>Addition and Subtraction of Fractions>How to Find the Common Denominator Using Factorization

How to Find the Common Denominator Using Factorization

To use this method, you need to be comfortable with prime number factorization. This method works no matter what your denominators are.

Rule

Lowest Common Denominator, LCD

1.: Write down all the denominators below each other.
2.: Find the prime factors of each denominator.
3.: All the factors of the different denominators have to be included as the factors of the common denominator.
4.: The product of those factors is your lowest common denominator.

Example 1

Find $\frac{1}{16} + \frac{3}{20} + \frac{4}{30}$ .

First, you find the common denominator:


Denominator	Factors

$16$	$2 \cdot 2 \cdot 2 \cdot 2$

$20$	$2 \cdot 2 \cdot 5$

$30$	$2 \cdot 3 \cdot 5$

LCD	$2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$

The least common denominator is then:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 = 240

Then, you expand the fractions so all of them have the common denominator: $16$ is missing the factors $3$ and $5$ ; $20$ is missing the factors $2$ , $2$ and $3$ ; and $30$ is missing the factors $2$ , $2$ and $2$ to have all the factors in the LCD. That gives you

\begin{array}{l} \frac{1}{16} + \frac{3}{20} + \frac{4}{30} & = \frac{1 \cdot 3 \cdot 5}{16 \cdot 3 \cdot 5} + \frac{3 \cdot 2 \cdot 2 \cdot 3}{20 \cdot 2 \cdot 2 \cdot 3} \\ + \frac{4 \cdot 2 \cdot 2 \cdot 2}{30 \cdot 2 \cdot 2 \cdot 2} \\ = \frac{15}{240} + \frac{36}{240} + \frac{32}{240} \\ = \frac{15 + 36 + 32}{240} \\ = \frac{83}{240} . \end{array}

\begin{array}{l} \frac{1}{16} + \frac{3}{20} + \frac{4}{30} & = \frac{1 \cdot 3 \cdot 5}{16 \cdot 3 \cdot 5} + \frac{3 \cdot 2 \cdot 2 \cdot 3}{20 \cdot 2 \cdot 2 \cdot 3} + \frac{4 \cdot 2 \cdot 2 \cdot 2}{30 \cdot 2 \cdot 2 \cdot 2} \\ = \frac{15}{240} + \frac{36}{240} + \frac{32}{240} \\ = \frac{15 + 36 + 32}{240} \\ = \frac{83}{240} . \end{array}

Find the Common Denominator Using the Times Tables

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