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×
a c a c ac
In general: For any two rational numbers and , we define the product as: × =
b d b d bd
×
Example 11: Multiply the following.

1  − 3  15   − 3 −3  3  −14 5
(a) ×   (b)   ×   (c) ×   (d) ×
3  4   − 18  5  7  −  2 25 21

Solution: We have,
1  − 3 1 ×− ( 3) − 3 − 1
(a) ×   = = =
3  4  34 12 4
×
( )
3 15 ×− 3 1 1 3 ×− ( 1)
3
(b)    − 15   ×  − 5  = ( − 18 6) × 5 1 ( − 6 2) × 1 = − 1 = 1
=



18
2
− −2
9
33
−3  3  −× −× − ( ) 1 9
(c) ×   = = =
7  −  2 7 ×− ( ) 2 −14 × − ( ) 1 14
Think and Answer
2 1
−14 5 − 14 × 5 −2 Complete the following table by
(d) × = =
25 21 25 × 21 15 finding the product:
5 3 −2 −1 −4
 35 4   − 2 27 x
Example 12: Simplify:   7 × 7  −     9   × 8  5 7 13


4
 35 5 4    − 2 27 3
Solution: We have,  ×  −    ×  −5 5
 1 7 7    9  8 
6 42
 54   − 2 1  27 3  −3
× 
=   −    × 
 17 9 8  2
× 
   1  4 
20  −  3
= −
7  
 4

20 × 4  37
−× 
= −   [LCM of 7 and 4 is 28]
×
×
74  47 
80  −21  80 −− ( 21 ) 101 17
= −   = = = 3
28  28  28 28 28
Reciprocal of a Rational Number
If a rational number is multiplied by another rational number and the product thus obtained is
1, then these two rational numbers are the reciprocal to each other.

−2 3 −2 −3 6
For example: × = × = = 1
3 −2 3 2 6
−2 3
Here, and are the reciprocal to each other.
3 − 2

Hence, the product of a rational numbers with its reciprocal is always 1.

Mathematics-7 68