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Least Common Denominator (LCD)

A common denominator of two numbers is a number that can be divided by the
denominators of both numbers. For example, 1/6 and 4/9 have common denominators
of 18, 36, 54, 72, etc. The least common denominator, or LCD, is the
*lowest* number that can be divided by the denominators of both numbers.
For example, 18 is the least common denominator of 1/6 and 4/9.

The least common denominator of two fractions is the least common
multiple of their denominators. 18 is the LCM of 6 and
9.

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Uses of the Least Common Denominator

The least common denominator is a helpful tool in allowing you to take two
different fractions (ex. 3/4 and 7/11) and write them as equivalent fractions
with the same denominator (ex. 33/44 and 28/44). Such a tool is important in
comparing the size of fractions and because fractions can only be added and
subtracted from each other when they have the same denominator. The first step
in the process is to find the LCD. Then write each fraction as an equivalent
fraction with the LCD as a new denominator, using the two steps detailed in
the section on equivalent fractions.

*Example 1*: Write 3/14 and 4/21 as fractions with the same
denominator.

I. Find the LCD

1. Factor the denominators. 14 = 2×7 and 21 = 3×7.

2. Find the LCM of the denominators. 2×3×7 = 42 -or-
14×(21/7) = 42.

3. The LCD is 42.

II. Write each fraction as an equivalent fraction with the LCD (42) as the new
denominator.

(a) 14×**3** = 42. 3×**3** = 9.

(b) 21×**2** = 42. 4×**2** = 8.

Thus,

3/14 = 9/42 and

4/21 = 8/42.

**Note:** The number by which the numerator must be multiplied in Part II
will be the product of the factors of the other denominator that are not factors
of its denominator. Here, 3 was multiplied by 3, which is a factor of 21 but
not of 14, and 4 was multiplied by 2, which is a factor of 14 but not of 21.

*Example 2*: Write 2/5, 5/12, and 9/8 as fractions with the same
denominator.

I. Find the LCD.

1. Factor the denominators. 5 = 5, 12 = 2×2×3, and 8 = 2×2×2.
2. Find the LCM of the denominators. 2×2×2×3×5 = 120
3. The LCD is 120.

II. Write each fraction as an equivalent fraction with the LCD (120) as the new
denominator.

(a) 5×**2**4 = 120. 2×**2**4 = 48.

(b) 12×**1**0 = 120. 5×**1**0 = 50.

(c) 8×**1**5 = 120. 9×**1**5 = 135.

Thus,

2/5 = 48/120,

5/12 = 50/120, and

9/8 = 135/120.