Class 8 RD Sharma - Chapter 1 Rational Numbers - Exercise 1.7 | Set 1

Exercise 1.7 | Set 1 in Chapter 1 of RD Sharma's Class 8 mathematics textbook represents a culmination of the study of rational numbers, offering a comprehensive and challenging set of problems that integrate all the concepts learned throughout the chapter. This exercise is designed to test and reinforce students' understanding of rational numbers at an advanced level, pushing them to apply their knowledge in complex, multi-step problems and real-world scenarios. The problems in this set cover a wide range of topics, including operations with rational numbers, conversions between different representations, solving equations and inequalities involving rational numbers, and exploring advanced properties of these numbers.

Problem 1. Divide:

(i) 1 by 1/2

Solution:

= 1 / (1 / 2)

= 1 × (2 / 1)

= 2

ii) 5 by -5/7

Solution:

= 5 / (-5 / 7)

= 5 × (-7 / 5)

5 is the common factor

= -7

(iii) -3/4 by 9/-16

Solution:

= (-3 / 4) / (-9 / 16)

= -3 / 4 × (-16 / 9)

3 and 4 are the common factor

= (-3 × -16) / (4 × 9)

= 4 / 3

iv) -7/8 by -21/16

Solution:

= (-7 / 8) / (-21 / 16)

= -7 / 8 × (-16 / 21)

= (-7 × -16) / (8 × 21)

7 and 8 are the common factor

= 2 / 3

v) 7/-4 by 63/64

Solution:

= 7 / -4 / (63 / 64)

= -7 / 4 × (64 / 63)

= (-7 × 64) / (4 × 63)

4 and 7 are common factor

= -16 / 9

vi) 0 by -7/5

Solution:

= 0 / (-7 / 5)

= 0 × (-5 / 7)

= 0

(vii) -3/4 by -6

Solution:

= -3 / 4 / (-6 / 1)

= -3 / 4 × (-1 / 6)

= (-3 × -1) / (4 × 6)

3 is the common factor

= 1 / 8

(viii) 2/3 by -7/12

Solution:

= 2 / 3 / (-7 / 12)

= 2 / 3 × (12 / -7)

= (2 × 12) / (3 × -7)

3 is the common factor

= 8 / -7

= -8 / 7

(ix) -4 by -3/5

Solution:

= -4 / (-3 / 5)

= -4 × (5 / -3)

= (-4 × -5) / 3

= 20 / 3

(x) -3/13 by -4/65

Solution:

= -3 / 13 / (-4 / 65)

= -3 / 13 × (65 / -4)

= (-3 × -65) / (13 × 4)

13 is the common factor

= (-3 × -5) / 4

= 15 / 4

Problem 2. Find the value and express as a rational number in standard form:

(i) 2/5 ÷ 26/15

Solution:

= 2 / 5 / (26 / 15)

= 2 / 5 × 15 / 26

= (2 × 15) / (5 × 26)

5 and 2 are the common factor

= 3 / 13

(ii) 10/3 ÷ -35/12

Solution:

= 10 / 3 / (-35 / 12)

= 10 / 3 × -12 / 35

= (10 × -12) / (3 × 35)

Common factor is 5 and 3

= (2 × -4) / (7)

= -8 / 7

(iii) -6 ÷ -8/17

Solution:

= -6 / (-8 / 17)

= -6 × (-17 / 8)

= (-6 × -17) / (8 × 1)

2 is the common factor

= (-3 × -17) / 4

= 51 / 4

(iv) -40/99 ÷ -20

Solution:

= -40 / 99 / (-20 / 1)

= -40 / 99 × (-1 / 20)

= (-40 × -1) / (99 × 20)

20 is the common factor

= 2 / 99

(v) -22/27 ÷ -110/18

Solution:

= (-22 / 27) / (-110 / 18)

= (-22 / 27) × (18 / -110)

= (-22 × 18) / (27 × -110)

9 and 22 are the common factor

= 2 / (3 × 5)

= 2 / 15

(vi) -36/125 ÷ -3/75

Solution:

= (-36 / 125) / (-3 / 75)

= (-36 / 125) × (75 / -3)

= (-12 / 25) × (15 / -1)

= (-12 × 15) / (25 × -1)

= (-12 × -3) / 5

= 36 / 5

Problem 3. The product of two rational numbers is 15. If one of the numbers is -10, find the other.

Solution:

We know that the product of two rational numbers = 15

One of the number = -10

Let the other number be x

-10x = 15

x = 15 / -10

5 is the common factor

= -3 / 2

The other number is -3 / 2

Problem 4. The product of two rational numbers is -8/9. If one of the numbers is -4/15, find the other.

Solution:

We know that the product of two rational numbers = -8 / 9

One of the number = -4 / 15

Let the other number be x

(-4 / 15) x = -8 / 9

x = (-8 / 9) / (-4 / 15)

   = (-8 / 9) × (15 / -4)

3 and 4 are the common factor

= (-2 / 3) × (5 / -1)

= (-2 × 5) / (3 × -1)

= -10 / -3

= 10 / 3

The other number is 10 / 3

Problem 5. By what number should we multiply -1/6 so that the product may be -23/9?

Solution:

Let the number be x

So, x (-1 / 6) = -23 / 9

x = (-23 / 9) / (-1 / 6)

x = (-23 / 9) × (6 / -1)

= (-23 / 3) × (2 × -1)

= (-23 × -2) / (3 × 1)

= 46 / 3

It should be multiplied by 46 / 3

Problem 6. By what number should we multiply -15/28 so that the product may be -5/7?

Solution:

Let the number be x

So, x (-15 / 28) = -5 / 7

x = (-5 / 7) / (-15 / 28)

x = (-5 / 7) × (28 / -15)

   = (-5 ×28) / (7 × -15)

5 and 7 are the common factor

= -4 / -3

= 4 / 3

Problem 7. By what number should we multiply -8/13 so that the product may be 24?

Solution:

Let the number be x

So, x (-8 / 13) = 24

x = (24) / (-8 / 13)

   = (24) × (13 / -8)

   = (24 × 13) / (-8)

8 is the common factor

= -3 × 13

= -39

It should be multiplied by -39

Problem 8. By what number should -3/4 be multiplied in order to produce 2/3?

Solution:

Let the number be x

x (-3 / 4) = 2 / 3

x = (2 / 3) / (-3 / 4)

   = (2 / 3) × (4 / -3)

   = -8 / 9

It should be multiplied by -8 / 9

Problem 9. Find (x+y) ÷ (x-y), if

(i) x = 2/3, y = 3/2

Solution:

x + y = 2 / 3 + 3 / 2

LCM is 6

= (2 × 2 + 3 × 3) / 6

= (4 + 9) / 6

= 13 / 6

x - y = 2 / 3 - 3 / 2

LCM is 6

= (2 × 2 - 3 × 3) / 6

= (4 - 9) / 6

= -5 / 6

(x + y) ÷ (x - y) = (13 / 6) / (-5 / 6)

= (13 / 6) × (6 / -5)

= (13 × -6) / (6 × 5)

6 is the common factor

= -13 / 5

(ii) x = 2/5, y = 1/2

Solution:

x + y = 2 / 5 + 1 / 2

LCM is 10

= (2 × 2 + 1 × 5) / 10

= (4 + 5) / 10

= 9 / 10

x - y = 2 / 5 - 1 / 2

LCM is 10

(2 × 2 - 1 × 5) / 10

= (4 - 5) / 10

= -1 / 10

(x + y) ÷ (x - y) = (9 / 10) / (-1 / 10)

= (9 / 10) × (10 / -1)

= (9 × 10) / (10 × -1)

10 is the common factor

= -9

(iii) x = 5/4, y = -1/3

Solution:

x + y = 5 / 4 + -1 / 3

LCM is 12

= (5 × 3 - 1 × 4) / 12

= (15 - 4) / 12

= 11 / 12

x - y = 5 / 4 - (-1/3)

= 5 / 4 + 1 / 3

LCM is 12

= (5 × 3 + 1 × 4) / 12

= 19 / 12

(x + y) ÷ (x - y) = (11 / 12) / (19 / 12)

= (11 / 12) × (12 / 19)

= (11 × 12) / (12 × 19)

Common factor is 12

= 11 / 19

(iv) x = 2/7, y = 4/3

Solution:

x + y = 2 / 7 + 4 / 3

LCM is 21

= (2 × 3 + 4 × 7) / 21

= (6 + 28) / 21

= 34 / 21

x - y = 2 / 7 - 4 / 3

LCM is 21

= (2 × 3 - 4 × 7) / 21

= (6 - 28) / 21

= -22 / 21

(x + y) ÷ (x - y) = (34 / 21) / (-22 / 21)

= (34 / 21) × (21 / -22)

21 is the common factor

= -34 / 22

= -17 / 11

(v) x = 1/4, y = 3/2

Solution:

x + y = 1 / 4 + 3 / 2

LCM is 4

= (1 + 3 × 2) / 4

= 7 / 4

x - y = 1 / 4 - 3 / 2

LCM is 4

= (1 - 3 × 2) / 4

= -5 / 4

(x + y) ÷ (x - y) = (7 / 4) / (-5 / 4)

= (7 / 4) × (4 / -5)

4 is the common factor

= -7 / 5

Problem 10. The cost of 723 meters of rope is Rs 12 ¾. Find the cost per meter.

Solution:

23 / 3 meters of rope = Rs 51 / 4

Let us consider a number = x

So, x (23 / 3) = 51 / 4

x = (51 / 4) / (23 / 3)

   = (51 / 4) × (3 / 23)

   = (51 × 3) / (4 × 23)

   = 153 / 92

   = 1 61 / 92

Cost per meter is Rs 1 61 / 92

Summary

Exercise 1.7 | Set 1 in Chapter 1 of RD Sharma's Class 8 mathematics textbook offers a comprehensive and challenging collection of problems on rational numbers. This exercise integrates all the concepts covered in the chapter, presenting students with complex, multi-step problems that require a deep understanding of rational number operations, properties, and applications. The questions range from algebraic manipulations and proofs to practical word problems, encouraging students to apply their knowledge in various contexts. By solving these problems, students enhance their problem-solving skills, logical reasoning, and ability to apply mathematical concepts to real-world situations. This exercise serves as an excellent preparation for more advanced mathematical topics and helps solidify students' understanding of rational numbers.