Page 19 - Maths Skills - 8

Rational Numbers 17

8 5 7
3. Multiplication: For any three rational numbers, say, , and , you have to prove
9 11 3
8 5 7 8 5

9 11 3 9 11 3
LHS 8 5 280 and RHS 40 7 280
9 33 297 99 3 297
∴ LHS = RHS Hence, proved.
Therefore, multiplication is associative for rational numbers.
In general, for any three rational numbers a, b and c,

a × (b × c) = (a × b) × c

3 2 4
4. Division: For any three rational numbers, say, , and , you have to prove
5 7 9
3 2 4 3 2 4

5 7 9 5 7 9

9
2
LHS = 3 ÷  2 ÷  4  = 3 ÷  ×  = 3 ÷ 9 = 3 × 14 = 14

5  7  9 5 7 4  5 14 5 9 15

RHS 3 2 4 3 7 4 21 4 21 9 189

5 7 9 5 2 9 10 9 10 4 40
i.e., LHS ≠ RHS
Therefore, division is not associative for rational numbers. In general, for any three rational numbers a, b and c.
a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c
Distributive Properties
2 2 3
For any three rational numbers, say, , and , you have to prove
5 3 5
2 2 3 2 2 2 3

5 3 5 5 3 5 5



LHS 2 2 3 2 10 9 2 19 38 RHS =  ×  +  ×  = 4 + 6 = 20 + 18 = 38

2
2
3
2

5 3 5 5 15 5 15 75 ,  5 3   5 5  15 25 75 75
∴ LHS = RHS
2 2 3 2 2 2 3

5 3 5 5 3 5 5
Therefore, multiplication is distributive over addition for rational numbers. In general, for any three rational
numbers a, b and c,
a × (b + c) = (a × b) + (a × c) ...(i)
Similarly, you can prove that multiplication is distributive over subtraction for rational numbers, that is,
a × (b – c) = (a × b) – (a × c) ...(ii)
It may be noted that in equation (i) if a and b are positive and c is negative, then equation (i) becomes equation (ii).