Last modified on October 29th, 2024

chapter outline

 

Fractions

Fractions represent a part of the whole. It represents the number of parts of a certain number, size, or collection compared to the total number of equal parts. 

Notation

A number is written in the fractional form as ab 

NumeratorDenominator

Examples

If a pizza is cut into ten equal slices and one slice of the pizza is placed on a dish, then each dish is said to have 110 of the pizza. It is read as ‘one-tenth’ or ‘1 by 10.’

Fractions

Here are some more real-life examples involving fractions:

  • Dividing a cake into 5 equal parts means each part is 15 of the whole.
  • If the traveling time from your home to the office is 30 minutes, it means ‘half an hour’ is taken, which is written as 12.
  • Test scores are often represented as fractions, like 910 or 4850.
  • Food recipes use fractions, such as 14 teaspoon of sugar or 12 tablespoon of salt.

Parts

A fraction consists of two main parts: a numerator and a denominator, separated by a horizontal bar known as the fraction bar.

Numerator

The numerator is the top number above the fraction bar, indicating how many parts we have. 

Denominator

The denominator is the number below the fraction bar, which shows how many equal parts the whole is divided into.

The diagram below shows the different parts of the fraction 110.

Parts of a Fraction

Representing a Fraction

A fraction can be represented in three ways: a fraction, a percentage, or a decimal.

Fractional Form 

It is the most common form of representing fractions. A fraction in fractional form is denoted in pq form, where p is the numerator, q is the denominator, and they are separated by a fraction bar.

Example

110 represents 1 part out of 10 equal parts of a whole. Here, the numerator is 1, and the denominator is 10

Decimal Form

In this form, the fraction is represented as a decimal number.

Example 

The fraction 110 can be written in decimal form by dividing the numerator (1) by the denominator (10), which gives 0.1

Percentage Form

The fraction can also be represented as a percentage by multiplying the fraction by 100.

Example

The fraction 110 is multiplied by 100

110×100=10%

Thus,  110 is percentage form is 10%

Types

Fractions can be classified into several types:

Types of Fractions

On a Number Line

A number line is a useful tool for visualizing fractions. Let us express 110 on the number line. 

  1. Drawing a horizontal line: This line represents the number line.
  1. Marking whole numbers: We then divide the number line into whole numbers (here, 0 and 1) by placing points at equal intervals based on the denominator of the given fraction.
  1. Dividing the Intervals: We further divide the interval of whole numbers into equal parts based on the denominator. Here, we will divide the number line between 0 and 1 into 10 equal parts. The first interval is marked as 110, the second is 210, and so on, with the last mark being 1010, or 1. 
  1. Plotting the fraction: The fraction is finally plotted by marking a point on the number line corresponding to the numerator. 

Here is the number line.

Fractions on a Number Line

Here is a printable chart for visualizing equivalent fractions.

Fraction Chart

Properties

Here are some properties used to simplify problems involving fractions.

  1. Commutative and associative properties hold for addition and multiplication of fractions
  2. Fractions obey the distributive law of multiplication over addition
  3. The identity element is 0 for fractional addition and 1 for fractional multiplication 
  4. The multiplicative inverse of ab is ba, where a and b are non-zero elements 

Simplifying

Simplifying a fraction means to reduce the fraction to its simplest form. To simplify, we divide the numerator and denominator by the greatest common factor (GCF). 

Let us simplify 2030

The GCF of 20 and 30 is 10

Now, by dividing the numerator and denominator by 10, we get

2030

= 23

Adding and Subtracting

With Same Denominators

To add or subtract fractions with the same denominator, we follow the following steps:

Let us add 73 and 23

Step 1: Adding/subtracting the numerators together

73+23

= 7+23

Step 2: Simplifying

= 93

= 3

Now, if we subtract the fraction 23 from 73, then

7323

= 723

= 53

With Different Denominators

We follow the given steps to add or subtract fractions with different denominators. 

Let us add 12+34

Step 1: Finding the LCM of the denominators

Here, the denominators are 2 and 4

The LCM of 2 and 4 is 4

Step 2: Rationalizing the denominators

Now, for rationalizing, we will multiply the first fraction by 2 and the second fraction by 1

1×22×2+3×14×1

= 24+34, the fractions with the same denominators.

Step 3: Adding the numerators together

= 2+34

Step 4: Simplifying

= 54

= 114

Multiplying and Dividing

Multiplying

While multiplying fractions, we multiply all the numerators and all the denominators together.

On multiplying pq and rs 

ab×cd

= acbd

For example, 

37×511

= 3×57×11

= 1577

Dividing

While dividing the fractions, we multiply the first fraction by the reciprocal of the second fraction.

For example,

37÷511

= 37×115

= 3×117×5

= 3335

Fraction Rules Summary 

Here is a summary of the rules we follow for simplifying fractions.

Fraction Rules

Solved Examples

From the given fractions, determine the proper and improper fractions.
115, 38, 92, 1117, 133, 711, 48, and 32

Solution:

Here,
The proper fractions are 115, 38, 1117, 711, and 48
The improper fractions are 92, 133, and 32

In a class of 60 students, 15 of them went on a family trip during their summer break. How many students went on a trip?

Solution:

Here, the total number of students is 60
As we know,
The number of students who went on a family trip is 15 of the total number of students.
Thus, the number of students who went on a trip 
= 15 of 60
= 15×60
= 12

Last modified on October 29th, 2024