Least Common Denominator

... is the Least Common Multiple of the denominators ...

What is a Denominator?

fraction

The denominator is the bottom number in a fraction.

It shows how many equal parts the item is divided into

Fractions with Different Denominators

You can't add fractions with different denominators:

  1/3 + 1/6 = ?      
  1/3   1/6   1   1  

So what do you do? How can they be added?

Answer: You need to make the denominators the same.

Common Denominator

But what should the new denominator be?

One simple answer is to multiply the current denominators together:

3 × 6 = 18

So instead of having 3 or 6 slices, we will make both of them have 18 slices.

The pizzas now look like this (I will show calculations later):

  6/18 + 3/18 = 9/18      
  1/3   1/6   1   1  

(Read more about Common Denominators.)

Least Common Denominator

That is all fine, but 18 is a lot of slices ... can you do it with fewer slices?

Here is how to find out:

1/3 List the multiples of 3:   3, 6, 9, 12, 15, 18, 21, ...
1/6 List the multiples 6:   6, 12, 18, 24, ...

Then find the smallest number that is the same

  multiples of 3:   3, 6, 9, 12, 15, 18, 21, ...
  multiples 6:   6, 12, 18, 24, ...

The answer is 6, and that is the Least Common Denominator.

So let us try using it! We want both to have 6 slices.

  • When we multiply top and bottom of 1/3 by 2 we get 2/6
  • 1/6 already has a denominator of 6

And our question now looks like:

  2/6 + 1/6 = 3/6      
  2/6   1/6   3/6   1  

One last step is to simplify the fraction (if possible). In this case 3/6 is simpler as 1/2:

  2/6 + 1/6 = 3/6 = 1/2  
  2/6   1/6   3/6   3/6  

 

And that is what the Least Common Denominator is all about.

It lets you add (or subtract) fractions using the least number of slices.

What Did We Do?

The trick was to list the multiples of each denominator, then find the Least Common Multiple

In the previous example the Least Common Multiple of 3 and 6 was 6.

In other words the Least Common Denominator of 1/3 and 1/6 is 6.

Here are the steps to follow:

  • Find the Least Common Multiple of the denominators (which is called the Least Common Denominator).
  • Change each fraction (using equivalent fractions) to make their denominators the same as the least common denominator
  • Then add (or subtract) the fractions, as you wish!
   

Example: What is 1/6 + 7/15 ?

The Denominators are 6 and 15:

  multiples of 6:   6, 12, 18, 24, 30, 36, ...
  multiples 15:   15, 30, 45, 60, ...

So the Least Common Multiple of 6 and 15 is 30.

 

Now let's try to make the denominators the same.

Note: what you do to the bottom of the fraction,
you must also do to the top

When you multiply 6 × 5 you get 30, and when you multiply 15 × 2 you also get 30:

  × 5  
1  =  5
6 30
  × 5  
and
  × 2  
7  =  14
15 30
  × 2  

 

Now we can do the addition by adding the top numbers:

5/30 + 14/30 = 19/30

The fraction is already as simple as it can be, so that is the answer.

 

Least Common Multiple Tool

To find the least common denominator automatically, you can use our Least Common Multiple Tool - just put in the denominators, press the button, and the least common denominator is shown for you.

One More Example

Example: What is 3/8 + 5/12 ?

List the multiples of 8 and 12

  multiples of 8:   8, 16, 24, 32, 40, ...
  multiples 12:   12, 24, 36, 48, ...

The Least Common Multiple is 24

Let's try to make the denominators the same ... when you multiply 8 × 3 you get 24, and when you multiply 12 × 2 you also get 24. So, let's use that:

  × 3  
arrow
3  =  9
8 24
arrow
  × 3  
and
  × 2  
arrow
5  =  10
12 24
arrow
  × 2  

Now we can do the addition:

9/24 + 10/24 = 19/24

The fraction is already as simple as it can be, so that is the answer.