Let the numbers be ‘a’ and ‘b’

Given, sum of two numbers is 15. The sum of their reciprocals is 1/4

⇒ a + b = 15

⇒ b = 15 – a

Also, 1/a + 1/b = 3/10

⇒ 1/a + 1/(15 – a) = 3/10

⇒ 15 × 10 = 45a – 3a²

⇒ a² – 15a + 50 = 0

⇒ a² – 15a – 5a + 50 = 0

⇒ a(a – 10) – 5(a – 10) = 0

⇒ (a – 5)(a – 10) = 0

⇒ a = 5, 10

Numbers are are 5,10 or 10, 5