Algebra Demystified 2nd Ed

Chapter 1 FraCtions 17

Because 84 is the first number on each list, 84 is the LCD for 121 and 149. This
method works fine as long as the lists aren’t too long. But what if the denomi-
nators are 6 and 291 for example? The LCD for these denominators (which is
582) occurs 97th on the list of multiples of 6.
We can use the prime factors of the denominators to find the LCD more
efficiently. The LCD consists of every prime factor in each denominator (at its
most frequent occurrence). To find the LCD for 121 and 149 , we factor 12 and 14
into their prime factorizations: 12 = 2 ⋅ 2 ⋅ 3 and 14 = 2 ⋅ 7. There are two 2’s
and one 3 in the prime factorization of 12, so the LCD will have two 2’s and
one 3. There is one 2 in the prime factorization of 14, but this 2 is covered by
the 2’s from 12. There is one 7 in the prime factorization of 14, so the LCD
will also have a 7 as a factor. Once we have computed the LCD, we divide the
LCD by each denominator and then multiply the fractions by these numbers
over themselves.

LCD = 2⋅ 2 ⋅ 3 ⋅7 = 84

84 ÷ 12 = 7: multiply 121 by^77 84 ÷ 14 = 6: multiply 149 by^66

1

12

9

14

1

12

7

7

9

14

6

6

7

84

54

84

61

84

+=⋅









+⋅









=+=

EXAMPLE

Find the sum or difference after computing the LCD.

5

6

4

15

+

SOLUTION

We begin by factoring the denominators: 6 = 2 ⋅ 3 and 15 = 3 ⋅ 5. The

LCD is 2 ⋅ 3 ⋅ 5 = 30. Dividing 30 by each denominator gives us 30 ÷ 6 = 5

and 30 ÷ 15 = 2. Once we multiply^56 by^55 and 154 by^22 , we can add the

fractions.

5

6

4

15

5

6

5

5

4

15

2

2

25

30

8

30

33

30

+= ⋅^1









+⋅







=+==^11

10

EXAMPLE

Find the sum or difference after computing the LCD.

SOLUTION

We begin by factoring the denominators: 6 = 2

✔