## NCERT Class VI Mathematics Chapter 7 Fractions

National Council of Educational Research and Training (NCERT) Book for Class VI
Subject: Mathematics
Chapter: Chapter 7 – Fractions

Class VI NCERT Mathematics Text Book Chapter 7 Fractions is given below.

7.1 Introduction

Subhash had learnt about fractions in Classes IV and V, so whenever possible he would try to use fractions. One occasion was when he forgot his lunch at home. His friend Farida invited him to share her lunch. She had five pooris in her lunch box. So, Subhash and Farida took two pooris each. Then Farida made two equal halves of th  fifth poori and gave one-half to Subhash and took the other half herself. Thus, both Subhash and Farida had 2 full pooris and one-half poori.

one whole (Fig 7.2). So, each equal part is one-fourth of one whole poori and

4 parts together will be 4 /4 or 1 whole poori.

Farida said that we have learnt that a fraction is a number representing part of a whole. The whole may be a single object or a group of objects.
Subhash observed that the parts have to be equal

7.2 A Fraction

4. What fraction of a day is 8 hours?

5. What fraction of an hour is 40 minutes?

6. Arya, Abhimanyu, and Vivek shared lunch. Arya has brought two sandwiches, one made of vegetable and one of jam. The other two boys forgot to bring their lunch. Arya agreed to share his sandwiches so that each person will have an equal share of each sandwich.

(a) How can Arya divide his sandwiches so that each person has an equal share?

(b) What part of a sandwich will each boy receive?

7. Kanchan dyes dresses. She had to dye 30 dresses. She has so far finished 20 dresses. What fraction of dresses has she finished?

8. Write the natural numbers from 2 to 12. What fraction of them are prime numbers?

7.3 Fraction on the Number Line

You have learnt to show whole numbers like 0,1,2… on a number line.

We can also show fractions on a number line. Let us draw a number line and try to mark 1/ 2 on it?

We know that 1/2 is greater than 0 and less than 1, so it should lie between 0 and 1.

Since we have to show 1/2 , we divide the gap between 0 and 1 into two

equal parts and show 1 part as 1/2 (as shown in the Fig 7.5).

Suppose we want to show 1/3 on a number line. Into how many equal parts should the length between 0 and 1 be divided? We divide the length between
0 and 1 into 3 equal parts and show one part as 1 / 3 (as shown in the Fig 7.6)

Can we show 2/3 on this number line? 2 /3 means 2 parts out of 3 parts as shown (Fig 7.7).

7.4 Proper Fractions

You have now learnt how to locate fractions on a number line. Locate the fractions 3/4 , 1/2, 9/10, 0/3, 5/8  on separate number lines.

Does any one of the fractions lie beyond 1?

All these fractions lie to the left of 1as they are less than 1.

In fact, all the fractions we have learnt so far are less than 1. These are proper fractions. A proper fraction as Farida said (Sec. 7.1), is a number representing part of a whole. In a proper fraction the denominator shows the number of parts into which the whole is divided and the numerator shows the number of parts we have taken out. Therefore, in a proper fraction the numerator is always less than the denominator.

Ravi said, ‘In both the ways of sharing each of us would get the same share, i.e., 5 quarters. Since 4 quarters make one whole, we can also say that each of us would get 1 whole and one quarter. The value of each share would be five divided by four. Is it written as 5 ÷ 4?’ John said, ‘Yes the same as 5/4 ’.
Reshma added that in 5/4 , the numerator is bigger than the denominator. The fractions, where the numerator is bigger than the denominator are called improper fractions.

Thus, fractions like 3/2, 12/7, 18/5,   are all improper fractions.

1. Write five improper fractions with denominator 7.

2. Write five improper fractions with numerator 11.

Try (c) and (d) using both the methods for yourself.

Thus, we can express an improper fraction as a mixed fraction by dividing the numerator by denominator to obtain the quotient and the remainder. Then

7.6 Equivalent Fractions

Look at all these representations of fraction (Fig 7.10).

These fractions are 1/2, 2/4, 3/6   representing the parts taken from the total number of parts. If we place the pictorial representation of one over the other they are found to be equal. Do you agree?

These fractions are called equivalent fractions. Think of three more fractions that are equivalent to the above fractions.

Understanding equivalent fractions

To find an equivalent fraction of a given fraction, you may multiply both the numerator and the denominator of the given fraction by the same number.

What do we infer? The product of the numerator of the first and the denominator of the second is equal to the product of denominator of the first and the numerator of the second in all these cases. These two products are called cross products. Work out the cross products for other pairs of equivalent fractions. Do you find any pair of fractions for which cross products are not equal? This rule is helpful in finding equivalent fractions.

7.9 Comparing Fractions

In both the fractions the whole is divided into 8 equal parts. For 3/8  and 5/8 , we take 3 and 5 parts respectively out of the 8 equal parts. Clearly, out of 8 equal parts, the portion corresponding to 5 parts is larger than the portion

corresponding to 3 parts. Hence, 5/8  > 3/8 . Note the number of the parts taken is given by the numerator. It is, therefore, clear that for two fractions with the same denominator, the fraction with the greater numerator is greater. Between 4/5  and 3/5  , 4/5 is greater. Between 11/20 and 13/20 , 13/20 is greater and so on.

7.9.2 Comparing unlike fractions

Two fractions are unlike if they have different denominators. For example, 1/3 and 1/5 are unlike fractions. So are 2/3 and 3/5 .

Unlike fractions with the same numerator :

Consider a pair of unlike fractions 1/3 and 1/5 , in which the numerator is the

Suppose we want to compare 2/3  and 3/4 . Their numerators are different and so are their denominators. We know how to compare like fractions, i.e.
fractions with the same denominator. We should, therefore, try to change the denominators of the given fractions, so that they become equal. For this
purpose, we can use the method of equivalent fractions which we already know. Using this method we can change the denominator of a fraction without
changing its value.

The equivalent fractions with the same denominator are :

4/5 = 24/30   and  5/6 = 25/30

Since, 25/30 >24/30  so, 5/6 > 4/5

Note that the common denominator of the equivalent fractions is 30 which is 5 × 6. It is a common multiple of both 5 and 6.

So, when we compare two unlike fractions, we first get their equivalent fractions with a denominator which is a common multiple of the denominators
of both the fractions.

Example 7 : Compare 5/6   and 13/15.

Solution : The fractions are unlike. We should first get their equivalent fractions with a denominator which is a common multiple of 6 and 15.

Why LCM?

The product of 6 and 15 is 90; obviously 90 is also a common multiple of 6 and 15. We may use 90 instead of 30; it will not be wrong. But we know that
it is easier and more convenient to work with smaller numbers. So the common multiple that we take is as small as possible. This is why the LCM of the denominators of the fractions is preferred as the common denominator.

8. Ila read 25 pages of a book containing 100 pages. Lalita read  2/5  of the same  book. Who read less?

9. Rafiq exercised for 3/6 of an hour, while Rohit exercised for 3/4  of an hour.

Who exercised for a longer time?

10. In a class A of 25 students, 20 passed in first class; in another class B of 30 students, 24 passed in first class. In which class was a greater fraction of students getting first class?

7.10 Addition and Subtraction of Fractions

So far in our study we have learnt about natural numbers, whole numbers and then integers. In the present chapter, we are learning about fractions, a different type of numbers.

Whenever we come across new type of numbers, we want to know how to operate with them. Can we combine and add them? If so, how? Can we take
away some number from another? i.e., can we subtract one from the other? and so on. Which of the properties learnt earlier about the numbers hold now?Which are the new properties? We also see how these help us deal with our daily life situations.

Thus, we can say that the difference of two like fractions can be obtained as follows:

Step 1 Subtract the smaller numerator from the bigger numerator.

Step 2 Retain the (common) denominator.

We have learnt to add and subtract like fractions. It is also not very difficult to add fractions that do not have the same denominator. When we have to add or subtract fractions we first find equivalent fractions with the same denominator and then proceed.

What added to 1/5  gives   1/2 ? This means subtract 1/5  from 1/2  to get the required number.

Since 1/5 and 1/2 are unlike fractions, in order to subtract them, we first find their equivalent fractions with the same denominator. These are 2/10
and 5/10  respectively.

same denominator. This denominator is given by the LCM of 4 and 6. The required LCM is 12.

How do we add or subtract mixed fractions?

Mixed fractions can be written either as a whole part plus a proper fraction or entirely as an improper fraction. One way to add (or subtract) mixed fractions is to do the operation seperately for the whole parts and the other way is to write the mixed fractions as improper fractions and then directly add (or subtract) them.

What have we discussed?

1. (a) A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects.

(b) When expressing a situation of counting parts to write a fraction, it must be ensured that all parts are equal.

2. In  5/7  , 5 is called the numerator and 7 is called the denominator.

3. Fractions can be shown on a number line. Every fraction has a point associated with it on the number line.

4. In a proper fraction, the numerator is less than the denominator. The fractions, where the numerator is greater than the denominator are called improper fractions. An improper fraction can be written as a combination of a whole and a part, and such fraction then called mixed fractions.

5. Each proper or improper fraction has many equivalent fractions. To find an equivalent fraction of a given fraction, we may multiply or divide both the numerator and the denominator of the given fraction by the same number.

6. A fraction is said to be in the simplest (or lowest) form if its numerator and the denominator have no common factor except 1.

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