Question

The difference between two positive numbers x and y (x > y) is 4 and the difference between their reciprocals is `4/21`. Find the numbers.

Answer 1

Two numbers are x and y respectively such that x > y.

Then,

x - y = 4                    ...(i)

⇒ x = 4 + y

And,

`1/y - 1/x = 4/21`

⇒ `[ x - y ]/[ xy ] = 4/21`

⇒ `4/(xy) = 4/21`                 ...[From(1)]

⇒ xy = 21

⇒ (4 + y)y = 21

⇒ 4y + y² = 21

⇒ y² + 4y - 21 = 0

⇒ y² + 7y - 3y - 21 = 0

⇒ y(y + 7) - 3(y + 7) = 0

⇒ (y - 3)(y + 7) = 0

⇒ y = 3 and y = -7

We reject y = -7 since y is positive.

⇒ y = 3

⇒ x = 4 + y = 4 + 3 = 7

Thus, the two numbers are 7 and 3 respectively.