Question

The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.

Answer 1

Let the numbers be x and 8 - x

Given that the sum of these numbers is 8

And 15 times the sum of their reciprocals as 8

`rArr15(1/x+1/(8-x))=8`

`rArr15(((8-x)+x)/(x(8-x)))=8`

⇒ 15 ((8 - 𝑥) + 𝑥) = 8(𝑥(8 - 𝑥))

⇒ 15 [8 - 𝑥 + 𝑥] = 8𝑥(8 - 𝑥)

⇒ 120 = 64x - 8x²

⇒ 8𝑥² - 64𝑥 + 120 = 0

⇒ 8[𝑥² - 8𝑥 + 15] = 0

⇒ 𝑥² - 5𝑥 - 3𝑥 + 15 = 0

⇒ (𝑥 - 5)(𝑥 - 3) = 0

⇒ x = 5 or x = 3

∴ The two numbers are 5 and 3.