TITLE: How to solve an equation involving the floor of a number?
QUESTION [5 upvotes]: I'm looking for a solving procedure for this type of exercises.

If $[x]$ represents the floor of $x$, solve the equation:
$$\left[\frac{6x+5}8\right]=\frac{15x-7}5$$
Choose the correct answer:
a) {$\frac{4}5$}
b) {$\frac{3}4$}
c) {$\frac{7}{15}$,$\frac{4}5$} 
d) {$\frac{6}{15}$} 
e) {$\frac{1}{2}$,$\frac{3}4$} 
f) {$\frac{1}{2}$,$\frac{4}5$} 

Can someone please explain how I can solve this equation? Thank you very much!
ps. I asked a similar question, unfortunately, the same method doesn't seem to work.

REPLY [2 votes]: If $x$ is a solution we have
$$
0 \le \frac {6x + 5} 8 - \frac {15x - 7} 6 < 1
$$
and so
$41/90 < x \le 9/10$ and 
$$
87/90 < \frac {6x + 5} 8 \le 13/8
$$
Evidently $\lfloor \frac{6 x+ 5} 8 \rfloor$ can only be 0 or 1.
Now you have, only, to solve the equation replacing the lhs with those values.