The difference of two natural numbers is 3 and the difference of their reciprocals is . Find the numbers.

Let the larger natural number be x and the smaller natural number be y.

Given, x – y = 3 … (1)

And … [∵ If x > y, then 1/x < 1/y]

⇒

From (1),

⇒

Cross multiplying, we get

⇒ 84 = 3xy

⇒ y = 28/x … (2)

Substituting (2) in (1),

⇒ x – 28/x = 3

⇒ (x² – 28)/x = 3

⇒ x² – 28 = 3x

⇒ x² – 3x – 28 = 0

By factorization method,

⇒ x² + 7x - 4x – 28 = 0

⇒ x (x + 7) - 4 (x + 7) = 0

⇒ (x + 7) (x - 4) = 0

⇒ x + 7 = 0 (or) x - 4 = 0

⇒ x = -7 (or) x = 4

Since, x is a natural number, x = 4.

Putting the above x value in (2), we get

⇒ y = 28/4

⇒ y = 7

i.e. x, y = 4, 7

The numbers are 4, 7.