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

1.	The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.
Answer» Let the numbers be ‘a’ and ‘b’. Given, sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. ⇒ a + b = 8 ⇒ a = 8 – b Also, 15 × (1/a + 1/b) = 8 ⇒ 1/a + 1/b = 8/15 ⇒ 1/a + 1/(8 – a) = 8/15 ⇒ 15(8 – a + a) = 8(8a – a²) ⇒ a² – 8a + 15 = 0 ⇒ a² -5a – 3a + 15 = 0 ⇒ a(a – 5) -3(a – 5) = 0 ⇒ (a – 3)(a – 5) = 0 ⇒ a = 3, 5

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

Answer»

Let the numbers be ‘a’ and ‘b’.

Given, sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8.

⇒ a + b = 8

⇒ a = 8 – b

Also,

15 × (1/a + 1/b) = 8

⇒ 1/a + 1/b = 8/15

⇒ 1/a + 1/(8 – a) = 8/15

⇒ 15(8 – a + a) = 8(8a – a²)

⇒ a² – 8a + 15 = 0

⇒ a² -5a – 3a + 15 = 0

⇒ a(a – 5) -3(a – 5) = 0

⇒ (a – 3)(a – 5) = 0

⇒ a = 3, 5