Question

The sum of two natural numbers is 5 and the sum of their reciprocals is `5/6`, the numbers are ______.

Options

• 2 and 5

• 4 and 2

• 2 and 3

• 3 and 4

Answer 1

The sum of two natural numbers is 5 and the sum of their reciprocals is `5/6`, the numbers are 2 and 3.

Explanation:

Let the two natural numbers are x and 5 – x

Also, `1/x + 1/(5 - x) = 5/6`

`\implies (5 - x + x)/(x(5 - x)) = 5/6`

`implies 5/(x(5 - x)) = 5/6` 

`implies` 6 = x(5 – x)

`implies` 6 = 5x – x²

`implies` x² – 5x + 6 = 0

`implies` x² – 3x – 2x + 6 = 0

`implies` x(x – 3) – 2(x – 3) = 0

`implies` (x – 3)(x – 2) = 0

Either x – 3 = 0 or x – 2 = 0

`implies` x = 3 or x = 2

Thus, the required natural numbers are 2 and 3.