1. 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

2. 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).

3. Evaluating Algebraic Expressions 2-6 Adding and Subtracting with Unlike Denominators Find a common denominator: 8(7)=56. 2727 + 1818 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. 2727 + 1818 78 = 16 56 + 7 56 = 7 + 16 56 =

4. Evaluating Algebraic Expressions 2-6 Adding and Subtracting with Unlike Denominators 1616 1 – 1 5858 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: – 7676 13 8 = – 767643 = 39 24 – 28 24 = 28 – 39 24 =

5. Evaluating Algebraic Expressions 2-6 Adding and Subtracting with Unlike Denominators Find a common denominator: 3(8)=24. 5858 + 1313 Multiply by fractions equal to 1. Rewrite with the LCD. Check It Out! Example 1A Add or subtract. Method 1: 23 24 = 5858 + 131383 = 15 24 + 8 24 = = 8 + 15 24 Add numerators. Keep the denominator.

6. Evaluating Algebraic Expressions 2-6 Adding and Subtracting with Unlike Denominators 1616 2 + 3434 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 = 2 35 12 = Check It Out! Example 1B Add or subtract. Method 2: 3434 + 13 6 = 3434 + 23 = 9 12 + 26 12 = 26 + 9 12 = Add numerators. Keep the denominator.

7. 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.

8. 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 8432 = + 75 168 74 168 = 149 168 Add numerators. Keep the denominator.

9. 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 = 160 The LCD is 160. 19 32 9 80

10. 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 8052 = + 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

11. 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 7878 3434 The LCD is 8. There will be 9 inches left. 3838 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? 7878 3434 103 8 55 4 36 1 –– = 103 8 288 8 –– 110 8 =, or 9 3838 75 8 = Simplify.

12. 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 3434 5 12 The LCD is 12. Simplify. There will be foot left. 5656 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 3434 27 4 53 12 12 1 –– = 81 12 144 12 –– 53 12 =, or 5656 10 12 =

13. Evaluating Algebraic Expressions 2-6 Adding and Subtracting with Unlike Denominators Evaluate t – for t =. Additional Example 4: Evaluating Expressions with Rational Numbers 4545 – 5656 24 30 – 25 30 = Multiply by fractions equal to 1. 4545 5656 Substitute for t. 5656 Rewrite with a common denominator: 6(5) = 30. t – = 4545 4545 – 5656 56 = 1 30 = Simplify.

14. 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 =–. 5959 7 12 Substitute – for h. 7 12 Rewrite with the LCD. Simplify.