Chapter 7

Fractions

7.1 Introduction
Subhash had learnt about fractions in Classes IV and V, however he
was not very confident, 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 the 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.

2 pooris + half-poori–Subhash 2 pooris + half-poori–Farida

Where do you come across situations with fractions in your life?
1
Subhash knew that one-half is written as . While
2
eating he further divided his half poori into two equal
parts and asked Farida what fraction of the whole poori
Fig 7.1
was that piece? (Fig 7.1)
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Without answering, Farida also divided her portion
of the half puri into two equal parts and kept them
beside Subhash’s shares. She said that these four equal
parts together make one whole (Fig 7.2). So each equal
part is one-fourth of one whole poori and 4 parts
4 Fig 7.2
together will be or 1 whole poori.
4
As they ate they discussed what they had learnt earlier. Three parts out
3 3
of 4 equal parts is . Similarly is obtained when we divide a whole
4 7
1
into seven equal parts and take three parts (Fig 7.3). For , we divide
8
a whole into eight equal parts and take one part out of it (Fig 7.4).

Fig 7.3 Fig 7.4

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
Let us recapitulate the discussion
A fraction means a part of a group or of a region.
5
is a fraction. We read it as “five -twelfths”.
1 2
What does “12” stand for? It is the number of equal parts into which the
whole has been divided.
What does “5” stand for? It is the number of equal parts which have been
taken out.
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Here 5 is called the numerator and 12 is called the
denominator.
3
Name the numerator of the fraction . What is the
7
4
denominator of ?
1 5
2
Play this Game:
You can play this game with your friends.
Take many copies of the grid as shown here.
1
Consider any fraction, say .
2

1
Each one of you should shade of the grid.
2
7 The condition is that no two of you should shade same pattern.

EXERCISE 7.1
1. Write the fraction representing the shaded portion.

( i ( )i i ) ( i ( ii v ) i )

(v) ( v i( v) i i( )v i i i )
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( i x ) (x) ( x i )

( x (xiii) i i )

2. Colour the part according to the fraction given :

(( 1 ii(( 2 ii ii )) )(
)( 1 ii ii ii
(( ii(( i i ) )( ii ii ii
6 4 3

(iv) 3 (v) 4 (vi) 1

(iv) (v) (vi)
(iv)
(iv) 4 (v) 9 (vi)
(vi) 4
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3. Identify the error, if any

1 1 3
This is This is This is
2 4 4
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 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?
7 (b) What part of a sandwich will each boy receive?
7. Kanchan has three frocks that she wears while playing. The material is good,
but the colours are faded. Her mother buys some blue dye and uses it on two of
the frocks. What fraction of the Kanchan’s frocks did her mother dye?
8. Write the natural numbers from 2 to 12. What fraction of them are prime numbers?
9. Write the natural numbers from 102 to 113. What fraction of them are prime
numbers?
10. What fraction of these circles have X’s in them?

( a ( ) b ( ) c )

11. Dinesh, Sumit, Ram, Joy, Marshal, Imran, Jayant, Babu, Kabir and Rohan
decide to play basketball. The first five boys are in the first team and the rest are
in the second team. What fraction of the boys is in the first team?
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12. Kristin received a CD player for her birthday. She has been collecting CDs
since then. She bought 3 CDs and received 5 others as gifts. What fraction of
her total CDs did she buy?

7.3 Fraction on the Number Line

You have learnt to show whole numbers like 0,1,2... on a number line.
Can you show fractions on a number line? Let us draw a number line.
1
Can we show on it?
2
1
We know that is greater than 0 and less than 1, so it should lie between
2
0 and 1.
1
Since we have to show , we divide the gap between 0 and 1 into two
2

7
1
equal parts and show 1 part as (as shown in the Fig 7.5).
2

0 1 1
2
Fig 7.5
1
To show on a number line, into how many equal parts should the gap
3
between 0 and 1 be divided? We divide the gap between 0 and 1 into 3 equal
1
parts and show one part as (as shown in the Fig 7.6)
3

0 1 1
3
Fig 7.6
2 2
Can we show on this number line? means 2 parts out of 3 parts as
3 3
shown (Fig 7.7).
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0 1 2 1
3 3
Fig 7.7

0 3
Similarly, how would you show and on this number line?
3 3
0 3
is the point zero whereas is 1 whole, which can be shown by the
3 3
point 1 (as shown in Fig 7.8)

0 1 2 3
( = 0 ) ( = 1 )
3 3 Fig 7.8
3 3

3
7 So if we have to show on a number line, then, into how many equal
7
3
parts should the gap between 0 and 1 be divided? If P shows then how
7
0 7
many equal divisions lie between 0 and P? Where do and lie?
7 7

3
1. Show on a number line.
5
1 0 5 1 0
2. Show , , and on a number line.
1 0 1 0 1 0 1 0
3. Can you show any other fraction between 0 and 1?
Write five more fractions that you can show and depict them on the
number line.
4. How many fractions lie between 0 and 1? Think, discuss and write
your answer?
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7.4 Proper Fractions
You have now learnt how to locate fractions on a number line. Locate the
3 1 9 0 5
fractions , , , , on separate number lines.
4 2 1 0 3 8
Does any one of the fractions lie beyond 1?
All these fractions lie to the left of 1as they are less than 1. Why?
In a proper fraction, the numerator is less than the denominator.

1. Give a proper fraction :

(a) whose numerator is 5 and denominator is 7.
(b) whose denominator is 9 and numerator is 5.
(c) whose numerator and denominator add up to 10. How many
fractions of this kind can you make?
(d) whose denominator is 4 more than the numerator. 7
(Make any five. How many more can you make?)
2. A fraction is given.
How will you decide, by just looking at it, whether, the fraction is
(a) less than 1?
(b) equal to 1?
3. Fill up, using one of these : ‘>’, ‘<’ or ‘=’
1 3 7
(a) 1 (b) 1 (c) 1
2 5 8
4 0 2 0 0 5
(d) 1 (e) 1 (f) 1
4 6 2 0 0 5

7.5 Improper and Mixed Fractions

Anagha, Ravi, Reshma and John shared their tiffin. Anagha had brought
pooris, Ravi had brought rotis, Reshma had brought mixed vegetable and
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John sandwiches. They
shared the rotis and the
pooris as well as the
vegetables and sand-
wiches. Among them they
had brought 5 apples. After
eating the other things, the
four friends wanted to eat
apples.
How can they share five apples among four of them?
Anagha said, let each of us have one full apple and a quarter of the fifth
apple.

Anagha Ravi Reshma John

7 Reshma said that is fine but we can also divide each of the five apples
into 4 equal parts and take one-quarter from each apple.

Anagha Ravi Reshma John

Anagha said yes both the ways of sharing the apple are same. Each share
can be written as five divided by 4. Ravi said you mean 5 ÷ 4? John said yes,
5
it is also written as .
4

5
Anagha said in , the numerator is bigger than the denominator. The
4
fractions, where the numerator is bigger than the denominator are called
improper fractions.
3 1 2 1 8
Thus, fractions like , , are all improper fractions.
2 7 5
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1. Write five improper fractions with denominator 7.
2. Write five improper fractions with numerator 11.
5
The fraction of apples received by each of the friends is . However, as
4
Anagha said, it is also one apple and one fourth more as shown in the picture
(Fig 7.9).

This is 1 1
Each of these is
(one) 4
(one fourth)

7
Fig 7.9

1 1 5
Thus, 1 is written as 1 and is the same as .
4 4 4

1
Recall the pooris eaten by Farida. She got 2 poories (Fig 7.10).
2
i.e.,

This is 1 1
Fig 7.10
This is 2
2

1
How many shaded halves are there in 2 ? There are 5 shaded halves.
2
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5 5
So, the fraction is . Obviously this is not .
2 4

1 1
Fractions such as 1 and 2 are Do you know?
4 2
The grip-sizes of tennis racquets are often
called Mixed Fractions. A mixed in mixed numbers. For example one size
fraction has a combination of a
7 3
whole and a part. is “3 inches” and “4 inches” is another..
8 8
Where do you come across
mixed fractions? Give some
examples.
Example 1 : Express the following as mixed fractions :
1 7 1 1 2 7 7
(a) (b) (c) (d)
4 3 5 3

7 Solution : (a)
1 7
4 1 7
4
4
1 6
1

1 1
i.e., 4 whole and more, or 4
4 4

1 1 3
(b) 3 1 1
3
9
2

2 2
i.e., 3 whole and more, or 3
3 3
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2 7 5
(c) 5 2 7
5
2 5
2

2 2
i.e., 5 whole and more, or 5 .
5 5

7 2
(d) 3 7
3
6
1

7 1
We write =2
3 3
7
7 6 1 1 1
A l t e r n a t e l y , 2 2
3 3 3 3 3
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 the mixed fraction will
Remainder
be written as Q u o t i e n t .
Diviser
Example 2 : Express the following mixed fractions as improper
fractions :
3 1 3
(a) 2 (b) 7 (c) 5
4 9 7
3 ( 2 4 ) 3 1 1
Solution : (a) 2 o r
4 4 4
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180
1 ( 7 9 ) 1 6 4
(b) 7 o r
9 9 9

3 ( 5 7 ) 3 3 8
(c) 5 o r
7 7 7
Thus, we can express a mixed fraction as an improper
fraction by multiplying the whole with the denominator and
adding the numerator to it. Then the improper fraction will
be
(Whole × Denominator) + Numerator
Denominator

EXERCISE 7.2
1. Draw number lines and locate the points on them.

7 (a)
1 1
, ,
3
,
4 1 2 3 7
(b) , , , (c)
2 3 8 4
, , ,
2 4 4 4 8 8 8 8 5 5 5 5
2. Express the following as mixed fractions:
2 0 1 1 1 7
(a) (b) (c)
3 5 7
2 8 1 9 3 5
(d) (e) (f )
5 6 9
3. Express the following as improper fractions:
3 6 5
(a) 7 (b) 5 (c) 2
4 7 6
3 3 4
(d) 1 0 (e) 9 (f ) 8
5 7 9
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7.6 Equivalent Fractions
Look at all these representations of fraction (Fig 7.11).

Fig 7.11
1 2 3
These fractions are , , representing the parts taken from the total
2 4 6
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.
7
1 2
1. Are a n d equivalent? Give reason.
3 7
2 2
2. Are a n d equivalent? Give reason.
5 7
2 6
3. Are a n d equivalent? Give reason.
9 2 7
4. Give example of four equivalent fractions.
5. Identify the fractions in each.
Are these fractions equivalent?
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Understanding equivalent fractions
1 2 3 3 6
, , , . . . , . . are
. , all equivalent fractions.
2 4 6 7 2
They represent the same part of a whole.
Think, Discuss and Write
Why do the equivalent fractions represent the same part of a whole? How
can we obtain one from the other?
1 2
Consider a n d. The numerator of the second fraction is twice the
2 4
numerator of the first and the denominator of the second is also twice that
of the first.
What does this mean?
2 1 × 2
This means =
4 2 × 2

7 Similarly
3 1 × 3 1
= =
6 2 × 3 2
1 1 × 4 4
and = =
2 2 × 4 8
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.
1
Rajni says that equivalent fractions of are :
3
1 2 2 1 3 3 1 4 4
, , and many more.
3 2 6 3 3 9 3 4 1 2
Do you agree with her? Explain with reasons.
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1. Find five equivalent fractions each of :

2 1 3 5
(i) (ii) (iii) (iv)
3 5 5 9

Another way
Is there any other way to obtain equivalent fractions? Look at the (Fig 7.12) :

4 2
is shaded here. is shaded here.
6 Fig 7.12 3

They cover equal area i.e.,

4
6
2
and are equivalent fractions.
3
7
4 4 ÷ 2 2
= =
6 6 ÷ 2 3
To find an equivalent fraction, we may divide both the numerator and
the denominator by the same number.
1 2 1 2 ÷ 3 4
One equivalent fraction of is =
1 5 1 5 ÷ 3 5
9
Can you find the equivalent fraction of having denominator 5 ?
1 5
2
Example 3 : Find the equivalent fraction of with numerator 6.
5
2
Solution : The given fraction is . The required fraction has
5
numerator 6.
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184
We know 2 × 3 = 6. This means we need to multiply both
numerator and denominator of the first fraction by 3 to get
the equivalent fraction.
2 2 3 6
i.e.,
5 5 3 1 5

2 6
Thus, we get =
5 1 5
Can you show this pictorially?
1 5
Example 4 : Find the equivalent fraction of with denominator 7.
3 5

1 5
Solution : We have =
3 5 7
We observe the denominators. Since 35 ÷ 5 = 7, so we divide
7 the numerator of
1 5
also by 5.
3 5

1 5 1 5 ÷ 5 3
We have = =
3 5 3 5 ÷ 5 7

1 5 3
Thus, replacing by 3, we get =
3 5 7

An Interesting Fact
There is something quite interesting about equivalent fractions. Complete
the table given. The first two rows have been done. (Nr. is for numerator
and Dr. for denominator).
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Equivalent Product of Product of Are the
fractions the Nr. of the1st and the Nr. of the 2nd products
the Dr. of the 2nd and the Dr. of equal?
the 1st
1 3
= 1 × 9 = 9. 3 × 3 = 9. Yes
3 9
4 2 8
= 4 × 35 = 140 5 × 28 = 140 Yes
5 3 5
1 4
=
4 1 6
2 1 0
=
3 1 5
3 2 4
=
7 5 6

What do we infer? The product of the numerator of the first and the 7
denominator of the second is equal to the product of denominator of the
first and the numerator of the second in all these cases. You can try this
with other fractions as well. Did you find any pair of fractions for which it
does not happen? This rule is sometimes helpful to find equivalent fractions.
2
Example 5 : Find the equivalent fraction of with denominator of 63.
9
2
Solution : We have =
9 6 3
For this, we should have, 9 2 6 3 .

9 2 6 3
S o ,
9 9
2 6 3 2 1 4
T h e r e f o r e , 1 4 i . e . ,
9 9 6 3
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7.7 Simplest Form of a Fraction
3 6
Given the fraction , let us try to get an equivalent fraction in which the
5 4
numerator and the denominator have no common factor except 1.
How do we do it? We see that both 36 and 54 are divisible by 2.
3 6 3 6 2 1 8
5 4 5 4 2 2 7
But 18 and 27 also have common factors other than one.
The common factors are 1, 3, 9.
1 8 1 8 9 2
Therefore,
2 7 2 7 9 3

2
Since 2 and 3 have no common factor, we get the fraction in the
3
7 simplest form.
A fraction is said to be in the simplest (or lowest) form if its
numerator and denominator have no common factor except 1.
The shortest way :
The shortest way to find the equivalent fraction
in the simplest form is to find the HCF of the
numerator and denominator. Then divide both
of them by the HCF and get the equivalent
fraction in the simplest form.
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3 6 A game
Consider .
2 4 The equivalent fractions given here are quite
The HCF of 36 and 24 interesting. Each one of them uses all the digits from
is 12. 1 to 9 once!

3 6 1 2 3 2 3 5 8
= =
Therefore, 6 9 1 7 4
2 4 1 2 2
2 3 7 9
Thus the idea of HCF = =
4 6 1 5 8
helps us to reduce a
fraction to its lowest form. Can you find two more such equivalent fractions?

1. Write the simplest form of :

1 5 1 6 1 7 4 2 8 0
(i) (ii) (iii) (iv) (v)
7 5 7 2 5 1 2 8 2 4

2. Is
1 6 9
in its simplest form? 7
2 8 9

EXERCISE: 7.3
1. Write the fractions. Are all these fractions equivalent?

(a)

(b)
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188
2. Write the fractions and pair up the equivalent fractions from each row.

(a) (b) (c) (d) (e)

(i) (ii) (iii) (iv) (v)

3. Replace in each of the following by the correct number:

7 2 8 5 1 0 3
(a) 7 = (b) (c)
8 5 2 0

4 5 1 5 1 8
(d) (e)
6 0 2 4 4

3
4. Find the equivalent fraction of having
5
(a) denominator 20 (b) numerator 9
(c) denominator 30 (d) numerator 27
3 6
5. Find the equivalent fraction of with
4 8
(a) numerator 9 (b) denominator 4
6. Check whether the given fractions are equivalent:
5 3 0 3 1 2 7 5
(a) , (b) , (c) ,
9 5 4 1 0 5 0 1 3 1 1
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7. Reduce the following fractions to simplest form :
4 8 1 5 0 8 4
(a) (b) (c)
6 0 6 0 9 8
1 2 7
(d) (e)
5 2 2 8
8. Ramesh had 20 pencils, Sheelu had 50 pencils and Jamaal had 80 pencils. After
4 months, Ramesh used up 10 pencils, Sheelu used up 25 pencils and Jamaal
used up 40 pencils. What fraction did each use up? Check if each has used up an
equal fraction of their pencils?
9. Match the equivalent fractions and write two more for each :

52 0 2
( ) i
( a )
4 0 0 3
1 8 0 2
( i i ) (b )
2 0 0 5

7
6 6 0 1
( i i i ) (c )
9 9 0 2
1 8 0 5
( i v ) (d )
3 6 0 8
2 2 0 9
( v ) ( e)
5 5 0 1 0

7.8 Like Fractions

Fractions with same denominators are called like fractions.
1 2 3 8
Thus , , , are all like fractions.
1 5 1 5 1 5 1 5
7 7
Are and like fractions?
2 7 2 8
Their denominators are different therefore, they are not like fractions.
They are called unlike fractions.
Write five pairs of like fractions and five pairs of unlike fractions.
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7.9 Comparing Fractions
1 1 0
Sohni has 3 pencils. Rita has pencils. Who has more pencils? The
2 4
1 0
answer to this seems simple since Rita has pencils which is equivalent
4
2
to 2 pencils. Clearly since Sohni has 3 full pencils and more and Rita has
4
less than 3 full pencils, so Sohni has more pencils.
1 1 1 1
Now consider some other fractions , , a n d . (Fig 7.13).
2 3 4 5

Fig 7.13

Can you tell which fraction is the biggest?

1
Is it ? Why not?
5
1
If is the largest fraction then what can we conclude?
2
The smaller the denominator, the greater the fraction. But is this always
1
true that larger the denominator the smaller is the fraction? Think about
5
3
and . Which is bigger?
1 0
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1. You get one-fifth of a bottle of juice and your sister gets one-third
of a bottle of juice. Who gets more?

7.9.1 Comparing Like Fractions

Like fractions are fractions with the same denominator. Which of these
are like fractions?
2 3 1 7 3 4 4
, , , , , ,
5 4 5 2 5 5 7
As the denominators in the like fractions are the same so in order to
compare them we only see which of them have a bigger numerator.
For example, which is smaller?
2 8
or ?
8 8
2 8
7
8 8
2
Obviously is smaller..
8
5 2 5
Similarly between and , is greater and so on.
8 8 8
To compare two like fractions it is just enough to compare their
numerators!
7 1 5
So, which is bigger : or ?
1 0 1 0

1. Which is the larger fraction?

7 8 1 1 1 3 1 7 1 2
(i) o r (ii) o r (iii) o r
1 0 1 0 2 4 2 4 1 0 2 1 0 2
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Why are these comparisons easy to make?

2. Write these in ascending and also in descending order
1 3 5 1 1 1 3
(a) , , , ,
8 8 8 8 8
1 3 4 7 1 1
(b) , , , ,
5 5 5 5 5
1 3 2 1 1 1 3 1 5
(c) , , , , ,
7 7 7 7 7 7

7.9.2 Comparing Unlike Fractions

1 1 1 1 1
Among , , , , which is the greatest fraction? It is obvious that
2 3 5 6 9
1
is the greatest number, because the equal part of a whole which this
7 2
fraction represents would be larger than the equal parts represented by the
1
other fractions. On the other hand, the fraction is the smallest number
9
as it represents the smallest equal part. In each case only one equal part has
been taken.
2 2 2 2 2 2 2
Let us now look at these fractions, , , , , , , .
1 1 3 9 7 5 1 0 3
Which is the smallest fraction of these?
Which is the largest among them?
In each case here, two equal parts have been taken. The largest number of
equal parts are taken in the first fraction, where each part taken is the whole
2
itself. The smallest number of equal parts are in .
1 3
2 2
It is therefore obvious that is the smallest and the largest.
1 3 1
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1. Arrange the following in ascending and descending order :

1 1 1 1 1 1 1
(a) , , , , , ,
1 2 2 3 5 7 5 0 9 1 7
3 3 3 3 3 3 3
(b) , , , , , ,
7 1 1 5 2 1 3 4 1 7
(c) Write 3 more similar examples and arrange them in ascending
and descending order.

How do we compare fractions that have different denominators and

numerators?
2 3
Suppose we want to compare two unlike fractions like and . It would
3 4
be possible to compare them if somehow we can make the number of parts
in the denominator of the two fractions equal i.e., make the equal parts 7
taken in either case exactly matching each other. Once we do that, then the
number of parts in the numerator can be easily compared.
2 3
Consider and . Which of them is larger? How do we make the
3 4
denominator same? We find equivalent fractions to both.
2 4 6 8 1 0
= ....
3 6 9 1 2 1 5
3 6 9 1 2
Similarly, = ....
4 8 1 2 1 6
2 3
The equivalent fractions of and with the same denominator
3 4
8 9
12 are a n d repectively..
1 2 1 2
2 8 3 9
i.e., and .
3 1 2 4 1 2
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194
9 8
Since,
1 2 1 2

3 2
Therefore, >
4 3

EXERCISE 7.4
1. Write each fraction. Arrange them in ascending and descending order using correct
sign ‘<’, ‘=’, ‘>’ between the fractions:

(a)

7 (b)

2 4 8 6
(c) Show , , and on the number line. Put appropriate signs between
6 6 6 6
the fractions given
5 2 3 0 1 6 8 5
, , ,
6 6 6 6 6 6 6 6
2. Compare the fractions and put an appropriate sign.
3 5 1 1
(a) (b)
6 6 7 4

4 0 3 4
(c) (d)
5 5 2 0 2 0
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3. Make five more such pairs and put appropriate signs.
4. Name the fractions and arrange them in ascending order :

5. Look at the figures and write ‘<’ or ‘>’, ‘=’ between the pairs of fractions.

0 1
1 1

0 1 2
2 2 2
7
0 1 2 3
3 3 3 3

0 1 2 3 4
4 4 4 4 4

0 1 2 3 4 5
5 5 5 5 5 5

0 1 2 3 4 5 6
6 6 6 6 6 6 6

1 1 3 2 2 2
(a) (b) (c)
6 3 4 6 3 4

6 3 0 0 5 5
(d) (e) (f)
6 3 1 6 6 5
Make five more such problems and solve them with your friends.
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6. How quickly can you do this? Fill appropriate sign. ( <, =, >)
1 1 2 3 3 2
(a) (b) (c)
2 5 4 6 5 3

3 2 3 6 7 3
(d) (e) (f)
4 8 5 5 9 9

1 2 6 4 3 7
(g) (h) (i)
4 8 1 0 5 4 8

6 4 5 1 5
( j) (k)
1 0 5 7 2 1
7. The following fractions represent just three different numbers. Separate them into
three groups of equivalent fractions, by changing each one to its simplest form.
2 3 8
(a) (b) (c)
1 2 1 5 5 0
7 (d)
1 6
(e)
1 0
(f )
1 5
1 0 0 6 0 7 5
1 2 1 6 1 2
(g) (h) (i)
6 0 9 6 7 5
1 2 3 4
( j) (k) (l)
7 2 1 8 2 5
8. Find answers to the following. Write and indicate how you solved them.
5 4 9 5
(a) Is equal to ? (b) Is equal to ?
9 5 1 6 9
4 1 6 1 4
(c) Is equal to ? (d) Is equal to ?
5 2 0 1 5 3 0
1
9. Ila read 25 pages of a book containing 100 pages. Lalita read of the same
2
book. Who read less?
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197
3 3
10. Rafiq exercised for of an hour, while Rohit exercised for of an hour. Who
6 4
exercised for a longer time?
11. 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 were there more fraction of
students getting first class?

7.10 Addition and Subtraction of Fractions

Whenever we find new kind 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? That is, 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.
Look at the following example. A tea stall owner consumes in her shop

2
1 1
litre of milk in the morning and 1 litre of milk in the evening. What
7
2 2
is the total amount of milk she uses in the stall?
1
Or Shekhar ate 2 chapatis for lunch and 1 chapati for dinner. What is
2
the total number of chapatis he ate?
Clearly both the situations require the fractions to be added. Some of
these additions can be done orally and the sum can be found quite
easily.

1. My mother divided an apple into 4 equal parts. She gave me 2 parts

and my brother one part. How much apple did she give to both
of us?
2. Mother asked Neelu and her brother to pick stones from the
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198
1
wheat. Neelu picked t hof the total stones in it and her brother also
4
1
picked up t hof the stones. What fraction of the stones did both
4
pick up together?
1
3. Sohan was making a table. He made t hof the table by Monday. He
4
1
made another t h
on Tuesday and the remaining on Wednesday. What
4
fraction of the table was made on Wednesday?

In all these you can also find out the remaining part. In the first problem
the apple remaining, in the second the fraction of the stones left to be
picked and in the third the remaining fraction of the table to be made.

7
Make ten such problems with your friends and solve them.
7.10.1 Adding or Subtracting like Fractions
All fractions cannot be added orally. We need to know how they can be
added in different situations and learn the procedure for it. We begin by
looking at addition of like fractions.
Take a 7 × 4 grid sheet (Fig 7.14). The sheet has seven boxes in each row
and four boxes in each column .
How many boxes are there in total?
Colour five of its boxes in green.
What fraction of the whole is the green region?
Now colour another four of its boxes in yellow.
What fraction of the whole is this yellow region?
What fraction of the whole is coloured
altogether? Fig 7.14
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199
5 4 9
Does this explain that ?
2 8 2 8 2 8
Look at more examples
In Fig 7.15 (i) we have 2 quarter parts of the figure shaded. This means we
1
have 2 parts out of 4 shaded or of the figure shaded.
2

Fig. 7.15 (i) Fig. 7.15 (ii)

That is, 1 1 2 1.
4 4 4 2
Look at Fig 7.15 (ii)
7
1 1 1 3 1
Fig 7.15 (ii) demonstrates .
9 9 9 9 3
What do we learn from the above examples? The sum of two or more
like fractions can be obtained as follows:
Step 1 Add the numerators.
Step 2 Retain the (common) denominator.
Result of Step 1
Step 3 Write the fraction as:
Result of Step 2

3 1 3 1 3 1 4
Let us thus add and . We have
5 5 5 5 5 5

7 3
So, what will be the sum of and ?
1 2 1 2
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200

1. Add, with the help of a diagram:

1 1 2 3 1 1 1
(i) (ii) (iii)
8 8 5 5 1 2 1 2 1 2

1 1 1
2. What do we get when we do this
1 2 1 2 1 2
How will we show this pictorially? Using paper folding?
3. Make 10 more examples of problems given in 1 and 2 above.
Solve them with your friends.

Think, Discuss and Write

Can you think of solving these sums in some other way?

7 Write how you could do them?

Finding the balance

In one of the problems above we found the amount of apple distributed. We
also considered the amount of apple left and asked how much of the table
3
is yet to be made. The apple left is one whole minus the t happle
4
1
distributed and the table left to be built is one whole minus of the table.
2
These answers could be easily found.
Consider this example :
5 2
Sharmila had of a cake. She gave out of that to her younger brother..
6 6
How much cake is left with her?
 Fractions
201
A diagram can explain the situation. (Note that, here the given fractions
are like fractions) (Fig 7.16).

Fig 7.16
5 2 5 2 3 1
We find that o r
6 6 6 6 2
(Is this not similar to the method of adding like fractions?)
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.

7
Result of Step 1
Step 3 Write the fraction as:
Result of Step 2
3 8
Can we now subtract from ?
1 0 1 0

7 3
1. Find the difference between and .
4 4
2. Mother made a gud patti in a round shape. She divided it into 5 parts.
Seema ate one piece from it. If I eat another piece then how much
would be left?
3. My elder sister divided the watermelon into 18 parts. I ate 7 out
them. My friend ate 4. How much did we eat between us? How much
more of water melon did I eat compared to my friend? What amount
of water melon remained?
4. Make five problems of this type and solve them with friends.
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202

EXERCISE 7.5
1. Write these fractions appropriately as additions or subtractions :

(a) .... =

. . . .
(b) =

. . . . =
(c)

2. Complete the statements given in each Figure.

7 + =
(a)

+ + =
(b)

3. Solve :
1 1 8 3 7 5
(a) (b) (c)
1 2 1 2 1 5 1 5 7 7

1 2 1 5 6 1 2 7
(d) (e) (f)
2 2 2 2 9 9 1 5 1 5

5 3 3 4 2 3
(g) (h) (i) 1 1
8 8 5 5 3 3
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203
1 0 0 0 1 6 7
( j) (k) (l)
4 4 2 2 5 5

1 2 1 2
(m) 2 1 (n) 3 –
3 3 5
2
4. Shubham painted of the wall space in his room. His sister Madhavi helped
3
1
and painted of the wall space. How much did they paint together?
3
1 1
5. Kamlesh bought 3 kg sugar whereas Anwar bought 2 kg of sugar. Find the
2 2
total amount of sugar bought by both of them.
6. Fill in the missing fractions.
7 3 3 5
(a) (b)
1 0 1 0 2 1 2 1

3 3 5 1 2
7
(c) – (d)
6 6 2 7 2 7
3 1
7. The teacher taught of the book, Mahesh revised more on his own. How
5 5
much does he still have to revise?
5
8. Javed was given of a basket of oranges. What fraction of oranges was left in
7
the basket?

7.10.2 Adding and Subtracting all Fractions

We have learnt to do addition and subtraction with like fractions. It is also
not very difficult to add fractions that do not have the same denominator.
Remember, to compare unlike fractions we converted them into equivalent
fractions with the same denominators. We did this so that we could compare
the number of equal parts between the two fractions. When we add or
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204
subtract fractions we have to do the same. First find equivalent fractions
with the same denominator and then proceed.
1 1 1 1
What added to gives ? This means subtract from to get the
5 2 5 2
required number.
1 1
Since and are unlike fractions, therefore, in order to subtract them,
5 2
we find equivalent fractions for them which are like fractions. Equivalent
1 1 5 2
fractions of and with the same denominator are and respectively..
2 5 1 0 1 0
1 1 × 5 5 1 1 × 2 2
This is because = = a n d = =
2 2 × 5 1 0 5 5 × 2 1 0

1 1 5 2 5 – 2 3
Therefore, – –
7 2 5 1 0 1 0 1 0 1 0

3 5
Example 6 : Let us now subtract from .
4 6
Solution : We have, LCM (4, 6) = 12 [Remember we have done this to
find equivalent fractions].
5 3 5 2 3 3 1 0 9 1
Therefore, – – –
6 4 6 2 4 3 1 2 1 2 1 2

2 3
1. Add and .
5 7
2 5
2. Subtract from .
5 7

How do we add mixed fractions?

Mixed fractions can be seen to have two parts.
 Fractions
205
They can be either written as a whole part plus a proper fraction or entirely
as an improper fraction. In either case for adding such fractions the
denominators of the proper parts or of the entire improper fraction have to
be made the same.

4 5
Example 7 : Add 2 and 3
5 6

4 5 4 5 4 5
Solution : 2 3 2 3 5
5 6 5 6 5 6
Here again we need to make equivalent fractions. You can
use any of the methods that you have learnt.
4 2 4 4 × 6
Equivalent fraction for,
5 3 0 5 × 6

5 2 5 5 × 5
and for
6 3 0 6 × 5 7
4 5
The sum is therefore 5
5 6

1
2 4 2 5 4 9 3 0 4 9
= 5 5
3 0 3 0
3 0
1 9

1 9 1 9
=5 1 =6
3 0 3 0
Think, Discuss and Write
Can you find some other procedures to solve this problem?
2 1
Example 8 : Find 4 2
5 5
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206
Solution : The whole numbers 4 and 2 and the fractional numbers
2 1
a n d can be subtracted separately. (Note that 4 > 2 and
5 5
2 1
)
5 5
2 1 2 1 1 1
S o , 4 2 4 2 2 2
5 5 5 5 5 5
1 5
Example 9 : Simplify: 8 2 .
4 6
1 5
Solution : Here 8 > 2 but . It can be solved as follows:
4 6
1 38 ×3 4 +5 1 2 × 6 + 5 1 7
8 = = a n d 2 = =
4 4 4 6 6 6

7 Now,
3 3 1 7 3 3 3 1 7 2
(Since LCM (4,5)=12)
4 6 1 2 1 2
9 9 3 4 6 5 5
5
1 2 1 2 1 2

1 2 2 6
1. Add 2 and 3 2. Subtract 2 from 5
5 6 3 7

EXERCISE 7.6
1. Solve
2 1 3 7 4 2 5 1
(a) (b) (c) (d)
3 7 1 0 1 5 9 7 7 3
2 1 4 2 3 1 5 1
(e) (f ) (g) – (h) –
5 6 5 3 4 3 6 3
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207
7 2 1 1 1 1 6 1
(i) – (j) – (k) – (l) –
1 0 5 2 3 2 6 8 3

2 3 1 1 1 1 1 2 2 1
(m) (n) (o) 1 3 (p) 4 3
3 4 2 2 3 6 3 3 3 4

1 6 7 4 1 1 2 2 2
(q) – (r) – (s) 2 – 1 (t) 3 – 1
5 5 3 2 3 3 3 3
2 3
2. Sarita bought metre of ribbon and Lalita metre of ribbon. What was the
5 4
total length of the ribbon they bought?
1 1
3. Naina was given 1 piece of cake and Najma was given 1 piece of cake.
2 3
Find the total amount of cake given to both of them.
5 1 1 1 1 1
4. Fill in the boxes : (a) (b) (c)
8 4 5 2 2 6 7
5. Complete the addition – subtraction box.
+

2 4 1 1
3 3 2 3

– 1 2 – 1 1
(a) 3 3 (b) 3 4

7 1
6. A piece of wire metre long broke into two pieces. One piece was metre
8 4
long. How long is the other piece?
9
7. Nandini’s house is km from her school. She walked some distance and then
1 0
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208
1
took a bus forkm to reach the school. How far did she walk?
2
8. Asha and Samuel have bookshelves of the same size. Asha’s
5 2
shelf is full of books and Samuel’s shelf is full. Whose
6 5
bookshelf is more full? By what fraction?
1 7
9. Jaidev takes 2 minutes to walk across the school ground. Rahul takes
5 4
minutes to do the same. Who takes less time and by what fraction?

What have we discussed?

1. (a) A fraction is a number representing a part of a whole or is an operation on
a number. 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.
7 5
2. In , 5 is called the numerator and 7 is called the denominator. Denominator
7
and numerator can be identified for any fraction.
3. Fractions can be shown on a number line. Any 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 bigger than the denominator are called improper fractions.
Improper fraction can be written as a combination of a whole and a part, and
are then called mixed fractions.
5. Two fractions are said to be equivalent if they represent the same quantity. Each
proper or improper fraction has infinite 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.