# Columbus State Community College

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Columbus State Community College
Chapter 4 Section 4 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.

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

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 a b a + c = c + and a – c 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.

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

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

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

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

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

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. 6 1 9 1

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. 4 1 8 1

Finding the LCD by Inspection
EXAMPLE Finding the LCD by Inspection (a) Find the LCD for and 5 7 3 14 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 5 7 3 14

Finding the LCD by Inspection
EXAMPLE Finding the LCD by Inspection (b) Find the LCD for and 1 6 2 9 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. 1 6 2 9

Using Prime Factors to Find the LCD
EXAMPLE Using Prime Factors to Find the LCD (a) What is the LCD for and 1 12 5 20 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 5 20 1 12

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.

Using Prime Factors to Find the LCD
EXAMPLE Using Prime Factors to Find the LCD (b) What is the LCD for and 3 40 8 15 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 8 15 3 40

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.

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

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

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

EXAMPLE Adding and Subtracting Unlike Fractions (c) Find the difference. 8 63 11 42 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

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

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

1 4 3 8 + 5 8 c A b + c A b = 3 8 1 4 (b) 7 8 5 6 1 24 c A b c A b = 5 6 7 8 (c) 8 63 11 42 c A b c A b = 17 126 11 42 8 63

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

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.

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

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. 7 n m 4 5 6 1 8 + 3 4 2 5 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.