NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Exercise 2.1

1. Solve:

(i) 2 - 3/5

Answer: 7/5

Explanation: For addition or subtraction, the denominators of the fraction should be equal. It helps is easy calculation.

We can write the above fraction as:

2/1 - 3/5

Multiplying the first fraction by 5, we get:

(2 × 5)/ (1 × 5)

= 10/5

Now subtracting,

10/5 - 3/5

= (10 - 3)/5

= 7/5

(ii) 4 + 7/8

Answer: 39/8

Explanation: For addition or subtraction, the denominators of the fraction should be equal. It helps is easy calculation.

We can write the above fraction as:

4/1 + 7/8

Multiplying the first fraction by 8, we get:

(4 × 8)/ (1 × 8)

= 32/8

Now adding,

32/8 + 7/8

= (32 + 7)/8

= 39/8

In terms of mixed fraction, we can write it as:

4 + 7/8

= 4 7/8

(iii) 3/5 + 2/7

Answer: 31/35

Explanation: For addition or subtraction, the denominators of the fraction should be equal. It helps is easy calculation.

To make the denominators equal, we need to multiply the first fraction by 7 and the second fraction by 5. It is because both the numbers in the denominator does not have any common factor except 1.

Multiplying the first fraction by 7, we get:

(3 × 7)/ (5 × 7)

= 21/35

Multiplying the second fraction by 5, we get:

(2 × 5)/ (7 × 5)

= 10/35

Now adding,

21/35 + 10/35

= (21 + 10)/35

= 31/35

(iv) 9/11 - 4/15

Answer: 91/165

Explanation: For addition or subtraction, the denominators of the fraction should be equal. It helps is easy calculation.

To make the denominators equal, we need to multiply the first fraction by 15 and the second fraction by 11. It is because both the numbers in the denominator does not have any common factor except 1.

Multiplying the first fraction by 15, we get:

(9 × 15)/ (11 × 15)

= 135/165

Multiplying the second fraction by 11, we get:

(4 × 11)/ (15 × 11)

= 44/165

Now subtracting,

135/165 - 44/165

= (135 - 44)/ 165

= 91/165

(v) 7/10 + 2/5 + 3/2

Answer: 13/5

Explanation: For addition or subtraction, the denominators of the fraction should be equal. It helps is easy calculation.

Now, the denominators of the three fractions have some common factors. In such cases, we need to find the LCM, which acts as a common multiple.

LCM (10, 5, 2) = 10

To make the denominators equal, we need to multiply the second fraction by 2 and the third fraction by 5. The first fraction is already present in the correct form with a denominator 10.

Multiplying the second fraction by 2, we get:

(2 × 2)/ (5 × 2)

= 4/10

Multiplying the third fraction by 5, we get:

(3 × 5)/ (2 × 5)

= 15/10

Now adding,

7/10 + 4/10 + 15/10

= (7 + 4 + 15)/10

= 26/10

To convert to its lowest form, divide the fraction by 2 (common factor)

Dividing 26/10 by 2, we get:

= 13/5

We can also write the above fraction in the mixed form.

= 2 3/5

(vi) 2 2/3 + 3 1/2

Answer: 37/6 or 6 1/6

Explanation: 2 2/3 + 3 1/2

The given fraction is a mixed fraction. For easy calculation, we will convert the given fraction to the whole fraction.

2 2/3 = (2 × 3 + 2)/ 3

= 8/3

3 1/2 = (3 × 2 + 1)/ 2

= 7/2

Now, the fraction becomes

8/3 + 7/2

For addition or subtraction, the denominators of the fraction should be equal. It helps is easy calculation.

To make the denominators equal, we need to multiply the first fraction by 2 and the second fraction by 3.

Multiplying the first fraction by 2, we get:

(8 × 2)/ (3 × 2)

= 16/6

Multiplying the second fraction by 3, we get:

(7 × 3)/ (2 × 3)

= 21/6

Now adding,

16/6 + 21/6

= (16 + 21)/6

= 37/6

The fraction in the mixed form can be written as:

6 1/6

(vii) 8 1/2 - 3 5/8

Answer: 39/8 or 4 7/8

Explanation: 8 1/2 - 3 5/8

The given fraction is a mixed fraction. For easy calculation, we will convert the given fraction to the whole fraction.

8 1/2 = (8 × 2 + 1)/ 2

= 17/2

3 5/8 = (3 × 8 + 5)/ 8

= 29/8

Now, the fraction becomes

17/2 - 29/8

For addition or subtraction, the denominators of the fraction should be equal. It helps is easy calculation.

Now, the denominators of these two fractions have some common factors. In such cases, we need to find the LCM, which acts as a common multiple.

LCM (2, 8) = 8

To make the denominators equal, we need to multiply the first fraction by 4. The second fraction already has a denominator 8.

Multiplying the first fraction by 4, we get:

(17 × 4)/ (2 × 4)

= 68/8

Now subtracting,

68/8 - 29/8

= (68 - 29)/ 8

= 39/8

The fraction in the mixed form can be written as:

4 7/8

2. Arrange the following in descending order:

(i) 2/9, 2/3, 8/21

Answer: 2/3 > 8/21 > 2/9

Explanation: To compare, the denominators of the fraction should be equal. It helps is easy comparison.

Now, the denominators of these three fractions have some common factors. In such cases, we need to find the LCM, which acts as a common multiple.

LCM (9, 3, 21) = 63

Now,

63/9 = 7

63/3 = 21

63/21 = 3

To make the denominators equal, we will multiply the first fraction by 7, the second fraction by 21, and the third fraction by 3.

Multiplying the first fraction by 7, we get:

(2 × 7)/ (9 × 7)

= 14/63

Multiplying the second fraction by 21, we get:

(2 × 21)/ (3 × 21)

= 42/63

Multiplying the third fraction by 3, we get:

(8 × 3)/ (21 × 3)

= 24/63

The given fractions can be written as:14/63, 42/63, 24/63

Decreasing order is the order of the fractions from largest to the smallest.

42/63 > 24/63 > 14/63

Or

2/3 > 8/21 > 2/9

(ii) 1/5, 3/7, 7/10

Answer: 7/10 > 3/7 > 1/5

Explanation: To compare, the denominators of the fraction should be equal. It helps is easy comparison.

Now, the denominators of these three fractions have some common factors. In such cases, we need to find the LCM, which acts as a common multiple.

LCM (5, 7, 10) = 70

Now,

70/5 = 14

70/7 = 10

70/10 = 7

To make the denominators equal, we will multiply the first fraction by 14, the second fraction by 10, and the third fraction by 7.

Multiplying the first fraction by 14, we get:

(1 × 14)/ (5 × 14)

= 14/70

Multiplying the second fraction by 10, we get:

(3 × 10)/ (7 × 10)

= 30/70

Multiplying the third fraction by 7, we get:

(7 × 7)/ (10 × 7)

= 49/70

The given fractions can be written as:

14/70, 30/70, 49/70

Decreasing order is the order of the fractions from largest to the smallest.

49/70 > 30/70 > 14/70

Or

7/10 > 3/7 > 1/5

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?

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Answer: Yes, it is a magic square.

Explanation: Let's find the sum of each row, column, and diagonal given in the square.

Row 1:

Sum = 4/11 + 9/11 + 2/11

Sum = (4 + 9 + 2)/11

Sum = 15/11

Row 2:

Sum = 3/11 + 5/11 + 7/11

Sum = (3 + 5 + 7)/11

Sum = 15/11

Row 3:

Sum = 8/11 + 1/11 + 6/11

Sum = (8 + 1 + 6)/11

Sum = 15/11

Sum of all the three rows is equal.

Column 1:

Sum = 4/11 + 3/11 + 8/11

Sum = (4 + 3 + 8)/11

Sum = 15/11

Column 2:

Sum = 9/11 + 5/11 + 1/11

Sum = (9 + 5 + 1)/11

Sum = 15/11

Column 3:

Sum = 2/11 + 7/11 + 6/11

Sum = (2 + 7 + 6)/11

Sum = 15/11

Sum of all the three columns is equal.

Diagonal 1:

Sum = 4/11 + 5/11 + 6/11

Sum = (4 + 5 + 6)/11

Sum = 15/11

Diagonal 2:

Sum = 2/11 + 5/11 + 8/11

Sum = (2 + 5 + 8)/11

Sum = 15/11

The sum of two diagonals is also equal.

Hence, it is a magic square.

4. A rectangular sheet of paper is 12 1/2 cm long and 10 2/3 cm wide. Find its perimeter.

Answer: 139/3 or 46 1/3

Explanation: Perimeter refers to the sum of all sides of a given figure.

Perimeter of a rectangle = 2 × (Length + Width)

Length of the rectangle = 12 1/2 cm

= (12 × 2 + 1)/2

= 25/2 cm

Width of the rectangle = 10 2/3 cm

= (10 × 3 + 2)/3

= 32/3 cm

Perimeter of a rectangle = 2 × (25/2 + 32/3)

To add, we need to convert the denominators to the same value.

Multiplying the first fraction by 3 and the second fraction by 2, we get:

Perimeter of a rectangle = 2 × (75/6 + 64/6)

Perimeter of a rectangle = 2 × (139/6)

= 139/3

The perimeter in the mixed fraction can be represented as:

46 1/3

5. Find the perimeters of

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

(i) ∆ABE

Answer: 8 17/20 cm

Explanation: Perimeter refers to the sum of all sides of a given figure.

Perimeter of a triangle = Side 1 + Side 2 + Side 3

Side 1 = 5/2 cm

Side 2 = 2 3/4 cm

= 11/4 cm

Side 3 = 3 3/5 cm

= 18/5 cm

We can take the sides in any order. The answer will be the same in each case.

To add, the denominators of the three fractions should be the same.

LCM (2, 4, 5) = 20

Multiplying the first fraction by 10, the second fraction by 5, and the third fraction by 4, we get:

Side 1 = 50/20 cm

Side 2 = 55/20 cm

Side 3 = 72/20 cm

Perimeter of a triangle = Side 1 + Side 2 + Side 3

Perimeter of a triangle = 50/20 + 55/20 + 72/20

Perimeter of a triangle = (50 + 55 + 72)/20

Perimeter of a triangle =177/20

In terms of mixed fraction, the perimeter can be represented as;

8 17/20

(ii) The rectangle BCDE in this figure.

Answer: 7 5/6 cm

Explanation: Perimeter refers to the sum of all sides of a given figure.

Perimeter of a rectangle = 2 × (Length + Width)

Length of the rectangle = 2 3/4 cm

= 11/4 cm

Width of the rectangle = 7/6 cm

To add, we need to convert the denominators to the same value.

LCM (4, 6) = 12

Multiplying the first fraction by 3 and the second fraction by 2, we get:

Length = 33/12 cm

Width = 14/12 cm

Perimeter of a rectangle = 2 × (33/12 + 14/12)

Perimeter of a rectangle = 2 × (47/12)

= 47/6

The perimeter in the mixed fraction can be represented as:

7 5/6

Whose perimeter is greater?

Answer: Triangle ∆ABE

Explanation: Perimeter of a triangle = 177/20 cm

Perimeter of a rectangle = 47/6 cm

To compare, we will again convert the denominators to the same value.

LCM (20, 6) = 60

Multiplying the first fraction by 3, and the second fraction b 10, we get:

Perimeter of a triangle = 531/60

Perimeter of a rectangle = 470/60

531/60 > 470/60

Thus,

The perimeter of triangle is greater than the perimeter of rectangle.

6. Salil wants to put a picture in a frame. The picture is 7 3/5 cm wide. To fit in the frame the picture cannot be more than 7 3/10 cm wide. How much should the picture be trimmed?

Answer: 3/10 cm

Explanation: Width of the picture = 7 3/5 cm

= 38/5 cm

Multiplying the fraction by 2, we get:

Width of the picture = 76/10 cm

Width of the frame = 73/10 cm

= 73/10 cm

The above two values signifies that the width of the picture is greater than the width of the frame. To fit, the width of the picture and the frame should be equal.

Thus,

Part to be trimmed = Width of the picture - width of the frame

= 76/10 - 73/10

= (76 - 73)/10

= 3/10 cm

3/10 part of the picture needs to be trimmed to make the width equal to the width of the frame.

7. Ritu ate 3/5 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?

Answer: 2/5, Ritu, 1/5

Explanation: Fraction of apple eaten by Ritu = 3/5

The whole part of an apple is considered as 1 or 1/1

The remaining fraction of apple = 1 - 3/5

= 1/1 - 3/5

Multiplying the first fraction by 5,

= 5/5 - 3/5

= (5 - 3)/5

= 2/5

Thus, the 2/5 part of the apple was eaten by her brother Somu.

Share of Ritu = 3/5

Share of Somu = 2/5

Thus,

Ritu has the larger share.

Difference = 3/5 - 2/5

= 1/5

Ritu has the larger share by 1/5.

8. Michael finished colouring a picture in 7/12 hour. Vaibhav finished colouring the same picture in 3/4 hour. Who worked longer? By what fraction was it longer?

Answer: Vaibhav; 1/6

Explanation: Time taken by Michael to colour a picture = 7/12 hour

Time taken by Vaibhav to colour a picture = 3/4 hour

To compare, let's make the denominators equal.

Multiplying the fraction by 3, we get:

Time taken by Vaibhav to colour a picture = (3 × 3)/ (4 × 3)

= 9/12

Time taken by Vaibhav > Time taken by Michael

Thus,

Vaibhav worked longer.

Difference = Time taken by Vaibhav - Time taken by Michael= 9/12 - 7/12

= 2/12

= 1/6

Exercise 2.2

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

i. 2 × 1/5

Answer: (d)

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation: 1/5 fraction represents:

Shaded sections/ Total number of sections

Shaded section = 1

Total sections = 5

The above figure also has 1 shaded section and 5 total sections.

2 × 1/5 represents twice the figures.

We can also write it as:

2 × 1/5 = 1/5 + 1/5

ii. 2 × 1/2

Answer: (b)

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation: 1/2 fraction represents:

Shaded sections/ Total number of sections

Shaded section = 1

Total sections = 2

The above figure also has 1 shaded section and 2 total sections.

2 × 1/2 represents twice the figures.

We can also write it as:

2 × 1/2 = 1/2 + 1/2

iii. 3 × 2/3

Answer: (a)

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation: 2/3 fraction represents:

Shaded sections/ Total number of sections

Shaded section = 2

Total sections = 3

The above figure also has 2 shaded section and 3 total sections.

3 × 2/3 represents thrice the figures.

We can also write it as:

3 × 2/3 = 2/3 + 2/3 + 2/3

iv. 3 × 1/4

Answer: (c)

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation: 1/4 fraction represents:

Shaded sections/ Total number of sections

Shaded section = 1

Total sections = 4

The above figure also has 1 shaded section and 4 total sections.

3 × 1/4 represents twice the figures.

We can also write it as:

3× 1/4 = 1/4 + 1/4 + 1/4 + 1/4

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

(i) 3 × 1/5 = 3/5

Answer: (c)

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation:

LHS

1/5 fraction represents:

Shaded sections/ Total number of sections

Shaded section: 1

Total sections: 5

3 × 1/5 shows that there are three blocks with the fraction (1/5).

3 × 1/5 can also be written as:

1/5 + 1/5 + 1/5

RHS

3/5 fraction represents:

Shaded sections/ Total number of sections

Shaded section: 3

Total sections: 5

1/5 + 1/5 + 1/5 = (1 + 1 + 1)/5

= 3/5

ii. 2 × 1/3 = 2/3

Answer: (a)

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation:

LHS

1/3 fraction represents:

Shaded sections/ Total number of sections

Shaded section: 1

Total sections: 3

2 × 1/3 shows that there are two blocks with the fraction (1/3).

2 × 1/3 can also be written as:

1/3 + 1/3

RHS

2/3 fraction represents:

Shaded sections/ Total number of sections

Shaded section: 2

Total sections: 3

1/3 + 1/3 = (1 + 1)/3

= 2/3

iii. 3 × 3/4 = 2 1/4

Answer: (b)

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation:

LHS

3/4 fraction represents:

Shaded sections/ Total number of sections

Shaded sections: 3

Total sections: 4

3 × 3/4 shows that there are three blocks with the fraction (3/4).

3 × 3/4 can also be written as:

¾ + ¾ + ¾

RHS

2 1/4 fractions represent:

Shaded sections/ Total number of sections

Shaded section: 2 full shaded figure and a ¼ shaded figure

Total sections: 3 figures

2 ¼ = 9/4 in terms of whole fractions

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

i. 7 × 3/5

Answer: 4 1/5

Explanation: 7 × 3/5

= (7× 3)/5

= 21/5

The above fraction can be written as:

= 20/5 + 1/5

= 4 + 1/5

= 4 1/5

If there are any common factors in the numerator and the denominator, it can be cancelled to bring it to their lowest form.

The above fraction does have any common factors except 1. It is already present in its lowest form.

ii. 4 × 1/3

Answer: 1 1/3

Explanation: 4 × 1/3

= (4 × 1)/3

= 4/3

The above fraction can be written as:

= 3/3 + 1/3

= 1 + 1/3

= 1 1/3

If there are any common factors in the numerator and the denominator, it can be cancelled to bring it to their lowest form.

The above fraction does have any common factors except 1. It is already present in its lowest form.

iii. 2 × 6/7

Answer: 1 5/7

Explanation: 2 × 6/7

= (2× 6)/7

= 12/7

The above fraction can be written as:

= 7/7 + 5/7

= 1 + 5/7

= 1 5/7

If there are any common factors in the numerator and the denominator, it can be cancelled to bring it to their lowest form.

The above fraction does have any common factors except 1. It is already present in its lowest form.

iv. 5 × 2/9

Answer: 1 1/9

Explanation: 5 × 2/9

= (5× 2)/9

= 10/9

The above fraction can be written as:

= 9/9 + 1/9

= 1 + 1/9

= 1 1/9

If there are any common factors in the numerator and the denominator, it can be cancelled to bring it to their lowest form.

The above fraction does have any common factors except 1. It is already present in its lowest form.

v. 2/3 × 4

Answer: 2 2/3

Explanation: 2/3 × 4

= (2× 4)/3

= 8/3

The above fraction can be written as:

= 6/3 + 2/3

= 2 + 2/3

= 2 2/3

If there are any common factors in the numerator and the denominator, it can be cancelled to bring it to their lowest form.

The above fraction does have any common factors except 1. It is already present in its lowest form.

vi. 5/2 × 6

Answer: 15

Explanation: 5/2 × 6

= (5× 6)/2

= (5× 3 × 2)/2

The above fraction has a common factor 2, which can be cancelled to bring it to its lowest form.

= 15

vii. 11 × 4/7

Answer: 6 2/7

Explanation: 11 × 4/7

= (11× 4)/7

= 44/7

The above fraction can be written as:

= 42/7 + 2/7

= 6 + 2/7

= 6 2/7

If there are any common factors in the numerator and the denominator, it can be cancelled to bring it to their lowest form.

The above fraction does have any common factors except 1. It is already present in its lowest form.

viii. 20 × 4/5

Answer: 16

Explanation: 20 × 4/5

= (20× 4)/5

= (4× 5 × 4)/5

The above fraction has a common factor 5, which can be cancelled to bring it to its lowest form.

= 16

ix. 13 × 1/3

Answer: 4 1/3

Explanation: 13 × 1/3

= (13× 1)/3

= 13/3

The above fraction can be written as:

= 12/3 + 1/3

= 4 + 1/3

= 4 1/3

If there are any common factors in the numerator and the denominator, it can be cancelled to bring it to their lowest form.

The above fraction does have any common factors except 1. It is already present in its lowest form.

x. 15 × 3/5

Answer: 9

Explanation: 15 × 3/5

= (15× 3)/5

= (3× 5 × 3)/5

The above fraction has a common factor 5, which can be cancelled to bring it to its lowest form.

= 9

4. Shade:

(i) 1/2 of the circles in box

Answer:

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation: 1 refers to the full figure, i.e., all the number of holes given inside the figure.

1/2 means shading the half amount of circles.

Total circles = 12

½ circles = 12/2

= 6

Thus, we will shade 6 circles in the figure.

(ii) 2/3 of the triangles in box

Answer:

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation: 1 refers to the full figure, i.e., all the number of triangles given inside the figure.

Total triangles = 9

2/3 triangles = 2/3 × 9

= (2 × 9)/3

= 18/3

= 6

Thus, we will shade 6 triangles in the figure.

(iii) 3/5 of the squares in box

Answer:

NCERT Solutions for class 7 Maths Chapter 2: Fractions and Decimals

Explanation: 1 refers to the full figure, i.e., all the number of squares given inside the figure.

Total squares = 15

3/5 squares = 3/5 × 15

= (3 × 15)/5

= (3 × 3 × 5)/5

= 9

Thus, we will shade 9 squares in the figure.

5. Find:

(a) 1/2 of

(i) 24

Answer: 12

Explanation: Of signifies multiplication.

So, we will multiply 1/2 with 24

1/2 × 24

= 24/2

= (12 × 2)/2

= 12

(ii) 46

Answer: 23

Explanation: Of signifies multiplication.

So, we will multiply 1/2 with 46

1/2 × 46

= 46/2

= (23 × 2)/2

= 23

(b) 2/3 of

(i) 18

Answer: 12

Explanation: Of signifies multiplication.

So, we will multiply 2/3 with 18

2/3 × 18

= (2 × 18)/3

18 = 6 × 3

= (2 × 6 × 3)/3

Cancelling out the common factors (3), we get:

= 12

(ii) 27

Answer: 18

Explanation: Of signifies multiplication.

So, we will multiply 2/3 with 27

2/3 × 27

= (2 × 27)/3

27 = 3 × 3 × 3

= (2 × 3 × 3 × 3)/3

Cancelling out the common factors (3), we get:

= 18

(c) 3/4 of

(i) 16

Answer: 12

Explanation: Of signifies multiplication.

So, we will multiply 3/4 with 16

3/4 × 16

= (3 × 16)/4

16 = 4 × 4

= (3 × 4 × 4)/4

Cancelling out the common factors (4), we get:

= 3 × 4

= 12

(ii) 36

Answer: 27

Explanation: Of signifies multiplication.

So, we will multiply 3/4 with 36

3/4 × 36

= (3 × 36)/4

36 = 4 × 9

= (3 × 4 × 9)/4

Cancelling out the common factors (4), we get:

= 3 × 9

= 27

(d) 4/5 of

(i) 20

Answer: 16

Explanation: Of signifies multiplication.

So, we will multiply 4/5 with 20

4/5 × 20

= (4 × 20)/5

20 = 4 × 5

= (4 × 4 × 5)/5

Cancelling out the common factors (5), we get:

= 4 × 4

= 16

(ii) 35

Answer: 28

Explanation: Of signifies multiplication.

So, we will multiply 4/5 with 35

4/5 × 35

= (4 × 35)/5

35 = 5 × 7

= (4 × 5 × 7)/5

Cancelling out the common factors (5), we get:

= 4 × 7

= 28

6. Multiply and express as a mixed fraction:

Note: To multiply a mixed fraction to a whole number, we need to first convert the mixed fraction to an improper fraction and then multiply. We will convert the result back to the mixed fraction after multiplication.

a. 3 × 5 1/5

Answer: 15 3/5

Explanation:3 × 5 1/5

3 × 26/5

= (3 × 26)/5

= 78/5

= 15 3/5

b. 5 × 6 3/4

Answer: 33 3/4

Explanation: 5 × 6 ¾

= 5 × 27/4

= (5 × 27)/4

= 135/4

= 33 3/4

c. 7 × 2 1/4

Answer: 15 3/4

Explanation: 7 × 2 ¼

= 7× 9/4

= (7 × 9)/4

= 63/4

= 15 3/4

d. 4 × 6 1/3

Answer: 25 1/3

Explanation: 4 × 6 1/3

= 4× 19/3

= (4 × 19)/3

= 76/3

= 25 1/3

e. 3 ¼ × 6

Answer: 19 1/2

Explanation: 3 ¼ × 6

= 13/4× 6

= (13 × 6)/4

The numerator and the denominator have 1 common factor, i.e., 2.

So, cancelling out the common factors, we get:

= (13 × 2 × 3)/(2 × 2)

= 39/2

= 19 1/2

f. 3 2/5 × 8

Answer: 27 1/5

Explanation: 3 2/5 × 8

= 17/5× 8

= (17 × 8)/5

= (17 × 8)/5

= 136/5

= 27 1/5

7. Find:

(a) 1/2 of

To multiply a mixed fraction to a whole number, we need to first convert the mixed fraction to an improper fraction and then multiply. We will convert the result back to the mixed fraction after multiplication.

(i) 2 3/4

Answer: 1 3/8

Explanation:

2 ¾ = 11/4

Of signifies multiplication.

So, we will multiply ½ with 11/4

1/2 × 11/4

= (1 × 11)/ (2 × 4)

= 11/8

= 1 3/8

(ii) 4 2/9

Answer: 2 1/9

Explanation: 4 2/9 = 38/9

Of signifies multiplication.

So, we will multiply ½ with 38/9

1/2 × 38/9

= (1 × 38)/ (2 × 9)

= 38/18

The numerator and the denominator have 1 common factor, i.e., 2.

So, cancelling out the common factors, we get:

= 19/9

= 2 1/9

(b) 5/8 of

(i) 3 5/6

Answer: 2 19/48

Explanation: 3 5/6 = 23/6

Of signifies multiplication.

So, we will multiply 5/8 with 23/6

= 5/8× 23/6

= (5 × 23)/ (8 × 6)

= 115/48

= 2 19/48

(ii) 9 2/3

Answer: 6 1/24

Explanation: 9 2/3 = 29/3

Of signifies multiplication.

So, we will multiply 5/8 with 29/3

= 5/8× 29/3

= (5 × 29)/ (8 × 3)

= 145/24

= 6 1/24

8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed 2/5 of the water. Pratap consumed the remaining water.

(i) How much water did Vidya drink?

Answer: 2 litres

Explanation: Total water in the water bottle = 5 litres

Water consumed by Vidya = 2/5th of 5 litres

= 2/5 × 5

= 2 litres

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

Answer: 3/5

Explanation: Total water in the water bottle = 5 litres

Water consumed by Vidya = 2/5th of 5 litres

= 2/5 × 5

= 2 litres

The remaining water was consumed by Pratap.

Remaining water = Total water - water consumed by Vidya

= 5 - 2

= 3 litres

Fraction of water consumed by Vidya = 2/5

Remaining fraction of water consumed by Pratap = 1 - 2/5

= (5 - 2)/5

= 3/5

Exercise 2.3

1. Find:

(i) 1/4 of

(a) 1/4

Answer: 1/16

Explanation: Of means multiplication.

So,

1/4 of 1/4

= 1/4 × 1/4

= (1 × 1)/ (4 × 4)

= 1/16

(b) 3/5

Answer: 3/20

Explanation: Of means multiplication.

So,

1/4 of 3/5

= 1/4 × 3/5

= (1 × 3)/ (4 × 5)

= 3/20

(c) 4/3

Answer: 1/3

Explanation: Of means multiplication.

So,

1/4 of 4/3

= 1/4 × 4/3

= (1 × 4)/ (4 × 3)

Cancelling out the common term (4), we get:

= 1/3

(ii) 1/7 of

(a) 2/9

Answer: 2/63

Explanation: Of means multiplication.

So,

1/7 of 2/9

= 1/7 × 2/9

= (1 × 2)/ (7 × 9)

= 2/ 63

(b) 6/5

Answer: 6/35

Explanation: Of means multiplication.

So,

1/7 of 6/5

= 1/7 × 6/5

= (1 × 6)/ (7 × 5)

= 6/ 35

(c) 3/10

Answer: 3/70

Explanation: Of means multiplication.

So,

1/7 of 3/10

= 1/7 × 3/10

= (1 × 3)/ (7 × 10)

= 3/ 70

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

(i) 2/3 × 2 2/3

Answer: 1 7/9

Explanation: 2/3 × 2 2/3

Let's convert the mixed fraction into improper fraction.

2 2/3 = 8/3

Substituting the value, we get:

2/3 × 2 2/3

= 2/3 × 8/3

= (2 × 8)/ (3 × 3)

= 16/9

16/9 in terms of mixed fraction can be represented as:

1 7/9

(ii) 2/7 × 7/9

Answer: 2/9

Explanation: 2/7 × 7/9

= (2 × 7)/ (7 × 9)

Cancelling out the common term (7), we get:

2/9

(iii) 3/8 × 6/4

Answer: 9/16

Explanation: 3/8 × 6/4

= (3 × 6)/ (8 × 4)

Factorizing 6 and 8,

= (3 × 3 × 2)/ (2 × 4 × 4)

Cancelling the common term (2), we get:

= (3 × 3)/ (4 × 4)

= 9/16

(iv) 9/5 × 3/5

Answer: 27/25

Explanation: 9/5 × 3/5

= (9 × 3)/ (5 × 5)

= 27/ 25

(v) 1/3 × 15/8

Answer: 5/8

Explanation: 1/3 × 15/8

= (1 × 15)/ (3 × 8)

15 = 3 × 5

Substituting the value of 15,

= (1 × 3 × 5)/ (3 × 8)

= 5/8

(vi) 11/2 × 3/10

Answer: 1 13/20

Explanation: 11/2 × 3/10

= (11 × 3)/ (2 × 10)

= 33/20

= 1 13/20

(vii) 4/5 × 12/7

Answer: 1 13/35

Explanation: 4/5 × 12/7

= (4 × 12)/ (5 × 7)

= 48/35

= 1 13/35

3. Multiply the following fractions:

(i) 2/5 × 5 1/4

Answer: 2 1/10

Explanation: 2/5 × 5 1/4

2/5 × 21/4

= (2 × 21)/ (5 × 4)

4 = 2 × 2

Substituting the value of 4,

= (2 × 21)/ (5 × 2 × 2)

= 21/10

=2 1/10

(ii) 6 2/5 × 7/9

Answer: 4 44/45

Explanation: 6 2/5 × 7/9

32/5 × 7/9

= (32 × 7)/ (5 × 9)

= 224/45

= 4 44/45

(iii) 3/2 × 5 1/3

Answer: 8

Explanation: 3/2 × 5 1/3

= 3/2 × 16/3

= (3 × 16)/ (2 × 3)

= 16/2

= 8

(iv) 5/6 × 2 3/7

Answer: 2 1/42

Explanation: 5/6 × 2 3/7

= 5/6 × 17/7

= (5 × 17)/ (6 × 7)

= 85/42

= 2 1/42

(v) 3 2/5 × 4/7

Answer: 1 33/35

Explanation: 3 2/5 × 4/7

= 17/5 × 4/7

= (17 × 4)/ (5 × 7)

= 68/35

= 1 33/35

(vi) 2 3/5 × 3

Answer: 7 4/5

Explanation: 2 3/5 × 3

= 13/5 × 3/1

= (13 × 3)/ (5 × 1)

= 39/5

= 7 4/5

(vii) 3 4/7 × 3/5

Answer: 2 1/7

Explanation: 3 4/7 × 3/5

= 25/7 × 3/5

= (25 × 3)/ (7 × 5)

= (5 × 5 × 3)/ (7 × 5)

= 15/7

= 2 1/7

4. Which is greater?

(i) 2/7 of 3/4 or 3/5 of 5/8

Answer: 3/5 of 5/8

Explanation: 2/7 of 3/4

= 2/7 × 3/4

= (2 × 3)/ (7 × 4)

= 3/14

3/5 of 5/8

= 3/5 × 5/8

= (3 × 5)/ (5 × 8)

= 3/8

To make the denominators of both the fraction equal, let's multiply 3/14 by 4 and 3/8 by 7.

2/7 of 3/4 = 3/14 = (3 × 4)/ (14 × 4) = 12/56

3/5 of 5/8 = 3/8 = (3 × 7)/ (14 × 7) = 21/56

21/56 > 12/56

Thus,

3/5 of 5/8 is greater.

(ii) 1/2 of 6/7 or 2/3 of 3/7

Answer: 1/2 of 6/7

Explanation: 1/2 of 6/7

= 1/2 × 6/7

= (1 × 6)/ (2 × 7)

= 3/7

2/3 of 3/7

=2/3 × 3/7

= (2 × 3)/ (3 × 7)

= 2/7

3/7 > 2/7

Thus,

1/2 of 6/7 is greater.

5. Saili plants 4 saplings, in a row, in her garden. The distance between two adjacent saplings is 3/4 m. Find the distance between the first and the last sapling.

Answer: 2 1/4 m

Explanation: Total saplings in a row in Saili's garden = 4

Distance between the two adjacent saplings = 3/4 m

Let the four saplings be 1, 2, 3, and 4.

Distance between 1 and 2 = 3/4 m

Distance between 2 and 3 = 3/4 m

Distance between 3 and 4 = 3/4 m

So, the distance between the first and last sapling = 3/4 + 3/4 + 3/4

= (3 + 3 + 3)/ 4

= 9/4 m

= 2 ¼ m

6. Lipika reads a book for 1 3/4 hours everyday. She reads the entire book in 6 days. How many hours in all were required by her to read the book?

Answer: 10 ½ hours

Explanation: The time taken by Lipika to read a book a day = 1 3/4 hours

Total days = 6 days

The time required by Lipika to read the book = No. of days × time taken per day

= 6 × 1 3/4

= 6 × 7/4

= 21/2

= 10 ½ hours

7. A car runs 16 km using 1 litre of petrol. How much distance will it cover using 2 3/4 litres of petrol.

Answer: 44 km

Explanation: Distance covered by a car using 1 litre of petrol = 16 km

Distance covered by a car using 2 3/4 litres of petrol = 16 km× 2 ¾

= 16 × 11/4

= 4 × 11

= 44 km

8.

(a)

(i) Provide the number in the box, such that 2 / 3 × ___ = 10/30

Answer: 5/10

Explanation: Let the blank box be A/B

2/3 × A/B = 10/30

2 × A = 10

A = 5

3 × B = 30

B = 10

Thus,

A/B = 5/10

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

Answer: 1/2

Explanation: A/B = 5/10

10 = 5 × 2

Substituting the value of 10,

A/B = 5/ (2 × 5)

= 1/2

(b)

(i) Provide the number in the box, such that 3/5 × __ = 24/75

Answer: 8/15

Explanation: Let the blank box be A/B

3/5 × A/B = 24/75

Let's compare the numerator.

3 × A = 24

A = 24/3

A = 8

Now, let's compare the denominator.

5 × B = 75

B = 75/5

B = 15

So,

A/B = 8/15

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

Answer: 8/15

Explanation: A/B = 8/15

8 and 15 does not have any common factor except 1. Thus, it is already present in its simplest or lowest form.

Exercise 2.4

1. Find:

(i) 12 ÷ 3/4

Answer: 16

Explanation: 12 ÷ 3/4

= 12 × 4/3

= (12 × 4)/3

12 = 4 × 3

Substituting the value of 12,

(4 × 3 × 4)/ 3

Cancelling the common term (3), we get:

4 × 4

= 16

(ii) 14 ÷ 5/6

Answer: 84/5

Explanation: 14 ÷ 5/6

= 14 × 6/5

= (14 × 6)/ 5

= 84/5

(iii) 8 ÷ 7/3

Answer: 24/7

Explanation: 8 ÷ 7/3

= 8 × 3/7

= (8 × 3)/ 7

= 24/7

(iv) 4 ÷ 8/3

Answer: 3/2

Explanation: 4 ÷ 8/3

= 4 × 3/8

= (4 × 3)/ 8

8 = 4 × 2

Substituting the value of 8,

= (4 × 3)/ (4 × 2)

Cancelling out the common term, we get:

3/2

(v) 3 ÷ 2 1/3

Answer: 9/7

Explanation: 3 ÷ 2 1/3

2 1/3 = 7/3

3 ÷ 7/3

= 3 × 3/7

= (3 × 3)/ 7

= 9/7

(vi) 5 ÷ 3 4/7

Answer: 7/5

Explanation: 5 ÷ 3 4/7

3 4/7 = 25/7

: 5 ÷ 25/7

= 5 × 7/ 25

= (5 × 7)/ 25

25 = 5 × 5

Substituting the value of 25, we get:

= (5 × 7)/ (5 × 5)

Cancelling out the common term (5), we get:

= 7/5

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

(i) 3/7

Answer: 7/3, Improper fraction

Explanation: Reciprocal refers to the reverse of a number. It means that the value numerator is replaced by the denominator and the value of denominator is replaced by the numerator.

Or

The non-zero numbers, whose product with each other is 1, are called the reciprocals of each other.

Reciprocal of 3/7= 7/3

Improper fraction is the fraction whose numerator is greater or equal than the numerator.

7 > 3

Hence, it is an improper fraction.

(ii) 5/8

Answer: 8/5, Improper fraction

Explanation: The non-zero numbers, whose product with each other is 1, are called the reciprocals of each other.

Reciprocal of 5/8= 8/5

Improper fraction is the fraction whose numerator is greater or equal than the numerator.

8 > 5

Hence, it is an improper fraction.

(iii) 9/7

Answer: 7/9, proper fraction

Explanation: The non-zero numbers, whose product with each other is 1, are called the reciprocals of each other.

Reciprocal of 9/7 = 7/9

Proper fraction is the fraction whose numerator is less than the numerator.

7 < 9

Hence, it is a proper fraction.

(iv) 6/5

Answer: 5/6, proper fraction

Explanation: The non-zero numbers, whose product with each other is 1, are called the reciprocals of each other.

Reciprocal of 6/5 = 5/6

Proper fraction is the fraction whose numerator is less than the numerator.

5 < 6

Hence, it is a proper fraction.

(v) 12/7

Answer: 7/12, proper fraction

Explanation: The non-zero numbers, whose product with each other is 1, are called the reciprocals of each other.

Reciprocal of 12/7 = 7/12

Proper fraction is the fraction whose numerator is less than the numerator.

7 < 12

Hence, it is a proper fraction.

(vi) 1/8

Answer: 8, whole number

Explanation: The non-zero numbers, whose product with each other is 1, are called the reciprocals of each other.

Reciprocal of 1/8 = 8/1 or 8

Any number with 1 as the denominator is considered as the whole number.

(vii) 1/11

Answer: 11, whole number

Explanation: The non-zero numbers, whose product with each other is 1, are called the reciprocals of each other.

Reciprocal of 1/11 = 11/1 or 11

Any number with 1 as the denominator is considered as the whole number.

3. Find:

(i) 7/3 ÷ 2

Answer: 7/6

Explanation: 7/3 ÷ 2

= 7/3 × 1/2

= (7 × 1)/ (3 × 2)

= 7/6

(ii) 4/9 ÷ 5

Answer: 4/45

Explanation: 4/9 ÷ 5

= 4/9 × 1/5

= (4 × 1)/ (9 × 5)

= 4/ 45

(iii) 6/ 13 ÷ 7

Answer: 6/91

Explanation: 6/ 13 ÷ 7

= 6/ 13 × 1/7

= (6 × 1)/ (13 × 7)

= 6/91

(iv) 4 1/3 ÷ 3

Answer: 13/9

Explanation: 4 1/3 ÷ 3

= 13/3 ÷ 3

= 13/3 × 1/3

= (13 × 1)/ (3 × 3)

= 13/ 9

(v) 3 1/2 ÷ 4

Answer: 7/8

Explanation: 3 1/2 ÷ 4

= 7/2 ÷ 4

= 7/2 × 1/4

= (7 × 1)/ (2 × 4)

= 7/8

(vi) 4 3/7 ÷ 7

Answer: 31/49

Explanation: 4 3/7 ÷ 7

= 31/7 ÷ 7

= 31/7 × 1/7

= (31 × 1)/ (7 × 7)

= 31/49

4. Find:

(i) 2/5 ÷ 1/2

Answer: 4/5

Explanation: 2/5 ÷ 1/2

= 2/5 × 2/1

= (2 × 2)/ (5 × 1)

= 4/5

(ii) 4/9 ÷ 2/3

Answer: 2/3

Explanation: 4/9 ÷ 2/3

= 4/9 × 3/2

= (4 × 3) / (9 × 2)

Factorizing 4 and 9,

= (2 × 2 × 3) / (3 × 3 × 2)

= 2/3

(iii) 3/7 ÷ 8/7

Answer: 3/8

Explanation: 3/7 ÷ 8/7

= 3/7 × 7/8

= (3 × 7)/ (7 × 8)

Cancelling out the common terms, we get:

3/8

(iv) 2 1/3 ÷ 3/5

Answer: 35/9

Explanation: 2 1/3 ÷ 3/5

7/3 ÷ 3/5

= 7/3 × 5/3

= (7 × 5)/ (3 × 3)

= 35/ 9

(v) 3 1/2 ÷ 8/ 3

Answer: 21/ 16

Explanation: 3 1/2 ÷ 8/ 3

= 7/2 ÷ 8/ 3

= 7/2 × 3/8

= (7 × 3)/ (2 × 8)

= 21/ 16

(vi) 2/5 ÷ 1 1/2

Answer: 4/15

Explanation: 2/5 ÷ 1 1/2

= 2/5 ÷ 3/2

= 2/5 × 2/3

= (2 × 2)/ (5 × 3)

= 4/ 15

(vii) 3 1/5 ÷ 1 2/3

Answer: 48/25

Explanation: 3 1/5 ÷ 1 2/3

= 16/5 ÷ 5/3

= 16/5 × 3/5

= (16 × 3)/ (5 × 5)

= 48/25

(viii) 2 1/5 ÷ 1 1/5

Answer: 11/6

Explanation: 2 1/5 ÷ 1 1/5

= 11/5 ÷ 6/5

= 11/5 × 5/6

= (11 × 5)/ (5 × 6)

Cancelling out the common term (5), we get:

= 11/6

Exercise 2.5

1. Which is greater?

(i) 0.5 or 0.05

Answer: 0.5

Explanation:

0.5 = 5/10

0.05 = 5/100

To compare, let's convert the denominator of the two fractions equal.

Multiplying the fraction 5/10 by 10, we get:

50/100

Now,

50/100 or 5/100

Thus,

50/100 > 5/100

0.5 > 0.05

(ii) 0.7 or 0.5

Answer: 0.7

Explanation: 0.7 = 7/10

0.5 = 5/10

7/10 > 5/10

Thus,

0.7 > 0.5

(iii) 7 or 0.7

Answer: 7

Explanation: The decimal value is always less than the whole number value.

Or

0.7 = 7/10

Multiplying the fraction (7/1) by 10, we get:

70/10

70/10 > 7/10

Or

7 > 0.7

(iv) 1.37 or 1.49

Answer: 1.49

Explanation: 1.37 = 137/100

= 149/100

137/100 < 149/100

Or

1.37 < 1.49

(v) 2.03 or 2.30

Answer: 2.30

Explanation: 2.03 = 203/100

2.30 = 230/100

203/100 < 230/100

Thus,

2.03 < 2.30

(vi) 0.8 or 0.88.

Answer: 0.88

Explanation: 0.8 = 80/100

0.88 = 88/100

80/100 < 88/100

Thus,

0.8 < 0.88

2. Express as rupees using decimals:

(i) 7 paise

Answer: ₹ 0.07

Explanation: 1 rupee = 100 paise

1 paise = 1/100 rupee

7 paise = 1/100 × 7

= 7/100

= 0.07

(ii) 7 rupees 7 paise

Answer: ₹ 7.07

Explanation: 1 rupee = 100 paise

1 paise = 1/100 rupee

7 paise = 1/100 × 7

= 7/100

= 0.07

7 rupees 7 paise = 7 rupees 0.07 rupees

= 7 + 0.07

= 7.07 rupees

(iii) 77 rupees 77 paise

Answer: ₹ 77.77

Explanation: 1 rupee = 100 paise

1 paise = 1/100 rupee

77 paise = 1/100 × 77

= 77/100

= 0.77

77 rupees 77 paise = 77 rupees 0.77 rupees

= 77 + 0.77

= 77.77 rupees

(iv) 50 paise

Answer: ₹ 0.50

Explanation: 1 rupee = 100 paise

1 paise = 1/100 rupee

50 paise = 1/100 × 50

= 50/100

= 0.50

(v) 235 paise

Answer: ₹ 2.35

Explanation: 1 rupee = 100 paise

1 paise = 1/100 rupee

235 paise = 1/100 × 235

= 235/100

= 2.35

3.

(i) Express 5 cm in metre and kilometre

Answer: 0.05 m, 0.00005 km

Explanation: 1m = 100 cm

1 cm = 1/100 m

5 cm = 1/100 × 5

= 5/100

= 0.05 m

1 km = 1000 m

1m = 1/1000 km

0.05 m = 1/1000 × 0.05

= 0.05/1000

= 0.00005 km

(ii) Express 35 mm in cm, m and km

Answer: 3.5 cm, 0.035 m, 0.000035 km

Explanation: 1 cm = 10mm

1mm = 1/10 cm

35 mm = 1/10 × 35

= 35/10

= 3.5 cm

1m = 100 cm

1 cm = 1/100 m

3.5 cm = 1/100 × 3.5

= 3.5 /100

= 0.035 m

1 km = 1000 m

1m = 1/1000 km

0.035 m = 1/1000 × 0.035

= 0.035/1000

= 0.000035 km

4. Express in kg:

(i) 200 g

Answer: 0.2 kg

Explanation: 1 kg = 1000 g

1 g = 1/1000 kg

200 g = 1/1000 × 200

= 200/1000

= 0.2 kg

(ii) 3470 g

Answer: 3.470 kg

Explanation: 1 kg = 1000 g

1 g = 1/1000 kg

3470 g = 1/1000 × 3470

= 3470 /1000

= 3.470 kg

(iii) 4 kg 8 g

Answer: 4.008 kg

Explanation: 1 kg = 1000 g

1 g = 1/1000 kg

8 g = 1/1000 × 8

= 8 /1000

= 0.008 kg

4 kg 8 g = 4kg 0.008 kg

= 4 + 0.008

= 4.008 kg

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

(i) 20.03

Answer: 2 × 10 + 0 × 1 + 0 × 1/10 + 3 × 1/100

(ii) 2.03

Answer: 2 × 1 + 0 × 1/10 + 3 × 1/100

(iii) 200.03

Answer: 2 × 100 + 0 × 10 + 0 × 1 + 0 × 1/10 + 3 × 1/100

(iv) 2.034

Answer: 2 × 1 + 0 × 1/10 + 3 × 1/100 + 4 × 1/1000

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

(i) 2.56

Answer: Ones

Explanation: 2.56 = 2 × 1 + 5 × 1/10 + 6 × 1/100

Thus, the place value of 2 is ones.

(ii) 21.37

Answer: Tens

Explanation: 21.37 = 2 × 10 + 1 × 1 + 3 × 1/10 + 7 × 1/100

(iii) 10.25

Answer: Tenths

Explanation: 10.25 = 1 × 10 + 0 × 1 + 2 × 1/10 + 5 × 1/100

(iv) 9.42

Answer: Hundredths

Explanation: 9.42 = 9 × 1 + 4 × 1/10 + 2 × 1/100

(v) 63.352

Answer: thousandths

Explanation: 63.352 = 6 × 10 + 3 × 1 + 3 × 1/10 + 5 × 1/100 + 2 × 1/1000

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?

Answer: Ayub travelled more by 900m / 0.9 km

Explanation: Distance travelled by Dinesh from place A to place B to place C = 7.5 km + 12.7 km

= 20.2 km

Distance travelled by Ayub from place A to place D to place C = 9.3 km + 11.8 km

= 21.1 km

Thus, Ayub travelled more.

Difference = 21.1 km - 20.2 km

= 0.9 km

1 km = 1000 m

0.9km = 0.9 × 1000

= 900 m

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?

Answer: Sarala bought more fruits

Explanation: Apples bought by Shyama = 5 kg 300 g

Mangoes bought by Shyama = 3 kg 250 g

Total fruits bought by Shyama = 5 kg 300 g + = 3 kg 250 g

= (5 + 3) kg + (300 + 250) g

= 8 kg 550 g

Oranges bought by Sarala = 4 kg 800 g

Bananas bought by Sarala = 4 kg 150 g

Total fruits bought by Sarala = 4 kg 800 g + 4 kg 150 g

= (4 + 4) kg + (800 + 150) g

= 8 kg 950 g

The weight of fruits bought by Sarala is greater than the weight of fruits bought by Shyama. Thus, Sarala bought more fruits.

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

Answer: 14.6 km

Explanation: 42.6 km - 28 km

= 14.6 km

Exercise 2.6

1. Find:

(i) 0.2 × 6

Answer: 1.2

Explanation: 0.2 × 6

= 2/10 × 6

= (2 × 6)/ 10

= 12/10

= 1.2

(ii) 8 × 4.6

Answer: 36.8

Explanation: 8 × 4.6

= 8 × 46/10

= (8 × 46)/10

= 368/10

= 36.8

(iii) 2.71 × 5

Answer: 13.55

Explanation: 2.71 × 5

= 271/100 × 5

= (271 × 5)/100

= 1355/100

= 13.55

(iv) 20.1 × 4

Answer: 80.4

Explanation: 20.1 × 4

= 201/10 × 4

= (201 × 4)/ 10

= 804/10

= 80.4

(v) 0.05 × 7

Answer: 0.35

Explanation: 0.05 × 7

= 5/100 × 7

= (5 × 7)/100

= 35/100

= 0.35

(vi) 211.02 × 4

Answer: 844.08

Explanation: 211.02 × 4

= 21102/100 × 4

= (21102 × 4)/ 100

= 84408 /100

= 844.08

(vii) 2 × 0.86

Answer: 1.72

Explanation: 2 × 0.86

= 2 × 86/ 100

= (2 × 86)/ 100

= 172/100

= 1.72

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

Answer: 17.1 cm2

Explanation: Length of the rectangle = 5.7 cm

Breadth of the rectangle = 3 cm

Area of rectangle = Length × Breadth

Area of rectangle = 5.7 cm × 3 cm

= 17.1 cm2

Thus, the area of rectangle whose length is 5.7cm and breadth is 3 cm is 17.1 cm2.

2. Find:

(i) 1.3 × 10

Answer: 13

Explanation: 1.3 × 10

= 13/10 × 10

= (13 × 10)/10

Cancelling the common multiple (10), we get:

13

(ii) 36.8 × 10

Answer: 368

Explanation: 36.8 × 10

= 368 /10 × 10

= (368 × 10)/10

Cancelling the common multiple (10), we get:

368

(iii) 153.7 × 10

Answer: 1537

Explanation: 153.7 × 10

= 1537/10 × 10

= (1537 × 10)/10

Cancelling the common multiple (10), we get:

1537

(iv) 168.07 × 10

Answer: 1680.7

Explanation: 168.07× 10

= 16807/100 × 10

= (16807 × 100)/10

Cancelling the common multiple (10), we get:

= 16807 /10

= 1680.7

(v) 31.1 × 100

Answer: 3110

Explanation: 31.1 × 100

= 311/10 × 100

= (311 × 100)/ 10

Cancelling the common multiple (10), we get:

100 = 10 × 10

= 311 × 10

= 3110

(vi) 156.1 × 100

Answer: 15610

Explanation: 156.1 × 100

= 1561/10 × 100

= (1561 × 100)/ 10

Cancelling the common multiple (10), we get:

100 = 10 × 10

= 1561 × 10

= 15610

(vii) 3.62 × 100

Answer: 362

Explanation: 3.62 × 10

= 362/100 × 100

= (362 × 100)/100

Cancelling the common multiple (100), we get:

362

(viii) 43.07 × 100

Answer: 4307

Explanation: 43.07 × 100

= 4307/100 × 100

= (4307× 100) / 100

Cancelling the common multiple (100), we get:

4307

(ix) 0.5 × 10

Answer: 5

Explanation: 0.5 × 10

= 5/10 × 10

= (5 × 10)/ 10

= 50/10

= 5

(x) 0.08 × 10

Answer: 0.8

Explanation: 0.08 × 10

= 8/100 × 10

= (8 × 10)/ 100

= 80/100

= 0.8

(xi) 0.9 × 100

Answer: 90

Explanation: 0.9 × 100

= 9/10 × 100

= (9 × 100)/ 10

= 900/10

= 90

(xii) 0.03 × 1000

Answer: 30

Explanation: 0.03× 1000

= 3/100 × 1000

= (3 × 1000)/ 100

= 3000/100

= 30

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?

Answer: 553 km

Explanation: Distance covered by a two-wheeler in one litre of petrol = 55.3 km

Distance covered in 10 litres of petrol = 55.3 × 10

= 553 km

5. Find:

(i) 2.5 × 0.3

Answer: 0.75

Explanation: 2.5 × 0.3

= 25/10 × 3/10

= (25 × 3)/ (10 × 10)

= 75/100

=0.75

(ii) 0.1 × 51.7

Answer: 5.17

Explanation: 0.1 × 51.7

= 1/10 × 517/10

= (1 × 517)/ (10 × 10)

= 517/100

= 5.17

(iii) 0.2 × 316.8

Answer: 63.36

Explanation: 0.2 × 316.8

= 2/10 × 3168/10

= (2 × 3168)/ (10 × 10)

= 6336/ 100

= 63.36

(iv) 1.3 × 3.1

Answer: 4.03

Explanation: 1.3 × 3.1

= 13/10 × 31/10

= (13 × 31)/ (10 × 10)

= 403/ 100

= 4.03

(v) 0.5 × 0.05

Answer: 0.025

Explanation: 0.5 × 0.05

= 5/10 × 5/100

= (5 × 5)/ (10 × 100)

= 25/1000

= 0.025

(vi) 11.2 × 0.15

Answer: 1.68

Explanation: 11.2 × 0.15

= 112/10 × 15/100

= (112 × 15)/ (10 × 100)

= 1680/ 1000

= 1.68

(vii) 1.07 × 0.02

Answer: 0.0214

Explanation: 1.07 × 0.02

= 107/100 × 2/100

= (107 × 2)/ (100 × 100)

= 214/10000

= 0.0214

(viii) 10.05 × 1.05

Answer: 10.5525

Explanation: 10.05 × 1.05

= 1005/100 ×105/100

= (1005 × 105)/ (100 × 100)

= 105525/10000

= 10.5525

(ix) 101.01 × 0.01

Answer: 1.0101

Explanation: 101.01 × 0.01

= 10101/100 × 1/100

= (10101 × 1)/ (100 × 100)

= 10101/10000

= 1.0101

(x) 100.01 × 1.1

Answer: 110.011

Explanation: 100.01 × 1.1

= 10001/100 × 11/10

= (10001 × 11)/ (100 × 10)

= 110011/1000

= 110.011

Exercise 2.7

1. Find:

(i) 0.4 ÷ 2

Answer:

Explanation: 0.4 ÷ 2

= 4/10 ÷ 2

= (4/10) × (1/2)

= (4 × 1)/ (10 × 2)

= 4/20

20 = 4 × 5

= (4 × 1) (4 × 5)

= 1/5

(ii) 0.35 ÷ 5

Answer: 0.07

Explanation: 0.35 ÷ 5

= 35/100 ÷ 5

= 35/100 × 1/5

= (35 × 1)/ (100 × 5)

= 35/500

= 7/100

= 0.07

(iii) 2.48 ÷ 4

Answer: 0.62

Explanation: 2.48 ÷ 4

= 248/100 ÷ 4

= 248/100 × 1/4

= (248 × 1)/ (100 × 4)

248 = 4 × 62

Substituting the value of 248,

= (4 × 62 × 1)/ (100 × 4)

Cancelling out the common factors, we get:

62/100

= 0.62

(iv) 65.4 ÷ 6

Answer: 10.9

Explanation: 65.4 ÷ 6

= 654/10 ÷ 6

= 654/10 × 1/6

= (654 × 1)/ (10 × 6)

654 = 6 × 109

Substituting the value of 654,

= (6 × 109 × 1)/ (10 × 6)

Cancelling out the common factors, we get:

109/10

= 10.9

(v) 651.2 ÷ 4

Answer: 162.8

Explanation: 651.2 ÷ 4

= 6512/10 ÷ 4

= 6512/10 × 1/4

= (6512 × 1)/ (10 × 4)

6512 = 4 × 1628

Substituting the value of 6512,

= (4 × 1628 × 1)/ (10 × 4)

Cancelling out the common factors, we get:

= 1628/10

= 162.8

(vi) 14.49 ÷ 7

Answer: 2.07

Explanation: 14.49 ÷ 7

= 1449/100 ÷ 7

= 1449/100 × 1/7

= (1449 × 1)/ (100 × 7)

1449 = 7 × 207

Substituting the value of 1449,

= (7 × 207 × 1)/ (100 × 7)

Cancelling out the common factors, we get:

207/100

= 2.07

(vii) 3.96 ÷ 4

Answer: 0.99

Explanation: 3.96 ÷ 4

= 396/100 ÷ 4

= 396/100 × 1/4

= (396 × 1)/ (100 × 4)

396 = 4 × 99

Substituting the value of 396,

= (4 × 99 × 1)/ (100 × 4)

Cancelling out the common factors, we get:

99/100

= 0.99

(viii) 0.80 ÷ 5

Answer: 0.16

Explanation: 0.80 ÷ 5

= 80/100 ÷ 5

= 80/100 × 1/5

= (80 × 1)/ (100 × 5)

80 = 16 × 5

Substituting the value of 80,

= (16 × 5 × 1)/ (100 × 5)

Cancelling out the common factors, we get:

= 16/100

= 0.16

2. Find:

(i) 4.8 ÷ 10

Answer: 0.48

Explanation: 4.8 ÷ 10

= 4.8/10

= 0.48

The decimal value of the number moves one decimal point to the left when divided by ten.

(ii) 52.5 ÷ 10

Answer: 5.25

Explanation: 52.5 ÷ 10

= 52.5/10

= 5.25

(iii) 0.7 ÷ 10

Answer: 0.07

Explanation: 0.7 ÷ 10

= 0.7/10

= 0.07

(iv) 33.1 ÷ 10

Answer: 3.31

Explanation: 33.1 ÷ 10

= 3.31

(v) 272.23 ÷ 10

Answer: 27.223

(vi) 0.56 ÷ 10

Answer: 0.056

(vii) 3.97 ÷10

Answer: 0.397

3. Find:

The decimal value of the number moves two decimal points to the left when divided by hundred.

(i) 2.7 ÷ 100

Answer: 0.0027

(ii) 0.3 ÷ 100

Answer: 0.003

(iii) 0.78 ÷ 100

Answer: 0.0078

(iv) 432.6 ÷ 100

Answer: 4.326

(v) 23.6 ÷100

Answer: 0.236

(vi) 98.53 ÷ 100

Answer: 0.9853

4. Find:

The decimal value of the number moves three decimal points to the left when divided by thousand.

(i) 7.9 ÷ 1000

Answer: 0.0079

(ii) 26.3 ÷ 1000

Answer: 0.0263

(iii) 38.53 ÷ 1000

Answer: 0.03853

(iv) 128.9 ÷ 1000

Answer: 0.1289

(v) 0.5 ÷ 1000

Answer: 0.0005

5. Find:

(i) 7 ÷ 3.5

Answer:

Explanation: 7 ÷ 3.5

= 7 ÷ 35/10

= 7 × 10/35

(A fraction is reversed when changes from division to multiplication).

= (7 × 10)/35

35 = 7 × 5

Substituting the value of 35, we get:

= (7 × 10)/ (7 × 5)

= 10/5

= 2

(ii) 36 ÷ 0.2

Answer: 180

Explanation: 36 ÷ 0.2

= 36 ÷ 2/10

= 36 × 10/2

(A fraction is reversed when changes from division to multiplication).

= (36 × 10)/2

36 = 2 × 18

Substituting the value of 36, we get:

= (2 × 18 × 10)/2

= 18 × 10

= 180

(iii) 3.25 ÷ 0.5

Answer: 6.5

Explanation: 3.25 ÷ 0.5

= 325/100 ÷ 5/10

= 325/100 × 10/5

(A fraction is reversed when changes from division to multiplication).

= (325 × 10)/ (100 × 5)

Factorizing the numbers in the numerator,

= (5 × 65 × 10)/ (100 × 5)

= 650/100

= 6.5

(iv) 30.94 ÷ 0.7

Answer: 44.2

Explanation: 30.94 ÷ 0.7

= 3094/100 ÷ 7/10

= 3094/100 × 10/7

= (3094 × 10) / (100 × 7)

3094 = 7 × 442

Substituting the value of 3094, we get:

= (7 × 442 × 10) / (100 × 7)

= 4420/100

= 44.2

(v) 0.5 ÷ 0.25

Answer: 2

Explanation: 0.5 ÷ 0.25

= 5/10 ÷ 25/100

= 5/10 × 100/25

= (5 × 100)/ (10 × 25)

Factorizing 100 and 25,

= (5 × 5 × 2 × 10)/ (10 × 5 × 5)

= 2

(vi) 7.75 ÷ 0.25

Answer: 31

Explanation: 7.75 ÷ 0.25

= 775/100 ÷ 25/100

= 775/100 × 100/25

= (775 × 100/ (100 × 25)

775 = 25 × 31

Substituting the value of 775, we get:

= (25 × 31 × 100/ (100 × 25)

= 31

(vii) 76.5 ÷ 0.15

Answer: 510

Explanation: 76.5 ÷ 0.15

= 765/10 ÷ 15/100

= 765/10 × 100/15

= (765 × 100)/ (10 × 15)

765 = 15 × 51

Substituting the value of 765, we get:

= (15 × 51× 100)/ (10 × 15)

= 510

(viii) 37.8 ÷ 1.4

Answer: 27

Explanation: 37.8 ÷ 1.4

= 378/10 ÷ 1.4/10

= 378/10 × 10/14

= (378 × 10)/ (10 × 14)

378 = 14 × 27

Substituting the value of 378, we get:

= (14 × 27 × 10)/ (10 × 14)

= 27

(ix) 2.73 ÷ 1.3

Answer: 2.1

Explanation: 2.73 ÷ 1.3

= 273/100 ÷ 13/10

= 273/100 × 10/13

= (273 × 10)/ (100 × 13)

273 = 13 × 21

Substituting the value of 273, we get:

= (13 × 21× 10)/ (100 × 13)

= 21/10

= 2.1

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?

Answer: 18 km

Explanation: Distance covered in 2.4 litres of petrol = 43.2 km

Distance covered in 1 litre of petrol = 43.2 ÷ 2.4

= 432/10 ÷ 24/10

= 432/10 × 10/24

= (432 × 10) / (10 × 24)

432 = 24 × 18

Substituting the value of 432, we get:

= (24 × 18 × 10) / (10 × 24)

= 18

Thus, a vehicle covers a distance of 18 km in 1 litre of petrol.






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