Question

The sum of two natural numbers is 9 and the sum of their reciprocals is `1/2`. Find the numbers.

Answer 1

Let the required natural numbers be x and (9 – x).

According to the given condition, 

`1/x + 1/(9 - x) = 1/2` 

⇒ `(9 - x + x)/(x(9 - x)) = 1/12` 

⇒ `9/(9x - x^2) = 1/2` 

⇒ 9x – x² = 18 

⇒ x² – 9x + 18 = 0 

⇒ x² – 6x – 3x + 18 = 0 

⇒ x(x – 6) – 3(x – 6) = 0 

⇒ x – 3 = 0 or x – 6 = 0 

⇒ x = 3 or x = 6 

When x = 3, 

9 – x = 9 – 3

= 6 

When x = 6, 

9 – x = 9 – 6

= 3 

Hence, the required natural numbers are 3 and 6.