Step 4: Adding More than Two Fractions

Adding more than two fractions

Adding more than two fractions is simple. You can add them the same way as previously described.

If they all have the same denominator, just add all of the numerators and put the result over the denominator.

For example: 1/9 + 2/9 + 1/9 + 2/9 = 6/9

Just like before, if you add the numbers and the numerator is larger than the denominator, you can change the fraction to a mixed number following the same steps.

Finding a Common Denominator

If the denominators are not the same, you still need to convert them before adding.

For example: 1/2 + 1/4 + 1/3 + 1/12 =

You can convert them all at once or you can convert them a pair at a time.

Since it's typically easiest to convert them all at once, we'll do that with our example. In this case, it's easiest to find a common multiple, or a number that is a multiple of all of the denominators.

In our example, the multiples of:
2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...
4 are 4, 8, 12, 16, 20, 24, ...
3 are 3, 6, 9, 12, 15, 18, 21, 24, ...
12 are 12, 24, ...

So, looking for a multiple that's common to all four of our denominators, we see that 12 and 24 both appear on all four lists. We can use either, but typically the smallest, or lowest, is used, which is why it's often called the lowest common multiple.

Once we've found a common multiple, we'll use 12 in our case, you need to convert each of our fractions to an equivalent one with that denominator.

So for 1/2, to get 12 as our denominator, we need to multiply both the numerator and denominator by 6 (multiplying the top and bottom both by the same number gives us an equivalent fraction since its the same as multiplying the fraction by 1 (e.g. 2/2 = 1)).

This gives 1/2 = 6/12.

Doing the same with 1/4 (multiplying top and bottom by 3 to give us a denominator of 12) gives:
1/4 = 3/12.

And subsequently with 1/3 (multiplying top and bottom by 4 to give us a denominator of 12) gives:
1/3 = 4/12.

With 1/12, we're all set since it's denominator is already 12.

This makes our problem:

1/2 + 1/4 + 1/3 + 1/12 =

6/12 + 3/12 + 4/12 + 1/12 =

14/12 = 1 2/12

Note that you can use the lowest common multiple technique to add 2 fractions, as well. But it's often easier to just multiply as we did in Step 3. Multiplying the two denominators automatically gives you a common multiple of the two numbers. It's easy and will always work. The only minor drawback is that your answer may require a little more simplification which won't be a problem as soon as you complete Step 5.
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croutonicus5 years ago
or 1 1/6 i thought you were simplifying all these :P