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−5
9
Again,    7 ÷  3   4 ÷ 7 =  −5 ×  4   3 ÷ 7 = −20 ÷ 7 = −20 × = −60

21
9
9
 7
9
21
7
49
−20 −60 −5  3 7   −5  3 7
Clearly, ≠ . Hence, ÷  ÷  ≠  ÷  ÷ .
27 49 7  4  9  7  4 9
This shows that division is not associative for rational numbers. In other words, the associative
property does not hold for the division of rational numbers.
create and solve
Create a table as given below on a sheet of paper and ask your benchmate to complete it.
Rational numbers Do as directed Property Satisfied (Yes/No)

1 1 1 1 2
a = , b = a + b = +=
3 3 3 3 3
5 1 a ÷ b =
a = , b = b ÷ a =
7 3
4 2 a × b =
a = , b = b × a =
7 3

2 1 5 a + (b + c) =
a = , b = and c = (a + b) + c =
3 3 7
4 1 2 a × (b × c) =
a = , b = and c = 1
7 3 5 (a × b) × c =

3 a + b =
a = 1 , b = 0
8 b + a =
3 a × b =
a = , b = 1
8 b × a =

4 4 a – b =
a = , b = − b – a =
7 7
1 1 5 a × (b + c) =
a = , b = and c =
3 5 7 (a × b) + (a × c) =

After completing the table, discuss with your teacher and classmates about the properties of arithmetic operations
(+, –, ×, ÷) for rational numbers verified.

Practice Time 1E

1. Divide the following.

(a) 2 by − 4 (b) 11 by − 5 (c) −3 by −6 (d) −3 by 15
3 15 12 8 10 11 16 8
2. Simplify the following.

(a) 21 ÷   − 30  (b) 5 ÷ 7 (c) 18 ÷ 72
36  44  8 10 49 − 35

27 Rational Numbers