Fractions and Chapter 2


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FRACTIONS AND DECIMALS

29

Chapter 2

Fractions and Decimals

d 2.1 INTRODUCTION T he You have learnt fractions and decimals in earlier classes. The study of fractions included R lis proper, improper and mixed fractions as well as their addition and subtraction. We also
studied comparison of fractions, equivalent fractions, representation of fractions on the
E b number line and ordering of fractions. C u Our study of decimals included, their comparison, their representation on the number
N ep line and their addition and subtraction. r We shall now learn multiplication and division of fractions as well as of decimals.
© be 2.2 HOW WELL HAVE YOU LEARNT ABOUT FRACTIONS? to A proper fraction is a fraction that represents a part of a whole. Is 7 a proper fraction?
t 4 o Which is bigger, the numerator or the denominator?
n An improper fraction is a combination of whole and a proper fraction. Is 7 an 4
improper fraction? Which is bigger here, the numerator or the denominator?

The improper fraction 7 can be written as 1 3 . This is a mixed fraction.

4

4

Can you write five examples each of proper, improper and mixed fractions?

EXAMPLE 1 Write five equivalent fractions of 3 .
5

SOLUTION

One of the equivalent fractions of 3 is 5

3 = 3× 2 = 6 . Find the other four. 5 5× 2 10

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EXAMPLE 2

Ramesh solved 2 part of an exercise while Seema solved 4 of it. Who

7

5

solved lesser part?

SOLUTION In order to find who solved lesser part of the exercise, let us compare

24 7 and 5 .

Converting them to like fractions we have, 2 = 10 , 4 = 28 . 7 35 5 35

Since10 < 28 , so 10 < 28 . 35 35

T hed Thus,

2<4 . 75

R lis Ramesh solved lesser part than Seema.

NCEepub EXAMPLE 3

Sameera purchased 3 1 kg apples and 4 3 kg oranges. What is the

2

4

total weight of fruits purchased by her?

© be r SOLUTION

 1 3 The total weight of the fruits =  3 2 + 4 4 kg

to  7 19 14 19 t =  2 + 4  kg =  4 + 4  kg

no = 33 kg = 8 1 kg

4

4

EXAMPLE 4

Suman studies for 5 2 hours daily. She devotes 2 4 hours of her time

3

5

for Science and Mathematics. How much time does she devote for

other subjects?

SOLUTION

Total time of Suman’s study = 5 2 h = 17 h 3 3

Time devoted by her for Science and Mathematics = 2 4 = 14 h 5 5

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Thus, time devoted by her for other subjects = 137 − 154 h

=

 

17

×

5



14

×

3

 

h

 15 15 

=

 851–542

h

= 43 h = 213 h

15

15

EXERCISE 2.1

1. Solve:

d (i) 2 − 3 5

(ii) 4 + 7 8

(iii) 3 + 2 57

(iv) 9 − 4 11 15

T he (v) 7 + 2 + 3 R lis 10 5 2

(vi) 2 2 + 3 1 32

(vii) 8 1 − 3 5 28

2. Arrange the following in descending order:

CE ub (i) 2 , 2 , 8 (ii) 1 , 3 , 7 .

N p 9 3 21

5 7 10

e 3. In a “magic square”, the sum of the numbers in each row, in each column and along

r the diagonals is the same. Is this a magic square?

© be 4 9 2
11 11 11
to 3 5 7 t 11 11 11 o 8 1 6 n 11 11 11

(Along the first row 4 + 9 + 2 = 15 ). 11 11 11 11

4. A rectangular sheet of paper is 12 1 cm long and 10 2 cm wide.

2

3

Find its perimeter.

5. Find the perimeters of (i) ∆ ABE (ii) the rectangle BCDE in this

figure. Whose perimeter is greater?

6. Salil wants to put a picture in a frame. The picture is 7 3 cm wide. 5

5 cm

3 3 cm

2 2 3 cm 5

4

7 cm 6

To fit in the frame the picture cannot be more than 7 3 cm wide. How much should 10
the picture be trimmed?

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3 7. Ritu ate 5 part of an apple and the remaining apple was eaten by her brother Somu.
How much part of the apple did Somu eat? Who had the larger share? By how much?
7 8. Michael finished colouring a picture in 12 hour.Vaibhav finished colouring the same
3 picture in 4 hour. Who worked longer? By what fraction was it longer?

2.3 MULTIPLICATION OF FRACTIONS

d You know how to find the area of a rectangle. It is equal to length × breadth. If the length e and breadth of a rectangle are 7 cm and 4 cm respectively, then what will be its area? Its T h area would be 7 × 4 = 28 cm2.

R lis 1 E b What will be the area of the rectangle if its length and breadth are 7 2 cm and

C u 1

1 1 15 7

15

N p 3 2 cm respectively? You will say it will be 7 2 × 3 2 = 2 × 2 cm2. The numbers 2

re 7 © e and 2 are fractions. To calculate the area of the given rectangle, we need to know how to b multiply fractions. We shall learn that now.

to 2.3.1 Multiplication of a Fraction by a Whole Number ot 1 nObserve the pictures at the left (Fig 2.1). Each shaded part is 4 part of

a circle. How much will the two shaded parts represent together? They

Fig 2.1

will represent 1 + 1 = 2× 1 .

44

4

Combining the two shaded parts, we get Fig 2.2 . What part of a circle does the

2 shaded part in Fig 2.2 represent? It represents 4 part of a circle .

Fig 2.2
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The shaded portions in Fig 2.1 taken together are the same as the shaded portion in Fig 2.2, i.e., we get Fig 2.3.

=

Fig 2.3

or 2× 1 = 2 . 4 4

Can you now tell what this picture will represent? (Fig 2.4)

And this? (Fig 2.5)
Let us now find 3× 1 . 2
We have

d = RT lishe Fig 2.4 NCErepub= © be Fig 2.5 not to 3× 1 = 1 + 1 + 1 = 3
2 222 2

We also have

1 + 1 + 1 = 1+1+1 = 3×1 = 3

222 2

22

So 3× 1 = 3×1 = 3 2 2 2

Similarly

2×5 = 2×5 = ?

3

3

Can you tell

3× 2 = ? 7

4×3 =? 5

1223 3 The fractions that we considered till now, i.e., , , ,and were proper fractions.
2375 5

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For improper fractions also we have, 2 × 5 = 2 × 5 = 10 33 3

Try,

3× 8 = ? 4× 7 = ?

7

5

Thus, to multiply a whole number with a proper or an improper fraction, we

multiply the whole number with the numerator of the fraction, keeping the

denominator same.

TRY THESE

1. Find: (a) 2 × 3
d 7

(b) 9 × 6 7

(c) 3 × 1 8

(d) 13 × 6 11

e If the product is an improper fraction express it as a mixed fraction.

RT lish 2. Represent pictorially : 2× 2 = 4 55

NCE pub TRY THESE re Find: (i) 5× 2 3
© 7 be (ii) 14 × 6
to 9

To multiply a mixed fraction to a whole number, first convert the mixed fraction to an improper fraction and then multiply.

Therefore,

3× 2 5 = 3× 19 = 57 = 8 1 .

7

77 7

Similarly,

2× 4 2 = 2× 22 = ?

5

5

ot Fraction as an operator ‘of ’ n Observe these figures (Fig 2.6)

The two squares are exactly similar.

1 Each shaded portion represents 2 of 1.

1 So, both the shaded portions together will represent 2 of 2.

1 Combine the 2 shaded 2 parts. It represents 1.

1

1

So, we say 2 of 2 is 1. We can also get it as 2 × 2 = 1.

1

1

Thus, 2 of 2 = 2 × 2 = 1

Fig 2.6

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Also, look at these similar squares (Fig 2.7).

1 Each shaded portion represents 2 of 1.

1 So, the three shaded portions represent 2 of 3. Combine the 3 shaded parts.

13 It represents 1 2 i.e., 2 .

1

3

1

3

So, 2 of 3 is 2 . Also, 2 × 3 = 2 .

Fig 2.7

1

1

3

d Thus, 2 of 3 = 2 × 3 = 2 .

e So we see that ‘of’represents multiplication.

RT lish Farida has 20 marbles. Reshma has 1 th of the number of marbles what 5
E b Farida has. How many marbles Reshma has?As, ‘of ’indicates multiplication, NC pu so, Reshma has 1 × 20 = 4 marbles.
e 5

© e r Similarly, we have 1 of 16 is 1 ×16 = 16 = 8.

2

2

2

to b TRY THESE

t 1

1

2

o Can you tell, what is (i) 2 of 10?, (ii) 4 of 16?, (iii) 5 of 25?

n EXAMPLE 5 In a class of 40 students 1 of the total number of studetns like to study 5

2 English, 5 of the total number like to study Mathematics and the remaining students like to study Science.

(i) How many students like to study English?

(ii) How many students like to study Mathematics?

(iii) What fraction of the total number of students like to study Science?

SOLUTION Total number of students in the class = 40.

1 (i) Of these 5 of the total number of students like to study English.

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Thus, the number of students who like to study English = 1 of 40 = 1 × 40 = 8.

5

5

(ii) Try yourself.

(iii) The number of students who like English and Mathematics = 8 + 16 = 24. Thus, the number of students who like Science = 40 – 24 = 16.

Thus, the required fraction is 16 . 40

EXERCISE 2.2

1. Which of the drawings (a) to (d) show :

d (i) 2× 1 5
RT lishe (a)

(ii) 2× 1 2

(iii) 3× 2 3
(b)

(iv) 3× 1 4

NCEepub (c)

(d)

© e r 2. Some pictures (a) to (c) are given below. Tell which of them show:

to b (i) 3× 1 = 3 55

(ii) 2× 1 = 2 33

(iii) 3× 3 = 2 1 44

not =

(a)

(b)

=

(c)

3. Multiply and reduce to lowest form and convert into a mixed fraction:

(i) 7× 3 5

(ii) 4× 1 3

(iii) 2× 6 7

(iv) 5× 2 9

(v) 2 × 4 3

(vi) 5 × 6 2

(vii) 11× 4 7

(viii) 20 × 4 5

(ix) 13× 1 3

(x) 15× 3 5

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1 4. Shade: (i) 2 of the circles in box (a)

2 (ii) 3 of the triangles in box (b)

3 (iii) 5 of the squares in box (c).

(a)

(b)

(c)

5. Find:

ed 1 T h (a) 2 of (i) 24 (ii) 46

2 (b) 3 of (i) 18

(ii) 27

R lis 3 E b (c) 4 of (i) 16 (ii) 36

4 (d) 5 of

C u 6. Multiply and express as a mixed fraction :

(i) 20

(ii) 35

N ep (a) 3×51 r 5

(b) 5× 6 3 4

(c) 7 × 2 1 4

© e (d) 4×61 b 3

(e) 3 1 × 6 4

(f) 3 2 ×8 5

to 7. Find: (a) 1 of (i) 2 3 (ii) 4 2

t 2

4

9

(b) 5 of (i) 3 5

8

6

(ii) 9 2 3

o 8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that

n2
contained 5 litres of water. Vidya consumed 5 of the water. Pratap consumed the

remaining water.

(i) How much water did Vidya drink?

(ii) What fraction of the total quantity of water did Pratap drink?

2.3.2 Multiplication of a Fraction by a Fraction

Farida had a 9 cm long strip of ribbon. She cut this strip into four equal parts. How did she do it? She folded the strip twice. What fraction of the total length will each part represent?
9 Each part will be 4 of the strip. She took one part and divided it in two equal parts by

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Fig 2.8
A
Fig 2.9

19 folding the part once. What will one of the pieces represent? It will represent 2 of 4 or 19 2 × 4.

Let us now see how to find the product of two fractions like 1 × 9 . 2 4

To do this we first learn to find the products like 1 × 1 . 2 3

1 (a) How do we find 3 of a whole? We divide the whole in three equal parts. Each of
d 1 T he the three parts represents 3 of the whole. Take one part of these three parts, and R lis shade it as shown in Fig 2.8.

E b 1

1

C u (b) How will you find 2 of this shaded part? Divide this one-third ( 3 ) shaded part into

N p 1 1 11 re two equal parts. Each of these two parts represents 2 of 3 i.e., 2 × 3 (Fig 2.9).

© e 11 b Take out 1 part of these two and name it ‘A’. ‘A’ represents 2 × 3 .

to 1 t (c) What fraction is ‘A’of the whole? For this, divide each of the remaining 3 parts also
o in two equal parts. How many such equal parts do you have now? n There are six such equal parts. ‘A’ is one of these parts.

1

11 1

So, ‘A’ is 6 of the whole. Thus, 2 × 3 = 6 .

1 How did we decide that ‘A’ was 6 of the whole? The whole was divided in 6 = 2 × 3 parts and 1 = 1 × 1 part was taken out of it.

Thus,

1 1 1 1×1 2 × 3 = 6 = 2×3

1 1 1×1 or 2 × 3 = 2×3

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Fractions and Chapter 2