**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.

Before we move ahead with the understanding of dividing fractions, it is important to recall some of the terms that are relevant to the division of 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 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.

Now let us understand what we mean by reducing a fraction to its lowest 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.

Now, the Division of a fraction is not complete without having the knowledge of multiplication of fractions. Therefore, let us understand what are the general steps involved in the multiplication of fractions –

**General Steps Involved in multiplying two or more Fractions**

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

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

Therefore, we can say that

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

Suppose we have two improper fractions, $\frac{3}{2}$ and $\frac{5}{4}$

We can see that both the fractions are improper fractions as in both cases the numerator is greater than the denominator.

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

Therefore,

$\frac{3}{2} x \frac{5}{4} = \frac{3 x 5}{2 x 4} = \frac{15}{8}$

**Multiplication of Mixed Fractions**

We know that mixed fractions are the fractions that have a whole number mixed as a fraction. They are an extended form of improper fractions where the numerator is greater than the denominator. For example, $2\frac{1}{3}$ is a mixed fraction which in improper form is equal to $\frac{7}{3}$.

So, how do we multiply two mixed fractions?

Multiplication of mixed fractions is done the same manner as we do for any other fractions but before that, we need to perform one action. The action is that we must first convert the mixed fraction into an improper fraction. Now that we have the fraction in the form of an improper fraction, we can multiply it using the same technique as we did in the previous section. For 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,

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

**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.

Now, let us learn about the division of fractions.

**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.

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

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

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-

- 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.
- Take the reciprocal of the fraction from which the dividend is being dividend.
- Replace the divisor with its reciprocal and change the sign of division between the two fractions to the sign of multiplication.
- Proceed with the multiplication of fractions.
- Multiply your numerators to get your new numerator
- Multiply your denominators to get your new denominator
- Simplify the final fraction, if possible
- 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, 23 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, **$\frac{4}{9} ÷ \frac{2}{3} = \frac{2}{3}$

**Division of Mixed Fractions**

**In the above example, we can see that the fractions are proper fractions. The process for division and proper fractions is the same. However, if we have mixed fractions, we will first need to convert the mixed fraction into an improper fraction and then only we can proceed ahead with the division of fractions. **

Let us understand it by an example.

**Example**

Suppose we have two fractions $15\frac{3}{7}$ and $1\frac{23}{49}$ and we want to find the value of $15\frac{3}{7} ÷ 1\frac{23}{49}$. Let us see how will we find its value.

**Solution**

We have been given two fractions $15\frac{3}{7}$ and $1\frac{23}{49}$ and we want to find the value of $15\frac{3}{7} ÷ 1\frac{23}{49}$.

We can see that the fractions given here are both mixed fractions. Therefore, we will first convert them into proper fractions. We will get –

$15\frac{3}{7} = \frac{15 x 7+3}{7} = \frac{105+3}{7} = \frac{108}{7}$

Similarly,

$1\frac{23}{49} = \frac{49 x 1+23}{49} = \frac{49+23}{49} = \frac{72}{49}$

Now, we will replace the mixed fractions in the given question with the improper fractions we have obtained, we will get,

$15\frac{3}{7} ÷ 1\frac{23}{49} = \frac{108}{7} ÷ \frac{72}{49}$

Here, $\frac{72}{49}$ is the divisor. Therefore we will take its reciprocal. We will get $\frac{49}{72}$

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,

$\frac{108}{7} x \frac{49}{72}$

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,

$\frac{108 \:3}{7 \:1} x \frac{49 \:7}{72 \:2} = \frac{3 x 7}{2} = \frac{21}{2}$

**Hence, **$15\frac{3}{7} ÷ 1\frac{23}{49} = \frac{21}{2}$

**Solved Examples**

**Example 1** The product of two fractions is $20\frac{5}{7}$. If one of the fractions is $6\frac{2}{3}$, find the other.

**Solution **We have been given that the product of two fractions is $20\frac{5}{7}$. Also, one of the fractions is $6\frac{2}{3}$. We need to find the other fractions.

Now, we know that the product of two fractions, say and b can be written as a x b. if their product is c, we write the equation as a x b = c ………………. ( 1 )

Similarly, we have been given the product of two fractions as $20\frac{5}{7}$. Also, one of the fractions is $6\frac{2}{3}$.

We can see that both of the given fractions are in the form of mixed fractions. Therefore, we shall first convert these fractions into improper fractions. We will get –

$20\frac{5}{7} = \frac{20 x 7+5}{7} = \frac{140+5}{7} = \frac{145}{7}$

$6\frac{2}{3} = \frac{6 x 3 +2}{3} = \frac{18+2}{3} = \frac{20}{3}$

Now, we have been given that one of the fractions is $\frac{20}{3}$. Let the other number be “ b”. Therefore, substituting the given values in equation ( 1 ), we get,

$\frac{20}{3} x b = \frac{145}{7}$

Therefore,

b = $\frac{145}{7} ÷ \frac{20}{3}$

Here, $\frac{20}{3}$ is the divisor. Therefore we will take its reciprocal. We will get $\frac{3}{20}$.

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,

$\frac{145}{7} x \frac{3}{20}$

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,

$\frac{145 \:29}{7} x \frac{3}{20 \:4} = \frac{29 x 3}{7 x 4} = \frac{87}{28} = 3\frac{3}{28}$

**Hence, the other number is **$3\frac{3}{28}$

**Example 2** In mid-day meal scheme $\frac{5}{10}$ litres of milk is given to each student of a primary school. If 50 litres of milk is distributed every day in the school, how many students are there in the school?

**Solution** We have been given that in mid-day meal scheme $\frac{5}{10}$ litres of milk is given to each student of a primary school and 50 litres of milk is distributed every day in the school. Let us categorise this information first.

We have –

Litres of milk is given to each student of a primary school = $\frac{5}{10}$ litres

Litres of milk is distributed every day in the school = 50 litres

Now, we are required to find the number of students that are there in the school. Let the number of students in the school be “p”.

From the given information we can say that –

(Litres of milk is given to each student of a primary school ) x p = (Litres of milk is distributed every day in the school ) ……………….. ( 1 )

Now, substituting the given information in the equation ( 1 ) we will get,

p x $\frac{5}{10}$ = 50

Therefore,

$p = 50 ÷ \frac{5}{10}$

Here $\frac{5}{10}$ is the divisor. Therefore we will take its reciprocal. We will get $\frac{10}{5}$.

p = 50 x $\frac{10}{5}$

p = $\frac{50 x 10}{5} = \frac{50 x 10 \:2}{5 \:1} = 100$

**Hence, the number of students in the school is 100**

**Key Facts and Summary**

- A fraction is a number representing a part of a whole.
- Fractions that have the same denominator are called like fractions.
- Fractions with different denominators are called unlike fractions.
- Fractions with the numerator less than the denominator are called proper fractions.
- Fractions with the numerator either equal or greater than the denominator are called improper fractions
**.** - A combination of a whole number and a proper fraction is called a mixed fraction.
- 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.
- Two fractions are said to be the reciprocal or multiplicative inverse of each other if their product is 1.
- In order to find the reciprocal of a fraction, we need the swap its numerator and the denominator.
- 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 $\frac{c}{d}$.
- The process for division and proper fractions is the same. However, if we have mixed fractions, we will first need to convert the mixed fraction into an improper fraction and then only we can proceed ahead with the division of fractions.

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