Adding and Subtracting Fractions

Finding “Lowest Common Denominators”

Before two fractions can be added, they may have to be converted to equivalent fractions so that both have the same denominator, a so-called common denominator. Let's describe this process.

When attempting a sum such as

we need to first convert both fractions to equivalent fractions which have the same denominator. The method for obtaining equivalent fractions is to multiply the numerator and denominator of the given fraction by the same number:

where ‘a’ and ‘b’ stand for some whole numbers which have values so that

4a = 3b = LCD

Actually, any values of ‘a’ and ‘b’ that satisfy this condition will work, but there are advantages to picking ‘a’ and ‘b’ so that the resulting number, LCD, is as small as possible. Since this smallest number will be the new denominator for both fractions, it is called the lowest common denominator or LCD.

From the condition above, we see that LCD = 4a, so LCD must be evenly divisible by 4 (since ‘a’ is supposed to be a whole number). Similarly, LCD = 3b, so LCD must be divisible by 3 (since ‘b’ is supposed to be a whole number).

The systematic procedure for finding LCD in this case is as follows:

step (i): write each of the original denominators as a product of prime factors: Here:

4 = 2 ²

3 = 3

a very simple situation.

step (ii): A power of every distinct prime factor that occurs in any of the denominators will be a factor of the LCD.

Here:

LCD = 2 ^x · 3 ^y

since the only distinct prime factors in the original denominators are 2 and 3 in this case.

step (iii): The powers of each factor in the LCD are just the highest power to which that factor is raised in any of the original denominators.

So here,

x = 2, since 4 contains 2 ² and 3 contains 2 ⁰.

Therefore, the highest power of 2 that occurs in either of the original denominators is 2.

y = 1, since 4 contains 3 ⁰ and 3 contains 3 ¹.

Therefore the highest power of 3 occurring in either of the original denominators is 1.

Thus, for this simple example,

LCD = 2 ² · 3 = 12

(This happens to be equal to the product, 4 × 3, of the original denominators in this case. However, often the LCD is much smaller than the product of the original denominators because of common prime factors between them. The fact that the LCD can be smaller or simpler than the product of the original denominators is why this procedure for finding the LCD is worthwhile. Using the LCD is particularly important when working with algebraic fractions.)

step (iv): Convert the original fractions to equivalent fractions with this LCD as denominator and do the arithmetic. The numbers ‘a’ and ‘b’ involved in this conversion, as illustrated above, are just LCD divided by the original denominators of each fraction.

So, in our simple example, the fraction 3 / 4 must be multiplied top and bottom by

LCD / 4 = 12 / 4 = 3, and the fraction 2 / 3 must be multiplied top and bottom by

LCD / 3 = 12 / 3 = 4:

Sometimes people write this conversion to equivalent fractions as multiplication by fractions with identical numerators and denominators, as in:

but because of the rule for multiplication of fractions, the result is the same.