Page 25 - ICSE Math 8

6 3
Example 25: Divide by .
7 4
6 3 6 4 64× 24 8  3 4 
Solution: ÷ = × = = =   Reciprocal of = 
7 4 7 3 73× 21 7  4 3 

−8 −4
Example 26: The product of two rational numbers is . If one of the numbers is , find the other.
9 15
−4 −8
Solution: One number = , Product =
15 9
Let the other rational number be x.

−4 −8 −8  4 −8  15  −8  −15 
− 
×=x ⇒ x = ÷   = ×   = ×  
15 9 9  15  9  4 9  4 
− 
(−×8 ) (−15 ) (−×2 ) (−5 ) 10
= = =
×
×
94 31 3
2 −4
Example 27: 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 5 −5
⇒ = ÷ = × = = = ⇒ x = =
x 5 3 5 2 52× 51× 5 −6 6
−5
Hence, the required number is .
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Properties of Division of Rational Numbers
p r r  p r 
I. Closure Property: “If and are any two rational numbers such that ≠ 0, then  ÷  is also a
q s s  q s 
rational number that is the set of all non-zero rational numbers is closed under division.”
Example 28: Show that the following are rational numbers.
2 5 −7 3
(a) ÷ (b) ÷
7 3 8 4
×
2 5 2 3 23 6
Solution: (a) ÷ = × = = , is a rational number.
7 3 7 5 75× 35
−7 3 −7 4 ( −7) × 4 ( −7) × 1() −7
(b) ÷ = × = = = , is a rational number.
×
8 4 8 3 83 23 6
×
II. Commutative Property: “Division of rational numbers is not commutative.”
p r p r r p
Thus, if and are two non-zero rational numbers, then ÷ ≠ ÷ .
q s q s s q
−5 3 3  − 5
Example 29: Show that ÷ ≠ ÷   .
4 7 7  4 
−5 3 −5 7 −×57 −35 3  − 5  3 4 3 − 4 3×−( 4) − 12
Solution: ÷ = × = = and ÷   = × = × = =
4 7 4 3 43 12 7  4  7 − 5 7 5 75× 35
×
–5 3 3  5
− 
So, ÷ ≠ ÷  
4 7 7  4 

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