Subtracting Fractions (With The Same Or Different Denominators)

Mathematically, when we want to refer to only a part of something whole, we use fractions. For example, when someone says they drank half a glass of milk, mathematically they mean that they drank glass of milk and is a fraction. That was a brief refresher of what fractions are and now let’s move on to the topic of this article. We will learn all about subtraction of fractions here and let’s start with a story.

Mike invited his friends over to play some board games and have fun on the weekend. He ordered a pizza for them to enjoy and there were three of them and each one of them took one slice when Mike’s little sister walked in and demanded to be included in the group.

With 3 slices gone, only pizza was left and Mike offered his little sister to take as much as she want.

She took 3 slices i.e th portion of the original pizza. What fraction of the pizza is left? Its pretty easy question and you probably already guessed the answer. It’s a simple matter of counting the slices. Out of five slices, the little girl took three slices and obviously two were left. This means that fraction of pizza is left. This fraction can be simplified and written as (If you are not sure how simplification is done, see our article on fraction simplification). So, in mathematical terms:

Subtracting from can be easily done when we have figures available but when there is no pizza to count slices from and no story to help us, then what? How to do we subtract fractions then? Let’s learn the mathematical procedure of subtracting fractions.

p.s. if you use the follwing links to learn about simplifying fractions or adding fractions or to use our fraction calculator.

If you have read our article on adding fractions, this going to be easy for you because basic concept is the same. Let’s see how:

First step is to find LCM (Least Common multiple) of denominators of both fractions. If you are not familiar with the concept of LCM please see our article on GCF and LCM. Since both denominators are the same, the number 8, LCM will also be 8. Remember, when you encounter fractions with same denominator, no need to go into detail to find LCM, keep in mind that LCM in that case is always the denominator that is common to both fractions. If three or more fractions are involved, then check whether denominator of all the fractions is same, if yes then LCM will be that number otherwise you will have to calculate the LCM.

Step 1:

Find LCM

Step 2:

Once LCM is calculated, write it in the denominator.

Step 3:

This step is a tricky one but for our first example its pretty simple. We will tackle the tricky portion in the next section. For now, learn this short cut:

When the denominator and the LCM is same, the numerator will also remain same for adding or subtracting fractions.

Since denominator and LCM of first fraction is same, numerator will be written as it is:

Also, denominator and LCM of second fractions is also same, numerator will be written as it is:

So we have:

That is the same fraction that we got by counting the pizza slices!

As a rule of thumb, remember that, when denominators are same in case of addition or subtraction of fractions, simply add/subtract the numerators while keeping the denominator same. For example,

Subtracting fractions with different denominators:

Denominators of fractions are not necessarily same. In fact, in majority of the problems, denominator of fractions involved will be different. Let’s see the mathematical procedure to tackle that problem.

Solve:

Step 1:

Find LCM

Mini-Step 1.1: List prime factors of the denominator

Mini-Step 1.2: Write the factors in index form i.e. when a factor occurs more than once, write it raised to some power. For example, when 2 is multiplied four times it can be written as 2 raised to power 4: . When a number has no power, it means its power is 1.

Mini-Step 1.3: If a factor occurs for both numbers, their maximum power is chosen. if a factor occurs for only one number it is always chosen.

Step 2:

Once LCM is calculated, write it in the denominator.

Step 3:

Since two fractions are being subtracted, numerator will be a difference of two numbers. First number is obtained by the following procedure:

  • Divide LCM by denominator of first fraction:
  • Multiply the answer by numerator of first fraction:

So,

The step is explained in the figure below to help you better understand by visualization.

Repeat same procedure by considering second fraction.

And we get:

This concept is diagrammatically explained below:

Finally, we have:

The final answer is .

Subtracting more than two fractions

As in case of addition, subtracting more than two fraction involves the same procedure. Here is a solved example:

Solve:

Step 1:

Find LCM

Mini-Step 1.1:

Mini-Step 1.2:

Mini-Step 1.3: Choose highest powers of each factor for LCM.

Step 2:

Once LCM is calculated, write it in the denominator.

Step 3:

Since three fractions are being subtracted, numerator will be a difference of three numbers. First number is obtained by the following steps:

  • Divide LCM by denominator of first fraction:
  • Multiply the answer by numerator of first fraction:

So,

Repeat the same procedure by considering the second fraction.

This gives us,

Again, repeat the same procedure by considering the third fraction.

Finally, we have:

So, the final answer is

Subtraction of fraction and integers

So far, we have learned all about subtracting fractions. Sometimes, while solving mathematical problems, you might come across problems where you need to find the difference of fractions and integers. This section will explain how to do that.

The single most important thing that you need to remember is: integers are also fractions with denominator equal to 1! Here are some examples to help you understand what we are talking about:

Using this information, now you can treat integers as fractions solve the problems using the same procedure as outlined above.

Solve:

This question can be rewritten as:

Step 1:

Find LCM

Mini-Step 1.1:

Mini-Step 1.2: There are no factors with powers. 1 always remains 1 no matter its power.

Mini-Step 1.3:

Step 2:

Once LCM is calculated, write it in the denominator.

Step 3:

Since three fractions are being subtracted, numerator will be a difference of three numbers. First number is obtained by doing the following steps

  • Divide LCM by denominator of first fraction:
  • Multiply the answer by numerator of first fraction:

So,

Repeat same procedure by considering the second fraction.

Now, we have:

Again, repeat same procedure by considering third fraction.

Finally, we have:

Final Thoughts!

Learning about fractions can be dry and boring sometimes, but we have tried to explain everything in an interesting and easy way. This article was all about subtracting fractions. We have another article on addition of fractions and if you have not already seen it, do have a look!