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

What is a Common Denominator?

When the denominators of two or more fractions are the same, they have Common Denominators.

What is the Least Common Denominator?

The "Least Common Denominator" is the smallest of all the possible common denominators.

We will see how to find it soon, but first let's investigate why common denominators are needed.

Different Denominators

We can't add fractions with different denominators:

1 3 + 1 6 = ?
1/3   1/6   1

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

Answer: We need to make the denominators the same.

Finding a 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:

6 18 + 3 18 = 9 18
1/3   1/6   1

(Read more about Common Denominators.)

Least Common Denominator

That is all fine, but 18 is a lot of slices ... can we 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 of 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 of 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 fractions 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 us 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 we 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 we do to the bottom of the fraction,
we must also do to the top

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

  × 5  
16 = 530
  × 5  
and
  × 2  
715 = 1430
  × 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 use the Least Common Multiple Tool. Just put in the denominators, press the button, and the least common denominator is shown.

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 we multiply 8 × 3 we get 24, and when we multiply 12 × 2 we also get 24. So, let's use that:

  × 3  
arrow
38 = 924
arrow
  × 3  
and
  × 2  
arrow
512 = 1024
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.