CG Board Class 8 Mathematics Chapter 18 Solutions: Operation on Rational Numbers (परिमेय संख्याओं पर संक्रियाएँ) – 2026-27

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CG SCERT (Chhattisgarh Board) Class 8 Mathematics Solutions | 2026-27 Official Edition

Complete 100% textbook-grounded CG Board Class 8 Mathematics Chapter 18 Solutions: Operation on Rational Numbers (परिमेय संख्याओं पर संक्रियाएँ) in Bilingual Format (English & Hindi Medium).

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  • Closure Property: Sum, difference, and product of two rational numbers is always a rational number.
  • Commutative Property:
  • Addition: p/q + r/s = r/s + p/q
  • Multiplication: p/q × r/s = r/s × p/q
  • Subtraction and Division are not commutative.
  • Associative Property:
  • Addition: a + (b + c) = (a + b) + c
  • Multiplication: a × (b × c) = (a × b) × c
  • Additive Identity & Inverse:
  • Additive Identity = 0 (p/q + 0 = p/q)
  • Additive Inverse of p/q = -p/q
  • Multiplicative Identity & Inverse:
  • Multiplicative Identity = 1
  • Multiplicative Inverse (Reciprocal) of p/q = q/p
  • Distributive Property: x × (y + z) = x × y + x × z.
SECTION / EXERCISE

EXERCISE 18.1 / 18.1

Question 1

Add the following rational numbers


📝 Question:

Add the following rational numbers /

💡 Solution:

(i) 3/2 and 13/17**

LCM of denominators 2 and 17 = 34.

3/2 + 13/17 = (3 × 17 + 13 × 2)/(34) = (51 + 26)/(34) = 77/34 = 29/34

(ii) -7/9 and -3/4

LCM of denominators 9 and 4 = 36.

(-7/9) + (-3/4) = (-7 × 4 + (-3) × 9)/(36) = (-28 – 27)/(36) = -55/36 = -119/36

(iii) -3/4 and 2/5

LCM of 4 and 5 = 20.

(-3/4) + 2/5 = (-3 × 5 + 2 × 4)/(20) = (-15 + 8)/(20) = -7/20

Question 2

Use commutative property to fill up the following…


📝 Question:

Use commutative property to fill up the following blanks /

💡 Solution:

(i) 5/9 + 4/9 = 4/9 + **

5/9

(ii) -7/15 + 3/7 = + (-7/15)

3/7

(iii) 5/13 + (-11/13) = -11/13 +

5/13

(iv) 2/3 + 5/6 = 5/6 +

2/3

Question 3

Write the name of property in the blanks


📝 Question:

Write the name of property in the blanks /

💡 Solution:

(i) 13/15 + 4/8 = 4/8 + 13/15**

Commutative Property /

(ii) -2/19 + (3/17 + 4/13) = (-2/19 + 3/17) + 4/13

Associative Property /

(iii) p/q + 0 = p/q

Additive Identity /

(iv) r/s + (-r/s) = 0

Additive Inverse /

Question 4

Find the value of


📝 Question:

Find the value of /

💡 Solution:

(i) 3/7 + 1/4 + 5/7**

Grouping terms with common denominators

(3/7 + 5/7) + 1/4 = 8/7 + 1/4 = (8 × 4 + 1 × 7)/(28) = (32 + 7)/(28) = 39/28 = 111/28

(ii) (-1/6) + (-2/3) + (-1/3) + 5/6

[(-1/6) + 5/6] + [(-2/3) + (-1/3)] = 4/6 + (-3/3) = 2/3 – 1 = -1/3

(iii) 3/8 + (-2/5) + 1/8 + (-1/5)

(3/8 + 1/8) + [(-2/5) + (-1/5)] = 4/8 + (-3/5) = 1/2 – 3/5 = (5 – 6)/(10) = -1/10

Question 5

Simplify 14 59 712


📝 Question:

Simplify / : **1/4 + 5/9 + (-7/12)

💡 Solution:

LCM of 4, 9, 12 = 36.

(1 × 9 + 5 × 4 + (-7) × 3)/(36) = (9 + 20 – 21)/(36) = 8/36 = 2/9

Question 6

Write the additive inverse of the following Rule…


📝 Question:

Write the additive inverse of the following / : *(Rule: Additive inverse of x is -x so that x + (-x) = 0)

💡 Solution:

(i) 5 ⇒ -5**

(ii) -7/9 ⇒ 7/9

(iii) 0 ⇒ 0

(iv) 42/17 ⇒ -42/17

(i) Additive inverse of 5/7 is

-5/7

(ii) Additive inverse of -9/11 is

9/11

(iii) 39/51 + (-39/51) =

0

(iv) Additive inverse of -x/y is

x/y

SECTION / EXERCISE

EXERCISE 18.2 / 18.2

Question 1

Subtract the first rational number from the second…


📝 Question:

Subtract the first rational number from the second rational number /

💡 Solution:

(i) 4/5 from 3/4**

3/4 – 4/5 = (3 × 5 – 4 × 4)/(20) = (15 – 16)/(20) = -1/20

(ii) -1/8 from 1/4

1/4 – (-1/8) = 1/4 + 1/8 = (2 + 1)/(8) = 3/8

(iii) -13/24 from 5/12

5/12 – (-13/24) = 5/12 + 13/24 = (10 + 13)/(24) = 23/24

(iv) -7/13 from -8/13

(-8/13) – (-7/13) = -8/13 + 7/13 = (-8 + 7)/(13) = -1/13

Question 2

Solve


📝 Question:

Solve /

💡 Solution:

(i) 2/9 + 1/3 – 5/9**

(2/9 – 5/9) + 1/3 = -3/9 + 1/3 = -1/3 + 1/3 = 0

(ii) 1/2 – 3/5 + 1/7

LCM of 2, 5, 7 = 70.

(1 × 35 – 3 × 14 + 1 × 10)/(70) = (35 – 42 + 10)/(70) = 3/70

(iii) 1/6 + 3/12 – 6/5

Simplifying 3/12 = 1/4. LCM of 6, 4, 5 = 60.

(10 + 15 – 72)/(60) = -47/60

Question 3

What should be added to 38 so that…


📝 Question:

What should be added to 3/8 so that the sum is 11/12? / 3/8 11/12 ?

💡 Solution:

Let the required number be x.

3/8 + x = 11/12 ⇒ x = 11/12 – 3/8

LCM of 12 and 8 = 24.

x = (11 × 2 – 3 × 3)/(24) = (22 – 9)/(24) = 13/24

Question 4

What should be subtracted from 1325 so that…


📝 Question:

What should be subtracted from 13/25 so that the difference is 19/25? / 13/25 19/25 ?

💡 Solution:

Let the required number be y.

13/25 – y = 19/25 ⇒ y = 13/25 – 19/25 = (13 – 19)/(25) = -6/25

Question 5

Write true or false and give right statements…


📝 Question:

Write true or false and give right statements for false statements / /

💡 Solution:

(i) Additive inverse of 3/5 is 5/3**

False /

  • Correct Statement: Additive inverse of 3/5 is -3/5.

(ii) 4/5 – 7/9 = 7/9 – 4/5

False /

  • Correct Statement: Subtraction is not commutative (4/5 – 7/9 ≠ 7/9 – 4/5).

(iii) The value of a number does not change after subtracting 0 from it.

True / (p/q – 0 = p/q).

(iv) Subtraction of a rational number means the addition of the additive inverse of that number.

True / (a – b = a + (-b)).

SECTION / EXERCISE

EXERCISE 18.3 / 18.3

Question 1

Substitute the values given below and check if…


📝 Question:

Substitute the values given below and check if p/q × r/s = r/s × p/q / p/q × r/s = r/s × p/q

💡 Solution:

(i) p/q = -3/7, r/s = 11/15**

  • LHS: (-3/7) × 11/15 = -33/105 = -11/35
  • RHS: 11/15 × (-3/7) = -33/105 = -11/35

LHS = RHS. Verified!

(ii) p/q = 2/3, r/s = 13/17

  • LHS: 2/3 × 13/17 = 26/51
  • RHS: 13/17 × 2/3 = 26/51

LHS = RHS. Verified!

(iii) p/q = -10/13, r/s = -5/8

  • LHS: (-10/13) × (-5/8) = 50/104 = 25/52
  • RHS: (-5/8) × (-10/13) = 50/104 = 25/52

LHS = RHS. Verified!

(iv) p/q = -16/3, r/s = 0

  • LHS: (-16/3) × 0 = 0
  • RHS: 0 × (-16/3) = 0

LHS = RHS. Verified!

Question 2

Check whether x y z x y x…


📝 Question:

Check whether x × (y + z) = x × y + x × z / x × (y + z) = x × y + x × z

💡 Solution:

(i) x = 1/2, y = -5/7, z = -7/4**

  • LHS: 1/2 × (-5/7 – 7/4) = 1/2 × ((-20 – 49)/(28)) = 1/2 × (-69/28) = -69/56
  • RHS: (1/2 × -5/7) + (1/2 × -7/4) = -5/14 – 7/8 = (-20 – 49)/(56) = -69/56

LHS = RHS. Verified!

(ii) x = -3/2, y = 8/5, z = 17/6

  • LHS: -3/2 × (8/5 + 17/6) = -3/2 × ((48 + 85)/(30)) = -3/2 × 133/30 = -133/20
  • RHS: (-3/2 × 8/5) + (-3/2 × 17/6) = -12/5 – 17/4 = (-48 – 85)/(20) = -133/20

LHS = RHS. Verified!

(iii) x = 1, y = 9/5, z = 0

  • LHS: 1 × (9/5 + 0) = 9/5
  • RHS: (1 × 9/5) + (1 × 0) = 9/5 + 0 = 9/5

LHS = RHS. Verified!

Question 3

Fill in the blanks using commutative property


📝 Question:

Fill in the blanks using commutative property /

💡 Solution:

(i) 4/3 × 2/5 = 2/5 × **

4/3

(ii) -11/19 × 1/2 = × (-11/19)

1/2

(iii) -7/9 × 3/17 = 3/17 ×

-7/9

Question 4

Multiply


📝 Question:

Multiply /

💡 Solution:

(i) (-9/11) × 22/27**

(-9 × 22)/(11 × 27) = (-1 × 2)/(1 × 3) = -2/3

(ii) 2/3 × (-6/7) × 8/15

(iii) (-5/8) × (-7/4)

((-5) × (-7))/(8 × 4) = 35/32 = 13/32

Question 5

Write the property name in each blank


📝 Question:

Write the property name in each blank /

💡 Solution:

(i) a × (b + c) = a × b + a × c**

Distributive Property over Addition /

(ii) a × (b – c) = a × b – a × c

Distributive Property over Subtraction /

(iii) 8/11 × 3/7 = 3/7 × 8/11

Commutative Property /

(iv) 5/3 × 3/5 = 1

Multiplicative Inverse /

(v) (-3/12) × 1 = 1 × (-3/12) = -3/12

Multiplicative Identity /

Question 6

Write the reciprocal of the following


📝 Question:

Write the reciprocal of the following /

💡 Solution:

(i) 4 ⇒ 1/4**

(ii) 17/5 ⇒ 5/17

(iii) -6/29 ⇒ -29/6

(iv) p/q ⇒ q/p

(i) The product of a rational number and its reciprocal is one.

True /

(ii) If the reciprocal of x is y, the reciprocal of y is 1/x.

False / (Reciprocal of y is x)

(iii) The multiplicative inverse of a positive rational number is a negative rational number.

False / (It is a positive rational number)

(iv) Zero has no reciprocal.

True /

SECTION / EXERCISE

EXERCISE 18.4 / 18.4

Question 1

Divide


📝 Question:

Divide /

💡 Solution:

(i) 1/6 by 3/4**

1/6 ÷ 3/4 = 1/6 × 4/3 = 2/9

(ii) -8/11 by 5/9

(-8/11) ÷ 5/9 = -8/11 × 9/5 = -72/55 = -117/55

(iii) -4/9 by 7

(-4/9) ÷ 7 = -4/9 × 1/7 = -4/63

(iv) -102/38 by -17/19

(-102/38) ÷ (-17/19) = (-51/19) × (-19/17) = 3

(v) 6/15 by -8/35

2/5 ÷ (-8/35) = 2/5 × (-35/8) = -7/4 = -13/4

(vi) -60/10 by -9

(-6) ÷ (-9) = -6/-9 = 2/3

Question 2

Simplify


📝 Question:

Simplify /

💡 Solution:

(i) 4/5 ÷ (-1) = -4/5**

(ii) 95/16 ÷ 19/8 = 95/16 × 8/19 = 5/2 = 21/2

(iii) (-7/8) ÷ (-2/15) = (-7/8) × (-15/2) = 105/16 = 69/16

(iv) 21/5 ÷ (-7/5) = 21/5 × (-5/7) = -3

(v) -6/7 ÷ (-15) = -6/7 × (-1/15) = 2/35

(vi) (-7) ÷ (-5) = 7/5 = 12/5

Let the second number be x.

x × 3/5 = 12 ⇒ x = 12 ÷ 3/5 = 12 × 5/3 = 20

Question 4

Which number multiplied by 95 gives the product…


📝 Question:

Which number multiplied by -9/5 gives the product -11? / -9/5 -11 ?

💡 Solution:

Let the number be y.

y × (-9/5) = -11 ⇒ y = (-11) ÷ (-9/5) = (-11) × (-5/9) = 55/9 = 61/9

Question 5

What should be multiplied by 2815 to get…


📝 Question:

What should be multiplied by -28/15 to get 7/3? / -28/15 7/3 ?

💡 Solution:

Let the multiplier be z.

z × (-28/15) = 7/3 ⇒ z = 7/3 ÷ (-28/15) = 7/3 × (-15/28) = -5/4 = -11/4

SECTION / EXERCISE

EXERCISE 18.5 / 18.5

Question 1

Fill up the blanks in following figures


📝 Question:

Fill up the blanks in following figures /

💡 Solution:

(i) Between 5/7 and 3/2:**

Middle Number = 1/2(5/7 + 3/2) = 1/2((10 + 21)/(14)) = 31/28 = 13/28

(ii) Between -5/9 and 3/7:

Middle Number = 1/2(-5/9 + 3/7) = 1/2((-35 + 27)/(63)) = 1/2(-8/63) = -4/63

(iii) Between -4/9 and -2/9:

Middle Number = 1/2(-4/9 – 2/9) = 1/2(-6/9) = -1/3

There are infinitely many (countless) rational numbers between any two rational numbers.

Explanation: By converting given rational numbers into equivalent fractions with larger common denominators, or by repeatedly taking the mean (a + b)/(2), we can insert as many rational numbers between them as desired.

Question 3

Write 5 rational numbers between 13 and 12…


📝 Question:

Write 5 rational numbers between 1/3 and 1/2 / 1/3 1/2 5

💡 Solution:

Converting to common denominator 60

1/3 = 20/60, 1/2 = 30/60

Five rational numbers between 20/60 and 30/60 are

21/60, 22/60, 23/60, 24/60, 25/60 (or 7/20, 11/30, 23/60, 2/5, 5/12)

Question 4

Write 4 rational numbers between 12 and 37…


📝 Question:

Write 4 rational numbers between -1/2 and 3/7 / -1/2 3/7 4

💡 Solution:

Converting to common denominator 14

-1/2 = -7/14, 3/7 = 6/14

Four rational numbers between -7/14 and 6/14 are

-6/14, -5/14, 0, 1/14 (or -3/7, -5/14, 0, 1/14)

Question 5

Write 6 rational numbers between 16 and 34…


📝 Question:

Write 6 rational numbers between -1/6 and 3/4 / -1/6 3/4 6

💡 Solution:

Converting to common denominator 12

-1/6 = -2/12, 3/4 = 9/12

Six rational numbers between -2/12 and 9/12 are

-1/12, 0, 1/12, 2/12, 3/12, 4/12 (or -1/12, 0, 1/12, 1/6, 1/4, 1/3)

Question 6

True or False


📝 Question:

True or False /

💡 Solution:

(i) 1/10 is the middle number of -1/2 and 3/5**

True /

(ii) There is no rational number between 4/5 and 6/5

False / (e.g. 5/5 = 1 lies between them).

(iii) There are only 3 rational numbers between 3 and 7

False / (There are infinite rational numbers).

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