Fractions

Introduction

In mathematics, we come across different kinds of numbers such as natural numbers, whole numbers, decimals, rational numbers and more. One such set of numbers is fractions. What are fractions and how do we perform various operations involving fractions? Let us find out. 

Definition

A fraction is a number representing a part of a whole. The whole may be a single or a group of objects.  This means that when one whole is divided into equal parts, each part is a fraction. The whole may be a single object or a group of objects. A whole one can be divided into 2 halves or 3 thirds or 4 quarters and so on. For example, in the below figure, the fraction of the shaded part is $\frac{3}{8}$ .

Types of Fractions

Before we start the discussion on operations on fractions, it is important to understand some key terms and definitions that are integral to fractions. Following are the types of fractions we come across when performing various operations on fractions. 

Unit Fractions

A fraction in which the numerator is always 1 is called a unit fraction. A unit fraction is the base unit of any fraction.

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. 

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. 

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 they are proper fractions.

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.

Mixed Fractions – A combination of a whole number and a proper fraction is called a mixed fraction. Example $4\frac{2}{3}$. 

Equivalent Fractions – A given fraction and various fractions obtained by multiplying or dividing its numerator and denominator by the same non-zero number are called equivalent fractions.

How to Compose Fractions

We can compose fractions by taking two more fractions and combining them to form a larger fraction.

Composing Fractions using Two Halves

Let us learn how to compose fractions using two halves. We know that a whole can be represented as – 

Now let us divide it into two equal halves. We will have,

We can see that each part of the two halves of the whole part is a fraction and is denoted by $\frac{1}{2}$. Hence, we can say that 2 halves when composed together make a whole. Hence, we can say that 2 halves when composed together make a whole. This means that – 

$\frac{1}{2}$ + $\frac{1}{2}$ = 1

In the above case, we can see that the numerator of both the fractions was 1 while the denominator was also the same. So, it was easy to add the two fractions and compose them into a larger fraction. 

Composing Fractions using 3 Thirds

Let us learn how to compose fractions using 3 thirds. Again, we know that a whole can be represented as – 

Now let us divide it into three equal halves. We will have,

We can see that each part of the three halves of the whole part is a fraction and is denoted by $\frac{1}{3}$. Hence, we can say that 3 thirds when composed together make a whole. This means that – 

$\frac{1}{3}$ + $\frac{1}{3}$ + $\frac{1}{3}$ = 1

Composing Fractions using 4 Quarters

Let us learn how to compose fractions using 4 quarters. Again, we know that a whole can be represented as – 

Now let us divide it into three equal halves. We will have,

We can see that each part of the three halves of the whole part is a fraction and is denoted by $\frac{1}{4}$. Hence, we can say that 4 quarters when composed together make a whole. This means that – 

$\frac{1}{4}$ + $\frac{1}{4}$ + $\frac{1}{4}$ + $\frac{1}{4}$  = 1

Addition and Subtraction of Fractions

Addition and subtraction of Fractions is done based on the equivalence of their denominators, i.e. whether they have same different denominators. Let us discuss both the cases. 

Fractions with Same Denominators

Recall that, fractions with the same denominator are called like fractions. Therefore, if the denominators of the fractions are the same, then would be called like fractions. Therefore, in order to add or subtract two proper fractions, we follow the following steps:

1. Obtain the numerators of the two given fractions and their common denominator.
2. Add or subtract the numerators obtained in the first step.
3. Write a fraction whose numerator is the sum obtained in the second step and the denominator is the common denominator of the given fractions.

Let us understand it by an example.

Example

Suppose we want to add the fractions $\frac{2}{9}$ and $\frac{5}{9}$

Solution

Here we can see that both the fractions have the same denominator, i.e. 9 and for both the fractions, the numerators as less than the denominators, which means they are proper fractions. 

Therefore, we go by the above-defined steps.

We check the numerators of both the fractions. They are 2 and 5

Then, we add these numerators and get 2 + 5 = 7.

Now, we write the sum of these fractions as $\frac{7}{9}$

Hence,  $\frac{2}{9}$ +  $\frac{5}{9}$ = $\frac{7}{9}$

Similarly, $\frac{2}{9}$ –  $\frac{5}{9}$ = $\frac{-3}{9}$ = $\frac{-1}{3}$

In the above example, we have seen how to add or subtract two fractions having the same denominator. What if their denominators are different? 

Fractions with Different Denominators

Fractions having different denominators are called unlike fractions. . For example, $\frac{1}{7}$ and $\frac{3}{8}$ are unlike fractions as both the fractions have different denominators. But they are proper functions as well as their numerators are less than their respective denominators.

Let us see how to add or subtract unlike fractions. We shall use the following steps to find the sum or difference of such fractions with different denominators.

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 (LCM) of their denominators. This step is exactly the same as finding the Least Common Denominator (LCD).

3. 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.

4. Since the fractions are now like fractions, add or subtract them as we do for like a fraction, i.e. add their numerators.

5. Reduce the fraction to its simplest form, if required.

Let us understand the above steps 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 $\frac{5}{6}$ – $\frac{3}{4}$

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

Multiplication of Fractions

how do we multiply two fractions? There is one general method of multiplication of fractions. But before understanding the method of multiplication, it is important to understand how to reduce a fraction in its lowest form or standard form.

Reducing a Fraction to its Simplest Form

As the name suggests, a fraction is said to be in the simplest form if the numerator ad denominator has no common factor other than 1 or we say that these are co-primes.

Let us understand this by an example – 

Suppose we have a fraction $\frac{24}{56}$ and we want to reduce it to its lowest form.

First of all, we will write the factors of both the numerator and the denominator. We have

The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24

The factors of 56 are 1, 2, 4, 7, 8, 14, 28 and 56

From above we can see that the Highest Common Factor (HCF) of 24 and 56 is 8. Therefore we will divide both the numerator and the denominator by 8 to make the numbers a pair of co-primes.

We will get

$\frac{24 ÷8}{56 ÷8} = \frac{3}{7}$

The fraction $\frac{24}{56}$ has now been reduced to $\frac{3}{7}$. 3 and 7 are prime numbers and do not have any common factors between them. Hence we can say that the simplest form of the fraction  $\frac{24}{56}$ is $\frac{3}{7}$.

We do not need to reduce a fraction to its simplest every time. There may be instances where the numerator and the denominator of a fraction are already in a form such that they do not have any factors in common, though they might not be prime numbers. For example, the numbers 8 and 81 both are not prime numbers, yet they do not have any factor in common. Therefore, we put them in a fraction, say $\frac{8}{81}$, they are already in the lowest form. Let us consider another example –

 Suppose we have a fraction $\frac{15}{34}$ and we want to reduce it to its lowest form.

First of all, we will write the factors of both the numerator and the denominator. We have

The factors of 15 are 1, 3, 5 and 15

The factors of 34 are 1, 2, 17 and 34

We can see that 15 and 34 have no common factor between them other than 1. Hence the said fraction is““““` said to be in its simplest form.

General Steps Involved in multiplying two or more Fractions

The following steps are involved in multiplying two or more fractions – 

1. First, we need to multiply all the numerators.
2. Next, we need to multiply all the denominators
3. Lastly, we need to simplify the fraction, if required

Therefore, we can say that

Product of fractions = $\frac{Product\: of\: the\: Numerators}{Product\: of\: the\: denominators}$

Let us understand it through an example.

Suppose we want to multiply 2$\frac{1}{3}$ by 2$\frac{1}{5}$

We can see that both the fractions are mixed. Therefore, we first convert them into improper fractions, we get,

2$\frac{1}{3} = \frac{7}{3}$ and

2$\frac{1}{5} = \frac{11}{5}$

Now we have two improper fractions, $\frac{7}{3}$ and $\frac{11}{5}$ which can be multiplied using the formula 

Product of fractions = $\frac{Product\: of\: the\: Numerators}{Product\: of\: the\: denominators}$

Therefore, we get,

$\frac{7}{3} x \frac{11}{5} = \frac{7 x 11}{3 x 5} = \frac{77}{15}$

Division of Fractions

The division of a fraction $\frac{a}{b}$ by a non-zero fraction $\frac{c}{d}$ is defined as the product of $\frac{a}{b}$  with the multiplicative inverse of $\frac{c}{d}$. Now, where what do we mean by the multiplicative inverse. The multiplicative inverse is the other name for the reciprocal of a fraction. So, how do we define the reciprocal of a fraction?

Reciprocal of a Fraction

Two fractions are said to be the reciprocal or multiplicative inverse of each other if their product is 1. For example,

$\frac{5}{7}$ and $\frac{7}{5}$ are the reciprocals of each other because $\frac{5}{7}$ x $\frac{7}{5}$ = 1

So, how do we obtain the reciprocal of a number? Let us find out.

If we observe the two fractions we have just discussed, i.e. $\frac{5}{7}$ and $\frac{7}{5}$ we can clearly see that the numerator of the first fraction is the denominator of the second one. Similarly, the denomination of the first fraction is the numerator of the second one.

This means that in order to find the reciprocal of a fraction, we need the swap its numerator and the denominator. In other words, in a reciprocal of a fraction, the numerator becomes the denominator and vice-versa. 

Therefore, if we two fractions say $\frac{a}{b}$ and $\frac{c}{d}$ and we want to divide the fraction $\frac{a}{b}$ by $\frac{c}{d}$, we will have 

$\frac{a}{b}$÷ $\frac{c}{d}$ = $\frac{a}{b}$ x  $\frac{d}{c}$ = $\frac{a d}{b c}$  

For example, suppose we two fractions, say $\frac{3}{5}$ and $\frac{5}{9}$ and we want to divide the $\frac{3}{5}$ by $\frac{5}{9}$, we will have 

$\frac{3}{5}$÷  $\frac{5}{9}$ = $\frac{3}{5}$ x $\frac{9}{5}$ = $\frac{27}{25}$  

So, the steps involved in the division of fractions can be defined as under-  

1. Identify the divisor and the dividend of the two fractions. This is important as we know that division is not commutative which means that a ÷ b is not equal to b ÷ a. 
2. Take the reciprocal of the fraction from which the dividend is being dividend.
3. Replace the divisor with its reciprocal and change the sign of division between the two fractions to the sign of multiplication.
4. Proceed with the multiplication of fractions.
5. Multiply your numerators to get your new numerator
6. Multiply your denominators to get your new denominator
7. Simplify the final fraction, if possible
8. The fraction thus obtained is your result.

Let us understand it with an example.

Example

Divide $\frac{4}{9}$  by $\frac{2}{3}$

Solution

We have been given two fractions $\frac{4}{9}$ and $\frac{2}{3}$ and we need to find $\frac{4}{9}$ by $\frac{2}{3}$. We will proceed in accordance with the steps we have defined above. We will have,

$\frac{4}{9} ÷ \frac{2}{3}$

Here, $\frac{2}{3}$ is the divisor. Therefore we will take its reciprocal. We will get $\frac{3}{2}$

Now, we will replace the divisor with its reciprocal and change the sign of division between the two fractions to the sign of multiplication. We will get,

$\frac{4}{9} x \frac{3}{2}$

Now, this is a case of multiplication of fractions, we will see if the numerators and the denominator have anything in common. We will get,

$\frac{4\: 2}{9\: 3}  x \frac{3}{2} = \frac{2}{3}$

Hence, $\frac{4}{9} ÷ \frac{2}{3} = \frac{2}{3}$

Key Facts and Summary

1. A fraction is a number representing a part of a whole.
2. A fraction in which the numerator is always 1 is called a unit fraction.
3. Fractions that have the same denominator are called like fractions.
4. Fractions with different denominators are called unlike fractions.
5. Fractions with the numerator less than the denominator are called proper fractions.
6. Fractions with the numerator either equal or greater than the denominator are called improper fractions.
7. In order to add two or more proper fractions with the same denominators, we add the fractions in the same manner as we do for like fractions.
8. In order to add two or more proper fractions with different denominators, 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.
9. A given fraction and various fractions obtained by multiplying or dividing its numerator and denominator by the same non-zero number are called equivalent fractions.
10. 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.
11. Two fractions are said to be the reciprocal or multiplicative inverse of each other if their product is 1.
12. In order to find the reciprocal of a fraction, we need the swap its numerator and the denominator.
13. The division of a fraction $\frac{a}{b}$ by a non-zero fraction $\frac{c}{d}$ is defined as the product of $\frac{a}{b}$ with the multiplicative inverse of $\frac{c}{d}$.

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