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Finding Common Denominators
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Introduction

In this lesson, you will learn how to find common denominators. Finding a common denominator is essential for adding fractions together.


These videos illustrate the lesson material below. Watching the videos is optional.


Finding Common Denominators

In order to add fractions, the denominators must be the same. This means each part of a whole is equal in size.

Example 1
Imagine that you have two circles, and they are each divided into a different number of pieces. One circle is divided into six pieces, while the other is divided into four pieces. That means that in the first circle, one piece is \(\frac{1}{6}\) of the circle. In the second circle, one piece is \(\frac{1}{4}\) of the circle.

If you take away all but one piece from each circle, how many total pieces do you have left? If you want to cut both pieces so all the remaining pieces are equal, how big should those pieces be?

Two circles. The first circle is divided into 6 parts, with one part shaded gold, and is labeled one-sixth. The other circle is divided into 4 parts, with one part shaded grey, and is labeled one-fourth. 

Figure 1

Since you are using fractions, the denominator, or the bottom part of the fraction, tells you the size of each piece. If the denominators for the two fractions are the same, it means that each piece of the circles are the same size. To equalize the pieces in the circles above, you'll need to find the least common multiple (LCM) of 6 and 4.

When you factor out 6 and 4, you find that \(2\times3=6\), and \(2\times2=4\). .

The least common multiple will need to have at least one 2 and one 3 to be a multiple of 6, and it will need to have at least two 2’s to be a multiple of 4. \(2\times2\times3=12\), so the least common multiple is 12.

Another way to determine the least common multiple is to list the multiples of each number and then determine what the LCM is:

  • LCM of 6: 6, 12, 18, 24, …
  • LCM of 4: 4, 8, 12, 16, 20, 24, …

12 is the least common multiple of 6 and 4.
The least common multiple between numbers in the denominator is called the least common denominator. In this example, the least common denominator of \(\frac{1}{6}\) and \(\frac{1}{4}\) is 12. If you want each piece of the circles to be equal in size, you need to turn each piece into a fraction that has 12 as the denominator.

In this example, you need to multiply \(\frac{1}{6}\) by 2, because \(6\times2=12\), and \(\frac{1}{4}\) by 3, because \(4\times3=12\):

\begin{align*} \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \;\;\;\;\;\;\;\;\;\;\;\; \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \end{align*}

Two circles. The first circle is divided into 12 parts, with two parts shaded gold, and is labeled one-sixth equals two-twelfths. The other circle is divided into 12 parts, with three parts shaded gray, and is labeled one-fourth equals three-twelfths. 

Figure 2

Note that both the numerator and the denominator must be multiplied in these situations. When you divide both circles into 12 pieces, \(\frac{1}{6}\) is converted into \(\frac{2}{12}\), and \(\frac{1}{4}\) is converted into \(\frac{3}{12}\).

Now that both fractions have equal denominators, they are equivalent. Equivalent fractions may not be in their simplest form, but because they have the same denominator, they can be added or subtracted together.

\begin{align*} \frac{1}{6}+ \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \end{align*}

Notice that \(\frac{5}{12}\) cannot be reduced, so this is the final answer.

Example 2
Find the common denominator of \(\frac{3}{5}\) and \(\frac{1}{3}\).

Begin by finding the least common multiple. Since both 3 and 5 are prime numbers, they cannot be further broken down by prime factorization. This means the LCM will need to be a multiple of both 5 and 3.

  • LCM of 5: 5, 10, 15, 20, 25, …
  • LCM of 3: 3, 6, 9, 12, 15, 18, …

15 is the least common multiple of 5 and 3, so both of the fractions need to be converted to have 15 as a denominator.

Now that both fractions have a common denominator, subtract these fractions:

\begin{align*} \frac{3}{5} - \frac{1}{5} = \frac{9}{15}-\frac{5}{15}=\frac{4}{15} \end{align*}

\(\frac{4}{15}\) cannot be reduced any further, so this is the final answer.

Adding Fractions with Different Denominators

An inch can be divided into several fractions. On a measuring tape or ruler, an inch is normally divided into halves, fourths, eighths, and sixteenths. It’s very common to need to add fractions with different denominators when working with inches.

This image shows a one-inch segment of a measuring tape or ruler. The inch is divided into sections between the 1- and 2-inch markers, with each section labeled from left to right: one-sixteenth, one-eighth, three-sixteenths, one-fourth, five-sixteenths, three-eighths, seven-sixteenths, one-half, nine-sixteenths, five-eighths, eleven-sixteenths, three-fourths, thirteen-sixteenths, seven-eighths, and fifteen-sixteenths. 

Figure 3

Example 3
Add \(\frac{1}{2}\) and \(\frac{1}{8}\) together.

On the number line above, notice that \(\frac{1}{2}\) is the same as \(\frac{4}{8}\). So, if you count over another \(\frac{1}{8}\), you get \(\frac{5}{8}\).

To solve this problem when you don't have a number line, you need to find a common denominator of the two fractions.

\begin{align*}\frac{1}{8} &+ \frac{1}{2} & \color{red}\small\text{Find a common denominator}\\\frac{1}{8} &+ \frac{4}{8} & \color{red}\small\text{LCD is 8, convert fractions}\\&\frac{5}{8} & \color{red}\small\text{Add the fractions}\\\end{align*}

Once again, you see that \(\frac{1}{8} + \frac{4}{8} = \frac{5}{8}\).


Things to Remember


  • In order to add or subtract two fractions, the fractions must have the same denominator.
  • The least common denominator is the least common multiple that two fractions can share as a denominator.
  • To find the new numerator, multiply the existing numerator by the same number multiplied to its denominator to get the LCM.

Practice Problems

  1. What is the common denominator you would use to add the fractions \(\dfrac{1}{4}\) and \(\dfrac{1}{3}\)? (
    Video Solution
    | Transcript)
  2. What is the common denominator you would use to add the fractions \(\dfrac{1}{6}\) and \(\dfrac{1}{9}\)? (
    Solution
    x
    Solution:
    18
    )
  3. What do you get when you add the fractions \(\dfrac{1}{4}\) and \(\dfrac{1}{3}\)? (
    Solution
    x
    Solution:
    \(\displaystyle \frac{1}{4}+\frac{1}{3}=\frac{3}{12}+\frac{4}{12}=\frac{7}{12}\)
    )
  4. Add: \(\displaystyle \frac{1}{6}+\frac{1}{9}\). (
    Solution
    x
    Solution: \(\displaystyle \frac{1}{6}+\frac{1}{9}=\frac{3}{18}+\frac{2}{18}=\frac{5}{18}\)

    Details:
    You are adding \(\dfrac{1}{6}\) to \(\dfrac{1}{9}\). You can represent that as \(\dfrac{1}{6}\) of a whole, added to \(\dfrac{1}{9}\) of a whole:
    Two pie charts. The first has one sixth shaded, the second has one ninth shaded, illustrating one-sixth plus one-ninth. 

    Step 1: Find the least common denominator.
    There are two ways to find the common denominator: skip counting using multiples or by factoring each denominator. For this example, use multiples:

    Multiples of 6: \(6, 12, {\color{red}18}, 24, etc.\)

    Multiples of 9: \(9, {\color{red}18}, 27, etc.\)

    The smallest number that both 6 and 9 divide into evenly is \(\color{red}18\)

    Step 2: Write both fractions as an equivalent fraction with a denominator of 18.
    Start with \(\dfrac{1}{6}\).
    \(6 \times {\color{red}3} = 18\)

    Multiply the numerator and denominator of \(\dfrac{1}{6}\) by 3:
    \(\displaystyle \dfrac{1\times\color{RED}3}{6\times\color{RED}3}=\dfrac{3}{18}\)

    Do the same for \(\dfrac{1}{9}\):
    \(9 \times {\color{red}2} = 18\)

    Multiply the numerator and the denominator by 2:
    \(\dfrac{1\times\color{RED}2}{9\times\color{RED}2}=\dfrac{2}{18}\)
    Two pie charts. The first has three eighteenths shaded, the second has two eighteenths shaded, illustrating 1 times 3 over 6 times 3, plus 1 times 2 over 9 times 2. 

    Which gives you \(\displaystyle \dfrac{3}{18}+\dfrac{2}{18}\):
    Two pie charts. The first has three eighteenths shaded, the second has two eighteenths shaded, illustrating 3 eighteenths plus 2 eighteenths. 

    Note that both circles now have 18 parts. \(\dfrac{1}{6}\) is the same amount as \(\dfrac{3}{18}\) and \(\dfrac{1}{9}\) is the same amount as \(\dfrac{2}{18}\).

    Step 3: Add the fractions:
    \(\displaystyle \dfrac{3}{18}+\dfrac{2}{18}=\frac{5}{18}\)
    Three pie charts. The first has three eighteenths shaded, the second has two eighteenths shaded, the third has five eighteenths shaded, illustrating 3 eighteenths plus 2 eighteenths equals five eighteenths. 
    )
  5. Subtract: \(\displaystyle \frac{1}{6}-\frac{1}{9}\). (
    Solution
    x
    Solution: \(\displaystyle \frac{1}{6}-\frac{1}{9}=\frac{3}{18}-\frac{2}{18}=\frac{1}{18}\)

    Details:
    You are subtracting \(\dfrac{1}{9}\) from \(\dfrac{1}{6}\). You can represent that as \(\dfrac{1}{6}\) of a whole, minus \(\dfrac{1}{9}\) of a whole:
    Two pie charts. The first has 1 sixth shaded, the second has 1 ninth shaded, illustrating 1 sixth minus 1 ninth. 

    Step 1: Find the least common denominator.

    There are two ways to find the least common denominator: skip counting using multiples or by factoring each denominator. For this example, use multiples.

    Multiples of 6: \(6, 12, {\color{red}18}, 24, etc.\)
    Multiples of 9: \(9, {\color{red}18}, 27, etc.\)

    The smallest number that both 6 and 9 divide into evenly is \(\color{red}18\)

    Step 2: Write both fractions as an equivalent fraction with a denominator of 18.

    Start with \(\dfrac{1}{6}\). \(6 \times 3 = 18\) so multiply the numerator and denominator of \(\dfrac{1}{6}\) by 3:
    \(\displaystyle \frac{1\times\color{RED}3}{6\times\color{RED}3}=\frac{3}{18}\)

    Do the same for \(\dfrac{1}{9}\). \(9 \times {\color{red} 2} = 18\), so multiply the numerator and the denominator by 2:
    \(\displaystyle \frac{1\times\color{RED}2}{9\times\color{RED}2}=\frac{2}{18}\)
    Two pie charts. The first has 3 eighteenths shaded, the second has 2 eighteenths shaded, illustrating 1 times 3 over 6 times 3, minus 1 times 2 over 9 times 2. 

    Which gives you the following:
    Two pie charts. The first has 3 eighteenths shaded, the second has 2 eighteenths shaded, illustrating 3 eighteenths minus 2 eighteenths. 

    Note that both circles now have 18 parts. \(\dfrac{1}{6}\) is the same amount as \(\dfrac{3}{18}\) and \(\dfrac{1}{9}\) is the same amount as \(\dfrac{2}{18}\).

    Step 3: Subtract the fractions.

    You are subtracting \(\dfrac{2}{18}\) from \(\dfrac{3}{18}\):
    A pie chart. It has three eighteenths shaded, with two of those eighteenths being removed from the pie chart, illustrating 3 eighteenths minus 2 eighteenths. 

    Which gives you \(\displaystyle \frac{3}{18}-\frac{2}{18}=\frac{1}{18}\):
    Three pie charts. The first has three eighteenths shaded, the second has 2 eighteenths shaded, the third has one eighteenth shaded, illustrating 3 eighteenths minus 2 eighteenths equals 1 eighteenth. 
    )
  6. Subtract: \(\displaystyle \frac{1}{4}-\frac{5}{8}\). (
    Video Solution
    x
    Solution: \(\displaystyle \frac{1}{4}-\frac{5}{8}=\frac{2}{8}-\frac{5}{8}=-\frac{3}{8}\)

    Details:

    (Finding Common Denominators #6 (02:58 mins) | Transcript)
    | Transcript)

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