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# Subtracting Fractions

## Introduction

We have learnt that a fraction is a number representing a part of a whole. The whole may be a single or a group of objects.  Let us now learn about the subtraction of fractions.

But, before that, we must recall what we mean by like and unlike fractions.

### Like Fractions

Fractions that have the same denominator are called like fractions. For example, the fractions, $\frac{4}{9}$, $\frac{13}{9}$, $\frac{1}{9}$ and  $\frac{5}{9}$ are all like fractions as they have a common denominator 9.

### Unlike Fractions

Fractions with different denominators are called unlike fractions. For example, $\frac{3}{7}$ and $\frac{5}{8}$ are unlike fractions as both the fractions have different denominators.

Now let us recall what is meant by reducing a fraction to its lowest form.

### Reducing a Fraction to Its Lowest Form

We know that a fraction is represented in the form of $\frac{p}{q}$ where p and q are integers. So, what is the lowest form of a fraction?

We should recall that a fraction is said to be in its lowest form if both the numerator as well as the denominator are positive and the numerator and the denominator have no common divisor other than 1.

In order to express a given fraction in the standard / simplest form, the following steps should be followed –

Step 1 – Check whether the given number is in the form of a fraction.

Step 2 – See whether the denominator of the fraction is positive or not. If it is negative, multiply or divide the numerator as well as the denominator by -1 so that the denominator becomes positive.

Step 3 – Find the greatest common divisor (GCD) of the absolute values of the numerator and the denominator.

Step 4 – Divide the numerator and the denominator of the given fraction by the GCD (HCF) obtained in step III. The fraction so obtained is the standard / simplest form of the given fraction.

This was important to understand because, after the addition or subtraction of fractions, we will need to reduce them in the simplest form.

Let us understand it using an example

Suppose we want to reduce the fraction $\frac{8}{28}$

We can see that the number in the denominator is 28 which, is a positive number. In order to express it in standard / simplest form, we must divide its numerator and the denominator by the greatest common divisor of 8 and 28.

The greatest common divisor of 8 and 28 is 4.

Therefore,

Dividing the numerator and the denominator of $\frac{8}{28}$ by 4, we get

$\frac{8}{28} = \frac{8 ÷4}{28 ÷4} = \frac{2}{7}$

Hence the simplest form of $\frac{8}{28}\:is\:\frac{2}{7}$.

Now, let us learn how to subtract two or more fractions.

## Subtraction of Like Fractions

In order to subtract like fractions, we follow the following steps:

1. Obtain the numerators of the two given fractions and their common denominator.
2. Subtract the subtrahend numerator from the minuend numerator obtained in the first step. Recall here that the number that is subtracted is called the subtrahend while the number from which the subtrahend is subtracted is called minuend. The result of this subtraction is called the difference.
3. Write a fraction whose numerator is the difference obtained in the second step and the denominator is the common denominator of the given fractions.

Let us understand this through an example.

Example

Suppose we want to find the difference between fractions $\frac{3}{7}$ and $\frac{1}{7}$

Solution

Here we can see that both the fractions have the same denominator, i.e. 7.

Therefore, we go by the above-defined steps.

We check the numerators of both the fractions. They are 3 and 1.

Then, we subtract 1 from 3, we will get 3 – 1 = 2

Now, we write the difference of these fractions as $\frac{2}{7}$

Hence, $\frac{3}{7} – \frac{1}{7} = \frac{2}{7}$

In order to understand the subtraction of like fractions in a clearer manner, let us check the graphical representation of subtraction of like fractions.

### Graphical Representation of Subtraction of Like Fractions

Suppose we have the fractions, $\frac{5}{8}$ and $\frac{1}{8}$ and we want to find the difference between these fractions. If we go by the steps that we have defined above for subtraction of like fractions, we can see that the two fractions have the same denominator, i.e. 8. So, in order to subtract these fractions, we will simply subtract the numerators, i.e. 5 and 1. We will get 5 – 1 = 4 Therefore,

$\frac{5}{8} – \frac{1}{8} = \frac{4}{8}$

Now, let us represent each of these fractions, graphically.

The fraction $\frac{5}{8}$  will graphically be represented as

We can see that the above fraction has 5 shaded parts out of a total of 8 equal parts.

The fraction $\frac{1}{8}$ will graphically be represented as

We can see that the above fraction has 1 shaded part out of a total of 8 equal parts.

If we subtract the two shaded parts in these fractions, we can see that we will have 5 – 1 = 4 shaded parts. This will be represented graphically as –

The above fraction will be written as $\frac{4}{8}$ which is the result that we had obtained after we subtracted the two fractions above.

## Subtraction of Unlike Fractions

Subtraction of unlike fractions is similar to their addition with the only difference being that instead of adding the equivalent fractions obtained, we will subtract them. The following steps are following for subtraction of unlike fractions –

1. Obtain the fractions and their denominators. The denominators of the fractions should be such that they are not the same.
2. Find the Least Common Multiple ( L.C.M ) of the denominators. In other words, make the denominators the same by finding the Least Common Multiple ( L. C. M. ) of their denominators. This step is exactly the same as finding the Least Common Denominator ( L. C. D. ).
3. Convert each fraction into an equivalent fraction having the same denominator equal to the L.C.M obtained in the previous step. This means that you need to rewrite each fraction into its equivalent fraction with a denominator that is equal to the Least Common Multiple that you found in the previous step.
4. Since the fractions are now like fractions, subtract them as we do for a like fraction, i.e. subtract their numerators.
5. Reduce the fraction to its simplest form, if required.

Let us understand the above steps through an example.

Example

Solve $\frac{33}{4} – \frac{17}{6}$

Solution

We have been given the fractions, $\frac{33}{4}\:and\:\frac{17}{6}$.

We can clearly see that the denominators of these fractions are different, hence they are unlike fractions. Therefore, we will proceed according to the steps defined above to obtain their difference.

We will first find the L.C.M of 4 and 6

L.C.M of 4 and 6 = 12

So, we will convert the given fractions into equivalent fractions with denominator 12.

We will get,

$\frac{33}{4} = \frac{33 x 3}{4 x 3} = \frac{99}{12}$

Similarly,

$\frac{17}{6} = \frac{17 x 2}{6 x 2} = \frac{34}{12}$

Now, we two fractions, $\frac{99}{12}$ and $\frac{34}{12}$ which have a common denominator 12 and are thus like fractions. So, we will subtract their numerators to get,

$\frac{99}{12} – \frac{34}{12} = \frac{99- 34}{12} = \frac{65}{12}$

Hence, $\frac{33}{4} – \frac{17}{6} = \frac{65}{12}$

Now, let us discuss subtraction of Improper Fractions

## Subtraction of Proper Fractions

Before learning about subtracting improper fractions let us recall what we mean by proper fractions.

Fractions with the numerator less than the denominator are called proper fractions. For example, consider the fractions $\frac{1}{2}$ and $\frac{2}{3}$. In both of these fractions, the numerators are less than their respective denominators. Hence these fractions are proper fractions. However, note that the fraction $\frac{3}{3}$ is not a proper fraction as in this case the numerator is equal to the denominator. So, we can say the condition for a fraction to be a proper fraction is that its numerator should be strictly less than its denominator. Let us now understand how to subtract proper fractions.

Subtraction of proper fractions is just similar to the subtraction of like and unlike fractions where the calculations are based on the equivalence of their denominators, i.e. whether they have the same or different denominators.

Let us understand it through an example.

Example

Subtract $\frac{3}{4}$ from $\frac{5}{6}$

Solution

We have been given the fractions $\frac{3}{4}$ and $\frac{5}{6}$ and we are required to find the value of $\frac{3}{4} – \frac{5}{6}$

Therefore,

To find the value of $\frac{5}{6} – \frac{3}{4}$

We can clearly see that the denominators of these fractions are different, hence they are unlike fractions. Therefore, we will proceed according to the steps defined above to obtain their difference.

We will first find the L.C.M of 4 and 6

LCM of 6 and 4 is 12.

So, we will convert the given fractions into equivalent fractions with denominator 12.

Therefore

$\frac{5}{6} = \frac{5 x 2}{6 x 2} = \frac{10}{12}$ and

$\frac{3}{4} = \frac{3 x 3}{4 x 3} = \frac{9}{12}$

Now, we two fractions, $\frac{10}{12}$  and $\frac{9}{12}$ which have a common denominator 12 and are thus like fractions. So, we will subtract their numerators to get,

Therefore,

$\frac{5}{6} – \frac{3}{4} = \frac{10}{12} – \frac{9}{12} = \frac{10-9}{12} = \frac{1}{12}$

Hence, $\frac{5}{6} – \frac{3}{4} = \frac{1}{12}$

## Subtraction of Improper Fractions

Before learning about subtracting improper fractions let us recall what we mean by improper fractions.

Fractions with the numerator either equal or greater than the denominator are called improper fractions. For example, consider the fractions $\frac{5}{2}$ and $\frac{7}{3}$. In both these fractions, the numerators are greater than their respective denominators. Hence they are improper fractions. Even the fraction $\frac{3}{3}$ is an improper fraction as in this case the numerator is equal to the denominator. So, we can say the condition for a fraction to be an improper fraction is that its numerator should be greater than or equal to its denominator

Subtraction of improper fractions is just similar to the subtraction of like and unlike fractions where the calculations are based on the equivalence of their denominators, i.e. whether they have the same or different denominators.

Let us understand it by an example

Example

Suppose we want to find the difference between fractions $\frac{11}{3}$ and $\frac{8}{3}$

Solution

In both the fractions, we can see that the numerators are greater than the denominators which means they are improper fractions. But, here we can also see that both the fractions have the same denominator, i.e. 3. Therefore, we go by the above-defined steps of subtracting like fractions.

We check the numerators of both the fractions. They are 11 and 8.

Then, we subtract 8 from 11, we will get 11 – 8 = 3

Now, we write the difference of these fractions as $\frac{3}{3} = 1$

Hence, $\frac{11}{3} – \frac{8}{3} = 1$

## Solved Examples

Example 1 Subtract $\frac{21}{25}$ from $\frac{18}{20}$

Solution We have been given the fraction, $\frac{21}{25}$ and $\frac{18}{20}$.

We need to find $\frac{18}{20} – \frac{21}{25}$

The denominators of the above fractions are different; therefore, we will find their L.C.M first.

L.C.M of 20 and 25 = 100

Now, we will convert the given fractions into equivalent fractions with denominator 100.

$\frac{21}{25} = \frac{21 x 4}{25 x 4} = \frac{84}{100}$

$\frac{18}{20} = \frac{18 x 5}{20 x 5} = \frac{90}{100}$

Now, we have two fractions, $\frac{84}{100}$ and $\frac{90}{100}$ who have a common denominator 100. We will now find the difference between the numerators to obtain their difference. Therefore,

$\frac{18}{20} – \frac{21}{25} = \frac{90}{100} – \frac{84}{100} = \frac{90-84}{100} = \frac{6}{100}$

We can see that the above fraction can be reduced to its simplest form if we divide both the numerator and the denominator by 2. We will get,

$\frac{6}{100} = \frac{6 ÷2}{100 ÷2} = \frac{3}{50}$

Hence, $\frac{18}{20} – \frac{21}{25} = \frac{3}{50}$

Example 2 A piece of wire $\frac{7}{8}$ metres long broke into two pieces. One piece was $\frac{1}{4}$ metre long. How long is the other piece?

Solution We have been given that a piece of wire $\frac{7}{8}$ metres long broke into two pieces and one piece was $\frac{1}{4}$ metre long.

We need to find the length of the other piece.

Let us first summarise the fractions given to us.

Length of the piece of wire = $\frac{7}{8}$ metres

Length of one of the broken pieces = $\frac{1}{4}$ metres

The length of the other piece can be obtained by finding the difference between the total length of the wire and one of the broken pieces.

Therefore,

Length of the other broken piece = $\frac{7}{8} – \frac{1}{4}$

We can clearly see that the fractions have different denominators. Therefore, first, we will calculate their L.C.M.

L.C.M of 8 and 4 = 8

Now, we will convert the given fractions into equivalent fractions with denominator 8.

Note, that the fraction $\frac{7}{8}$ already has the denominator as 8, so we need not convert it any further.

$\frac{1}{4} = \frac{1 x 2}{4 x 2} = \frac{2}{8}$

Now, we have two fractions, $\frac{7}{8}$  and $\frac{2}{8}$ who have common denominator 8. We will now find the difference between the numerators to obtain their difference. Therefore,

$\frac{7}{8} – \frac{2}{8} = \frac{5}{8}$

Therefore, the length of the other broken piece = $\frac{5}{8}$ m

Example 3 Find the difference of $\frac{17}{24}$ and $\frac{15}{16}$

Solution We have been given the fractions $\frac{17}{24}$ and $\frac{15}{16}$ and we need to find their difference.

Since the denominators of the above fractions are different; therefore, we will find their L.C.M first.

L.C.M of 24 and 16 = 48

Now, we will convert the given fractions into equivalent fractions with denominator 48.

$\frac{17}{24} = \frac{17 x 2}{24 x 2} = \frac{34}{48}$

$\frac{15}{16} = \frac{15 x 3}{16 x 3} = \frac{45}{48}$

Now, we have two fractions, $\frac{34}{48}$ and $\frac{45}{48}$ who have a common denominator 48. We will now find the difference between the numerators to obtain their difference. Therefore,

$\frac{45}{48} – \frac{34}{48} = \frac{45- 34}{48} = \frac{11}{48}$

We can see that the above fraction is already in its simplest form as 11 and 48 do not have any factors other than 1 in common.

Hence, $\frac{45}{48} – \frac{34}{48} = \frac{11}{48}$

## Key Facts and Summary

1. A fraction is a number representing a part of a whole. The whole may be a single or a group of objects.
2. Fractions that have the same denominator are called like fractions.
3. Fractions with different denominators are called unlike fractions.
4. A fraction is said to be in its lowest form if both the numerator as well as the denominator are positive and the numerator and the denominator have no common divisor other than 1.
5. For subtracting like fractions, we subtract the numerators while keeping the denominators the same.
6. For unlike fractions, we do not subtract the numerators and denominators directly.
7. In order to subtract two or more unlike fractions, we first convert them into the corresponding equivalent like fractions and then they are subtracted in the same manner as we do for like fractions.
8. Fractions with the numerator less than the denominator are called proper fractions.
9. Fractions with the numerator either equal or greater than the denominator are called improper fractions.
10. Subtraction of proper as well as improper fractions is similar to the subtraction of like and unlike fractions where the calculations are based on the equivalence of their denominators, i.e. whether they have the same different denominators.