Question

The difference of two numbers is 4. If the difference of their reciprocals is `4/21`, find the numbers.

Answer 1

Let the two numbers be x and x - 4

Given that the difference of two numbers is 4.

By the given hypothesis, we have

`1/(x-4)-1/x=4/21`

`rArr(x-x+4)/(x(x-4))=4/21`

⇒ 84 = 4x(x – 4)

⇒ 𝑥² - 4𝑥 - 21 = 0

⇒ 𝑥² - 7𝑥 + 3𝑥 - 21 = 0

⇒ 𝑥(𝑥 - 7) + 3(𝑥 - 7) = 0

⇒ (𝑥 - 7)(𝑥 + 3) = 0

⇒ 𝑥 = 7 𝑜𝑟 𝑥 = -3 and

If x = -3, x – 4 = -3 - 4 = -7

Hence, required numbers are 3, 7 and -3, -7