Fractions:

Finding Common Denominators

In order to add fractions, the denominators must be the same. In the example of pizza, this means each slice has to be the same size. If we are working with pizza slices that have been cut into different sizes, we need to find a way to cut all of them into slices that are the same size. This is done using Least Common Multiples. LCM’s are how we change the denominator of a fraction. We multiply by 1, but the 1 doesn’t look like a 1. It is in the form of something like $\frac{3}{3}$ or $\frac{7}{7}$. (Remember, anything divided by itself equals 1.) The following video will give more details and work through some examples.

Video Source (09:55 mins) | Transcript

The following video shows more examples using measurements of fractions of an inch instead of pizza slices:

Video Source (04:15 mins) | Transcript

To find the common denominator, find the LCM of the existing denominators. To find the new numerator, multiply the existing numerator by the same number multiplied to its denominator to get the LCM.

Additional Resources

Khan Academy: Finding Common Denominators (04:41 mins, Transcript)
Khan Academy: Adding Fractions with Unlike Denominators (07:23 mins, Transcript)
Khan Academy: Subtracting Fractions with Unlike Denominators (05:00 mins, Transcript)

Practice Problems

What is the common denominator you would use if you wanted to add the fractions $\frac{1}{4}$ and $\frac{1}{3}$?

What is the common denominator you would use if you wanted to add the fractions $\frac{1}{6}$ and $\frac{1}{9}$?

What do you get when you add the fractions $\frac{1}{4}$ and $\frac{1}{3}$?

Add: $\frac{1}{6}+\frac{1}{9}$

Subtract: $\frac{1}{6}-\frac{1}{9}$

Subtract: $\frac{1}{4}-\frac{5}{8}$

Solutions

12 (Solution Video | Transcript)

$\frac{1}{4}+\frac{1}{3}=\frac{3}{12}+\frac{4}{12}=\frac{7}{12}$

$\frac{1}{6}+\frac{1}{9}=\frac{3}{18}+\frac{2}{18}=\frac{5}{18}$ (Written Solution)

We are adding $\frac{1}{6}$ to $\frac{1}{9}$. We can represent that as $\frac{1}{6}$ of a whole added to $\frac{1}{9}$ of a whole:

Step 1: Find the least common denominator.

There are two ways that we can find the common denominator - skip counting using multiples or by factoring each denominator. For this example, we will use multiples

Multiples of 6: 6, 12, 18, 24, etc.

Multiples of 9: 9, 18, 27, etc.

The smallest number that both 6 and 9 divide into evenly is 18.

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

Let’s start with $\frac{1}{6}$, 6 × 3 = 18 so we multiply the numerator and denominator of $\frac{1}{6}$ by 3:

$\frac{1\times3}{6\times3}=\frac{3}{18}$

We do the same for $\frac{1}{9}$, 9 × 2 = 18, so we will multiply the numerator and the denominator by 2:

$\frac{1\times2}{9\times2}=\frac{2}{18}$

$This is a picture of 2 circles with a plus sign between them. The first has an area of three eighteenths shaded, with a caption underneath with the fraction 1 times 3 over 6 times 3. The second has an area of two eighteenths shaded, with a caption underneath with the fraction 1 times 2 over 9 times 2.$

Which gives us:

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

Step 3: Add fractions

$\frac{3}{18}+\frac{2}{18}=\frac{5}{18}$

$\frac{1}{6}-\frac{1}{9}=\frac{3}{18}-\frac{2}{18}=\frac{1}{18}$ (Written Solution)

We are subtracting $\frac{1}{9}$ from $\frac{1}{6}$. We can represent that as $\frac{1}{6}$ of a whole minus $\frac{1}{9}$ of a whole:

Step 1: Find the least common denominator.

There are two ways that we can find the common denominator - skip counting using multiples or by factoring each denominator. For this example, we will use multiples.

Multiples of 6: 6, 12, 18, 24, etc.

Multiples of 9: 9, 18, 27, etc.

The smallest number that both 6 and 9 divide into evenly is 18.

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

Let’s start with $\frac{1}{6}$, 6 × 3 = 18 so we multiply the numerator and denominator of $\frac{1}{6}$ by 3:

$\frac{1\times3}{6\times3}=\frac{3}{18}$

We do the same for $\frac{1}{9}$, 9 × 2 = 18, so we will multiply the numerator and the denominator by 2:

$\frac{1\times2}{9\times2}=\frac{2}{18}$

$This is a picture of 2 circles with a fraction under each circle and minus sign between the fractions. The first has an area of 3 eighteenths shaded, with a caption underneath with the fraction 1 times 3 over 6 times 3. The second has an area of 2 eighteenths shaded, with a caption underneath with the fraction 1 times 2 over 9 times 2.$

Which gives us:

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

Step 3: Subtract

We are subtracting $\frac{2}{18}$ from the $\frac{3}{18}$:

Which gives us:

Which gives us $\frac{3}{18}-\frac{2}{18}=\frac{1}{18}$

$\frac{1}{4}-\frac{5}{8}=\frac{2}{8}-\frac{5}{8}=-\frac{3}{8}$ (Solution Video | Transcript)