The difference of two natural numbers is 5 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 = 5 … (1)

And … [∵ If x > y, then 1/x < 1/y]

⇒

From (1),

⇒

Cross multiplying, we get

⇒ 70 = 5xy

⇒ y = 14/x … (2)

Substituting (2) in (1),

⇒ x – 14/x = 5

⇒ (x² – 14)/x = 5

⇒ x² – 14 = 5x

⇒ x² – 5x – 14 = 0

By factorization method,

⇒ x² - 7x + 2x – 14 = 0

⇒ x (x - 7) + 2 (x - 7) = 0

⇒ (x - 7) (x + 2) = 0

⇒ x - 7 = 0 (or) x + 2 = 0

⇒ x = 7 (or) x = -2

Since x is a natural number, x = 7

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

⇒ y = 14/7

⇒ y = 2

i.e. x, y = 7, 2

The numbers are 7, 2.