Question

the difference of two number is 5 and the difference their reciprocal is 1 by 10 find the number

Answer 1

Let numbers are x and ( x - 5 )

Difference of their reciprocal = 1/10

$\frac{1}{x - 5} - \frac{1}{x} = \frac{1}{10} \\ \\ \\ = > \frac{x - (x - 5)}{(x - 5)x} = \frac{1}{10} \\ \\ \\ => \frac{x - x + 5}{ {x}^{2} - 5x } = \frac{1}{10} \\ \\ \\ => \frac{5}{ {x}^{2} - 5x } = \frac{1}{10} \\ \\ \\ => 50 = {x}^{2} - 5x \\ \\ \\ => 0 = {x}^{2} - 5x - 50 \\ \\ \\ = > 0 = {x}^{2} - 10x + 5x - 50 \\ \\ \\ = > 0 = x(x - 10) + 5(x - 10) \\ \\ \\ = > 0 = (x - 10)(x + 5)$

By Zero product rule,

x = 10 or - 5

If we take x = 10,numbers are 10 and ( 10 - 5 = 5)

If we take x = -5, numbers are -5 and ( -5 - 5 = 10 )

Answer 2

Hey! ! !

Solution :-

☆ Let the two numbers be x and x - 5 . 

According to the question.

(1/x - 5 ) - 1/x = 1/10

x - ( x - 5 ) / x ( x - 5 )

⇒ x(x-5) = 50

⇒ x2 - 5x - 50 = 0 

⇒  x2 - 10x + 5x - 50 = 0 

⇒ x (x - 10) + 5 (x - 10) = 0

⇒  (x+5) (x-10) = 0

⇒   (x+5) (x-10) = 0 

⇒ x =  -5 or 10  

⇒ x = 10 (x = -5 , rejected) 

∴ Two numbers are 10 and (10-5) = 5.