Let the numbers be ‘a’ and ‘b’

Given, sum of two numbers is 16. The sum of their reciprocals is 1/3.

⇒ a + b = 16

⇒ a = 16 – b

Also, 1/a + 1/b = 1/3

⇒ 1/(16 – b) + 1/b = 1/3

⇒ 3(b + 16 – b) = 16b - b²

⇒ b² – 16b + 48 = 0

⇒ b² – 12b – 4b + 48 = 0

⇒ b(b – 12) – 4(b – 12) = 0

⇒ (b – 4)(b – 12) = 0

⇒ b = 4, 12

Numbers are 4 , 12