All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
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    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 1 of 4
From the graph (click to see), one looks for solutions between 1 and 2, between 2 and 3, etc., probably ending around 10 or 11.


First one:

9(1+f) / 10 = 1/f

where f is the fractional part.

9 + 9f = 10/f
9f + 9f^2 -10 = 0

f = (-9+sqrt(81+360))/18
  = 2/3
  
first solution: 5/3

Then

9(2+f) / 10 = 2/f
18 + 9f = 20/f
18f + 9f^2 -20 = 0
9f^2 + 18f - 20 = 0

f = (-18 + sqrt(324+720))/18
  ~= .7950549357115
  
second solution: approx. 2.7950549357115

In general

k + (sqrt(81k^2 + 360k) - 9k)/18

The valid values of k are 1 through 8, as the table shows:

 1             1.666666666666667 
 2             2.795054935711502 
 3             3.862907813126304 
 4             4.905932629027116 
 5             5.935921354681383 
 6             6.95811402901264 
 7             7.975240527365851 
 8             8.988876515698589 
 9             10 
 10            11.00925212577332 
 11            12.01707159867238 
 12            13.02376916856849 
 13            14.02957133205101 
 14            15.03464719546263 
 15            16.03912563829967 
 16            17.04310664416703 

as 10 is not valid as 9.99999... in this context, i.e., floor(10) <> 9; and, the more so, floor(11+) <> 10, etc.

table from:

DEFDBL A-Z
FOR k = 1 TO 16
    PRINT k,
    f = (SQR(81 * k * k + 360 * k) - 9 * k) / 18
    PRINT k + f
NEXT



  Posted by Charlie on 2015-01-30 14:36:09
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (9)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information