Rewriting Fractions
A fraction is a number that is written as a quotient, with a numerator and a denominator. The terms fraction and rational number are related, but are not exactly the same. The term fractionrefers to a number’s form, whereas the term rational numberrefers to its classification. For instance, the number 2 is a fraction when it is written as but it is a rational number regardless of how it is written.
All of the following fractions are positive and are equivalent to
All of the following fractions are negative and are equivalent to
In both arithmetic and algebra, it is often beneficial to write a fraction in simplest formor reduced form, which means that the numerator and denomina- tor have no common factors (other than 1). By finding the prime factors of the numerator and the denominator, you can determine what common factor(s) to divide out. The product of the common factors is the greatest common factor(or GCF).
2 3, 2
3 , 2 3, 2
3
23. 2
3, 2 3, 2
3 , 2 3
2 3.
2 1, Why You Should Learn It
Rational numbers are used to represent many real-life quantities. For instance, in Exercise 151 on page 44, you will use rational numbers to find the increase in the Dow Jones Industrial Average.
Lon C. Diehl/PhotoEdit
What You Should Learn
1 䉴Rewrite fractions as equivalent fractions.
2䉴Add and subtract fractions.
3䉴Multiply and divide fractions.
4䉴Add, subtract, multiply, and divide decimals.
Rules of Signs for Fractions
1. If the numerator and denominator of a fraction have like signs, the value of the fraction is positive.
2. If the numerator and denominator of a fraction have unlike signs, the value of the fraction is negative.
Writing a Fraction in Simplest Form To write a fraction in simplest form, divide both the numerator and denominator by their greatest common factor (GCF).
To find the GCF of two natural numbers, write the prime factorization of each number. For instance, from the prime factorizations
and
you can see that the common factors of 18 and 42 are 2 and 3.
So, it follows that the greatest common factor is 23,or 6.
42237
18233
Study Tip
1 䉴Rewrite fractions as equivalent fractions.
CHECKPOINT CHECKPOINT
EXAMPLE 1 Writing Fractions in Simplest Form
Write each fraction in simplest form.
a. Divide out GCF of 6.
b. Divide out GCF of 7.
c. Divide out GCF of 24.
Now try Exercise 15.
You can obtain an equivalent fraction by multiplying the numerator and denominator by the same nonzero number or by dividing the numerator and denominator by the same nonzero number. Here are some examples.
Equivalent
Fraction Fraction Operation
EXAMPLE 2 Writing Equivalent Fractions
Write an equivalent fraction with the indicated denominator.
a. b. c.
Solution
a. Multiply numerator and denominator by 5.
b. Multiply numerator and denominator by 6.
c.
Now try Exercise 25.
9
1533
35 37
572135
4
7 46
762442
2
325
351015
9
15䊏
35 4
7䊏
42 2
3䊏
15
2 3 8
12 2
121 2
2
121 3
12 10 6
5 62
52
3 4 9
123
1 3
31 4
24
72
1
2
121 2131
221 21 31 3 13
35
2157
371
1
5 3 18
24 2
1 31 3
222131 34
Divide numerator and denomi- nator by 3. (See Figure 1.31.) Multiply numerator and denominator by 2.
Divide numerator and denominator by GCF of 4.
Reduce first, then multiply numerator and denominator by 7.
9
3 12
4
Figure 1.31 Equivalent Fractions
CHECKPOINT
Adding and Subtracting Fractions
You can use models to add and subtract fractions, as shown in Figure 1.32.
Figure 1.32
The models suggest the following rules about adding and subtracting fractions.
EXAMPLE 3 Adding and Subtracting Fractions with Like Denominators
a. Add numerators.
b. Subtract numerators.
Now try Exercise 29.
To find a like denominator for two or more fractions, find the least common multiple (or LCM) of their denominators. For instance, the LCM of the denominators of and is 24. To see this, consider all multiples of 8 (8, 16, 24, 32, 40,48, . . .) and all multiples of 12 (12,24, 36,48, . . .). The numbers 24 and 48 are common multiples, and the number 24 is the smallest of the common multiples.
125 3
8
7 92
9 72
9 5
9 3
12 4
1234
12 7
12
− = − =
− = − =
1 2
1 4
2 4
1 4
1 4
+ =
+ =
1 3
1 3
2 3
Addition and Subtraction of Fractions Let and be integers with
1. With like denominators:
or
2. With unlike denominators:Rewrite the fractions so that they have like denominators. Then use the rule for adding and subtracting fractions with like denominators.
a cb
cab c a
cb
c ab c
c0.
c b, Adding fractions with unlike a,
denominators is an example of a basic problem-solving strategy that is used in mathematics—rewriting a given problem in a simpler or more familiar form.
Study Tip
2 䉴Add and subtract fractions.
CHECKPOINT CHECKPOINT
EXAMPLE 4 Adding Fractions with Unlike Denominators
LCM of 5 and 15 is 15.
Rewrite with like denominators.
Add numerators.
Now try Exercise 45.
EXAMPLE 5 Subtracting Fractions with Unlike Denominators
Evaluate Solution
To begin, rewrite the mixed number as a fraction.
Then subtract the two fractions as follows.
Rewrite as
LCM of 9 and 12 is 36.
Rewrite with like denominators.
Add numerators.
Now try Exercise 61.
You can add or subtract two fractions, without first finding a common denominator, by using the following rule.
31 36 64
36 33 36 16共4兲
9共4兲 11共3兲 12共3兲
16 9. 179
17 911
1216 9 11
12 17
917 99
9 7 916
9 179 17
911 12. 23
15 12
1511 15 4
5 11 154共3兲
5共3兲11 15
In Example 5, a common shortcut for writing as is to multiply 1 by 9, add the result to 7, and then divide by 9, as follows.
17
9 1共9兲7
9 16
9
16
179 9
Study Tip
Alternative Rule for Adding and Subtracting Two Fractions If a,b,c, and dare integers with and then
or a b c
d adbc bd . a
b c
dadbc bd
d0, b0
CHECKPOINT CHECKPOINT
Rewrite with like denominators.
Technology: Tip When you use a scientific or graphing calculator to add or subtract fractions, your answer may appear in decimal form. An answer such as 0.583333333 is not as exact as and may introduce roundoff error. Refer to the user’s manual for your calculator for instructions on adding and subtracting fractions and displaying answers in fraction form.
7 12
In Example 5, the difference between and was found using the least common multiple of 9 and 12. Compare those solution steps with the following steps, which use the alternative rule for adding or subtracting two fractions.
Apply alternative rule.
Multiply.
Simplify.
EXAMPLE 6 Subtracting Fractions
Add the opposite.
Apply alternative rule.
Multiply.
Simplify.
Now try Exercise 57.
EXAMPLE 7 Combining Three or More Fractions
Evaluate Solution
The least common denominator of 6, 15, and 10 is 30. So, you can rewrite the original expression as follows.
Add numerators.
Simplify.
Now try Exercise 69.
10 30
1 3 2514930
30 25
30 14
30 9
30 30 30 5
6 7 15 3
1015共5兲
6共5兲 共7兲共2兲
15共2兲 3共3兲
10共3兲共1兲共30兲 30 5
6 7 15 3
101.
262 480131
240 150112
480 5共30兲16共7兲
16共30兲 5
16冢307冣165 7 30 93
10831 36 19299
108 17
911
12 16共12兲9共11兲 9共12兲
11
179 12
CHECKPOINT CHECKPOINT
Multiplying and Dividing Fractions
The procedure for multiplying fractions is simpler than those for adding and subtracting fractions. Regardless of whether the fractions have like or unlike denominators, you can find the product of two fractions by multiplying the numerators and multiplying the denominators.
EXAMPLE 8 Multiplying Fractions
a. Multiply numerators and denominators.
Simplify.
b. Product of two negatives is positive.
Multiply numerators and denominators.
Divide out common factor.
Write in simplest form.
Now try Exercise 89.
EXAMPLE 9 Multiplying Three Fractions
Rewrite mixed number as a fraction.
Multiply numerators and denominators.
Divide out common factors.
Write in simplest form.
Now try Exercise 97.
56 9
共8兲(2兲共7兲(5兲 (5兲共3兲(2兲共3兲 16共7兲共5兲
5共6兲共3兲
冢315冣冢76冣冢53冣 冢165冣冢76冣冢53冣
5 27 7共5兲
9共3兲共7兲 7共5兲
9共21兲
冢79冣冢215冣 79215
15 16 5
8325共3兲 8共2兲
3 䉴Multiply and divide fractions.
Multiplication of Fractions
Let a,b,c, and dbe integers with and Then the product of and is
Multiply numerators and denominators.
a
bcd ac
bd.
c d a b
d0.
b0
CHECKPOINT
The reciprocal or multiplicative inverse of a number is the number by which it must be multiplied to obtain 1. For instance, the reciprocal of 3 is because Similarly, the reciprocal of is because
To divide two fractions, multiply the first fraction by the reciprocalof the second fraction. Another way of saying this is “invert the divisor and multiply.”
EXAMPLE 10 Dividing Fractions
a. Invert divisor and multiply.
Multiply numerators and denominators.
Divide out common factors.
Write in simplest form.
b. Invert divisor and multiply.
Multiply numerators and denominators.
Divide out common factors.
Write in simplest form.
c. Invert divisor and multiply.
Multiply numerators and denominators.
Write in simplest form.
Now try Exercise 107.
1 12 共1兲共1兲
共4兲共3兲 1
4共3兲 1