Let the numbers be ‘a’ and ‘b’

Given, sum of two numbers is 18. The sum of their reciprocals is 1/4

⇒ a + b = 18

⇒ b = 18 – a

Also, 1/a + 1/b = 1/4

⇒ 1/a + 1/(18 – a) = 1/4

⇒ 18 × 4 = 18a – a²

⇒ a² – 18a + 72 = 0

⇒ a² – 12a – 6a + 72 = 0

⇒ a(a – 12) – 6(a – 12) = 0

⇒ (a – 6)(a – 12) = 0

⇒ a = 6, 12

Numbers are are 6,12 or 12, 6