Least Common Denominator
... is the Least Common Multiple of the denominators ...
What is a Denominator?
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 haveCommon
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 = ?
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
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 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 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
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
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
1
6
5
30
and
7
15
14
30
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.
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
3
8
9
24
and
5
12
10
24
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.
