Question

Sum of two natural numbers is 8 and the difference of their reciprocal is 2/15. Find the numbers.

Answer 1

Let x and y be two numbers
Given that, x + y = 8 ……(i)
From equation (i), we have, y = 8 - x
Substituting the value of y in equation (ii),
we have,

`(1)/x - (1)/(8 - x) = (12)/(15)`

⇒ `(8 - x - x)/(x(8 - x)) = (2)/(15)`

⇒ `(8 - 2x)/(x(8 - x)) = (2)/(15)`

⇒ `(4 - x)/(x(8 - x)) = (1)/(15)`

⇒ 15(4 - x) = x(8 - x)
⇒ 60 - 15x = 8x - x²
⇒ x² - 15x - 8x + 60 = 0
⇒ x² - 23x + 60 = 0
⇒ x² - 20x - 3x + 60 = 0
⇒ x(x - 20) -3(x - 20) = 0
⇒ (x - 3)(x - 20) = 0
Either (x - 3) = 0
or 
(x - 20) = 0
⇒ x = 3
or
x = 20
Since sum of two natural numbers is 8 - x
i.e. 8 - 20 cannot be equal to 20
Thus x = 3
From equation (i), y = 8 - x = 8 - 3 = 5
Thus the values of x and y are 3 and 5 respectively.