Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

NS1.2 Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. NS2.2 Add and subtract fractions by using factoring to find a common denominator.

California Standards

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

To add and subtract fractions with unlike denominators, first find a common denominator using one of these methods:

Method 1 Find a common denominator by multiplying the denominators.

Method 2 Find the least common denominator (LCD).

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

Find a common denominator: 8(7)=56.

2 7

+1 8

Multiply by fractions equal to 1.Rewrite with a common denominator.

Additional Example 1A: Adding and Subtracting Fractions with Unlike Denominators

Add or subtract. Method 1:

= 23 56

Add numerators. Keep the denominator.

2 7

+1 8

7 7

8 8

=

16 56

+7 56

=

7 + 1656

=

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

1 61 – 1 5

8

List the multiples of each denominator and find the LCD.

Write as improper fractions.

Multiples of 6: 6, 12, 18, 24, 30

Multiples of 8: 8, 16, 24, 32

Rewrite with the LCD.

11 24= –

Multiply by fractions equal to 1.

Subtract numerators. Keep the denominator.

Additional Example 1B: Adding and Subtracting Fractions with Unlike Denominators

Add or subtract. Method 2:

–7 6

13 8

=

13 8

–7 6

4 4

3 3

=

39 24–28

24=

28 – 39 24

=

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

Find a common denominator: 3(8)=24.

5 8

+1 3

Multiply by fractions equal to 1.

Rewrite with the LCD.

Check It Out! Example 1AAdd or subtract. Method 1:

23 24

=

5 8

+1 3

8 8

3 3

=

15 24

+8

24=

=8 + 15

24Add numerators. Keep the denominator.

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

1 62 +

3 4

List the multiples of each denominator and find the LCD

Write as an improper fraction.

Multiples of 6: 6, 12, 18, 24, 30

Multiples of 4: 4, 8, 12, 16

Rewrite with the LCD.

Multiply by fractions equal to 1.

11 12

= 235 12

=

Check It Out! Example 1BAdd or subtract. Method 2:

3 4+

13 6

=

3 4+

13 6

2 2

33

=

9 12+26

12=

26 + 9 12

=Add numerators. Keep the denominator.

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

25 56

Write the prime factorization of each denominator. Circle the common factors.

Factors of 56: 2 ∙ 2 ∙ 2 ∙ 7

Factors of 84: 2 ∙ 2 ∙ 3 ∙ 7

List all the prime factors of the denominators, using the circled factors only once.

Multiply.

Additional Example 2: Using Factoring to Find the LCD

Find + . Write the answer in simplest form.37 84

2, 2, 2, 7, 3

2 ∙ 2 ∙ 2 ∙ 7 ∙ 3 = 168

The LCD is 168.

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

25 56

Multiply by fractions equal to 1 to get a common denominator.

Rewrite using the LCD.

Additional Example 2 Continued

Find + . Write the answer in simplest form.37 84

168 ÷ 56 = 3 168 ÷ 84 = 2

= +25 56

37 84

3 3

2 2

= +75 168

74 168

= 149 168

Add numerators. Keep the denominator.

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

Write the prime factorization of each denominator. Circle the common factors.

Factors of 32: 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2

Factors of 80: 2 ∙ 2 ∙ 2 ∙ 2 ∙ 5

List all the prime factors of the denominators, using the circled factors only once.

Multiply.

Check It Out! Example 2

Find + . Write the answer in simplest form.

2, 2, 2, 2, 2, 5

2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 5 = 160The LCD is 160.

19 32

9 80

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

Multiply by fractions that equal to 1 to get a common denominator.

Rewrite using the LCD.

160 ÷ 32 = 5 160 ÷ 80 = 2

= +19 32

9 80

5 5

2 2

= +95 160

18 160

= 113 160

Add numerators. Keep the denominator.

Check It Out! Example 2 Continued

Find + . Write the answer in simplest form.19 32

9 80

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

Additional Example 3: Consumer Application

Subtract both amounts from 36 to find the amount of ribbon left.

Write as improper fractions.

36 – 12 –13 7 8

3 4

The LCD is 8.

There will be 9 inches left.3 8

Two dancers are making necklaces from ribbon for their costumes. They need pieces measuring 13 inches and 12 inches. How much ribbon

will be left over after the pieces are cut from a 36-inch length?

7 8

3 4

103 8

55 4

36 1 – –=

103 8

288 8

– – 110 8

=

, or 9 3 8

75 8

=

Simplify.

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

Check It Out! Example 3

Subtract both amounts from 12 to find the amount of board left.

Write as improper fractions.

12 – 6 – 4 3 4

5 12

The LCD is 12.

Simplify.

There will be foot left.5 6

Fred and Jose are building a tree house. They need to cut a 6 foot piece of wood and a 4

foot piece of wood from a 12 foot board. How much of the board will be left?

5 12

3 4

27 4 53 12

12 1 – –=

81 12

144 12

– – 53 12

=

, or 5 6

10 12

=

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

Evaluate t – for t = .

Additional Example 4: Evaluating Expressions with Rational Numbers

4 5

–5 6

24 30

–25 30

=

Multiply by fractions equal to 1.

4 5

5 6

Substitute for t.5 6

Rewrite with a common denominator: 6(5) = 30.

t – = 4 5

4 5

–5 6

5 5

6 6

=

1 30

= Simplify.

Evaluating Algebraic Expressions

2-6 Adding and Subtracting with Unlike Denominators

Check It Out! Example 4

Multiply by fractions equal to 1.

Evaluate – h for h =– .5 9

7 12

Substitute – for h.7 12

Rewrite with the LCD.

Simplify.