Question

The difference of two natural numbers is 5 and the difference of their reciprocals is `5/14`. Find the numbers.

Answer 1

Let a and b be two natural number such that, a > b

Given,

a – b = 5   ...(1)

And `1/b - 1/a = 5/14`

⇒ `(a - b)/(ab) = 5/14`

⇒ `5/(ab) = 5/14`

⇒ ab = 14   ...(2)

Putting the value of a from equation (1) in equation (2), we get

(5 + b)b = 14

⇒ 5b + b² = 14

⇒ b² + 5b – 14 = 0

⇒ b² + 7b – 2b – 14 = 0

⇒ b(b + 7) – 2(b + 7) = 0

⇒ (b – 2) (b + 7) = 0

⇒ b = 2 or b = –7

Since, b is natural number, neglecting the value, b = –7, we get

b = 2

⇒ a = b + 5

= 2 + 5

= 7

So, the required natural numbers are 7 and 2.