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Find From Floor (Posted on 2015-01-30) Difficulty: 3 of 5
Find all possible values of a real number x that satisfy this equation:
9x       floor(x)
---- = --------------
10      x – floor(x)

No Solution Yet Submitted by K Sengupta    
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Solution Are you positive? | Comment 2 of 4 |
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
  Posted by Steve Herman on 2015-01-31 09:59:46

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