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Adding and Subtracting Fractions

You may have heard that "The whole is greater than the sum of its parts." Well, this is not true when adding and subtracting fractions! The whole exactly is the sum of its parts! Luckily for us, the task is a pretty straightforward and easy one.

Let's take a look:
1/3 + 1/3 = 2/3.

There's an example of the easiest type of fractions to add. All you have to do is add the top numbers (the numerators) and leave the denominators alone. And you're done! Even the "harder" ones are this easy, but they'll usually require that you get the denominator yourself.
Rule #1
This is the most important rule for adding and subtracting fractions. You can only add and subtract fractions that have a common denominator.

A "common denominator" just means that the denominators in each fraction are the same. If they are not the same, you need to make them the same!

The Easiest Ones:

If there is already a common denominator, the problem is very easy, as shown below:

3/13 + 1/13 + 7/13

Numerators: 3 + 1 + 7 = 11
Denominators: 13
Answer: 11/13.
(Be sure not to add the denominators!)


Finding Common Denominators
If the fractions you need to add do not have a common denominator, it's pretty easy to find one:

3/4 + 5/6

In these types of problems, you need to determine what you can multiply each denominator by in order to have both denominators equal.

Note: When you are looking for a common denominator, it should never be greater than each denominator multiplied together!

First, since the denominators, 4 and 6, are not equal, we look to their next multiple. For 4, the next multiple is 8. (4x2=8). For 6, the next is 12. Since 6 does not have any multiple equal to 8, the common denominator cannot be 8. However, since 8 is smaller than the next multiple of 6 (12), we just look for the next multiple of 4.

The next multiple of 4 is 12. (4x3=12). And that's the same as we found for 6! Since we've found a match (12), we convert the denominator to 12 - and we multiply the numerator of each fraction by the multiple to get to 12:

Fraction 1:
Fraction 2:

Basically, whatever it takes to multiply the denominator by to be equal to another fraction - be sure to multiply the numerator by the same number!

So now the problem is:

9/12 + 6/12

And we know all we need to do is add the numerator, and get the answer: 15/12. (Or 1¾ if you are using improper fractions.)

That's all that's to it!

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