# Chapter 7 - Fractions

## Introduction to Fractions

A fraction is represented with two numbers written one below the other, separated by a divider.

### Fractions, Numerator, Denominator

When a whole is divided into equal parts, the parts are represented using fractions. In a fraction,

• the number written at the top is known as the numerator, and
• the number written at the bottom is known as the denominator.

The total number of equal parts that a whole is divided into is shown by the denominator, while the number of parts is selected is shown by the numerator.

For example, if a shape is divided into 5 equal parts, of which 3 are selected, then the selected portion is ${3}/{5}$.

### Problem:

In a four-lane highway, if there are 3 lanes for cars and 1 for bikes, what are the fractions representing the cars and the bikes?

### Solution:

There are cars in 3 lanes. So, the fraction representing the cars = ${3}/{4}$ There is only 1 lane for bikes. So, the fraction representing the bikes = ${1}/{4}$

### Problem:

A pizza is divided into 6 equal parts. The pizza is equally divided between 2 friends. What is the fraction of pizza does each friend get? If the pizza is equally divided among 3 friends, what fraction of pizza does each friend get?

### Solution:

Total number of parts = 6 6 parts of pizza is equally divided between 2 friends. Then, each of them gets ${6}/{2}$ = 3 parts So the fraction of the pizza that each friend gets = ${3}/{6}$ If the pizza is divided among 3 friends, then each of them would get 2 parts. The fraction of the pizza that each friend gets = ${2}/{6}$

### Problem:

In a school, there was a 100 m race. Jack, Sam, and Jill ran the race. Jack ran 50 m, Sam ran 90 m, and Jill completed to win the race.

• What is the fraction of distance run by each of them?
• What is the fraction of the students who completed the race?
• What is the fraction of students who did NOT complete the race?

### Solution:

Total distance = 100 m Fraction of the distance run by Jack $$= {Distance\:run\:by\:Jack}/{Total\:distance} = {50}/{100} = {1}/{2}$$ Fraction of the distance run by Sam $$= {Distance\:run\:by\:Sam}/{Total\:distance} = {90}/{100} = {9}/{10}$$ Fraction of the distance run by Jill $$= {Distance\:run\:by\:Jill}/{Total\:distance} = {100}/{100} = {1}/{1}$$ Among the three students, only Jill completed the race. So, the fraction of students who completed the race = ${1}/{3}$ And, the fraction of students who did not complete the race = ${2}/{3}$

## Types of Fractions

Fractions can be of two types depending on whether the numerator is greater than the denominator or not.

### Proper Fractions

Fractions in which the numerator is less than the denominator are called proper fractions. These fractions have value less than 1.

For example, ${2}/{3}$, ${6}/{23}$, ${34}/{213}$, and so on.

### Improper Fractions

Fractions in which the numerator is greater than the denominator are called improper fractions. These fractions have value greater than 1.

For example, ${3}/{2}$, ${21}/{12}$, ${544}/{213}$, and so on.

### Mixed Fractions

Since improper fractions are greater than 1, they can be written as a combination of a whole number and a fraction. For example,

$${11}/{4} = 2 \: and \: {3}/{4} = 2{3}/{4}$$

Fractions in which the numerator and the denominator are equal represent 1 whole.

Such fractions (or numbers), which have a whole part as well as a fractional part, are known as mixed fractions (or mixed numbers).

Mixed fractions can be converted into improper fractions and vice versa.

## Conversion of Fractions

Improper fractions and mixed fractions are two different representations of the same quantity.

### Mixed Fractions

Mixed fractions (or mixed numbers) are another way of representing improper fractions. They are numbers that consist of a whole number and a proper fraction.

### Improper Fraction to Mixed Fraction

To convert an improper fraction into a mixed fraction, the division operation is used.

• The numerator is first divided by the denominator.
• The quotient becomes the number of wholes.
• The remainder becomes the numerator of the fraction and the denominator stays the same.

### Problem:

Convert ${65}/{7}$ into a mixed fraction.

### Solution:

${65}/{7}$ is an improper fraction as the numerator is greater than the denominator. Divide 65 by 7, 65 ÷ 7 = 9 (remainder 2) We can write 65 as, 65 = (7 × 9) + 2 So, the mixed fraction form of ${65}/{7}$ is 9${2}/{7}$.

### Mixed Fraction to Improper Fraction

To convert a mixed fraction into an improper fraction,

• First, multiply the whole part of the number with the denominator.
• Add the product to the numerator. This is the numerator of the improper fraction.
• The denominator remains unchanged.

### Problem:

Convert 4${7}/{9}$ into an improper fraction.

### Solution:

Multiply the whole part with the denominator. 4 × 9 = 36 Add the product with the numerator. 36 + 7 = 43 The denominator remains unchanged. So, the improper fraction of 4${7}/{9}$ = ${43}/{9}$.

## Representation of Fractions on Number Line

Just like whole numbers, fractions can also be represented on the number line.

### Proper Fraction on the Number Line

• Since a proper fraction is a part of a whole, the representation requires dividing the whole into equal parts.
• The number of parts depends on the denominator of the fraction.
• So the gap between 0 and 1 is divided into that many equal parts as the denominator.
• Then the numerator of the proper fraction is identified.
• The part that equals the value of the numerator equals the proper fraction.

### Problem:

Represent the proper fraction ${3}/{7}$ on the number line.

### Solution:

The denominator of the fraction = 7 So, divide the gap between 0 and 1 into seven equal parts. The part that equals the value of the numerator which is 3 equals the proper fraction. So, the fraction ${3}/{7}$ can be shown as follows: ### Improper Fraction on the Number Line

Mixed fractions are numbers that lie between two whole numbers. In other words, it is a whole number and a part of one whole.

To represent an improper fraction,

• First convert the improper fraction into a mixed fraction.
• In the mixed numbers, the whole part represents the whole number after which the number lies.

Example: Represent the fraction 3${4}/{7}$ on the number line.

Solution:

3${4}/{7}$ would lie in between the whole numbers 3 and 4. Next, the gap is divided into seven equal parts (same as the denominator): They are numbered as: So, 3${4}/{7}$ will be 3 wholes and the 4th part of the whole between 3 and 4. ## Equivalent Fractions

Equivalent fractions look different but have the same value or represent the same quantity.

For example, ${2}/{4}$ and ${3}/{6}$ represent the same point on the number line. They are equivalent fractions.

Similarly, ${3}/{7}$, ${6}/{14}$, ${9}/{21}$... are equivalent fractions and all of them represent the same quantity ${3}/{7}$.

### Finding Equivalent Fractions

To find the equivalent fractions of a given fraction, the numerator and the denominator of the original fraction are multiplied or divided by the same number.

Example: Find the equivalent fractions of ${1}/{4}$

Solution:

Some equivalent forms of ${1}/{4}$ are:

• ${1 × 2}/{4 × 2}$ = ${2}/{8}$
• ${1 × 3}/{4 × 3}$ = ${3}/{12}$
• ${1 × 4}/{4 × 4}$ = ${4}/{16}$
• ${1 × 5}/{4 × 5}$ = ${5}/{20}$

Example: Find the equivalent fractions of ${3}/{8}$

Solution:

Some equivalent forms of ${3}/{8}$ are:

• ${3 × 2}/{8 × 2}$ = ${6}/{16}$
• ${3 × 3}/{8 × 3}$ = ${9}/{24}$
• ${3 × 4}/{8 × 4}$ = ${12}/{32}$
• ${3 × 5}/{8 × 5}$ = ${15}/{40}$

## Simplification of Fractions

A fraction ${6}/{12}$ can be written in large numbers as ${100}/{200}$ or in small numbers as ${1}/{2}$. Simplification of fractions is the process of finding the simplest equivalent form of a fraction.

The simplest form of a fraction is defined as the fraction in which the numerator and denominator have no common factor other than 1. For example, ${2}/{3}$, ${5}/{9}$, ${23}/{97}$, etc.

### First Method

To find the simplest form of a fraction, its equivalent forms are found by dividing the numerator and the denominator by the same number.

Example: Simplify the fraction ${9}/{36}$.

Solution:

Since 9 and 36 have a common factor, 3, one simplification would be:

$${9}/{36} = {9}/{3} \: / \: {36}/{3} = {3}/{12}$$

3 and 12 have a common factor, 3, so there will be another simplification:

$${3}/{12} = {3}/{3} \: / \: {12}/{3} = {1}/{4}$$

Since 1 and 4 do not have any common factor except 1, this is the simplest form of the original fraction.

### Second Method

We can find the simplest form of a fraction is by dividing the numerator and the denominator by the highest common factor (HCF) of both the numbers.

Example: Simplify the fraction ${9}/{36}$

Solution:

Let us find the HCF of the numbers 9 and 36. Hence the HCF of 9 and 36 is 9.

Divide the numerator and the denominator by the HCF,

$${9}/{36} = {9}/{9} \: / \: {36}/{9} = {1}/{4}$$

${1}/{4}$ is the fraction in the simplest form.

## Like Fractions and Unlike Fractions

### Like Fractions

Fractions that have the same denominator are called Like Fractions. For example,

$${2}/{7}, {5}/{7}, {14}/{7}, {236}/{7}$$

are like fractions because they have the same denominator '7'.

### Unlike Fractions

Fractions that have different denominators are called Unlike Fractions. For example,

$${23}/{3}, {322}/{12}, {56}/{531}, {12}/{4}$$

are unlike fractions as each fraction has a different denominator.

Example: A cake is cut into 20 equal pieces in a birthday party. Three friends get 5 pieces, 2 pieces, and 1 piece respectively. What fractions of the cake did the friends get?

Solution:

The fractions of the cake the friends got were ${5}/{20}$, ${2}/{20}$ and ${1}/{20}$.

These are like fractions.

Example: If the fractions in previous example, i.e., ${5}/{20}$, ${2}/{20}$ and ${1}/{20}$ are reduced to their simplest form, one gets ${1}/{4}$, ${1}/{10}$ and ${1}/{20}$. What type of fractions are these?

Solution:

The denominators of the fractions ${1}/{4}$, ${1}/{10}$ and ${1}/{20}$ are different.

These are unlike fractions.

Example: Categorize the following fractions as like and unlike fractions.

$${1}/{2}, {7}/{4}, {3}/{2}, {9}/{11}, {5}/{2}, {13}/{6}$$

Solution:

Of the given fractions,

${1}/{2}$, ${3}/{2}$, ${5}/{2}$ are like fractions, as they have the same denominator 2.

${7}/{4}$, ${9}/{11}$, ${13}/{6}$ are unlike fractions, as they have different denominators.

## Comparison of Fractions

### Comparison of Like Fractions

Comparison of fractions is easy in case of like fractions as they have the same denominator. To compare two like fractions, just compare their numerators.

Example: Compare the fractions ${2}/{7}$ and ${3}/{7}$

Solution:

The fractions ${2}/{7}$ and ${3}/{7}$ have the same denominator. So, since 2 < 3, we can say that ${2}/{7}$ < ${3}/{7}$.

### Comparison of Unlike Fractions

To compare two Unlike Fractions,

• First convert the Unlike Fractions to their Like equivalent fractions with the same denominator.
• Then, compare the numerators of the equivalent fractions.

The denominators of the fractions are converted to their LCM by multiplication/division.

The respective numerators are multiplied by the same numbers, thus resulting in equivalent fractions with the same denominator.

Example: Compare the fractions ${2}/{3}$ and ${5}/{7}$

Solution:

We have two unlike fractions: ${2}/{3}$ and ${5}/{7}$

Let's convert them to like fractions using equivalent fractions.

The LCM of the two denominators, 3 and 7 = 21.

So, the equivalent forms will have 21 as the denominator.

Converting the first fraction:

$${2 × 7}/{3 × 7} = {14}/{21}$$

Converting the second fraction:

$${5 × 3}/{7 × 3} = {15}/{21}$$

Now, compare the numerators of the equivalent like fractions.

Since 14 < 15,

$${14}/{21} < {15}/{21} \: or \: {2}/{3} < {5}/{7}$$

Example: Compare ${4}/{5}$ and ${16}/{20}$

Solution:

We have two unlike fractions: ${4}/{5}$ and ${16}/{20}$

LCM of denominators, 5 and 20 = 20

So, the equivalent forms will have 20 as the denominator.

Converting the first fraction:

$${4 × 4}/{5 × 4} = {16}/{20}$$

Converting the second fraction:

$${16 × 1}/{20 × 1} = {16}/{20}$$

Now, compare the numerators of the equivalent like fractions.

Both the numbers have the same numerators and denominators, therefore

$${4}/{5} = {16}/{20}$$

## Addition and Subtraction of Fractions

### Add and Subtract Like Fractions

Addition of fractions is different from addition of integers. While adding or subtracting like fractions, the denominator stays the same and the numerators get added or subtracted.

Example: Add the fractions ${3}/{16}$ + ${7}/{16}$

Solution: While adding like fractions, just add the numerators, keeping the denominator unchanged.

$${3}/{16} + {7}/{16} = {3 + 7}/{16} = {10}/{16}$$

Example: Subtract the fractions ${8}/{19}$ − ${7}/{19}$

Solution: While subtracting like fractions, just subtract the numerators, keeping the denominator unchanged.

$${8}/{19} − {7}/{19} = {8 − 7}/{19} = {1}/{19}$$

### Add and Subtract Unlike Fractions

While adding or subtracting unlike fractions,

• they are first converted into like fractions,
• then the numerators are added or subtracted together,
• the denominator stays the same.

Example: Add ${3}/{8}$ + ${7}/{6}$

Solution:

We have two unlike fractions,

$${3}/{8} \: and \: {7}/{6}$$

Let's first convert them to like fractions.

LCM of denominators, 8 and 6 = 24

So, the common denominator will be 24.

Equivalent like fractions of ${3}/{8}$ and ${7}/{6}$ are:

$${3}/{8} = {(3 × 3)}/{(8 × 3)} = {9}/{24}$$

$${7}/{6} = {(7 × 4)}/{(6 × 4)} = {28}/{24}$$

Add the numerators together, keeping the denominator unchanged.

$${3}/{8} + {7}/{6} = {9}/{24} + {28}/{24} = {37}/{24}$$

Example: Subtract ${7}/{6}$ − ${3}/{8}$

Solution:

LCM of 6 and 8 = 24

The common denominator will be 24

Subtracting their equivalent like fractions,

$${7}/{6} − {3}/{8} = {(7 × 4)}/{(6 × 4)} − {(3 × 3)}/{(8 × 3)}$$

$$= {28}/{24} − {9}/{24} = {28 − 9}/{24}$$

$$= {19}/{24}$$

## Special Cases of Addition and Subtraction

### Whole Number Plus Proper Fraction

Any whole number divided by 1 is that number itself. For example,

$${12}/{1} = 12 \: and \: {4}/{1} = 4$$

So, any whole number can be expressed as a fraction by placing 1 as its denominator.

Example: Add 9 + ${2}/{7}$

Solution:

$$9 + {2}/{7} = {9}/{1} + {2}/{7}$$

But these are unlike fractions.

Let's convert them to like fractions.

$${9}/{1} = {9 × 7}/{1 × 7} = {63}/{7}$$

Similarly,

$${2}/{7} = {2 × 1}/{7 × 1} = {2}/{7}$$

$${63}/{7} + {2}/{7} = {63 + 2}/{7} = {65}/{7}$$

Converting the improper fraction into mixed fraction,

$${65}/{7} = 9{2}/{7}$$

### Alternate Method

To add a whole number and a proper fraction, remove the positive sign in the question and write it as a mixed fraction. For example,

$$9 + {2}/{7} = 9{2}/{7}$$

$$5 + {3}/{11} = 5{3}/{11}$$

### Whole Number Minus Proper Fraction

The whole number is expressed as a fraction with denominator 1.

The unlike fractions are converted to like fractions and then subtracted to get the result.

Example: Add 9 − ${2}/{7}$

Solution:

$$9 − {2}/{7} = {9}/{1} − {2}/{7}$$

The like fractions of ${9}/{1}$ and ${2}/{7}$ are ${63}/{7}$ and ${2}/{7}$

Subtracting the like fractions,

$${63}/{7} − {2}/{7} = {63 − 2}/{7} = {61}/{7}$$

Converting the improper fraction into mixed fraction,

$${61}/{7} = 8{5}/{7}$$ 