(Adding Or Subtracting Like Fractions). Suppose A,B and C Numbers and B Is Not 0, Then a B a + B a B a − B + = and − = C C C C C C

Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs) 1

Learning Objectives:

☞ Add or Subtract Like Fractions ☞ Add or Subtract Given Fractional Replacement Values ☞ Solve Problems by Adding & Subtracting Like Fractions ☞ Find the Least Common Denominator (LCD) of a List of Fractions ☞ Writing Equivalent Fractions

Deﬁnition 1. Fractions with the same denominator are called like fractions. Theorem 1 (Adding or Subtracting Like Fractions). Suppose a,b and c numbers and b is not 0, then a b a + b a b a − b + = and − = c c c c c c

✎ Add or subtract the given fractions. 9 99 −9 99 −9 + 99 90 9 · 10 9 10 9 9 1. − + = + = = = = · = · 1 = 100 100 100 100 100 100 10 · 10 10 10 10 10

23 1 2. − 11 11 2.

7 11 3. + 30 30 3.

7 3 4. − 10 10 4.

13x 13x 5. − − 28 28 5.

9x 1 6. + 11 11 6.

7x 15x 7. − 16 16 7. Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs) 2

Recall that the Natural numbers are 1,2,3,4,.... Deﬁnition 2. A multiple of a number is the product of that number and any Natural number. For example, the multiples of 3 are 3 · 1 = 3, 3 · 2 = 6, 3 · 3 = 9, 3 · 4 = 12, and so on.

☞ Least Common Denominators (LCDs)

To add or subtract fractions that have unlike, or different, denominators, we write the fractions as equivalent fractions with a common denominator. The smallest common denominator is called the least common denominator (LCD).

Deﬁnition 3. The least common denominator (LCD) of a list of fractions is the smallest positive number divisible by all the denominators in the list. (The least common denominator is also the least common multiple (LCM) of the denominators.) 1 17 For example, the LCD of and is 24 since 24 is the smallest positive number divisible by 6 8 both 6 and 8.

Finding the LCD: Method 1 Step 1: Write the multiples of the largest denominator (starting with the number itself) until a multiple common to all denominators in the list is found. Step 2: The multiple found in Step 1 is the LCD. 9 1 Example: Find the LCD of and . 8 16

Solution: Write multiples of 16 until one that is also a multiple of 8 is found. 16 · 1 = 16 a multiple of 8 The LCD is 16.

29 17 Example: Find the LCD of and . 30 25 Solution: Write multiples of 30 until one that is also a multiple of 25 is found.

30 · 1 = 30 not a multiple of 25 30 · 2 = 60 not a multiple of 25 30 · 3 = 90 not a multiple of 25 30 · 4 = 120 not a multiple of 25 30 · 5 = 150 is a multiple of 25 The LCD is 150. Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs) 3

✎ Find the LCD of each list of fractions.

8 17 8. and 9 12 8.

13 7 9. and 12 20 9.

1 7 17 10. − and 12, 24 36 10.

4 7 17 11. − and 25, 6 15 11. Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs) 4

29 17 Recall that the LCD of and must be a multiple of both 30 and 25 . Examine the prime 30 25 factorization of both numbers:

25 = 5 · 5

30 = 2 · 3 · 5

The LCD is 5 · 5 · 2 · 3 = 150. We circle (and then multiply) the greatest number of factors for each different prime number in the set of prime factorizations to ﬁnd the LCD. Remember, 150 is the smallest number that both 25 and 30 divide into evenly (remainder of 0).

Finding the LCD: Method 2 Step 1. Write the prime factorization of each denominator. Step 2. For each different prime factor, circle the greatest number of times that factor occurs in any one factorization. Step 3. The LCD is the product of the circled factors.

✎ Find the LCD of each list of fractions.

13 1 12. 108, 40 12.

1 1 1 13. 18, 24, 42 13. Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs) 5

✎ Find the LCD of each list of fractions.

3 1 13 14. 4, 14 , 20 14.

1 1 15. − 36 , 24 15.

11 7 17 16. 20, 24, 45 16.

2 3 4 17. 9, x2 , x4 17. Professor Busken – 4.4 Adding & Subtracting Like Fractions, and Least Common Denominators (LCDs) 6

✎ Writing Equivalent Fractions

Recall that equivalent fractions are fractions that represent the same part of a whole. We will need to know how to write equivalent fractions to add and subtract fractions with different, or unlike denominators.

To write an equivalent fraction, a a c a · c = · = b b c b · c where a,b and c are nonzero numbers.

7 Example: Write as an equivalent fraction with a denominator of 24. 8 7 = 8 24 7 3 Solution: Since 24 = 8 · 3, we multiply by 1 in the form of . 8 3 7 7 7 3 7 · 3 21 = · 1 = · = = 8 8 8 3 8 · 3 24 21 X To check that this result is correct, write in simplest form, or compute the cross products 24 7 21 between and are equal. 8 24

✎ Write an equivalent fraction with the given denominator.

18. 3 = 5 18.

3 19. = 4 48 19.

7x 20. = 9 27 20.

7 21. = 11 132x 21.