let the two numbers be 'x' and 'y', x<>0, y<>0 and let x>y then 1/x < 1/y

the sum of the reciprocals of the two numbers is 9

1/x + 1/y = 9

the larger reciprocal exceeds the smaller by 9/2

1/y = 9/2 + 1/x

by solving the system of equations

1/x + 1/y = 9

1/y = 9/2 + 1/x

we find

x = 4/9

y = 4/27

we are considering the non-zero solutions since x<>0 and y<>0

the two numbers are 4/9 and 4/27.