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

⇒ x^{2 −} 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.