When working with fractions, it is sometimes practical to put every fraction on the same denominator. Finding the common denominator is important when comparing fractions, ordering fractions, or performing mathematical operations such as adding and subtracting fractions. In each case, we refer to the common denominator. The following are a few methods for finding a common denominator in two or more fractions.
To find a common denominator, look for the lowest common multiple (LCM) of the fractions’ denominators. The LCM will correspond to a common denominator.
- Find the LCM of the fractions’ denominators with the list of multiples method or with the factor tree method.
- Find the equivalent fractions using the common denominator.
To do this, we will use the list of multiples method and the factor tree method.
Find the LCM by listing multiples of each denominator. The common denominator will be the smallest common multiple on the multiple lists. Next, find equivalent fractions for each fraction using the common denominator.
With 2 fractions
Find a common denominator for the following two fractions:
||\frac{1}{12} \qquad \text{and} \qquad \frac{5}{8}||
1. List the multiples of each denominator
Multiples of |12=\{12,\underbrace{\color{red}{24}}_\color{blue}{2^{nd} \ \text{multiple}},36,48,...\}|
Multiples of |8=\{8,16,\underbrace{\color{red}{24}}_\color{green}{3^{rd} \ \text{multiple}},32,40,...\}|
2. Find the equivalent fractions
||\frac{1}{12}^\color{blue}{\times 2}_\color{blue}{\times 2} = \frac{2}{\color{red}{24}} \qquad \text{and} \qquad \frac{5}{8}^\color{green}{\times 3}_\color{green}{\times 3} = \frac{15}{\color{red}{24}}||
With 3 fractions
Find a common denominator for the following three fractions:
||\frac{1}{4} \qquad \frac{2}{3} \qquad \frac{3}{8}||
1. List the multiples of each denominator
Multiples of |4=\{4,8,12,16,20,\underbrace{\color{red}{24}}_\color{blue}{6^{th} \ \text{multiple}},28,...\}|
Multiples of |8=\{8,16,\underbrace{\color{red}{24}}_\color{green}{3^{rd} \ \text{multiple}},32,40,...\}|
2. Find the equivalent fractions
||\frac{1}{4}^\color{blue}{\times 6}_\color{blue}{\times 6} = \frac{6}{\color{red}{24}}\ \qquad \frac{2}{3}^\color{fuchsia}{\times 8}_\color{fuchsia}{\times 8} = \frac{16}{\color{red}{24}}\ \qquad \frac{3}{8}^\color{green}{\times 3}_\color{green}{\times 3} = \frac{9}{\color{red}{24}}||
Use the following strategy to find equivalent fractions.
Determine the number by which to multiply the numerator and the denominator to obtain the equivalent fraction by referring to the rank of the common multiple for each of the denominators.
Find the LCM using the factor tree for each denominator. Next, find the equivalent fractions for each of the fractions.
With 2 fractions
Find a common denominator for the following two fractions:
||\frac{7}{12} \qquad \text{and} \qquad \frac{5}{9}||
1. Find the LCM according to the factor tree of each denominator
Carrying out the factor tree for the two denominators results in the following prime factorizations. ||12=\color{blue}{2} \times \color{green}{2} \times \color{fuchsia}{3}\qquad \qquad 9=\color{fuchsia}{3} \times \color{orange}{3}||
To determine the LCM, multiply all the different prime factors. If any are common to both numbers, only use one in your multiplication, as follows: ||\begin{align}\text{LCM}\{9,12\}&= \underbrace{\color{blue}{2}\times \color{green}{2} \times \color{fuchsia}{3}}_{\text{factors of}\ 12} \times \underbrace{\not\color{fuchsia}{3} \times \color{orange}{3}}_{\text{factors of} \ 9} \\
&= \color{blue}{2}\times \color{green}{2} \times \color{fuchsia}{3} \times \color{orange}{3} \\ \\
&= \color{red}{36}\end{align}|| 2. Find the equivalent fractions
||\frac{7}{12}^{\color{orange}{\times 3}}_{\color{orange}{\times 3}} = \frac{21}{\color{red}{36}} \qquad \text{and} \qquad \frac{5}{9}^{\color{blue}{\times 2}\color{green}{\times 2}}_{\color{blue}{\times 2}\color{green}{\times 2}} = \frac{20}{\color{red}{36}}||
With 3 fractions
Find a common denominator for the following three fractions:
||\frac{1}{10} \qquad \frac{3}{8} \qquad \frac{5}{6}||
1. Find the LCM according to the factor tree of each denominator
Carrying out the factor tree for the three denominators results in the following prime factorizations. ||10=\color{blue}{2} \times \color{green}{5}\qquad \qquad 8=\color{blue}{2} \times \color{fuchsia}{2} \times \color{orange}{2}\qquad \qquad 6=\color{blue}{2} \times \color{purple}{3}|| To determine the LCM, multiply all the different prime factors. If there are any that are common to at least two of the three numbers, only use one in your multiplication, as follows: ||\begin{align} \text{LCM}\{6,8,10\} &= \underbrace{\color{blue}{2} \times \color{green}{5}}_{\text{factors of}\ 10} \times \underbrace{\not\color{blue}{2} \times \color{fuchsia}{2} \times \color{orange}{2}}_{\text{factors of} \ 8}\times \underbrace{\not\color{blue}{2} \times \color{purple}{3}}_{\text{factors of} \ 6} \\
&= \color{blue}{2} \times \color{green}{5} \times \color{fuchsia}{2} \times \color{orange}{2} \times \color{purple}{3} \\ \\
&= \color{red}{120}\end{align}||
2. Find the equivalent fractions
||\frac{1}{10}^{\color{fuchsia}{\times 2}\color{orange}{\times 2}\color{purple}{\times 3}}_{\color{fuchsia}{\times 2}\color{orange}{\times 2}\color{purple}{\times 3}} = \frac{12}{\color{red}{120}} \qquad \ \frac{3}{8}^{\color{green}{\times 5}\color{purple}{\times 3}}_{\color{green}{\times 5}\color{purple}{\times 3}} = \frac{45}{\color{red}{120}} \qquad \ \frac{5}{6}^{\color{green}{\times 5}\color{fuchsia}{\times 2}\color{orange}{\times 2}}_{\color{green}{\times 5}\color{fuchsia}{\times 2}\color{orange}{\times 2}} = \frac{100}{\color{red}{120}}||
To find the equivalent fractions, use the following trick to learn by which number to multiply the fraction’s numerator and denominator.
To determine the number by which to multiply the numerator and the denominator for the fraction with the denominator of |\small 10,| use the LCM factors without |\small 10:|
||\begin{align}\text{Common denominator} &= \color{blue}{2} \times \color{green}{5} \times \color{fuchsia}{2} \times \color{orange}{2} \times \color{purple}{3}\\
\text{Factors of} \ 10 &= \color{blue}{2} \times \color{green}{5}\end{align}||
Therefore, the number by which we must multiply the numerator and the denominator is:
||\underbrace{\not\color{blue}{2} \times \not\color{green}{5}}_{\text{factors of} \ 10} \times \color{fuchsia}{2} \times \color{orange}{2} \times \color{purple}{3}=\color{fuchsia}{2} \times \color{orange}{2} \times \color{purple}{3}||
The same process is repeated for all of the other fractions.
To find a common denominator, multiply all the denominators together. Next, find the equivalent fractions for each of the fractions obtained using the common denominator.
The common denominator obtained is often a large number.
With 2 fractions
Find a common denominator for the following two fractions:
||\frac{1}{\color{green}{12}} \qquad \text{and} \qquad \frac{5}{\color{blue}{8}}||
Multiplying |\color{green}{12}| and |\color{blue}{8}| results in a common denominator of |\color{red}{96}.|
Thus,
||\frac{1}{12}^\color{blue}{\times 8}_\color{blue}{\times 8} = \frac{8}{\color{red}{96}} \qquad \text{and} \qquad \frac{5}{8}^\color{green}{\times 12}_\color{green}{\times 12} = \frac{60}{\color{red}{96}}||
With 3 fractions
Transform the following three fractions under the same denominator:
||\frac{1}{\color{blue}{4}} \qquad\ \frac{2}{\color{green}{3}} \qquad\ \frac{7}{\color{fuchsia}{9}}||
Multiplying |\color{blue}{4},\color{green}{3} \ \text{and} \ \color{fuchsia}{9}| , results in a common denominator of |\color{red}{108}.|
Thus,
||\frac{1}{4}^{\color{green}{\times 3}\color{fuchsia}{\times 9}}_{\color{green}{\times 3}\color{fuchsia}{\times 9}} = \frac{27}{\color{red}{108}} \qquad \ \frac{2}{3}^{\color{blue}{\times 4}\color{fuchsia}{ \times 9}}_{\color{blue}{\times 4}\color{fuchsia}{ \times 9}} = \frac{72}{\color{red}{108}} \qquad\ \frac{7}{9}^{\color{blue}{\times 4}\color{green}{\times 3}}_{\color{blue}{\times 4}\color{green}{\times 3}} = \frac{84}{\color{red}{108}}||
To find equivalent fractions, multiply the numerator and denominator of each fraction by the denominators of the other fractions.
When the denominators of the fractions are algebraic expressions, the method for determining a common denominator is similar to the factor tree method above.
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Factor and reduce each of the fractions.
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Determine the common denominator.
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Find the equivalent fractions.
Because of its similarity with the factor tree, there is an emphasis on factoring. Thus, it is essential to master the different methods of factoring a polynomial.
Find the common denominator of the following fractions:
||\frac{3x^2+6x}{x^2+5x+6} \ \ \text{and} \ \ \frac{2x-6}{6x^2+36x+54}||
1. Factor and reduce each of the fractions
||\begin{align} \small \frac{\color{blue}{3x^2+6x}}{\color{red}{x^2+5x+6}} &\Rightarrow \small \color{blue}{3x^2 + 6x} &&&& \small\color{red}{x^2+5x+6} \\
&= \small \color{blue}{3x(x+2)} && \small \text{factoring the GCF} && \small\color{red}{(x+3)(x+2)} && \small \text{product sum}\\
\small \frac{\color{blue}{3x^2+6x}}{\color{red}{x^2+5x+6}} &= \small \frac{\color{blue}{3x (x+2)}}{\color{red}{(x+3)(x+2)}} \\
&= \small \frac{\color{blue}{3x}}{\color{red}{(x+3)}} && \small \text{simplification}\\\\
\small \frac{\color{green}{2x-6}}{\color{orange}{6x^2+36x+54}} &\Rightarrow \small \color{green}{2x-6} &&&& \small\color{orange}{6x^2+36x+54} \\
&= \small \color{green}{2(x-3)} && \small \text{factoring the GCF} && \small\color{orange}{6(x^2+6x+9)} && \small \text{factoring the GCF}\\
&&&&& \small \color{orange}{6(x+3)(x+3)} && \small \text{perfect square}\\
\small \frac{\color{green}{2x-6}}{\color{orange}{6x^2+36x+54}} &= \small \frac{\color{green}{2(x-3)}}{\color{orange}{6(x+3)(x+3)}} \\
&= \small \frac{\color{green}{(x-3)}}{\color{orange}{3(x+3)(x+3)}} && \small \text{simplification} \end{align}||
2. Determine the common denominator
In this step, ensure each denominator’s factors are found in the common denominator. If part of the first denominator is identical ("pairs") to part of the second denominator, only one of the "pairs" is kept.
||\begin{align} \small\text{denominator} &= \small \color{red}{(x+3)} && \small\text{and} && \small \color{orange}{3(x+3)(x+3)} \\
\small \text{common denominator} &= \small \underbrace{\color{red}{(x+3)}}_{\small\text{pairs}} \ \color{orange}{3} \ \underbrace{\color{orange}{(x+3)}}_{\small\text{pairs}} \ \color{orange}{(x+3)} && \small \text{factoring common denominators}\\
&= \small\underbrace{\color{red}{(x+3)}}_{\small\text{1 of the pairs}}\ \small\color{orange}{3} \phantom{(x+3)} \color{orange}{(x+3)} && \small \text{eliminating one of the “pairs”} \\
&= \small 3 \ (x+3) \ (x+3) && \small \text{common denominator} \end{align}||
3. Find the equivalent fractions
Finally, multiply the numerators and denominators of the initial fractions by the missing elements of the common denominator |\small 3 \ (x+3) \ (x+3)| .
||\begin{align} \small \frac{\color{blue}{3x}}{\color{red}{(x+3)}} &\Rightarrow \small\frac{\color{blue}{3x}}{\underbrace{\color{red}{(x+3)}}_{\small\text{initial}}}\cdot \frac{3(x+3)}{\underbrace{3 \ (x+3)}_{\small\text{missing}}} && \small\underbrace{\phantom{(}3\phantom}_{\small\text{missing}}\small\underbrace{(x+3)}_{\small\text{common}}\ \ \small\underbrace{(x+3)}_{\small\text{missing}} \\
&= \small\frac{9x^2+27x}{3 (x+3)(x+3)} \\\\
\small \frac{\color{green}{(x-3)}}{\color{orange}{3(x+3)(x+3)}} &\Rightarrow \small \frac{\color{green}{(x-3)}}{\underbrace{\color{orange}{3 \ (x+3) \ (x+3)}}_{\small\text{initial}}} \cdot \underbrace{\phantom{\frac{(\small\text{rien})}{(\small\text{rien})}}}_{\small\text{missing}} && \small\underbrace{3 \ (x+3) \ (x+3)}_{\small\text{common}} \\
&=\small \frac{(x-3)}{3\ (x+3)\ (x+3)} \end{align}||
Since the second initial fraction has no missing elements, it remains unchanged.
Thus,
||\begin{align} \small \frac{3x^2+6x}{x^2+5x+6}&&& \text{and} && \small \frac{2x-6}{6x^2+36x+54} \\\\
\Rightarrow \small\frac{9x^2+27x}{3 (x+3)(x+3)} &&& \text{and} && \small \frac{(x-3)}{3(x+3)(x+3)} \end{align}||
The two fractions now have a common denominator, therefore, we can add or subtract them.
To review your understanding of fractions interactively, see the following Crash Lesson:
