The sum of two numbers is 15. The sum of their reciprocals is 518. Find the numbers.
9 and 6
Let the numbers be x and 15−x
1x+115−x=518
15x(15−x)=518
54=15x−x2
x2−15x+54=0
x2−9x−6x+54=0
(x−9)(x−6)=0
x = 9, 6
Taking x = 9, 15−x = 6
Taking x = 6, 15−x = 9
Hence, two numbers are 9 and 6.