How to solve equations involving multiple floors of the same variable? I want to solve equations of this type:
$$\lfloor1x\rfloor+\lfloor2x\rfloor+\lfloor3x\rfloor+\lfloor4x\rfloor+\lfloor5x\rfloor=10$$
 A: Think of it like there are five rotating spinners, where the $i^{th}$ spinner makes $i$ revolutions per minute.  All spinners start at the same position at $x=0$ minutes. The first spinner completes a revolution every $60$ seconds, the second every $30$ seconds, etc. The number of revolutions completed at time $x$ by the $i^{th}$ spinner is $\lfloor ix\rfloor$, so your problem is to find all of the times where the combined total of revolutions is $10.$ 
Make a schedule of all the times a revolution is completed (a "tick"):
Spinner 1 ...........................................................x
Spinner 2 .............................x.............................x
Spinner 3 ...................x...................x...................x
Spinner 4 ..............x..............x..............x..............x
Spinner 5 ...........x...........x...........x...........x...........x

Each . is one second, and each x is a tick. We see that the $10^{th}$ tick occurs when Spinner 5 makes its penultimate tick at $x=0.8$ minutes $=48$ seconds. Therefore, the set of times for which there are $10$ ticks total is $x\in [0.8,1)$.
A: You can start from the following.
$x=[x]+\{x\}$ and consider number of cases:


*

*$0\leq\{x\}<\frac{1}{5};$

*$\frac{1}{5}\leq\{x\}<\frac{2}{5};$
$.$
$.$
$.$
Good luck!
Also, the left side increases! 
