Question

The difference of two natural numbers is 5 and the difference of their reciprocals is `1/10`. Find the numbers.

Answer 1

Let the two natural numbers be x and y such that x > y.

Given:

Difference between the natural numbers = 5

∴ x – y = 5   ...(i)

Difference of their reciprocals = 110   ...(Given)

`1/y - 1/x = 1/10`

⇒ `(x - y)/(xy) = 1/10`

⇒ `5/(xy) = 1/10`

⇒ xy = 50   ...(ii)

Putting the value of x from equation (i) in equation (ii), we get

(y + 5)y = 50

⇒ y² + 5y – 50 = 0

⇒ y² + 10y – 5y – 50 = 0

⇒ y(y + 10) – 5(y + 10) = 0

⇒ (y – 5)(y + 10) = 0

⇒ y = 5 or –10

As y is a natural number, therefore y = 5

Other natural number = y + 5 = 5 + 5 = 10

Thus, the two natural numbers are 5 and 10.

Answer 2

Given, difference of two natural numbers is 5.

Let the x, (x + 5) are two natural numbers.

Reciprocals of the numbers are `1/x` and `1/(x + 5)`.

According to the question,

`1/x - 1/(x + 5) = 1/10`

⇒ `(x + 5 - x)/(x(x + 5)) = 1/10`

⇒ `5/(x^2 + 5x) = 1/10`

⇒ x² + 5x – 50 = 0

By splitting the middle term, we get

⇒ x² + 10x – 5x – 50 = 0

⇒ x(x + 10) – 5(x – 10) = 0

⇒ (x + 10)(x – 5) = 0

⇒ x = 5 and x = –10

But given two numbers are natural numbers.

Therefore, x = 5.

Here, the required natural numbers are x = 5 and x + 5 = 5 + 5 = 10.