8th Maths : Chapter 1 : Numbers : Properties of Rational Numbers : Exercise 1.3 : Numerical solved problems and Text book back Numerical problems Questions with Solution

**Exercise 1.3**

**1. Verify the closure property for addition
and multiplication for the rational numbers ****−****5/7 and 8/9 .**

**Solution:**

Closure property for addition.

Let *a* = −5/7 and *b*
= 8/9 be the given rational numbers.

*a + b* = −5/7 + 8/9

= [ (−5 × 9) + (8 × 7)] / [7 × 9]

= [−45 + 56] / 63 = 11 / 63
is in Q.

i.e *a + b* = [−5/7] + [8/9] = 11/63 is in Q.

∴ Closure property is true for addition of rational numbers.

Closure property for multiplication

Let * a* = −5/7 and b
= 8/9

*a *×* b* = −5/7 × 8/9 = −40/63 is
in Q.

∴ Closure property is true for multiplication of rational
numbers.

** **

**2. Verify the commutative property for
addition and multiplication for the rational numbers ****−****10/11
and ****−****8/33**

**Solution:**

Let *a* = −10/11 and *b* = −8/33 be the given rational
numbers.

Now *a + b* = −10/11 + (−8/33) = [ (−10 × 3) + (−8 × 1)] /
33 = [−30 + (−8)] / 33

*a + b* = −38/33 ………….(1)

*b + a* =
(−8/33) + (−10/11) = [ (−8 × 1) + ((−10) × 3)] / 33 = [−8 + (−30)] / 33

*b + a* = − 38 / 33 …………..(2)

From (1) and (2)

*a + b = b + a* and hence addition
is commutative for rational numbers.

Further *a* × *b* = −10/11 × (−8/33) = 80/363

*a* × *b* = 80/363 …………….(3)

*b* × *a* = −8/33 × (−10/11) =
80/363

*b* × *a* = 80/363 ………….(4)

From (3) and (4) *a* × *b* = *b* × *a*

Hence multiplication is commutative for rational numbers.

** **

**3. Verify the associative property for
addition and multiplication for the rational numbers ****−****7/9
, 5/6 and ****−****4/3.**

**Solution:**

Let *a* = −7/9, *b *= 5/6, * c* = −4/3 be the given rational numbers.

(*a + b*) + *c* = (−7/9 + 5/6) + (−4/3) = ( [−7 × 2 + 5
× 3] / 18 ) + (−4/3)

= ( [−14 + 15] / 18) + (−4/3) = 1/18 + (−4/3)

= [1 + (−4) × 6] / 18 = [ 1 + (−24) ] / 18 = −23/18 ……..(1)

*a + *(*b* + *c* ) = −7/9 + (5/6 + (−4)/3) = −7/9 + ( [5 + (−4) 2] /
6)

= −7/9 + ( [5 + (−8)] / 6) = −7/9 + (−3/6) = −7/9 + (−1/2)

= [−7 × 2 + (−1) × 9] / 18 = [ −14 + (−9) ] / 18 = −23/18 ……..(2)

From (1) and (2), (*a + b*) + *c *= *a *+ (*b +
c*) is true for rational numbers.

Now

(*a* × *b*) ×* c* = (−7/9 × 5/6) × (−4/3) = ( [−7
× 5] / [9 × 6] ) × (−4/3)

= −35/54 × −4/3 = [−35 × (−4)] / [54 × 3] = 70/81 ………(1)

*a* × (*b* x* c*) = −7/9 × ([5/6]
× [−4/3]) = −7/9 × [5 × (−2)] / [3 × 3]

= −7/9 × (−10)/9 = 70/81
………(2)

From (1) and (2) (*a* × *b*) ×* c* = *a* × (*b*
×* c*) is true for addition and multiplication for the rational numbers.

Thus associative property.

** **

**4. Verify the distributive property a **

**Solution:**

Given the rational number *a* = −1/2 ; *b* = 2/3 and *c*
= −5/6

*a* × (*b* +* c*) = −1/2 × (2/3
+ (−5/6)) = −1/2 × ( [(2 × 2) + (−5 × 1)] /6 )

= −1/2 × ( [4 + (−5)] / 6 ) = −1/2 × (−1/6)

*a* × (*b* +* c*) = 1/12
………(1)

(*a* × *b*) + (*a*
×* c*) = (−1/2 × 2/3) + ( −1/2 × (−5/6) )

= −2/6 + 5/12 = [(−2 × 2) + 5 × 1] / 12 = [−4 + 5] / 12

(*a* × *b*) + (*a* ×* c*) = 1/12 ………(2)

From (1) and (2) we have *a* × (*b* +* c*) = (*a* × *b*) + (*a* ×* c*)
is true.

Hence multiplication is distributive over addition for rational
numbers.

** **

**5. Verify the identity property for addition
and multiplication for the rational numbers 15/19 and ****−****18/25.**

**Solution:**

[15/19] + 0 = [15/19] + [0/19] = [15 + 0] / 19 = 15/19

[−18/25] + 0 = [−18/25] + [0/25] = [−18 + 0] / 25 = −18/25

Identify property for addition verified.

[15/19] × 1 = [15 × 1] / 19 = 15/19

[−18/25] × 1 = [−18 × 1] / 25 = −18/25

Identify property for multiplication verified.

** **

**6. Verify the additive and multiplicative
inverse property for the rational numbers ****−****7/17 and 17/27. _{ }**

**Solution:**

−7/17 + 7/17 = [−7 + 7] / 17 = 0/17 = 0

17/27 + (−17/27) = [17 + (−17)] / 27 = 0/27 = 0

Additive inverse for rational numbers verified.

−7/17 × 17/−7 = [−7 × 17] / [17 × (−7)] = 1

17/27 × 27/17 = [17 × 27] / [27 × 17] = 1

Multiplicative inverse for rational numbers verified.

** **

**Objective
Type Questions **

**7. Closure property is not true for division
of rational numbers because of the number**

(A) 1

(B) –1

(C) 0

(D) 1/2

**[Answer: (C) 0 ]**

**8. **** illustrates that subtraction does not
satisfy the ________**** ****property for rational numbers.**

(A) commutative

(B) closure

(C) distributive

(D) associative

**[Answer: (D) associative]**

**9. Which of the following illustrates
the inverse property for addition?**

**[Answer: (A) 1/8 – 1/8 = 0]**

**10. **** illustrates
that multiplication is distributive over **

(A) addition

(B) subtraction

(C) multiplication

(D) division

**[Answer: (B) subtraction]**

We know that different operations with the same pair of rational
numbers usually give different answers. Check the following calculations which are
some interesting exceptions in rational numbers.

(i) 13/4 + 13/9 = 13/4 × 13/9

(ii) 169/30 + 13/15 = 169/30 ÷ 13/15

Amazing …! Isn’t it? Try a few more like these, if possible.

** **

**Think**

Observe that,

Use your reasoning skills, to find the sum of the first 7 numbers
in the pattern given above.

**Solution:**

1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1/ 6.7 + 1/7.8
= 7 / 8

**Answer:**

**Exercise 1.3**

Verify yourself for Qns 1 to 6.

7. (C) 0

8. (D) associative

9. (A) 1/8 – 1/8 = 0

10. (B) subtraction

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8th Maths : Chapter 1 : Numbers : Exercise 1.3 (Properties of Rational Numbers) | Questions with Answers, Solution | Numbers | Chapter 1 | 8th Maths

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