Adding Improper Fractions

Students are taught about adding and subtracting fractions with like denominators at the 3rd grade level, and with unlike denominators around 5th grade.

Adding improper fractions is exactly the same as adding two proper fractions - there is no real difference, other than the value of each of the fractions is over 1 whole

Here are 2 easy steps to help you add two fractions together.

Step 1: Convert the fractions to the same denominator.

If the fractions already have the same denominator, you do not need to do anything!

Step 2: Add the two numerators together (and keep the denominator the same) to get the answer.

You should now have the answer to your fraction sum!

You may also need to simplify the fraction at the end.

See below for a step-by-step breakdown of how to add fractions.

Step 1: Convert the fractions to the same denominator.

If the fractions already have the same denominator, skip the rest of this part.

If the fractions have different denominators, keep reading!

We can only add fractions together if they have the same denominator.

It is a bit like when you are adding inches and centimetres together - you need to convert both measures to the same units - either centimetres or inches!

There are two different possibilities for the this step:

Step 1a) If one of the denominators is a multiple of the other denominator.

If this is the case, you just need to multiply the numerator and denominator of the fraction with the lower denominator so that the two denominators are the same.

Step 1b) If one of the denominators is not a multiple of the other denominator.

The easiest and simplest way to convert the two fractions to equivalent fractions with the same denominators is to multiply the numerator and denominator of each fraction by the other fraction's denominator.

Step 2: Add the two numerators together

Once the two fractions have the same denominator, we can add the numerators together.

This will give us the answer we are looking for.

Have a look at the examples below to see how it all works!

Example 1) Work out the fraction sum below giving your answer as a mixed number. \[ {14 \over 9} + {16 \over 9} \]

Step 1)

You will notice that the two denominators are identical so we can go straight on to Step 2).

Step 2)

We just need to add the two numerators together, and simplify the fraction at the end.

\[ {14 \over 9} + {16 \over 9} = {14 + 16 \over 9} = {30 \over 9}\]

Step 3)

First we need to simplify the fraction by dividing the numerator and denominator by 3.

\[ {30 \over 9} = {30 ÷ 3 \over 9 ÷ 3} = {10 \over 3}\]

Next we need to convert this improper fraction to a mixed number.

\[{10 \over 3} \; = \; 3 {1 \over 3} \]

Final answer: \[ {14 \over 9} + {16 \over 9} \; = \; 3 {1 \over 3}\]

Example 2) Work out \[ {13 \over 5} + {22 \over 15} \]

Step 1)

You will notice that the second denominator (15) is a multiple of the first denominator (5) so we need to follow Step 1a)

So we need to multiply the numerator and denominator of the first fraction by 3.

\[ {13 \over 5} = {13 \times 3 \over 5 \times 3} = {39 \over 15} \]

This gives us: \[ {13 \over 5} + {22 \over 15} = {39 \over 15} + {22 \over 15} \]

Step 2)

Now the denominators are equal, all we need to do now is to add the numerators together.

\[ {39 \over 15} + {22 \over 15} = {39 + 22 \over 15} = {61 \over 15}\]

Step 3)

This fraction is already in simplest form.

Final answer: \[ {13 \over 5} + {22 \over 15} = {61 \over 15} \; or \; 4 {1 \over 15}\]

Example 3) Work out this fraction sum, giving your answer as a mixed number: \[ {11 \over 7} + {9 \over 4} \]

Step 1)

The denominator of the first fraction is 7. The denominator of the second fraction is 4. These numbers are not multiples of one another.

We need to use Step 1b)

So we multiply the numerator and denominator of the first fraction by the second fraction's denominator (4).

This gives us: \[ {11 \over 7} = {11 \times 4 \over 7 \times 4} = {44 \over 28} \]

Next we multiply the second fraction's numerator and denominator by the first fraction's denominator (7).

This gives us: \[ {9 \over 4} = {9 \times 7 \over 4 \times 7 = {63 \over 28} \]

So we now have: \[ {11 \over 7} + {9 \over 4} = {44 \over 28} + {63 \over 28} \]

Step 2)

Add the numerators together. \[ {44 \over 28} + {63 \over 28} = {44 + 63 \over 28} = {107 \over 28} \]

Step 3)

This fraction is already in simplest form, so we need to convert it to a mixed number.

\[{107 \over 28} \; = \; 3{23 \over 28} \]

Final answer: \[ {11 \over 7} + {9 \over 4} \; = \; 3{23 \over 28}\]