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.