MATHEMATICS 24EXAMPLE 2 Ramesh solved part of an exercise while Seema solved of it. Who75solved lesser part? SOLUTION In order to find who solved lesser part of the exercise, let us compare 24and 5.7428Converting them to like fractions we have, 210 , = .=735 535 10 28Since10 < 28 , so < .35 35 24Thus, < .75Ramesh solved lesser part than Seema. 13EXAMPLE 3 Sameera purchased 3 kg apples and 4 kg oranges. What is the24total weight of fruits purchased by her? ⎛ 13⎞SOLUTION The total weight of the fruits =⎜3 + 4 ⎟ kg⎝ 24⎠ ⎛ 719⎞⎛14 19⎞ = ⎜+ ⎟ kg =⎜ +⎟ kg⎝ 24 ⎠⎝ 44 ⎠ 33 1 = kg = 8 kg44 24EXAMPLE 4 Suman studies for 5 hours daily. She devotes 2 hours of her time35for Science and Mathematics. How much time does she devote for other subjects? 2 17SOLUTION Total time of Suman’s study = 5 h = h334 14Time devoted by her for Science and Mathematics = 2 5 = 5 h MATHEMATICS 37. Ritu ate part of an apple and the remaining apple was eaten by her brother Somu.5How much part of the apple did Somu eat? Who had the larger share? By how much? 78. Michael finished colouring a picture in hour. Vaibhav finished colouring the same123picture in hour. Who worked longer? By what fraction was it longer?42.3 MULTIPLICATION OF FRACTIONS You know how to find the area of a rectangle. It is equal to length × breadth. If the length and breadth of a rectangle are 7 cm and 4 cm respectively, then what will be its area? Its area would be 7 × 4 = 28 cm2. 1What will be the area of the rectangle if its length and breadth are 7 2 cm and 1 1 1157 153 cm respectively? You will say it will be 7 × 3 = × 2 cm2. The numbers22222 7and are fractions. To calculate the area of the given rectangle, we need to know how to2multiply fractions. We shall learn that now. 2.3.1 Multiplication of a Fraction by a Whole Number 1Observe the pictures at the left (Fig 2.1). Each shaded part is part of4a circle. How much will the two shaded parts represent together? They 11 1will represent + = 2× .444 Fig 2.1 Combining the two shaded parts, we get Fig 2.2 . What part of a circle does the 2shaded part in Fig 2.2 represent? It represents part of a circle .4Fig 2.2 FRACTIONS AND DECIMALS 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 12× 4 = 2 4 . Can you now tell what this picture will represent? (Fig 2.4)Fig 2.4 And this? (Fig 2.5) Fig 2.5 1Let us now find 3× . 2 1 111 3 We have 3× = ++= 2222 2 1 1 1 1+1+1 3×1 3We also have ++ = = = 2222 22 1 3×1 3 So 3× = = 222 2 2×5Similarly 3×5 = 3 = ? 23Can you tell 3×7 =? 4×5 =? 122 3 The fractions that we considered till now, i.e., ,, and were proper fractions.2375FRACTIONS AND DECIMALS 124. Shade: (i) of the circles in box (a) (ii) of the triangles in box (b)233(iii) of the squares in box (c).5(a) (b) (c) 5. Find: 12(a) of (i) 24 (ii) 46 (b) of (i) 18 (ii) 2723 34(c) of (i) 16 (ii) 36 (d) of (i) 20 (ii) 35456. Multiply and express as a mixed fraction : 13 1× 56 (c) ×(a) 35 (b) × 72 54 4 11 2×(d) 46 (e) 3 ×6 (f) 3 ×8 34 5 132 5527. Find: (a) of (i) 2 (ii) 4 (b) of (i) 3 (ii) 9 8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that 249863 2contained 5 litres of water. Vidya consumed of the water. Pratap consumed the5remaining 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? 9Each part will be of the strip. She took one part and divided it in two equal parts by4MATHEMATICS Fig 2.8 A Fig 2.9 19folding the part once. What will one of the pieces represent? It will represent of 4 or219 × 4.219Let us now see how to find the product of two fractions like × . 24 11To do this we first learn to find the products like × . 23 1(a) How do we find of a whole? We divide the whole in three equal parts. Each of3 1the three parts represents of the whole. Take one part of these three parts, and3 shade it as shown in Fig 2.8. 11(b) How will you find of this shaded part? Divide this one-third ( ) shaded part into23 11 11two equal parts. Each of these two parts represents of i.e., × (Fig 2.9).2323 11Take out 1 part of these two and name it ‘A’. ‘A’ represents × 3.21(c) What fraction is ‘A’of the whole? For this, divide each of the remaining parts also3in two equal parts. How many such equal parts do you have now? There are six such equal parts. ‘A’is one of these parts. How did we decide that ‘A’was of the whole? The whole was divided in 6 = 2 × 3 So, ‘A’ is 1 6 of the whole. Thus, 1 2 × 1 3 = 1 6 . 1 6parts and 1 = 1 × 1 part was taken out of it. 111 1×1 Thus, × = = 2362×3 1 1 1×1 or × = 232×3 MATHEMATICS 2. Multiply and reduce to lowest form (if possible) : (i) 2 3 × 2 2 3 (ii) 27 79× (iii) 36× 84 (iv) 93 55 × (v) 115 38 × (vi) 11 3 210 × (vii) 412 57 × 3. Multiply the following fractions: (i) 2 15 5 4 × (ii) 276 59 × (iii) 3 15 2 3 × (iv) 5 32 6 7 × (v) 243 57 × (vi) 32 3 5 × (vii) 433 75 × 4. Which is greater: (i) 2 7 of 3 4 or 3 5 of 5 8 (ii) 1 2 of 6 7 or 2 3 of 3 7 5. Saili plants 4 saplings, in a row, in her garden. The distance between two adjacent 3saplings is m. Find the distance between the first and the last sapling.4 36. Lipika reads a book for 1 hours everyday. She reads the entire book in 6 days.4How many hours in all were required by her to read the book? 37. A car runs 16 km using 1 litre of petrol. How much distance will it cover using 2 4 litres of petrol. 2 10 = .8. (a) (i) Provide the number in the box , such that × 3 30 (ii) The simplest form of the number obtained in is _____.3 24 = .(b) (i) Provide the number in the box , such that × 5 75 (ii) The simplest form of the number obtained in is _____. 2.4 DIVISION OF FRACTIONS John has a paper strip of length 6 cm. He cuts this strip in smaller strips of length 2 cm each. You know that he would get 6 ÷ 2 =3 strips. FRACTIONS AND DECIMALS 3John cuts another strip of length 6 cm into smaller strips of length cm each. How2 3many strips will he get now? He will get 6 ÷ strips.215 3A paper strip of length cm can be cut into smaller strips of length cm each to give22 15 3 ÷ pieces.22So, we are required to divide a whole number by a fraction or a fraction by another fraction. Let us see how to do that. 2.4.1 Division of Whole Number by a Fraction 1Let us find 1÷.2 We divide a whole into a number of equal parts such that each part is half of the whole. 11The number of such half ( ) parts would be 1÷ . Observe the figure (Fig 2.11). How22 many half parts do you see? There are two half parts. 12 12So, 1 ÷ = 2. Also, 1× = 1 × 2 = 2. Thus, 1 ÷ = 1 ×21 21 11Similarly, 3 ÷= number of parts obtained when each of the 3 whole, are divided44 1into equal parts = 12 (From Fig 2.12)4 MATHEMATICS Reciprocal of a fraction 2The number can be obtained by interchanging the numerator and denominator of1 113 1or by inverting . Similarly, is obtained by inverting 3.2 21 Let us first see about the inverting of such numbers.Observe these products and fill in the blanks : 17 7 × = 1 54 45 × = -------- 1 9 9 × = ----- 2 7 × -------= 1 23 32 × = 23 32 × × = 6 6 = 1 ------5 9 × = 1 Multiply five more such pairs. The non-zero numbers whose product with each other is 1, are called the 59 95 reciprocals of each other. So reciprocal of is and the reciprocal of is . What9559 12is the receiprocal of ? ?97 23 You will see that the reciprocal of is obtained by inverting it. You get 2.3 (i) Will the reciprocal of a proper fraction be again a proper fraction? (ii) Will the reciprocal of an improper fraction be again an improper fraction? Therefore, we can say that 12 11 ÷ 2 =1×1 = 1× reciprocal of 2. 14 13 ÷ 4 =3×1 = 3× reciprocal of 4 . 1 3 ÷ = ------= ----------------------.23 34So, 2 ÷ = 2 × reciprocal of = 2× .443 25 ÷ = 5 × -------------------= 5 × ------------9MATHEMATICS 4. Express in kg: (i) 200 g (ii) 3470 g (iii) 4 kg 8 g 5. Write the following decimal numbers in the expanded form: (i) 20.03 (ii) 2.03 (iii) 200.03 (iv) 2.034 6. Write the place value of 2 in the following decimal numbers: (i) 2.56 (ii) 21.37 (iii) 10.25 (iv) 9.42 (v) 63.352. 7. Dinesh went from place A to place B and from there to place C. A is 7.5 km from B and B is 12.7 km from C.Ayub went from placeAto place D and from there to place C. D is 9.3 km from A and C is 11.8 km from D. Who travelled more and by how much? 8. Shyama bought 5 kg 300 g apples and 3 kg 250 g mangoes. Sarala bought 4 kg 800 g oranges and 4 kg 150 g bananas. Who bought more fruits? 9. How much less is 28 km than 42.6 km? 2.6 MULTIPLICATION OF DECIMAL NUMBERS Reshma purchased 1.5kg vegetable at the rate of ` 8.50 per kg. How much money should she pay? Certainly it would be ` (8.50 × 1.50). Both 8.5 and 1.5 are decimal numbers. So, we have come across a situation where we need to know how to multiply two decimals. Let us now learn the multiplication of two decimal numbers. First we find 0.1 × 0.1. 1 11 1×1 1Now, 0.1 = . So, 0.1 ×0.1 = × = = = 0.01.10 10 1010×10100Let us see it’s pictorial representation (Fig 2.13) 1 The fraction represents 1 part out of 10 equal parts.101 The shaded part in the picture represents 10 . th part into 10 equal parts and take one part We know that, 1 10 1× 10 means 1 10 of 1 10 . So, divide this 1 10 out of it. Fig 2.13 FRACTIONS AND DECIMALS Thus, we have, (Fig 2.14). Fig 2.14 1 The dotted square is one part out of 10 of the th part. That is, it represents10 11× or 0.1 × 0.1.10 10Can the dotted square be represented in some other way? How many small squares do you find in Fig 2.14? There are 100 small squares. So the dotted square represents one out of 100 or 0.01. Hence, 0.1 × 0.1 = 0.01. Note that 0.1 occurs two times in the product. In 0.1 there is one digit to the right of the decimal point. In 0.01 there are two digits (i.e., 1 + 1) to the right of the decimal point. Let us now find 0.2 × 0.3. 23 We have, 0.2 ×0.3 = ×10 10 11 As we did for , let us divide the square into 1010 10 3 equal parts and take three parts out of it, to get . Again10 divide each of these three equal parts into 10 equal parts and 23take two from each. We get × . 10 10 23 The dotted squares represent × or 0.2 × 0.3. (Fig 2.15)10 10Since there are 6 dotted squares out of 100, so they also reprsent 0.06. Fig 2.15 FRACTIONS AND DECIMALS Take 31.5 ÷ 10 = 3.15. In 31.5 and 3.15, the digits are same i.e., 3, 1, and 5 but the decimal point has shifted in the quotient. To which side and by how many digits? The decimal point has shifted to the left by one place. Note that 10 has one zero over 1. Consider now 31.5 ÷ 100 = 0.315. In 31.5 and 0.315 the digits are same, but what about the decimal point in the quotient? It has shifted to the left by two places. Note that 100 has two zeros over1. So we can say that, while dividing a number by 10, 100 or 1000, the digits of the number and the quotient are same but the decimal point in the quotient shifts to the left by as many places as there are zeros over 1. Using this observation let us now quickly find: 2.38 ÷ 10 = 0.238, 2.38 ÷ 100 = 0.0238, 2.38 ÷ 1000 = 0.00238 2.7.2 Division of a Decimal Number by a Whole Number Let us find 6.4 . Remember we also write it as 6.4 ÷ 2. 2 64 641So, 6.4 ÷ 2 = ÷ 2 = × as learnt in fractions..10102 64 ×1 × 1164 641 32 = ==× × 32 == 3210 × 2 10 × 2 10 2= 10 10 . Or, let us first divide 64 by 2. We get 32. There is one digit to the right of the decimal point in 6.4. Place the decimal in 32 such that there would be one digit to its right. We get 3.2 again. To find 19.5 ÷ 5, first find 195 ÷5. We get 39. There is one digit to the right of the decimal point in 19.5. Place the decimal point in 39 such that there would be one digit to its right. You will get 3.9. 1296 1296 1 1 1296 1 Now, 12.96 ÷ 4 = ÷4 = ×4= × = ×324 = 3.24100 100 100 4 100 Or, divide 1296 by 4. You get 324. There are two digits to the right of the decimal in 12.96. Making similar placement of the decimal in 324, you will get 3.24. Note that here and in the next section, we have considered only those divisions in which, ignoring the decimal, the number would be completely divisible by another number to give remainder zero. Like, in 19.5 ÷ 5, the number 195 when divided by 5, leaves remainder zero. However, there are situations in which the number may not be completely divisible by another number, i.e., we may not get remainder zero. For example, 195 ÷ 7. We deal with such situations in later classes. Thus, 40.86 ÷ 6 = 6.81 MATHEMATICS 5. (a) The product of two proper fractions is less than each of the fractions that are multiplied. (b) The product of a proper and an improper fraction is less than the improper fraction and greater than the proper fraction. (c) The product of two imporper fractions is greater than the two fractions. 6. A reciprocal of a fraction is obtained by inverting it upside down. 7. We have seen how to divide two fractions. (a) While dividing a whole number by a fraction, we multiply the whole number with the reciprocal of that fraction. 3 510For example, 2 ÷=2× = 5 33 (b) While dividing a fraction by a whole number we multiply the fraction by the reciprocal of the whole number. 2 1 2For example, 2 ÷=7× = 3 37 21 (c) While dividing one fraction by another fraction, we multuiply the first fraction by 25 2714the reciprocal of the other. So, ÷=× = .37 3515 8. We also learnt how to multiply two decimal numbers. While multiplying two decimal numbers, first multiply them as whole numbers. Count the number of digits to the right of the decimal point in both the decimal numbers.Add the number of digits counted. Put the decimal point in the product by counting the digits from its rightmost place. The count should be the sum obtained earlier. For example, 0.5 × 0.7 = 0.35 9. To multiply a decimal number by 10, 100 or 1000, we move the decimal point in the number to the right by as many places as there are zeros over 1. Thus 0.53 × 10 = 5.3, 0.53 × 100 = 53, 0.53 × 1000 = 530 10. We have seen how to divide decimal numbers. (a) To divide a decimal number by a whole number, we first divide them as whole numbers. Then place the decimal point in the quotient as in the decimal number. For example, 8.4 ÷ 4 = 2.1 Note that here we consider only those divisions in which the remainder is zero. (b) To divide a decimal number by 10, 100 or 1000, shift the digits in the decimal number to the left by as many places as there are zeros over 1, to get the quotient. So, 23.9 ÷ 10 = 2.39,23.9 ÷ 100 = 0 .239, 23.9 ÷ 1000 = 0.0239 (c) While dividing two decimal numbers, first shift the decimal point to the right by equal number of places in both, to convert the divisor to a whole number. Then divide. Thus, 2.4 ÷ 0.2 = 24 ÷ 2 = 12.

### 2.1 Introduction

You have learnt fractions and decimals in earlier classes. The study of fractions included 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 number line and ordering of fractions.

Our study of decimals included, their comparison, their representation on the number line and their addition and subtraction.

We shall now learn multiplication and division of fractions as well as of decimals.

### 2.2 How Well have You Learnt about Fractions?

A proper fraction is a fraction that represents a part of a whole. Is a proper fraction? Which is bigger, the numerator or the denominator?

An improper fraction is a combination of whole and a proper fraction. Is an improper fraction? Which is bigger here, the numerator or the denominator?

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

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

**Example 1** Write five equivalent fractions of .

Solution One of the equivalent fractions of is

. Find the other four.

**Example 2** Ramesh solved part of an exercise while Seema solved of it. Who solved lesser part?

Solution In order to find who solved lesser part of the exercise, let us compare and .

Converting them to like fractions we have, , .

Since 10 < 28 , so .

Thus, .

Ramesh solved lesser part than Seema.

**Example 3** Sameera purchased kg apples and kg oranges. What is the total weight of fruits purchased by her?

Solution The total weight of the fruits

=

=

Example 4 Suman studies for hours daily. She devotes hours of her time for Science and Mathematics. How much time does she devote for other subjects?

Solution Total time of Suman’s study = h = h

Time devoted by her for Science and Mathematics = = h

Thus, time devoted by her for other subjects = h

= = = h = h

### Exercise 2.1

**1.** Solve:

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

(v) (vi) (vii)

**2.** Arrange the following in descending order:

(i) (ii) .

**3.** In a “magic square”, the sum of the numbers in each row, in each column and along the diagonals is the same. Is this a magic square?

**4.** A rectangular sheet of paper is cm long and cm wide. 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 cm wide. To fit in the frame the picture cannot be more than cm wide. How much should the picture be trimmed?

**7.** Ritu ate 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?

**8.** Michael finished colouring a picture in hour. Vaibhav finished colouring the same picture in hour. Who worked longer? By what fraction was it longer?

### 2.3 Multiplication of Fractions

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

What will be the area of the rectangle if its length and breadth are cm and

cm respectively? You will say it will be × = × cm2. The numbers and are fractions. To calculate the area of the given rectangle, we need to know how to multiply fractions. We shall learn that now.

#### 2.3.1 Multiplication of a Fraction by a Whole Number

Observe the pictures at the left (Fig 2.1). Each shaded part is part of a circle. How much will the two shaded parts represent together? They will represent = .

**Fig 2.1**

Combining the two shaded parts, we get Fig 2.2 . What part of a circle does the shaded part in Fig 2.2 represent? It represents part of a circle .

**Fig 2.2**

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 = .

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

**Fig 2.4**

And this? (Fig 2.5)

**Fig 2.5**

Let us now find .

We have =

We also have

So = =

Similarly = = ?

Can you tell = ?

The fractions that we considered till now, i.e., and were proper fractions.

For improper fractions also we have,

= =

Try, = ? = ?

*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) (b) (c) (d)

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

**2.** Represent pictorially :

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

Therefore, = = = .

Similarly, = = ?

### Try These

Find: (i)

(ii)

Fraction as an operator ‘of’

Observe these figures (Fig 2.6)

The two squares are exactly similar.

Each shaded portion represents of 1.

**Fig 2.6**

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

Combine the 2 shaded parts. It represents 1.

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

Thus, of 2 = × 2 = 1

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

**Fig 2.7**

Each shaded portion represents of 1.

So, the three shaded portions represent of 3.

Combine the 3 shaded parts.

It represents 1 i.e., .

So, of 3 is . Also, × 3 = .

Thus, of 3 = × 3 = .

So we see that **‘of’ represents multiplication.**

Farida has 20 marbles. Reshma has of the number of marbles what Farida has. How many marbles Reshma has? As, ‘of’ indicates multiplication, so, Reshma has = 4 marbles.

Similarly, we have of 16 is = = 8.

### Try These

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

**Example 5** In a class of 40 students of the total number of studetns like to study English, 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.

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

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

(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 .

### Exercise 2.2

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

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

(c) (d)

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

(i) (ii) (iii)

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

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

(vi) (vii) (viii) (ix) (x)

**4.** Shade: (i) of the circles in box (a) (ii) of the triangles in box (b)

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

**5.** Find:

(a) of (i) 24 (ii) 46 (b) of (i) 18 (ii) 27

(c) of (i) 16 (ii) 36 (d) of (i) 20 (ii) 35

6. Multiply and express as a mixed fraction :

(a) (b) (c)

(d) (e) (f)

7. Find: **(a)** of (i) (ii) **(b)** of (i) (ii)

8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed 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? Each part will be of the strip. She took one part and divided it in two equal parts by folding the part once. What will one of the pieces represent? It will represent of or × .

Let us now see how to find the product of two fractions like × .

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

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

**Fig 2.8**

(b) How will you find of this shaded part? Divide this one-third () shaded part into two equal parts. Each of these two parts represents of i.e., × (Fig 2.9).

Take out 1 part of these two and name it ‘A’. ‘A’ represents × .

A

**Fig 2.9**

(c) What fraction is ‘A’ of the whole? For this, divide each of the remaining parts also in two equal parts. How many such equal parts do you have now?

There are six such equal parts. ‘A’ is one of these parts.

So, ‘A’ is of the whole. Thus, × = .

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

Thus, × = =

or × =

The value of × can be found in a similar way. Divide the whole into two equal parts and then divide one of these parts in three equal parts. Take one of these parts. This will represent × i.e., .

Therefore × = = as discussed earlier.

Hence × = × =

Find × and × ; × and × and check whether you get

× = × ; × = ×

### Try These

Fill in these boxes:

(i) × = = (ii) × = =

(iii) × = = (iv) × = =

**Example 6** Sushant reads part of a book in 1 hour. How much part of the book will he read in hours?

Solution The part of the book read by Sushant in 1 hour = .

So, the part of the book read by him in hours = ×

= × =

Let us now find ×. We know that = × 5 .

So, × = × × 5 = =

Also, = . Thus, × = = .

This is also shown by the figures drawn below. Each of these five equal shapes (Fig 2.10) are parts of five similar circles. Take one such shape. To obtain this shape we first divide a circle in three equal parts. Further divide each of these three parts in two equal parts. One part out of it is the shape we considered. What will it represent? It will represent × = . The total of such parts would be 5 × = .

**Fig 2.10**

Similarly × = = .

We can thus find × as × = = .

So, we find that we multiply two fractions as .

Value of the Products

You have seen that the product of two whole numbers is bigger than each of the two whole numbers. For example, 3 × 4 = 12 and 12 > 4, 12 > 3. What happens to the value of the product when we multiply two fractions?

Let us first consider the product of two proper fractions.

We have,

### Try These

Find: × ; ×

*You will find that when two proper fractions are multiplied, the product is less than each of the fractions. Or, we say the value of the product of two proper fractions is smaller than each of the two fractions.*

Check this by constructing five more examples.

Let us now multiply two improper fractions.

*We find that the product of two improper fractions is greater than each of the two fractions.*

### Try These

Find: × ; × .

*Or, the value of the product of two improper fractions is more than each of the two fractions.*

Construct five more examples for yourself and verify the above statement.

Let us now multiply a proper and an improper fraction, say and .

We have × = . Here, < and >

The product obtained is less than the improper fraction and greater than the proper fraction involved in the multiplication.

Check it for × , × .

### Exercise 2.3

**1.** Find:

(i) of (a) (b) (c)

(ii) of (a) (b) (c)

**2.** Multiply and reduce to lowest form (if possible) :

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

(v) (vi) (vii)

**3.** Multiply the following fractions:

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

(v) (vi) (vii)

**4.** Which is greater:

(i) of or of (ii) of or of

**5.** Saili plants 4 saplings, in a row, in her garden. The distance between two adjacent saplings is m. Find the distance between the first and the last sapling.

**6.** Lipika reads a book for hours everyday. She reads the entire book in 6 days. How many hours in all were required by her to read the book?

**7.** A car runs 16 km using 1 litre of petrol. How much distance will it cover using litres of petrol.

**8.** (a) (i) Provide the number in the box , such that .

(ii) The simplest form of the number obtained in is _____.

(b) (i) Provide the number in the box , such that .

(ii) The simplest form of the number obtained in is _____.

### 2.4 Division of Fractions

John has a paper strip of length 6 cm. He cuts this strip in smaller strips of length 2 cm each. You know that he would get 6 ÷ 2 =3 strips.

John cuts another strip of length 6 cm into smaller strips of length cm each. How many strips will he get now? He will get 6 ÷ strips.

A paper strip of length cm can be cut into smaller strips of length cm each to give ÷ pieces.

So, we are required to divide a whole number by a fraction or a fraction by another fraction. Let us see how to do that.

#### 2.4.1 Division of Whole Number by a Fraction

Let us find 1÷.

We divide a whole into a number of equal parts such that each part is half of the whole. The number of such half () parts would be 1÷. Observe the figure (Fig 2.11). How many half parts do you see?

**Fig 2.11**

There are two half parts.

So, 1 ÷ = 2. Also, = 1 × 2 = 2. Thus, 1 ÷ = 1 ×

Similarly, 3 ÷ = number of parts obtained when each of the 3 whole, are divided into equal parts = 12 (From Fig 2.12)

**Fig 2.12**

Observe also that,

Find in a similar way, 3 ÷ and .

Reciprocal of a fraction

The number can be obtained by interchanging the numerator and denominator of or by inverting . Similarly, is obtained by inverting .

Let us first see about the inverting of such numbers.

Observe these products and fill in the blanks :

Multiply five more such pairs.

**The non-zero numbers whose product with each other is 1, are called the reciprocals of each other****.** So reciprocal of is and the reciprocal of is . What is the receiprocal of ? ?

You will see that the reciprocal of is obtained by inverting it. You get .

### Think, Discuss and Write

**(i)** Will the reciprocal of a proper fraction be again a proper fraction?

**(ii)** Will the reciprocal of an improper fraction be again an improper fraction?

Therefore, we can say that

1 ÷ = = 1× reciprocal of .

3 ÷ = = 3× reciprocal of .

3 ÷ = ------ = ----------------------.

So, 2 ÷ = 2 × reciprocal of = .

5 ÷ = 5 × ------------------- = 5 × -------------

*Thus, to divide a whole number by any fraction, multiply that whole number by the reciprocal of that fraction.*

### Try These

Find : (i) 7 ÷ (ii) 6 ÷ (iii) 2 ÷

- While dividing a whole number by a mixed fraction, first convert the mixed fraction into improper fraction and then solve it.

Thus, 4 ÷ = 4 ÷ = ? Also, 5 ÷ 3 = 3 ÷ = ?

### Try These

Find: (i) 6 ÷

(ii) 7 ÷

#### 2.4.2 Division of a Fraction by a Whole Number

What will be ÷ 3?

Based on our earlier observations we have: ÷ 3 = ÷ = × = =

So, ÷ 7 = × = ? What is ÷ 6 , ÷ 8 ?

While dividing mixed fractions by whole numbers, convert the mixed fractions into improper fractions. That is,

#### 2.4.3 Division of a Fraction by Another Fraction

We can now find ÷.

÷ = × reciprocal of = × = .

Similarly, × reciprocal of = ? and, ÷ = ?

### Try These

Find: (i) (ii) (iii) (iv)

### Exercise 2.4

**1.** Find:

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

**2.** Find the reciprocal of each of the following fractions. Classify the reciprocals as proper fractions, improper fractions and whole numbers.

(i) (ii) (iii) (iv) (v) (vi) (vii)

**3.** Find:

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

**4.** Find:

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

### 2.5 How Well have you Learnt about Decimal Numbers

You have learnt about decimal numbers in the earlier classes. Let us briefly recall them here. Look at the following table and fill up the blank spaces.

In the table, you wrote the decimal number, given its place-value expansion. You can do the reverse, too. That is, given the number you can write its expanded form. For example,

John has ₹ 15.50 and Salma has ₹ 15.75. Who has more money? To find this we need to compare the decimal numbers 15.50 and 15.75. To do this, we first compare the digits on the left of the decimal point, starting from the leftmost digit. Here both the digits 1 and 5, to the left of the decimal point, are same. So we compare the digits on the right of the decimal point starting from the tenths place. We find that 5 < 7, so we say

15.50 < 15.75. Thus, Salma has more money than John.

If the digits at the tenths place are also same then compare the digits at the hundredths place and so on.

Now compare quickly, 35.63 and 35.67; 20.1 and 20.01; 19.36 and 29.36.

While converting lower units of money, length and weight, to their higher units, we are required to use decimals. For example,

= 0.005 kg, 7 cm = 0.07 m.

Write 75 paise = ₹ ______, 250 g = _____ kg, 85 cm = _____m.

We also know how to add and subtract decimals. Thus, 21.36 + 37.35 is

What is the value of 0.19 + 2.3 ?

The difference 29.35 − 4.56 is

Tell the value of 39.87 − 21.98.

### Exercise 2.5

**1.** Which is greater?

(i) 0.5 or 0.05 (ii) 0.7 or 0.5 (iii) 7 or 0.7

(iv) 1.37 or 1.49 (v) 2.03 or 2.30 (vi) 0.8 or 0.88.

**2.** Express as rupees using decimals :

(i) 7 paise (ii) 7 rupees 7 paise (iii) 77 rupees 77 paise

(iv) 50 paise (v) 235 paise.

**3.** (i) Express 5 cm in metre and kilometre (ii) Express 35 mm in cm, m and km

**4.** Express in kg:

(i) 200 g (ii) 3470 g (iii) 4 kg 8 g

**5.** Write the following decimal numbers in the expanded form:

(i) 20.03 (ii) 2.03 (iii) 200.03 (iv) 2.034

**6.** Write the place value of 2 in the following decimal numbers:

(i) 2.56 (ii) 21.37 (iii) 10.25 (iv) 9.42 (v) 63.352.

**7.** Dinesh went from place A to place B and from there to place C. A is 7.5 km from B and B is 12.7 km from C. Ayub went from place A to place D and from there to place C. D is 9.3 km from A and C is 11.8 km from D. Who travelled more and by how much?

**8.** Shyama bought 5 kg 300 g apples and 3 kg 250 g mangoes. Sarala bought 4 kg 800 g oranges and 4 kg 150 g bananas. Who bought more fruits?

**9.** How much less is 28 km than 42.6 km?

### 2.6 Multiplication of Decimal Numbers

Reshma purchased 1.5kg vegetable at the rate of ` 8.50 per kg. How much money should she pay? Certainly it would be ` (8.50 × 1.50). Both 8.5 and 1.5 are decimal numbers. So, we have come across a situation where we need to know how to multiply two decimals. Let us now learn the multiplication of two decimal numbers.

First we find 0.1 × 0.1.

Let us see it’s pictorial representation (Fig 2.13)

The fraction represents 1 part out of 10 equal parts.

The shaded part in the picture represents .

We know that,

means of . So, divide this th part into 10 equal parts and take one part out of it.

Thus, we have, (Fig 2.14).

**Fig 2.14**

The dotted square is one part out of 10 of the th part. That is, it represents or 0.1 × 0.1.

Can the dotted square be represented in some other way?

How many small squares do you find in Fig 2.14?

There are 100 small squares. So the dotted square represents one out of 100 or 0.01.

Hence, 0.1 × 0.1 = 0.01.

Note that 0.1 occurs two times in the product. In 0.1 there is one digit to the right of the decimal point. In 0.01 there are two digits (i.e., 1 + 1) to the right of the decimal point.

Let us now find 0.2 × 0.3.

We have, 0.2 × 0.3 =

As we did for , let us divide the square into 10 equal parts and take three parts out of it, to get . Again divide each of these three equal parts into 10 equal parts and take two from each. We get .

The dotted squares represent or 0.2 × 0.3. (Fig 2.15)

Since there are 6 dotted squares out of 100, so they also reprsent 0.06.

Thus, 0.2 × 0.3 = 0.06.

Observe that 2 × 3 = 6 and the number of digits to the right of the decimal point in 0.06 is 2 (= 1 + 1).

Check whether this applies to 0.1 × 0.1 also.

Find 0.2 × 0.4 by applying these observations.

While finding 0.1 × 0.1 and 0.2 × 0.3, you might have noticed that first we multiplied them as whole numbers ignoring the decimal point. In 0.1 × 0.1, we found 01 × 01 or 1 × 1. Similarly in 0.2 × 0.3 we found 02 × 03 or 2 × 3.

Then, we counted the number of digits starting from the rightmost digit and moved towards left. We then put the decimal point there. The number of digits to be counted is obtained by adding the number of digits to the right of the decimal point in the decimal numbers that are being multiplied.

Let us now find 1.2 × 2.5.

Multiply 12 and 25. We get 300. Both, in 1.2 and 2.5, there is 1 digit to the right of the decimal point. So, count 1 + 1 = 2 digits from the rightmost digit (i.e., 0) in 300 and move towards left. We get 3.00 or 3.

Find in a similar way 1.5 × 1.6, 2.4 × 4.2.

While multiplying 2.5 and 1.25, you will first multiply 25 and 125. For placing the decimal in the product obtained, you will count 1 + 2 = 3 (Why?) digits starting from the rightmost digit. Thus, 2.5 × 1.25 = 3.225

### Try These

Find 2.7 × 1.35.

1. Find: (i) 2.7 × 4 (ii) 1.8 × 1.2 (iii) 2.3 × 4.35

2. Arrange the products obtained in (1) in descending order.

**Example 7** The side of an equilateral triangle is 3.5 cm. Find its perimeter.

Solution All the sides of an equilateral triangle are equal.

So, length of each side = 3.5 cm

Thus, perimeter = 3 × 3.5 cm = 10.5 cm

**Example 8** The length of a rectangle is 7.1 cm and its breadth is 2.5 cm. What is the area of the rectangle?

Solution Length of the rectangle = 7.1 cm

Breadth of the rectangle = 2.5 cm

Therefore, area of the rectangle = 7.1 × 2.5 cm2 = 17.75 cm2

#### 2.6.1 Multiplication of Decimal Numbers by 10, 100 and 1000

Reshma observed that 2.3 = whereas 2.35 = . Thus, she found that depending on the position of the decimal point the decimal number can be converted to a fraction with denominator 10 or 100. She wondered what would happen if a decimal number is multiplied by 10 or 100 or 1000.

Let us see if we can find a pattern of multiplying numbers by 10 or 100 or 1000.

Have a look at the table given below and fill in the blanks:

Observe the shift of the decimal point of the products in the table. Here the numbers are multiplied by 10,100 and 1000. In 1.76 × 10 = 17.6, the digits are same i.e., 1, 7 and 6. Do you observe this in other products also? Observe 1.76 and 17.6. To which side has the decimal point shifted, right or left? The decimal point has shifted to the right by one place. Note that 10 has one zero over 1.

In 1.76×100 = 176.0, observe 1.76 and 176.0. To which side and by how many digits has the decimal point shifted? The decimal point has shifted to the right by two places.

Note that 100 has two zeros over one.

Do you observe similar shifting of decimal point in other products also?

So we say, when a decimal number is multiplied by 10, 100 or 1000, the digits in the product are same as in the decimal number but the decimal point in the product is shifted to the right by as, many of places as there are zeros over one.

Based on these observations we can now say

0.07 × 10 = 0.7, 0.07 × 100 = 7 and 0.07 × 1000 = 70.

Can you now tell 2.97 × 10 = ? 2.97 × 100 = ? 2.97 × 1000 = ?

Can you now help Reshma to find the total amount i.e., ₹ 8.50 × 150, that she has to pay?

### Exercise 2.6

**1.** Find:

(i) 0.2 × 6 (ii) 8 × 4.6 (iii) 2.71 × 5 (iv) 20.1 × 4

(v) 0.05 × 7 (vi) 211.02 × 4 (vii) 2 × 0.86

**2.** Find the area of rectangle whose length is 5.7cm and breadth is 3 cm.

**3.** Find:

(i) 1.3 × 10 (ii) 36.8 × 10 (iii) 153.7 × 10 (iv) 168.07 × 10 (v) 31.1 × 100 (vi) 156.1 × 100

(vii) 3.62 × 100 (viii) 43.07 × 100

(ix) 0.5 × 10 (x) 0.08 × 10 (xi) 0.9 × 100 (xii) 0.03 × 1000

**4.** A two-wheeler covers a distance of 55.3 km in one litre of petrol. How much distance will it cover in 10 litres of petrol?

**5.** Find:

(i) 2.5 × 0.3 (ii) 0.1 × 51.7 (iii) 0.2 × 316.8 (iv) 1.3 × 3.1

(v) 0.5 × 0.05 (vi) 11.2 × 0.15 (vii) 1.07 × 0.02

(viii) 10.05 × 1.05 (ix) 101.01 × 0.01 (x) 100.01 × 1.1

### 2.7 Division of Decimal Numbers

Savita was preparing a design to decorate her classroom. She needed a few coloured strips of paper of length 1.9 cm each. She had a strip of coloured paper of length 9.5 cm. How many pieces of the required length will she get out of this strip? She thought it would be cm. Is she correct?

Both 9.5 and 1.9 are decimal numbers. So we need to know the division of decimal numbers too!

#### 2.7.1 Division by 10, 100 and 1000

Let us find the division of a decimal number by 10, 100 and 1000.

Consider 31.5 ÷ 10.

Let us see if we can find a pattern for dividing numbers by 10, 100 or 1000. This may help us in dividing numbers by 10, 100 or 1000 in a shorter way.

Take 31.5 ÷ 10 = 3.15. In 31.5 and 3.15, the digits are same i.e., 3, 1, and 5 but the decimal point has shifted in the quotient. To which side and by how many digits? The decimal point has shifted to the left by one place. Note that 10 has one zero over 1.

Consider now 31.5 ÷ 100 = 0.315. In 31.5 and 0.315 the digits are same, but what about the decimal point in the quotient? It has shifted to the left by two places. Note that 100 has two zeros over1. Using this observation let us now quickly find:

### Try These

Find: (i) 235.4 ÷ 10

(ii) 235.4 ÷100

(iii) 235.4 ÷ 1000

So we can say that, *while dividing a number by 10, 100 or 1000, the digits of the number and the quotient are same but the decimal point in the quotient shifts to the left by as many places as there are zeros over 1.* Using this observation let us now quickly find: 2.38 ÷ 10 = 0.238, 2.38 ÷ 100 = 0.0238, 2.38 ÷ 1000 = 0.00238

#### 2.7.2 Division of a Decimal Number by a Whole Number

Let us find. Remember we also write it as 6.4 ÷ 2.

### Try These

(i) 35.7 ÷ 3 = ?;

(ii) 25.5 ÷ 3 = ?

Or, let us first divide 64 by 2. We get 32. There is one digit to the right of the decimal point in 6.4. Place the decimal in 32 such that there would be one digit to its right. We get 3.2 again.

To find 19.5 ÷ 5, first find 195 ÷5. We get 39. There is one digit to the right of the decimal point in 19.5. Place the decimal point in 39 such that there would be one digit to its right. You will get 3.9.

### Try These

(i) 43.15 ÷ 5 = ?;

(ii) 82.44 ÷ 6 = ?

Or, divide 1296 by 4. You get 324. There are two digits to the right of the decimal in 12.96. Making similar placement of the decimal in 324, you will get 3.24.

Note that here and in the next section, we have considered only those divisions in which, ignoring the decimal, the number would be completely divisible by another number to give remainder zero. Like, in 19.5 ÷ 5, the number 195 when divided by 5, leaves remainder zero.

### Try These

Find: (i) 15.5 ÷ 5

(ii) 126.35 ÷ 7

However, there are situations in which the number may not be completely divisible by another number, i.e., we may not get remainder zero. For example, 195 ÷ 7. We deal with such situations in later classes.

**Example 9** Find the average of 4.2, 3.8 and 7.6.

Solution The average of 4.2, 3.8 and 7.6 is = 5.2.

#### 2.7.3 Division of a Decimal Number by another Decimal Number

Let us find i.e., 25.5 ÷ 0.5.

We have Thus, 25.5 ÷ 0.5 = 51

What do you observe? For , we find that there is one digit to the right of the decimal in 0.5. This could be converted to whole number by dividing by 10. Accordingly 25.5 was also converted to a fraction by dividing by 10.

Or, we say the decimal point was shifted by one place to the right in 0.5 to make it 5. So, there was a shift of one decimal point to the right in 25.5 also to make it 255.

Thus, 22.5 ÷ 1.5 = = = 15

Find and in a similar way.

### Try These

Find: (i) (ii) (iii)

Let us now find 20.55 ÷ 1.5.

We can write it is as 205.5 ÷ 15, as discussed above. We get 13.7. Find , .

Consider now, . We can write it as (How?) and we get the quotient as 134.9. How will you find ? We know that 27 can be written as 27.00.

So, = 900

**Example 10 ** Each side of a regular polygon is 2.5 cm in length. The perimeter of the polygon is 12.5cm. How many sides does the polygon have?

Solution The perimeter of a regular polygon is the sum of the lengths of all its equal sides = 12.5 cm.

Length of each side = 2.5 cm. Thus, the number of sides = = = 5

The polygon has 5 sides.

**Example 1****1 ** A car covers a distance of 89.1 km in 2.2 hours. What is the average distance covered by it in 1 hour?

Solution Distance covered by the car = 89.1 km.

Time required to cover this distance = 2.2 hours.

So distance covered by it in 1 hour = = = 40.5 km.

### Exercise 2.7

**1.** Find:

(i) 0.4 ÷ 2 (ii) 0.35 ÷ 5 (iii) 2.48 ÷ 4 (iv) 65.4 ÷ 6

(v) 651.2 ÷ 4 (vi) 14.49 ÷ 7 (vii) 3.96 ÷ 4 (viii) 0.80 ÷ 5

**2.** Find:

(i) 4.8 ÷ 10 (ii) 52.5 ÷ 10 (iii) 0.7 ÷ 10

(iv) 33.1 ÷ 10 (v) 272.23 ÷ 10 (vi) 0.56 ÷ 10 (vii) 3.97 ÷10

**3.** Find:

(i) 2.7 ÷ 100 (ii) 0.3 ÷ 100 (iii) 0.78 ÷ 100

(iv) 432.6 ÷ 100 (v) 23.6 ÷100 (vi) 98.53 ÷ 100

**4.** Find:

(i) 7.9 ÷ 1000 (ii) 26.3 ÷ 1000 (iii) 38.53 ÷ 1000

(iv) 128.9 ÷ 1000 (v) 0.5 ÷ 1000

**5.** Find:

(i) 7 ÷ 3.5 (ii) 36 ÷ 0.2 (iii) 3.25 ÷ 0.5

(iv) 30.94 ÷ 0.7 (v) 0.5 ÷ 0.25 (vi) 7.75 ÷ 0.25 (vii) 76.5 ÷ 0.15 (viii) 37.8 ÷ 1.4 (ix) 2.73 ÷ 1.3 (ix) 2.73 ÷ 1.3

**6.** A vehicle covers a distance of 43.2 km in 2.4 litres of petrol. How much distance will it cover in one litre of petrol?

**What have We Discussed?**

**1.** We have learnt about fractions and decimals alongwith the operations of addition and subtraction on them, in the earlier class.

**2.** We now study the operations of multiplication and division on fractions as well as on decimals.

**3.** We have learnt how to multiply fractions. Two fractions are multiplied by multiplying their numerators and denominators seperately and writing the product as .

**4.** A fraction acts as an operator ‘of’. For example, of 2 is × 2 = 1.

**5.** (a) The product of two proper fractions is less than each of the fractions that are multiplied.

(b) The product of a proper and an improper fraction is less than the improper fraction and greater than the proper fraction.

(c) The product of two imporper fractions is greater than the two fractions.

**6.** A reciprocal of a fraction is obtained by inverting it upside down.

**7.** We have seen how to divide two fractions.

(a) While dividing a whole number by a fraction, we multiply the whole number with the reciprocal of that fraction.

For example,

(b) While dividing a fraction by a whole number we multiply the fraction by the reciprocal of the whole number.

For example,

(c) While dividing one fraction by another fraction, we multuiply the first fraction by the reciprocal of the other. So,

**8.** We also learnt how to multiply two decimal numbers. While multiplying two decimal numbers, first multiply them as whole numbers. Count the number of digits to the right of the decimal point in both the decimal numbers. Add the number of digits counted. Put the decimal point in the product by counting the digits from its rightmost place. The count should be the sum obtained earlier.

For example, 0.5 × 0.7 = 0.35

**9.** To multiply a decimal number by 10, 100 or 1000, we move the decimal point in the number to the right by as many places as there are zeros over 1.

Thus 0.53 × 10 = 5.3, 0.53 × 100 = 53, 0.53 × 1000 = 530

**10.** We have seen how to divide decimal numbers.

(a) To divide a decimal number by a whole number, we first divide them as whole numbers. Then place the decimal point in the quotient as in the decimal number.

For example, 8.4 ÷ 4 = 2.1

Note that here we consider only those divisions in which the remainder is zero.

(b) To divide a decimal number by 10, 100 or 1000, shift the digits in the decimal number to the left by as many places as there are zeros over 1, to get the quotient.

So, 23.9 ÷ 10 = 2.39, 23.9 ÷ 100 = 0 .239, 23.9 ÷ 1000 = 0.0239

(c) While dividing two decimal numbers, first shift the decimal point to the right by equal number of places in both, to convert the divisor to a whole number. Then divide. Thus, 2.4 ÷ 0.2 = 24 ÷ 2 = 12.