Steps for adding fractions with the same denominator:

1. The denominator of the answer is the same as the denominators in the fractions being added.
2. Add the numerators of the fractions to make the numerator of the answer.

These steps can be more easily interpreted to mean:

1. The number on the bottom of the fractions is kept the same.
2. Add the numbers on the tops of the fractions.

To understand how we add fractions with the same denominator and why this method works, we will begin by looking at an example.

Below, we have

  ¹ ⁄ ₂   +   ¹ ⁄ ₂ .

The denominators are the numbers on the bottoms of the fractions. Both fractions have a denominator of ‘2’ and so, we are adding fractions with the same denominators.

When fractions have the same numbers on the bottom, they are said to have a common denominator. This is another way of saying that the fractions have the same denominators. They have a denominator in common.

The denominator tells us how many equal parts to divide a fraction into. A denominator of 2 means that the whole amount will be divided into 2 equal parts, or halves.

If we take the half from the fraction on the right and combine it with the fraction on the left, we can see that the whole shape is now shaded in.

We have added the two fractions together to make a total.

We have    ² ⁄ ₂    parts of the shape shaded in.

So, we can say that

 ¹ ⁄ ₂  +  ¹ ⁄ ₂    =   ² ⁄ ₂  .

To get this result, we kept the denominator, on the bottom of the fraction the same. The denominator was ‘2’.

We then added the numerators of the two fractions: ‘1’ + ‘1’ and made ‘2’.

Because the whole shape is shaded in, we can also say that

  ¹ ⁄ ₂   +  ¹ ⁄ ₂    = 1.

  ² ⁄ ₂    is the same value as 1 whole and is just a simpler way to write it.

Another way to interpret this fraction is that we have two out of two equal parts. We have two halves.

Another way to interpret the fraction is to remember that the line between the numerator and denominator means to perform a division.

The fraction means 2 ÷ 2, which equals 1.

Below is another example of adding two fractions that have the same denominator.

The fractions of  ² ⁄ ₄  and  ¹ ⁄ ₄   both have the same digits on the bottom, below the dividing line. They have a common denominator of four.

Fractions with a denominator of 4 are describing a division into four equal parts. These are called quarters.

When adding these fractions, we can think of the process as counting up the total number of quarters that we have.

And since we are counting the number of quarters we have, the answer will be in quarters too.

The denominator in our answer will therefore be a ‘4’.

In the example above we have:

² ⁄ ₄  +  ¹ ⁄ ₄ .

Remember that the method for adding fractions with the same denominator is to keep the same denominator and add the numerators.

The denominator in the answer will be a ‘4’.

The numerators are ‘2’ and ‘1’. 2 + 1 = 3.

² ⁄ ₄  +  ¹ ⁄ ₄  = ³ ⁄ ₄  .

We have combined the quarters to make a total of three quarters.

The shapes were divided into four equal parts and we are counting how many of these four equal parts we have in total.

There are two parts shaded in on the left and there is one part shaded in on the right, we have a total of three parts shaded in. And so, 3 out of 4 parts are shaded in.

We will look at the following example without using any bar model diagrams.

We will follow the two steps in our method for adding fractions with the same denominator.

We keep the denominator in our answer the same as the denominators in the fractions that we are adding.

We add the numerators of the fractions that we are adding.

The denominators of both   ¹ ⁄ ₅   and   ³ ⁄ ₅   are both 5.

The denominators are the same.

This means that the denominator in our answer will also be 5.

To add fractions with the same denominators, we simply add the numerators.

1 + 3 = 4.

Therefore,

¹ ⁄ ₅   +   ³ ⁄ ₅   =   ⁴ ⁄ ₅ .

Provided that the fractions that are being added have the same denominators, then this method can always be used to add the fractions.

Here is a final example, involving an answer that is an improper fraction.

The fractions being added are identical.

Both fractions are two-thirds.

² ⁄ ₃   and   ² ⁄ ₃  both have a denominator of three.

Therefore the denominator in the answer will also be three.

To find the numerator of the fraction in the answer, add the numerators of the two fractions that you are adding.

2 + 2 = 4

Therefore:

² ⁄ ₃   +   ² ⁄ ₃   =   ⁴ ⁄ ₃ .

When a fraction has a numerator larger than its denominator, it is known as an improper fraction.

An alternative name for an improper fraction is a top-heavy fraction.

It simply means that the number on the top of the fraction is larger than the number on the bottom of the fraction.

When adding fractions, you may obtain an answer that is an improper fraction.

The addition of the two fractions given in this example is really asking us ‘if we have two thirds and add two more thirds, how many thirds do we have in total’?

The image above shows that we have four thirds shaded in. Each of the larger rectangles were divided equally into three parts and four of them are shaded in total.

One full rectangle is shaded, along with one extra third of a rectangle.

We have one whole and one third in total.

  ⁴ ⁄ ₃ can be written as 1   ¹ ⁄ ₃ .

When writing fractions it is often considered better to write the final answer as a whole number alongside a fraction, such as in this example.

A whole number alongside a fraction is called a mixed number.

When teaching adding fractions, it is best to start with this lesson on adding fractions that have a common denominator and mastering this skill before progressing onto adding fractions that have different denominators.

Only move on to adding or subtracting fractions with unlike denominators once this process is well-understood.

It is best to start by showing the process visually, using bar models such as in this lesson, or pizza slices. Eventually you want to be able to add two fractions with the same denominator quickly and mentally, without the aid of a diagram.

The most common mistake with this method is adding the denominators together along with the numerators.

It is worthwhile pointing this mistake out in advance and frequently reminding that the denominators remain the same and that it is just the numerators that we add.

It may help to return to the visual rectangular bar models to help remember this if this mistake occurs.