Least Common Denominator
... is the Least Common Multiple of the denominators ...
What is a Denominator?
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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 | = | ? | ||||
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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 | ||||
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(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:
| 2/6 | + | 1/6 | = | 3/6 | ||||
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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:
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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:
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Now we can easily do the addition by adding the top numbers:
5/30 + 14/30 = 19/30
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:
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Now we can do the addition:
9/24 + 10/24 = 19/24