Find all possible values of a real number x that satisfy this equation:

9x floor(x)
---- = --------------
10 x – floor(x)

Let x = a + b, where a is an integer and b in the range [0,1).

Then 9(a+b)/10 = a/b, so b cannot be zero.

Solving for a gives a = 9b^2/(10-b), so a must be positive.

And the the numerator is maximized and the denominator minimized as b increases, so the maximum a is slightly below 9 (because b is at most slightly below 1).

So, by inspection, a can only be between 1 and 8.

Substitute all 8 values in the above, and solve a quadratic equation to get 8 solutions, matching Charlie's values.

*Edited on ***February 2, 2015, 1:48 pm**