Columbus State Community College

1. Columbus State Community College Chapter 4 Section 4 Adding and Subtracting Signed Fractions

2. Adding and Subtracting Signed Fractions • Add and subtract like fractions. • Find the lowest common denominator for unlike fractions. • Add and subtract unlike fractions. • Add and subtract unlike fractions that contain variables.

3. and and and and 1 3 7 6 4 4 9 2 8 m 5 m 5 5 a 9 Fractions Like Fractions Unlike Fractions Common denominator Different denominators Common denominator Different denominators

4. and a c a–c – a c a+c = + = b b b b b b Adding and Subtracting Like Fractions Adding and Subtracting Like Fractions You can add or subtract fractions only when they have a common denominator. If a, b, and c are numbers (and b is not 0), then In other words, add or subtract the numerators and write the result over the common denominator. Then check to be sure that the answer is in lowest terms.

5. 1 2 (a) + 9 9 1 3 = = 1 2 1 + 2 3 9 + = 9 9 9 Adding and Subtracting Like Fractions EXAMPLE 1 Adding and Subtracting Like Fractions Find each sum or difference. 1 3 Common denominator

6. 1 2 1 + 2 3 not + = = 9 9 9 + 9 18 1 2 1 + 2 + = 9 9 9 Adding Fractions CAUTION Add only the numerators. Do not add the denominators. In part (a) we kept the common denominator. Incorrect

7. 4 1 (b) + 5 5 4 1 4 + 1 3 3 + = = = 5 5 5 5 5 Adding and Subtracting Like Fractions EXAMPLE 1 Adding and Subtracting Like Fractions Find each sum or difference. Common denominator

8. 2 6 – (c) 7 7 2 6 2 – 6 4 4 – = = = 7 7 7 7 7 Adding and Subtracting Like Fractions EXAMPLE 1 Adding and Subtracting Like Fractions Find each sum or difference. Common denominator

9. 3 2 (d) + k k 3 2 3 + 2 5 + = = k k k k Adding and Subtracting Like Fractions EXAMPLE 1 Adding and Subtracting Like Fractions Find each sum or difference. Common denominator

10. 1 1 6 9 A Common Denominator for Unlike Fractions A Common Denominator for Unlike Fractions To find a common denominator for two unlike fractions, find a number that is divisible by both of the original denominators. For example, a common denominator for and is 18 because 6 goes into 18 evenly and 9 goes into 18 evenly.

11. 1 1 4 8 Least Common Denominator (LCD) Least Common Denominator (LCD) The least common denominator (LCD) for two fractions is the smallest positive number divisible by both denominators of the original fractions. For example, both 8 and 16 are common denominators for and , but 8 is smaller, so it is the LCD.

12. 3 3 (a) Find the LCD for and . 14 14 5 5 7 7 Finding the LCD by Inspection EXAMPLE 2 Finding the LCD by Inspection Check to see if 14 (the larger denominator) will work as the LCD. Is 14 divisible by 7 (the other denominator)? Yes, so 14 is the LCD for and .

13. 2 (b) Find the LCD for and . 9 1 1 6 6 2 9 Finding the LCD by Inspection EXAMPLE 2 Finding the LCD by Inspection Check to see if 9 (the larger denominator) will work as the LCD. Is 9 divisible by 6 (the other denominator)? No, 9 is not divisible by 6. So start checking numbers that are multiples of 9, that is, 18, 27, and 36. Notice that 18 will work because it is divisible by 6 and 9. The LCD for and is 18.

14. 1 1 5 5 (a) What is the LCD for and . 12 12 20 20 Using Prime Factors to Find the LCD EXAMPLE 3 Using Prime Factors to Find the LCD Write 20 and 12 as the product of prime factors. Then use prime factors in the LCD that “cover” both 20 and 12. 60 is divisible by 20 and by 12, it is the LCD for and . 20 = 2 • 2 • 5 12 = 2 • 2 • 3 Factors of 20 Factors of 12 LCD = 2 • 2 • 3 • 5 = 60

15. LCD CAUTION When finding the LCD, notice that we did not have to repeat the factors that 20 and 12 have in common. If we had used all the 2s and 3s, we would get a common denominator, but not the smallest one.

16. 3 3 8 8 (b) What is the LCD for and . 40 40 15 15 Using Prime Factors to Find the LCD EXAMPLE 3 Using Prime Factors to Find the LCD Write 15 and 40 as the product of prime factors. Then use prime factors in the LCD that “cover” both 15 and 40. 120 is divisible by 15 and by 40, it is the LCD for and . 15 = 3 • 5 40 = 2 • 2 • 2 • 5 Factors of 15 Factors of 40 LCD = 2 • 2 • 2 • 3 • 5 = 120

17. Adding and Subtracting Unlike Fractions Adding and Subtracting Unlike Fractions Step 1 Find the LCD, the smallest number divisible by both denomi- nators in the problem. Step 2 Rewrite each original fraction as an equivalent fraction whose denominator is the LCD. Step 3 Add or subtract the numerators of the like fractions. Keep the common denominator. Step 4 Write the sum or difference in lowest terms.

18. 3 3 3 1 5 1 5 2 (a) Find the sum. + 8 8 8 4 8 8 4 8 3 1 • 2 2 1 = = already has the LCD and 8 4 • 2 8 4 3 + 2 = = = + + 8 Adding and Subtracting Unlike Fractions EXAMPLE 4 Adding and Subtracting Unlike Fractions Step 1 The larger denominator ( 8 ) is the LCD. Step 2 Step 3 Add the numerators. Write the sum over the denominator. Step 4 is in lowest terms.

19. 5 5 7 7 – (b) Find the difference. 6 6 8 8 20 5 • 4 7 • 3 21 5 7 = = = = 24 6 • 4 8 • 3 24 8 6 20 21 1 20 – 21 – – – = = = 24 24 24 24 Adding and Subtracting Unlike Fractions EXAMPLE 4 Adding and Subtracting Unlike Fractions Step 1 The LCD is 24. Step 2 Step 3 Subtract the numerators. Write the difference over the common denominator.

20. 5 7 – (b) Find the difference. 6 8 1 – 24 Adding and Subtracting Unlike Fractions EXAMPLE 4 Adding and Subtracting Unlike Fractions Step 4 is in lowest terms.

21. 11 8 – (c) Find the difference. 42 63 Adding and Subtracting Unlike Fractions EXAMPLE 4 Adding and Subtracting Unlike Fractions Step 1 Use prime factorization to find the LCD. 42 = 2 • 3 • 7 63 = 3 • 3 • 7 Factors of 42 Factors or 63 LCD = 2 • 3 • 3 • 7 = 126

22. 11 8 – (c) Find the difference. 42 63 11 • 3 33 8 • 2 16 8 11 = = = = 42 • 3 126 63 • 2 126 63 42 33 – 16 17 11 8 33 16 = = – – = 126 126 42 63 126 126 Adding and Subtracting Unlike Fractions EXAMPLE 4 Adding and Subtracting Unlike Fractions Step 2 Step 3 Subtract the numerators. Write the difference over the common denominator.

23. 11 8 – (c) Find the difference. 42 63 17 126 Adding and Subtracting Unlike Fractions EXAMPLE 4 Adding and Subtracting Unlike Fractions Step 4 is in lowest terms.

24. a a b b (a) Find the sum. + 2 2 3 3 3a b• 2 a• 3 2b b a = = = = 6 3 • 2 2 • 3 6 3 2 3a 3a + 2b 2b = = + + 6 6 6 3a + 2b 6 Adding Unlike Fractions with Variables EXAMPLE 5 Adding Unlike Fractions with Variables Step 1 The LCD is 6. Step 2 Step 3 Add the numerators. Keep the common denominator. Step 4 is in lowest terms.

25. Combining Terms CAUTION In the previous problem, we could not add 3a + 2b in the numerator of the answer because 3a and 2b are not like terms. We could add 3a + 2a or 3b + 2b but not 3a + 2b. Variable parts match. Variable parts match.

26. m 7 – (b) Find the difference. 4 n m• n mn 7 • 4 28 m 7 = = = = 4 • n 4n n• 4 4n n 4 m 7 mn mn – 28 28 – – = = 4 n 4n 4n 4n mn – 28 4n Subtracting Unlike Fractions with Variables EXAMPLE 5 Subtracting Unlike Fractions with Variables Step 1 The LCD is 4 • n, or 4n. Step 2 Step 3 Subtract the numerators. Keep the common denominator. Step 4 is in lowest terms.

27. 2 m 1 7 3 5 8 4 5 6 n 4 – – + Common Denominators NOTE Notice in Example 5 (b) that we found the LCD for by multiplying the two denominators. The LCD is 4 • n or 4n. Multiplying the two denominators will always give you a common denominator, but it may not be the smallest common denominator. Here are more examples. If you multiply the denominators, 8 • 6 = 48 and 48 will work. But you’ll save some time by using the smallest common denominator, which is 24. If you multiply the denominators, 5 • 4 = 20 and 20 is the LCD.

28. Adding and Subtracting Signed Fractions Chapter 4 Section 4 – Completed Written by John T. Wallace