lowest terms when the The fraction is in simplest form.
3
numerator and the denominator have no common
factors other than 1.
30
To write a fraction in simplest form, write the prime Write in simplest form:
36
factorizations of the numerator and the denominator
# #
and then divide both by all common factors. 30 2 3 5 2 3 5 5 5
# #
= = # # = 1 1 =
# # #
#
36 2 2 3 3 2 3 2 3 6 6
1 1
# #
30 2 3 5 5
or = =
# # #
36 2 2 3 3 6
1 1
7 21
Two fractions are equivalent if Determine whether and are equivalent.
8 24
7
Method 1. They simplify to the same fraction. is in simplest form.
8
1
#
#
Method 2. Their cross products are equal. 21 3 7 1 7 7
= # = # =
#
#
24 7 8 21 24 3 8 1 8 8
1
7 21
= 168 = = 168 7 7 21
8 24 Since both simplify to , then = .
8 8 24
7 21
Since 168 = 168, =
8 24
Section 3.3 Multiplying and Dividing Fractions
To multiply two fractions, multiply the numerators and Multiply.
multiply the denominators.
#
2 5 2 5 10
# = =
#
3 7 3 7 21
1
#
#
3 1 3 1 3 1 1
# = = =
# #
#
4 6 4 6 4 3 2 8
1
3 5
To find the reciprocal of a fraction, interchange its The reciprocal of is .
numerator and denominator. 5 3 Copyright 2012 Pearson Education, Inc.
To divide two fractions, multiply the first fraction by the Divide.
reciprocal of the second fraction. 3 7 3 9 3 9 27
#
- , =- # =- =-
#
10 9 10 7 10 7 70
262