Today we’re going to find out why, **when we’re adding and subtracting fractions, they need to have the same denominator.**

If you didn’t already know, when we’re adding and subtracting fractions, they must be **homogeneous**. You can read more about homogeneous and heterogeneous fractions in this post.

It’s really easy to understand using visual aids, which we’ll take a look at below. The real reason is due to the definition of the fraction itself, which is a representation of parts of a total which must be **the same size**.

When you add or subtract fractions, you can’t express the result as a fraction if you do not divide the total into equal parts.

##### Adding fractions

For example, if you want to add ^{1}/_{2} + ^{1}/_{3}

We have:

- 1 of 2 equal parts of a whole unit (in green in the image).
- 1 of 3 equal parts of a unit (purple in the image).

To do the addition, we have to take the colored parts into account. Since each part is a different size, we can’t express this quantity in the form of a fraction.

We have **3 parts** (1 represented by a green rectangle and 2 represented by purple rectangles), but they are **not the same size**.

So what can we do? We can express the fractions we want to add in the form of a fraction that allows us to consider them parts **of the same size**.

As you can see in the following images, you can express the fraction ^{1}/_{2} as ^{3}/_{6} and the fraction ^{1}/_{3} as ^{2}/_{6}.

Now we’ve got the quantities that we want to add expressed in the form of fractions that have **parts of the same size**!

Now we can count the colored parts and express them in the form of a fraction. There are five equal parts: 5/6.

So ^{1}/_{2} + ^{1}/_{3} = ^{5}/_{6}.

##### Subtracting fractions

Now, if we to try subtract, for example, ^{1}/_{2} and ^{1}/_{3}, we get the same problem. To subtract ^{1}/_{3} from ^{1}/_{2}, **we need to take away parts that are the same size as the ones we have**.

So, we need to express both fractions homogeneously, and then we can take away the parts indicated by the subtraction.

If we express ^{1}/_{2} as ^{3}/_{6} and ^{1}/_{3} as ^{2}/_{6}, to subtract ^{1}/_{2} – ^{1}/_{3}, we take away 2 of the 3 equal parts of ^{3}/_{6}, and we get 1 part, or ^{1}/_{6}. So, we find that ^{1}/_{2} – ^{1}/_{3} = ^{1}/_{6}.

It’s easy to understand why the denominators must be the same when we’re adding and subtracting fractions, isn’t it?

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### Learn More:

- What Are Decimal Numbers?
- Understand What a Fraction Is and When It Is Used
- Homogeneous and Heterogeneous Fractions
- Singapore Math Bar Model for Multiplying Fractions
- Adding Fractions with the Visual Aid of Rectangles