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# Simplifying 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 the simplification of fractions.  But, before that, we must recall some basic terms related to fractions that are integral to understanding the process of simplifying them.

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 these fractions, the numerators 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$\frac{2}{3}$ is a mixed fraction.

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, $\frac{3}{5} = \frac{3 x 2}{5 x 2}$ = $\frac{6}{10}$. Here, $\frac{3}{5}$ and $\frac{6}{10}$ 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 understand how to simplify fractions.

## 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.

## How to Simplify Fractions

Let us now understand the steps involved in reducing a fraction to its simplest form. 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. If the fraction is in mixed form, we first convert it into an improper fraction.

Step 2 – Find the greatest common divisor (GCD) of the absolute values of the numerator and the denominator.

Step 3 – 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.

Let us understand the above steps through an example.

Example

Reduce $\frac{144}{180}$ to lowest terms

Solution

We have been given the fraction $\frac{144}{180}$ and we are required to reduce it to lowest terms. We can clearly see that both the numerators as well as the denominator are even numbers which means that they would certainly have some common factors. Therefore, our first steps are to find the common factors between them and eliminate them. Hence, in order to reduce it to the lowest terms, let us first find the Highest Common Factor ( H. C. F ) of both the numerator and the denominator. For finding the H. C. F of the two numbers, we will have to find their factors. Therefore,

The factors of 144 are – 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72 and 114. Similarly,

The factors of 180 are – 1, 2, 3, 4, 5, 6, 10, 12, 15, 18, 30, 36, 45, 60, 90 and 180

From above we can see that the common factors of 144 and 180 are 1, 2, 3, 4, 6, 12, 18 and 36. This means that the H. C. F of 144 and 180 is 36.

Now, we shall divide both the numerator as well as the denominator by 36 to get

$\frac{144}{180} = \frac{144 ÷36}{180 ÷36} = 45$

Hence, the simplified form of $\frac{144}{180}$ will be 45

### Alternative Method

The above process of finding factors first to find the H. C F and then reducing the fraction can be quite cumbersome for large numbers. Therefore, an alternative to this method is to start reducing the fractions by dividing both the numerator as well as the denominator by the common factors as we spot them.

Let us reduce the fraction $\frac{144}{180}$ through this method. As stated above, we can clearly see that both the numerators as well as the denominator are even numbers which means that they would certainly have some common factors. The first common factor that is easily identifiable is 2. Therefore, let us divide both the numerator as well as the denominator of the given fraction by 2 to get

$\frac{144}{180} = \frac{144 ÷2}{180 ÷2} = \frac{72}{90}$

Again, we can see that still both the numerators as well as the denominator are even numbers which means that they would certainly have some common factors. Let us again divide both the numerator as well as the denominator of the given fraction by 2 to get

$\frac{72}{90} = \frac{72 ÷2}{90 ÷2} = \frac{36}{45}$

Now, we can see that while the numerator is an even number, we have an odd number at the denominator which means that the numerator and the denominator do not have the factor 2 in common anymore. Let us check for the number 3. We can see that both 36 and 45 have 3 as a common factor. Therefore, this fraction can further be reduced by dividing both the numerator as well as the denominator by 3. We will now get,

$\frac{36}{45} = \frac{36 ÷3}{45 ÷3} = \frac{12}{15}$

Again, we can see that while the numerator is an even number, we have an odd number at the denominator. Also, we know that both 12 and 15 have 3 as a common factor. Therefore, this fraction can further be reduced by dividing both the numerator as well as the denominator by 3. We will now get,

$\frac{12}{15} = \frac{12 ÷3}{15 ÷3} = \frac{4}{5}$

Now the fraction obtained is $\frac{4}{5}$ where there no common factors left between the numerator and the denominator. Hence, the simplified form of is $\frac{4}{5}$ which is the same answer that we got using the H. C F method.

## Important points regarding simplification of fractions

Given below are some important points that need to be considered while simplification of fractions or reducing them to their lowest form.

1. The numerator and denominator of a fraction are called its terms. If we simplify a fraction, then we are reducing it to the lowest terms.
2. Reducing a fraction to the lowest terms will not change its value; the reduced fraction will be an equivalent fraction. All we need to do is divide the numerator and the denominator by the same nonzero whole number.
3. If the fraction is in mixed form, we first convert it into an improper fraction.
4. The result of all four operations on fractions, i.e. addition, subtraction, multiplication and division need to be checked whether the result obtained is in simplified form or not. If not, the fraction should be reduced to its simplest form.

## Solved Examples

Example 1 Simplify the fraction { 18 + ( 2 ½ + 4 / 5 ) } of 1 / 1000

Solution We have been given the fraction { 18 + ( 2 ½ + 4 / 5 ) } of 1 / 1000and we are required to simplify it.

We can see that we have a mixed term in the given fraction. Therefore, let us first convert it into an improper fraction.

Now, we know that 2 ½ = 5 / 2. Substituting this value in the given fraction, we have,

{ 18 + ( 2 ½ + 4 / 5 ) } of 1 / 1000

= { 18 + ( 5 / 2 + 4 / 5 ) } of 1 / 1000

Now, let us solve the innermost bracket of the given fraction. We can see that there are two fractions in the bracket which are unlike terms. We know that in order to add two unlike fractions, we will first have to find their L. C. M. therefore, we have,

5 / 2 + 4 / 5 = [ (5 x 5 ) + ( 4 x 2 ) ] / 10 = ( 25 + 8 ) / 10  = 33 / 10

Substituting this value in the given fraction, we have,

{ 18 + ( 5 / 2 + 4 / 5 ) } of 1 / 1000

= { 18 + 33 / 10 } of 1 / 1000

Again, let us solve the remaining bracket of the given fraction. We can see that there are two fractions in the bracket which are unlike terms. Again, finding their L. C. M, we will get,

18 + 33 / 10 = ( ( 18 x 10 ) + 33 ) / 10 = ( 180 + 33 ) / 10 = 213 / 10

Substituting this value in the given fraction, we have,

{ 18 + 33 / 10 } of 1 / 1000

= 213 / 10 of 1 / 1000

= ( 213 / 10 ) x  ( 1 / 1000 )

= 213 / 10000 or 0.0213

Hence, the simplified form of the fraction { 18 + ( 2 ½ + 4 / 5 ) } of 1 / 1000 will be 213 / 10000 or 0.0213

Example 2 Find the value of 15$\frac{3}{7}$ ÷ 1$\frac{23}{49}$

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$\frac{3}{7} ÷ 1\frac{23}{49} = \frac{21}{2}$

Example 3 Find the value of $\frac{18}{20} – \frac{21}{25}$ and simplify the result obtained.

Solution 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 4 Simplify $\frac{126}{90}$ to its lowest form

Solution We have been given the fraction $\frac{126}{90}$ and we are required to simplify it. We can clearly see that both the numerators as well as the denominator are even numbers which means that they would certainly have some common factors. The first common factor that is easily identifiable is 2. Therefore, let us divide both the numerator as well as the denominator of the given fraction by 2 to get,

$\frac{126}{90} = \frac{126÷2}{90 ÷2} = \frac{63}{45}$

Now, we can see that both the numerator and the denominator have 3 in common. Therefore, let us divide them by 3 to get,

$\frac{63}{45} = \frac{63 ÷3}{45 ÷3} = \frac{21}{15}$

Again, we can see that both the numerator and the denominator have 3 in common. Therefore, let us divide them by 3 to get,

$\frac{21}{15} = \frac{21 ÷3}{15 ÷3} = \frac{7}{5}$

Hence, the simplified form of the fraction $\frac{126}{90}$ is $\frac{7}{5}$

## Key Facts and Summary

1. Fractions that have the same denominator are called like fractions.
2. Fractions with different denominators are called unlike fractions.
3. Fractions with the numerator less than the denominator are called proper fractions.
4. Fractions with the numerator either equal or greater than the denominator are called improper fractions.
5. A combination of a whole number and a proper fraction is called a mixed fraction.
6. Fractions having the same value are called equivalent fractions.
7. 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.
8. In order to reduce a fraction to its lowest terms, we divide the numerator and the denominator by their H. C. F.
9. The numerator and denominator of a fraction are called its terms. If we simplify a fraction, then we are reducing it to the lowest terms.
10. Reducing a fraction to the lowest terms will not change its value; the reduced fraction will be an equivalent fraction. All we need to do is divide the numerator and the denominator by the same nonzero whole number.