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How to Subtract Fractions

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 a fraction of the glass of milk (i.e. 1/2 the glass of milk). 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 subtracting fractions. Let’s start with a story!

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

With 3 slices gone, only 5/8ths of the pizza was left and Mike told his little sister to take as much as she wanted.

She took 3 slices or 3/8ths of the original pizza. What fraction of the pizza is left? It’s a pretty easy question and you probably have 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 2/8ths of the pizza is left. This fraction can be simplified and written as 1/4th (If you are not sure how simplification is done, see our article on simplifying fractions). So, in mathematical terms: \(\frac{2}{8} = \frac{1}{4}\)

Subtracting 6 slices from this pizza can easily be 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 do we subtract fractions then? Let’s learn the mathematical procedure of subtracting fractions.

P.S. You can use the following links to learn about simplifying fractions or adding fractions or to use our fraction calculator.

First, let’s start with a “how to” video on adding and subtracting fractions:

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

\(\frac{8}{8} – \frac{6}{8}\)

 

The first step is to find the LCM (Least Common Multiple) of the 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, 8, the LCM will also be 8. Remember, when you encounter fractions with the same denominator, there is no need to find the LCM, keep in mind that the LCM in that case is always the denominator that is common to both fractions. If three or more fractions are involved, then check whether the denominator of all the fractions is same, if yes, then the LCM will be that number otherwise you will have to calculate the LCM.

Step 1:

Find the LCM: The LCM is 8 because both \(\frac{8}{8}\) and \(\frac{6}{8}\) have denominators of 8.

Step 2:

Once the LCM is calculated, write it in the denominator. Because our LCM stayed the same, our equation still looks like so: \(\frac{8}{8} – \frac{6}{8}\)

Step 3:

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

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

Since the denominator and LCM of the first fraction are the same, the numerator will be written as it is: \(\frac{8}{8}\)

Also, the denominator and LCM of the second fraction are also the same, so the numerator will be written as it is: \(\frac{6}{8}\)

So to solve this you subtract both numerators from each other, like so: 8 – 6 = 2

Then you put the denominators back so: \(\frac{8}{8} – \frac{6}{8} = \frac{2}{8}\)

Once that is done, you simplify the fraction and get: \(\frac{2}{8} = \frac{1}{4}\)

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

In the case of adding of subtracting fractions remember this rule of thumb: when the denominators are the same, simply add/subtract the numerators while keeping the denominator the same.

Subtracting fractions with different denominators:

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

Solve: \(\frac{7}{9}-\frac{3}{16}\)

Step 1:

Find LCM

Mini-Step 1.1: List prime factors of the denominator

9 = 1 × 3 × 3

16 = 1 × 2 × 2 × 2 × 2

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 the 4th power: 24. When a number has no power, it means its power is 1.

9 = 1 × 32

16 = 1 × 24

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.

1 × 32 × 24 = 144

Step 2:

Once the LCM is calculated, write it in the denominator.

Step 3:

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

  • Divide the LCM by the denominator of the first fraction: \(144 \div 9=16\) (red step)
  • Multiply the answer by the numerator of the first fraction: \(16 \times 7 = 112\) (blue step)

So, \(\frac{7}{9}=\frac{112}{144}\)

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

Repeat the same procedure with the second fraction.

And we get:\(\frac{3}{16}=\frac{27}{144}\)

This procedure is diagrammed below in further details:

Finally, we have: \(\frac{112-27}{144}=\frac{85}{144}\)

The final answer is \(\frac{85}{144}\).

Subtracting more than two fractions

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

Solve: \(\frac{3}{4}-\frac{2}{6}-\frac{1}{10}\)

Step 1:

Find LCM

Mini-Step 1.1:

4 = 1 × 2 × 2

6 = 1 × 2 × 3

10 = 1 × 2 × 5

Mini-Step 1.2:

4 = 1 × 22

6 = 1 × 2 × 3

10 = 1 × 2 × 5

Mini-Step 1.3: Choose the highest powers of each factor for LCM. **Remember: If a factor occurs in only one number it is always chosen.

1 × 22 × 3 × 5 = 60

Step 2:

Once LCM is calculated, write it in the denominator.

Step 3:

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

  • Divide the LCM by the denominator of the first fraction: \(60 \div 4=15\)
  • Multiply the answer by the numerator of the first fraction: \(15 \times 3=45\)

So, \(\frac{3}{4}=\frac{45}{60}\)

Repeat the same procedure with the considering the second fraction.

\(\frac{2}{6}=\frac{20}{60}\)

 

This gives us \(\frac{45-20-?}{60}\),

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

\(\frac{1}{10}=\frac{6}{60}\)

 

Finally, we have: \(\frac{45-20-6}{60}=\frac{19}{60}\)

So, the final answer is \(\frac{19}{60}\)

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 a denominator equal to 1! Here are some examples to help you understand what we are talking about:

\(8=\frac{8}{1}\),

\(5=\frac{5}{1}\),

\(20=\frac{20}{1}\).

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

Solve: \(4-\frac{7}{15}\)

This question can be rewritten as: \(\frac{4}{1}-\frac{7}{15}\)

Step 1:

Find LCM

Mini-Step 1.1:

1 = 1

15 = 1 × 3 × 5

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

1 = 1

15 = 1 × 3 × 5

Mini-Step 1.3:

1 × 3 × 5 = 15

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. The first number is obtained by doing the following steps

  • Divide the LCM by the denominator of the first fraction: \(15 \times 1=4\)
  • Multiply the answer by the numerator of the first fraction: \(15 \times 4=60\)

So, \(\frac{4}{1}=\frac{60}{15}\)

Repeat the same procedure with the second fraction.

\(\frac{7}{15}\)

 

Finally, we have: \(\frac{60-7}{15}=\frac{53}{15}\)

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!