9.1 Introduction

The picture below shows how a sister and a brother divided a Guava

into two equal parts.

The picture below shows how three people divided a cake into three

equal parts.

There are many such situations where a whole unit is divided into equal parts.

In the first situation, a person received one part out of the two equal parts of the Guava. If we numerically represent the Guava as 1 , then the numerical representation of the part a person receives is 1/2. This is read as “one half ”.

In the above second situation, a person received one part of the three equal parts the cake was divided. If we numerically represent the cake as 1, then the quantity one person receives is 1/3 . This is read as “one third”.

Let us consider further, the parts obtained by dividing a whole unit into equal parts as shown in the pictures below.

What we have done so far is to,

● numerically represent the quantity shown by a unit as 1.

● divide that unit into equal parts.

● numerically represent the quantity shown by one or several of those

parts.

Exercise 9.1

(1) Fill in the blanks given in the table.

(2) Consider each of the figures below as a whole unit. Now write down the coloured quantity as a fraction.

(3) Copy each of the pictures below and colour the quantity indicated by the fraction.

9.2 The denominator and the numerator of a fraction

Here, 7 is the number of parts a whole unit is divided equally into. We call it the denominator of the fraction. It is written below the line of the fraction.

4 is the number of parts considered. We call it the numerator of the fraction. It is written above the line of the fraction.

When we write a fraction numerically in this manner,

● the number written below the line is defined as the denominator of the fraction.

● the number written above the line is defined as the numerator of the fraction.

In a proper fraction, the numerator is always less than its denominator.

Such a fraction indicates the quantity of one part by dividing a whole unit into equal parts. These fraction are important because they can be used to explain other fractions.

Let us represent this by a figure.

Exercise 9.2

Let us consider equivalent fractions further.

This shows that, by multiplying both the numerator and the denominator of a fraction by the same whole number (except zero), a fraction which is equivalent to the first fraction can be obtained.

This shows that, by dividing both the numerator and the denominator of a fraction by the same whole number (where the division gives zero remainder), a fraction which is equivalent to the first fraction can be obtained.

 Comparison of fractions having the numerator as 1

In this manner, out of two unit fractions, the larger fraction is the fraction with the smaller denominator.

● Comparison of fractions having the same numerator

In this manner, out of two fractions having the same numerator, the larger fraction is the fraction with the smaller denominator.

● Comparison of fractions having the same denominator

Suppose a cake is cut into five equal parts and brother took three parts while sister took one part. Here, brother has taken a larger portion of the cake. Let us represent this by a figure.

Let us consider another example.

Fractions having 6 as their denominator are represented in the figure below.

According to the figures, it is clear that,

Out of two fractions having the same denominator, the larger fraction is the fraction with the larger numerator.

 More on comparison of fractions

Let us write these fractions as fractions having the same denominator using equivalent fractions. Thereafter, we can identify the larger fraction as in the earlier situation.

● Addition and subtraction of fractions having the same denominator.

In this manner, when adding two fractions having the same denominator, the denominator of the answer is the same as the denominators of the added fractions. The numerator of the answer is the addition of the numerators of the added fractions.

● Subtraction of fractions having the same denominator

When subtracting fractions having the same denominator, the denominator of the answer is same as the denominator of those fractions. The numerator of the answer is the value that is obtained by subtracting the numerator of the second fraction from the numerator of the first fraction.

(4) Write the relevant values in the boxes.

● More on addition of fractions

First, fractions having the same denominator, and which are equal to the given fractions are obtained in terms of equivalent fractions. Thereafter, the addition is carried out.

● More on subtraction of fractions

Here also, subtraction is carried out by obtaining fractions equivalent to the given fractions with the same denominator.

We already know to express parts of a whole unit as fractions. Now, let us express a part of a collection as a fraction.