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Addition & Subtraction Of Unlike 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 addition and subtraction of like fractions. But, before that, we must recall what we mean by 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 having the common denominator 9.

Before we start the discussion on how to add or subtract two or more unlike fractions, it is important to understand a few key terms and concepts that are vital to the addition or subtraction of unlike fractions.

Reducing a Fraction to its Standard Form

It is important to recall that a fraction is said to be in the standard form if the denominator is positive and the numerators have no common divisor other than 1.

In order to express a given fraction in the standard 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 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. 

Like Fractions

Fractions that have the same denominator are called like fractions. For example  $\frac{3}{7} and \frac{4}{7}$ are like fractions as both the fractions have the same denominators.

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. 

Equivalent Fractions

Fractions having the same value are called equivalent fractions. For example,

Let us consider the fraction $\frac{3}{5}$. Now, if we multiply both the numerator and the denominator of  $\frac{3}{5}$ by 2, we will get,

$\frac{3}{5} = \frac{3 x 2}{5 x 2}$ = 610

Here, 35 and 610 are equivalent fractions as the simplest of the fraction $\frac{6}{10} is \frac{3}{5}$.

Thus, an equivalent fraction of a given fraction can be obtained by multiplying its numerator and the denominator by the same number ( other than 0 ).

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

How to Add Unlike Fractions ?

We have learnt that unlike fractions are the fractions that do not have a common denominator. Therefore, we cannot add them in the same manner as we did for like fractions, where we simply added the numerators. In order to add two or more unlike fractions, we first convert them into the corresponding equivalent like fractions and then they are added in the same manner as we do for like fractions. The following steps are following for addition of unlike fractions – 

  1. Obtain the fractions and their denominators. The denominators of the fractions should be such that they are not the same. 
  1. 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 (LCM) of their denominators. This step is exactly the same as finding the Least Common Denominator (LCD).
  1. Convert each function 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.
  1. Since the fractions are now like fractions, add them as we do for like a fraction, i.e. add their numerators.
  1. Reduce the fraction to its simplest form, if required.

Let us understand the above steps through an example.

Example

Add $\frac{2}{3} and \frac{3}{7}$

Solution

We have been given the fractions, $\frac{2}{3} and \frac{3}{7}$. 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 sum.

Let us first find the L.C.M of 3 and 7.

The L.C.M of 3 and 7  = 21

So, we will convert the given fractions into equivalent fractions with denominator 21.

We will get,

$\frac{2}{3} = \frac{2 x 7}{3 x 7} = \frac{14}{21}$

Similarly,

$\frac{3}{7} = \frac{3 x 3}{7 x 3} = \frac{9}{21}$

Now, we two fractions, 1421 and 921 which have a common denominator 21 and are thus like fractions. So, we will add their numerators to get,

$\frac{14}{21}+ \frac{9}{21} = \frac{23}{21}$

Hence, 

$\frac{2}{3}$ + $\frac{3}{7}$ = $\frac{23}{21}$

How to Subtract 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. 
  1. 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 (LCM) of their denominators. This step is exactly the same as finding the Least Common Denominator (LCD).
  1. Convert each function 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.
  1. Since the fractions are now like fractions, subtract them as we do for a like fraction, i.e. subtract their numerators.
  1. 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, $\frac{33}{4} – \frac{17}{6}$. 

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

Solved Examples

Example 1 On Sunday, Fed read $\frac{5}{16}$ of the book. On Monday, he read $\frac{4}{8}$ of the book. What fraction of the book has Fed read?

Solution We have been given that, On Sunday, Fed read $\frac{5}{16}$ of the book. On Monday, he read $\frac{4}{8}$ of the book. We are required to find the fraction of the book that has been read by Fed. Let us first summarise what has been given to us – 

Fraction of book read by Fed on Sunday =  $\frac{5}{16}$

Fraction of book read by Fed on Monday = $\frac{4}{8}$

By adding the above fractions, we get the total fraction of the book that has been read by Fed. We can clearly see that the fractions have different denominators. Therefore, first, we will calculate their L.C.M.

L.C.M of 16 and 8 = 16

Now, we will convert the given fractions into equivalent fractions with denominator 16.

Note, that the fraction $\frac{5}{16}$ already has the denominator as 16, so we need not convert it any further. 

$\frac{4}{8} = \frac{4 x 2}{8 x 2} = \frac{8}{16}$

Now, we have two fractions, $\frac{5}{16} and \frac{8}{16}$ who have a common denominator 16. We will now add their numerators to obtain their sum. Therefore,

$\frac{5}{16} + \frac{8}{16} = \frac{5+8}{16} = \frac{13}{16}$

Hence, Fed has $\frac{13}{16}$ of the total book.

Example 2 Samuel is preparing for his final exam. He studied $\frac{9}{22}$ hours on Wednesday and $\frac{5}{11}$ hours on Sunday. How many hours did he study on two days?

Solution We have been given that Samuel is preparing for his final exam. He studied $\frac{9}{22}$ hours on Wednesday and $\frac{5}{11}$ hours on Sunday. We need to find out how many hours he studied altogether on the two days. Let us first summarise the fractions that have been given to us. We have,

Number of hours Samuel studied on Wednesday =  $\frac{9}{22}$ hours

Number of hours Samuel studied on Sunday = $\frac{5}{11}$ hours

By adding the above fractions, we get the total fraction of the number of hours studied by Samuel.

Total number of hours that Samuel studied on both days = $\frac{9}{22} + \frac{5}{11}$

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

L.C.M of 11 and 22 = 22

Now, we will convert the given fractions into equivalent fractions with denominator 22.

Note, that the fraction $\frac{9}{22}$ already has the denominator as 22, so we need not convert it any further. 

$\frac{5}{11} = \frac{5 x 2}{11 x 2} = \frac{10}{22}$

Now, we have two fractions, $\frac{9}{22} and \frac{10}{22}$ who have a common denominator 22. We will now add their numerators to obtain their sum. Therefore,

$\frac{9}{22} + \frac{10}{22} = \frac{9+10}{22} = \frac{19}{22}$

Therefore, Samuel studied a fraction of $\frac{19}{22}$ on two days combined. 

Example 3 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 4 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 5 A recipe needs 3/7 teaspoon black pepper and 1/4 teaspoon red pepper. How much more black pepper does the recipe need?

Solution We have been given that a recipe needs 3/7 teaspoon black pepper and 1/4 teaspoon red pepper. We need to find out how much more black pepper is required in the recipe as compared to red pepper.

First of all, we will summarise the fractions given to us. We have,

Fraction of black pepper needed in the recipe = $\frac{3}{7}$

Fraction of red pepper needed in the recipe = $\frac{1}{4}$

In order to find out how much more black pepper is required in the recipe as compared to red pepper, we will need to find the difference between the black pepper and the red pepper used.

Therefore, we need to find out the value of $\frac{3}{7} – \frac{1}{4}$

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

L.C.M of 7 and 4 = 28

Now, we will convert the given fractions into equivalent fractions with denominator 28.

$\frac{3}{7} = \frac{3 x 4}{7 x 4} = \frac{12}{28}$

$\frac{1}{4} = \frac{1 x 7}{4 x 7} = \frac{7}{28}$

Now, that the denominator of both the fractions is the same we will find the difference in their numerators. We have,

$\frac{3}{7} – \frac{1}{4} = \frac{12}{28} – \frac{7}{28} = \frac{12- 7}{28} = \frac{5}{28}$

Hence, the amount of more black pepper required in the recipe as compared to red pepper = $\frac{5}{28}$

Example 6 Add the fractions $\frac{2}{15} and \frac{3}{5}$

Solution We have been given the fractions $\frac{2}{15} and \frac{3}{5}$

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

L.C.M of 15 and 5 = 15

Now, we will convert the given fractions into equivalent fractions with denominator 15.

Now, we will convert the given fractions into equivalent fractions with denominator 15.

Note, that the fraction $\frac{2}{15}$ already has the denominator as 8, so we need not convert it any further. 

$\frac{3}{5} = \frac{3 x3}{5 x 3} = \frac{9}{15}$

Now, we have two fractions, $\frac{2}{15}$ and $\frac{9}{15}$ who have common denominator 15. We will now add the numerators to obtain their sum. Therefore,

$\frac{2}{15} + \frac{3}{5}  = \frac{2}{15} + \frac{9}{15} = \frac{2+ 9}{15} = \frac{11}{15}$

Therefore, $\frac{2}{15} + \frac{3}{5} = \frac{11}{15}$

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. A fraction is said to be in the standard form if the denominator is positive and the numerators have no common divisor other than 1.
  3. Fractions that have the same denominator are called like fractions.
  4. Fractions with different denominators are called unlike fractions.
  5. Fractions having the same value are called equivalent fractions.
  6. Thus, an equivalent fraction of a given fraction can be obtained by multiplying its numerator and the denominator by the same number ( other than 0 ).
  7. For unlike fractions, we do not add the numerators and denominators directly.
  8. In order to add two or more unlike fractions, we first convert them into the corresponding equivalent like fractions and then they are added in the same manner as we do for like fractions.

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