Page 25 - Start Up Mathematics_8 (Non CCE)

2 −4
Example 38: By what number should be divided to get ?
3 5
Solution: Let the required number be x.
2 − 4 2 1 − 4
So, ÷ x = ⇒ × =
3 5 3 x 5

1 − 4 2 − 4 3 − ( 4 ×) 3 − ( 2 ×) 3 − 6
⇒ = ÷ = × = = =
x 5 3 5 2 52× 51× 5
5 −5
⇒ x = =
−6 6
−5
Hence, the required number =
6

Properties of Division of Rational Numbers
I. Closure Property
“A set of rational numbers is not closed under division.”
p p
 q and 0 are both rational numbers and q ÷ 0 is not defined.

Example 39: Show that the following are rational numbers.
2 5 −7 3
(a) ∏ (b) ÷ (NCERT)
7 3 8 4
2 5 2 3 2 × 3 6
Solution: (a) ÷ = × = = , is a rational number.
7 3 7 5 7 × 5 35
−7 3 −7 4 ( −7) × 4 ( −7) × 1() −7
(b) ÷ = × = = = , is a rational number.
8 4 8 3 8 × 3 2 × 3 6

II. Commutative Property
“Division of rational numbers is not commutative.”
p r p r r p
So, if q and are two non-zero rational numbers, then q ∏ s ≠ s ∏ q .
s
−5 3 3  − 5
Example 40: Show that: ÷ ≠ ÷   (NCERT)
4 7 7  4 

−5 3 −5 7 −×57 −35
Solution: ÷ = × = =
4 7 4 3 43 12
×
3  − 5  3 4 3 − 4 3×−( 4) − 12
and ÷   = × = × = =
7  4  7 − 5 7 5 75× 35

−5 3 3  − 5
So, ÷ ≠ ÷  
4 7 7  4 
III. Associative Property
“Division of rational numbers is not associative.”
p r u  p r u p r  u
So, if q , and v are three non-zero rational numbers, then   q ÷ s   ÷ v ≠ q ÷   s ÷ v   .
s

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