Question

The difference of two natural numbers is 5 and sum of their reciprocals is `3/10`​. Find the two numbers.

Answer 1

Let the two natural numbers be x and x + 5.

Given: The sum of the reciprocals of two numbers is `3/10`.

⇒ `1/x + 1/(x + 5) = 3/10`

⇒ `(x + 5 + x)/(x(x + 5)) = 3/10`

⇒ `(2x + 5)/(x(x + 5)) = 3/10`

⇒ `10(2x + 5) = 3[x(x + 5)]`

⇒ 20x + 50 = 3(x² + 5x)

⇒ 20x + 50 = 3x² + 15x

⇒ 3x² + 15x − 20x − 50 = 0

⇒ 3x² − 5x − 50 = 0

⇒ 3x² +10x − 15x − 50 = 0

⇒ x(3x + 10) - 5(3x + 10) = 0

⇒ (x − 5) (3x + 10) = 0

⇒ (x − 5) = 0 or (3x + 10) = 0

⇒ x = 5 or 3x = −10

⇒ `x = 5 or x = -10/3`

Since the numbers are natural numbers, x cannot be negative.

Thus, x = 5

x + 5 = 5 + 5

= 10

Hence, the numbers are 5 and 10.