A Fractions can be defined as any part of a whole number, a small part or even a piece or amount. A fraction is made up of a numerator and a denominator, where the numerator would tell us how many parts of the whole there are, a while the denominator would give us an idea of how many parts are in the whole.

When and if the numerators are found to be smaller than the denominator then the fraction is considered to be a proper fraction.When and if the numerators are found to be greater than the denominators then the fraction is labelled as an improper fraction.When and if a fraction is accompanying a whole number then it is considered as a mixed number. In the recent past, the usage of fraction was done to describe shares of objects or group objects, but now these fractions have been replaced by decimals and the calculations are often completed with computers or calculators.

## What is a Fraction?

A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects and the parts have to be equal.

The general form of a fraction is a/b , where a and b are integers and b ? 0. Here a is called the numerator and b is called the denominator.

In the following figure, 4 out of 6 sectors are shaded. Hence the shaded region represents $\frac{4}{6}$.

### Examples on Fractions

In the figure (i) $\frac{3}{7}$ is obtained when we divide a whole into 7 equal parts and take 3 parts of it.

In figure (ii) $\frac{1}{8}$ we divide a whole into 8 equal parts and take 1 part out of it.

Now, $\frac{5}{12}$ is a fraction and we read it as "five-twelfths", where 12 is number of parts into which a whole has been divided and 5 is number of equal parts which have been taken out.
Here 5 is called the Numerator and 12 is called the Denominator.

## Types of Fractions

Depending on the relative values of the numerator and the denominator, fractions are classified as:

a) Proper fractions
b) Improper fractions
c) Mixed fractions

Proper fractions

Consider the fraction $\frac{(1)}{(2)}$ , $\frac{(3)}{(8)}$ , $\frac{(2)}{(9)}$ and $\frac{(10)}{(35)}$ . We can see that in all these fractions, the numerator is lesser than the denominator. Hence the value of these fractions is lesser than 1. Such fractions are called proper fractions.

Improper fractions

Unlike proper fractions, the numerator of an improper fraction will be greater than its denominator, and so the value of these fractions will be more than 1. Examples of improper fractions are $\frac{(4)}{(3)}$ , $\frac{(5)}{(2)}$ , $\frac{(29)}{(5)}$ , $\frac{(101)}{(11)}$ etc.

Mixed fractions

These types of fractions are a combination of a whole number and a part; that is a whole number and a proper fraction. Some examples are 5$\frac{(1)}{(2)}$ , 10$\frac{(3)}{(8)}$ , 9$\frac{(11)}{(13)}$ etc.

Adding fractions is an important concept in fractions. To add two fractions, we have to make sure the denominator is the same for both the fractions and then we have to add the numerator with out disturbing the denominator.

Given below are some examples for adding fractions

Find the sum $\frac{1}{5}$ + $\frac{2}{5}$

Solution:

Step 1: Add the numerators: 1 + 2 = 3
Step 2: Retain the common denominator, 5
Step 3: Sum of the fractions = $\frac{(Sum\ of\ the\ Numerators)}{(Common\ denominator)}$ =$\frac{3}{5}$

Based on this the addition of fractions is divided into two types:
• Adding fractions with like denominators
• Adding fractions with unlike denominators

## Subtracting Fractions

Subtracting fractions is an important concept. To subtract two fractions, we have to make sure the denominator is same for both the fractions and then we have to subtract the numerator with out disturbing the denominator.

Example for Subtracting a Fraction

Example for subtracting fraction:

Find $\frac{7}{11}$ - $\frac{3}{11}$

Solution:

$\frac{7}{11}$ - $\frac{3}{11}$ = $\frac{7-3}{11}$ = $\frac{4}{11}$

Based on this the subtraction of fractions divided into two types,
• Subtracting fractions with like denominators
• Subtracting fractions with unlike denominators

## Subtracting Negative Fractions

According to rule, a subtraction signs will applies to fraction as well. To subtract a negative fraction the denominator must be same. After getting common denominator we subtract numerators.

### Examples on Subtracting Negative Fractions

Below you could see example based up on subtracting negative fractions

Example: $-\frac{6}{4}$ - $(-\frac{3}{5})$

Solution: Given $-\frac{6}{4}$ - $(-\frac{3}{5}$)

Step 1: Denominators are not same, so find the LCM of the denominator, LCM of 4,5 =20.
Step 2: Express the fractions as equivalent fractions with the denominator as the LCM.

$-\frac{6}{4}$ = $-\frac{6\times5}{4\times5}$ = $-\frac{30}{20}$

$(-\frac{3}{5})$ = $(-\frac{3\times4}{5\times4})$ = $(-\frac{12}{20})$

Step 3: Now subtract negative fractions

$-\frac{30}{20}$ - $(-\frac{12}{20})$ = $-\frac{18}{20}$

Step 4: Simplify the fraction

$-\frac{18}{20}$ = $-\frac{9}{20}$

## Subtracting Fractions with Like Denominators

Subtract fractions with like denominators. We subtract the numerator with out disturbing denominators in this part.

In order to find the difference between two fractions with common denominators, use the following steps:
• Step 1: Find the difference between the numerators.
• Step 2: Retain the common denominator.
• Step 3: Difference between the fractions = $\frac{Difference\ between\ the\ Numerators}{The\ Common\ Denominator}$

### Examples of Subtracting Fractions with Like Denominators

Given are some examples based on subtracting fractions with like denominators

Find $\frac{3}{5}$ - $\frac{2}{5}$

Solution:

Step 1: Difference between the numerators = 3 -2 = 1.
Step 2: Retain the common denominator, 5.
Step 3: Difference between the fractions = $\frac{Difference\ between\ the\ Numerators}{The\ Common\ Denominator}$ = $\frac{1}{5}$

## Subtracting Fractions with Unlike Denominators

In order to subtract a fraction from another fraction with different denominators, we first find equivalent fractions with the same denominator. The common denominator is the least common multiple (LCM) of the denominators.

The difference of fractions with different denominators can be obtained through the following steps:
• Step 1: Find the LCM of the denominators
• Step 2: Express the fractions as equivalent fractions with the denominator as the LCM
• Step 3: Subtract the smaller numerator from the bigger one
• Step 4: Retain the common denominator (LCM)
• Step 5: Difference of fractions = Difference of numerators from step 3 / Common denominator (LCM)

### Examples for Subtracting Fractions with Unlike Denominators

Given are some examples based on subtracting fractions with unlike denominators

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

Solution: We have to find; $\frac{5}{6}$ - $\frac{3}{4}$

Step 1: Find the LCM of the denominator, LCM of 4,6 = 12
Step 2: Express the fractions as equivalent fractions with the denominator as the LCM
$\frac{5}{6}$ =$\frac{5\times2}{6\times2}$ = $\frac{10}{12}$ , and $\frac{3}{4}$ = $\frac{3\times3}{4\times3}$ = $\frac{9}{12}$
Step 3: Subtract the smaller numerator from the bigger one;
10-9 = 1
Step 4: Retain the common denominator, 12
Step 5: Difference of the fractions= Difference of numerators from step 3 / Common denominator (LCM) = $\frac{1}{12}$

2. Find $\frac{49}{8}$ - $\frac{11}{4}$

Solution:

The LCM of the denominators, 8 and 4 is 8

So we have $\frac{11}{4}$ = $\frac{11\times2}{4\times2}$ = $\frac{22}{8}$

Hence, $\frac{49}{8}$ - $\frac{11}{4}$ = $\frac{49}{8}$ - $\frac{22}{8}$

= $\frac{49-22}{8}$

= $\frac{27}{8}$

## Multiplying Fractions

In order to multiply two fractions, we multiply the numerators together and then multiply the denominators.

The product of two fractions can be obtained through the following steps.
Step 1: Multiply the numerators
Step 2: Multiply the denominators
Step 3: The product of the fractions = $\frac{Product\ of\ the\ Numerators}{Product\ of\ the\ Denominators}$
Step 4: Write the product in its simplest form
We can reduce the fractions into the simplest form by dividing both the numerator and the denominator by the greatest common factor.

### Examples for Multiplying Fractions

Below are some examples based on multiplying fractions

1. Find the product of $\frac{2}{5}$ and $\frac{3}{7}$

Solution:

Step 1: Multiply the numerators, 2 × 3=6
Step 2: Multiply the denominators, 5 × 7= 35
Step 3: The product of the factors = $\frac{Product\ of\ the\ Numerators}{Product\ of\ the\ Denominators}$
= $\frac{6}{35}$
Here we cannot reduce this fraction further as the greatest common factor of
the numerator and denominator is 1.

2. Find the product of $\frac{2}{7}$ and $\frac{1}{14}$

Solution:

Let us follow the steps to find the product,

$\frac{2}{7}$ × $\frac{1}{14}$ = $\frac{2\times1}{7\times14}$

= $\frac{2}{98}$

= $\frac{1}{49}$ , by dividing the numerator and denominator by 2

## Multiplying Fractions with Variables

When multiplying two or more fractions, the numerators are all multiplied together along with the denominators.

This is made possible by multiplying fractions that contains variables.

One method to carry out this operation is to reduce the multiplying fractions before the actual multiplication.

When we multiply a fraction, we have to look out for common numerator and denominator in the fractions and need not be in same fraction.

Example for multiplication of fraction with variables:

Let us multiply $\frac{10}{21}$ , $\frac{14}{35}$ , and $\frac{42}{32}$

As this would finally result into a huge value, so these are reduced further so that this operation becomes manageable.

The common factor for the numerator and the denominator work out and gives us a result of $\frac{1}{4}$ .

$\frac{10}{21}$ X $\frac{14}{35}$ X $\frac{42}{32}$ = $\frac{2\times2\times2}{32}$ = $\frac{1}{4}$

## Dividing Fractions

A fraction obtained by interchanging the numerator and denominator of a given fraction is called the Reciprocal of the fraction.

We can divide a fraction by another fraction through the following steps:
• Step 1: Find the reciprocal of the divisor.
• Step 2: Multiply the dividend by the fraction from step 1.
• Step 3: Write the reduced form of the fraction obtained.

Thus in general, we can write,

$\frac{a}{b}$ ÷ $\frac{c}{d}$ = $\frac{a}{b}$ × $\frac{d}{c}$ where a, b, c, and d are integers.

### Examples for Dividing Fractions

Some examples based on dividing fractions

1. Find $\frac{2}{15}$ ÷ $\frac{1}{3}$

Solution:

Step 1: Find the reciprocal of the divisor, reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$

Step 2: Multiply the dividend by the fraction from step 1, $\frac{2}{15}$ × $\frac{3}{1}$ = $\frac{6}{15}$

Step 3: Write the reduced form of the fraction obtained,

25$\frac{6}{15}$ = $\frac{2}{5}$, by removing the greatest common factor, 3, of the numerator and the denominator

Hence, $\frac{2}{15}$ ÷ $\frac{1}{3}$ = $\frac{2}{5}$

2. Divide $\frac{6}{17}$ by $\frac{24}{51}$

Solution:

$\frac{6}{17}$ ÷ $\frac{24}{51}$ = $\frac{6}{17}$ × $\frac{51}{24}$

= $\frac{3}{4}$ , by removing the greatest common factor, 6 × 17 , of the numerator and the denominator.

## Comparing Fractions

Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ can be compared as follows

Cross multiply as shown:

(i) If a $\geq$ d = b $\geq$ c, then $\frac{a}{b}$ = $\frac{c}{d}$

(ii) If a $\leq$ d = b $\leq$ c, then $\frac{a}{b}$ = $\frac{c}{d}$

(iii) If a = d = b = c, then $\frac{a}{b}$ = $\frac{c}{d}$

Note:

(i) If the numerator and the denominator of a fraction are both multiplied or divided by the same non-zero number, then the value of the fraction remains unchanged.

(ii) Every whole number is a fraction since every whole number can be written as $\frac{a}{1}$

(iii) A given fraction and the fraction obtained by multiplying (or dividing) both of the numerator and the denominator by the same non-zero number are called equivalent fractions.

### Examples for Comparing Fractions

Given below are some examples on comparing fractions:

1. Which of the given fractions $\frac{3}{4}$ , $\frac{5}{7}$ is greater ?

Solution: Cross multiply the given fractions as directed above

We get: 3 x 7 = 21 and 4 x 5 = 20

Clearly, 21 > 20

Therefore, $\frac{3}{4}$ is greater.

2. Compare the fractions $\frac{7}{8}$ , $\frac{8}{9}$ ?

Solution: Cross multiply the given fractions as directed above $\frac{7}{8}$ , $\frac{8}{9}$

We get: 9 x 7 = 63 and 8 x 8 = 64

Clearly, 64 > 63

Therefore $\frac{7}{8}$ = $\frac{8}{9}$

## Ordering Fractions

Arranging of fractions either in ascending (increasing) order or descending (decreasing) order is called the ordering of fractions.

If we pick up two like fractions the one with the greatest numerator is greater and the one with the least denominator is the smallest.

### Examples for Ordering Fractions

Given below are some examples on Ordering Fractions

1. Arrange the given fractions in ascending order of magnitude.

$\frac{7}{12}$ , $\frac{5}{12}$ , $\frac{17}{12}$ , $\frac{3}{12}$ and $\frac{13}{12}$

Solution:

The given fractions are $\frac{7}{12}$ , $\frac{5}{12}$ , $\frac{17}{12}$ , $\frac{3}{12}$ and $\frac{13}{12}$

Since the denominators of the fractions are the same, they are called like fractions

To arrange the fractions in the ascending order we see the values of the numerators of the fractions from the smallest to the greatest.

Therefore, the ascending order is $\frac{3}{12}$ , $\frac{5}{12}$ , $\frac{7}{12}$ , $\frac{13}{12}$ and $\frac{17}{12}$

2. Arrange the given fractions in descending order of magnitude.

$\frac{28}{140}$ , $\frac{60}{140}$ , $\frac{98}{140}$ , $\frac{65}{140}$ and $\frac{13}{140}$

Solution:

The given fractions are $\frac{28}{140}$ , $\frac{60}{140}$ , $\frac{98}{140}$ , $\frac{65}{140}$ and $\frac{13}{140}$

Since the denominators of the fractions are the same, they are called like fractions
To arrange the fractions in a descending order we see the values of the numerators of the fractions from the greatest to the smallest

Therefore, the ascending order is $\frac{13}{140}$ , $\frac{28}{140}$ , $\frac{60}{140}$ , $\frac{65}{140}$ and $\frac{98}{140}$

## Fraction to Decimal

Fractions can be represented as decimals. The accurate and the sure shot way of converting a fraction into decimal is by long form division. The quotient gives the required decimal.

Another method of converting fractions to decimals

For some fractions we can find a number such that if the denominator is multiplied by this number it will result in a power of 10. Such fractions can be converted to decimals by the following steps:

Step 1: Identify a number such that if the denominator of the fraction is multiplied by this number will result in a power of 10.

Step 2: Multiply the numerator and the denominator of the fraction with the same number as in Step 1.

Step 3: Place the decimal point such that there are the same number of digits to the right of the decimal point as in the denominator.

### Examples for converting fraction into decimal

Below are some examples based on converting fractions into decimals

1. Convert $\frac{2}{3}$ to a decimal number.

Solution:

We know that $\frac{2}{5}$ = 2 ÷ 5

Let us perform the long form division.

Hence, $\frac{2}{5}$ = 0.4

2. Convert $\frac{7}{8}$ to a decimal.

Solution:

We know that $\frac{7}{8}$ = 7 ÷ 8

Hence , $\frac{7}{8}$ = 0.875

In the above examples, the division resulted in zero remainder after a finite number of steps. But here are cases where the remainder does not become zero. This results in non-terminating decimals.

3. Convert $\frac{1}{3}$ to a decimal.

Solution:

Performing long term division we get,

The remainder does not become zero.

Hence , $\frac{1}{3}$ = 0.333...

The decimal does not terminate. Here 3 repeats in the decimal. This is represented as $\frac{1}{3}$ = 0.3

4. Convert $\frac{3}{5}$ to a decimal.

Solution:

Step 1: Identify a number such that if the denominator of the fraction is multiplied by this number it will result in a power of 10.

We can see that if we multiply the denominator by 2 we get a power of 10 in the denominator.

Step 2: Multiply the numerator and the denominator of the fraction with the same number in the Step 1.

= $\frac{3}{5}$ = $\frac{(3\times2)}{(5\times2)}$

= $\frac{6}{10}$

Step 3: Place the decimal point such that there are the same number of digits to the right of the decimal point as in the denominator.

We get, = $\frac{3}{5}$ = $\frac{6}{10}$ = 0.6

## Fraction to Percent

Percent literally means out of 100. A percent can be represented as a fraction and as a decimal number. We have learned how to write fractions as decimal numbers. The accurate and the sure shot way of converting a fraction to a decimal is by long form division. The quotient gives the required decimal. We have to multiply this decimal number by 100 and place the percent symbol with the answer.

If the denominator of the fraction is 100, we can write the equivalent percent by removing the denominator and place the percent symbol with the numerator.

Consider the following diagram

In this diagram, 3 out of 4 boxes are shaded. We write this as $\frac{3}{4}$ of the diagram is shaded.

We know $\frac{3}{4}$ = $\frac{(3\times25)}{(4\times25)}$ , writing an equivalent fraction by multiplying the numerator and denominator by 25

Here we made the denominator 100, by multiplying it by 25

Hence we get, $\frac{3}{4}$ = $\frac{75}{100}$

So we can say $\frac{75}{100}$ of the diagram is shaded. In other words 75% of the diagram is shaded.

### Examples on Fraction to Percent

Given are some examples on converting fraction to percent

1. Write the fraction; $\frac{4}{5}$ as a percent

Solution:

Performing long division, we get,

$\frac{4}{5}$ = 0. 8

Multiplying this number by 100, 0.8 100= 80.0

Hence 0.8 =80%

$\frac{4}{5}$ = 80%

Alternate method:

We make the denominator of the given fraction as 100.

$\frac{4}{5}$ = $\frac{(4\times20)}{(5\times20)}$ , multiplying the numerator and denominator by 20

$\frac{4}{5}$ = $\frac{80}{100}$

= 80%

2. Convert the fraction; $\frac{7}{8}$

Solution

By performing long division, we get,

$\frac{7}{8}$ = 0.875

0.875 x 100 = 87.5

Therefore, $\frac{7}{8}$ = 87.5%

## Simplifying Fractions

Simplifying fractions could also result in reducing the fractions to its lowest possible equivalent.

The steps for simplifying fractions

Step 1: Start with the lowest term which can divide both the numerator and the denominator until you cannot proceed further.

Step 2: Divide the fraction term by the greatest common factor.

### Examples for Simplifying Fractions

Given below are some examples on simplifying fractions

1. Simplify the fraction $\frac{4}{14}$

Solution :
Given fraction is $\frac{4}{14}$

4 = 2 × 2
14 = 2 × 7
Greatest common factor of 4 and 14 is 2.

$\frac{4}{14}$ × $\frac{2}{2}$ = $\frac{2}{7}$

The simplest form of the given fraction is $\frac{2}{7}$

2. Simplify the fraction $\frac{55}{66}$

Solution :
Given fraction is $\frac{55}{66}$

55 = 11 × 5
66 = 11 × 6
The greatest common factor of 55 and 66 is 11.

$\frac{55}{66}$ × $\frac{11}{11}$ = $\frac{5}{6}$

The simplest form of the given fraction is $\frac{5}{6}$