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Fractions

Math

Reducing Fractions and the Least Common Denominator

Summary Reducing Fractions and the Least Common Denominator

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.

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 = 9.
(b) 21×2 = 42. 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) 24 = 120. 24 = 48.
(b) 12×10 = 120. 10 = 50.
(c) 15 = 120. 15 = 135.
Thus, 2/5 = 48/120, 5/12 = 50/120, and 9/8 = 135/120.