How to solve an equation involving the floor of a number? 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.
 A: If you really want to solve it without using the fact that it's multiple-choice, notice that the fractional part of $\frac{6x+5}{8}$ is $Q=\frac{6x+5}{8}-\frac{15x-7}{5}$. Since $Q$ is a fractional part, $0 \leq Q < 1$; you should be able to work backwards to find bounds on $x$. Once you have those bounds, Ilya's observation that $3x-7/5$ is an integer means there are only finitely many values of $x$ you have to worry about checking, so just try them all (or write a program to do it for you).
A: Since $\frac{15x-7}{5}$ is required to be an integer, it is of the form $\frac{2+5j}{15}$ for $j$ an arbitrary integer. Thus, the equation to solve becomes
$$
\left\lfloor\frac{6\big(\frac{2+5j}{15}\big)+5}{8}\right\rfloor =
\frac{15\big(\frac{2+5j}{15}\big)-7}{5}\qquad\text{which simplifies to}\quad
\left\lfloor\frac{\frac{4}{5}+2j+5}{8}\right\rfloor = j-1 
$$ 
Now observe that the function on the right hand side increases quicker
than the function on the left hand side. So we can solve the question
by finding a maximal solution to the inequality $$
j-1\leq\left\lfloor\frac{\frac{4}{5}+2j+5}{8}\right\rfloor 
$$ 

A warning: But here we should be careful, because a maximal solution to this inequality is not necessarily a solution to our equality. Also, there might be more solutions than just the maximal solution $j_0$ to this inequality. But since the left hand side and the right hand side are linear in $j$, the set of solutions forms an interval in $\mathbb{Z}$. This means that to give all the solutions, it suffices to find the largest integer below $j_0$ for which the inequality is strict. 

Note that for any $n\in\mathbb{Z}$ and $a\in\mathbb{R}$, the
inequality $n\leq\lfloor a\rfloor$ holds if and only if $n\leq a$
holds. This helps, since now we can continue to compute 
$$
\begin{split} j-1 \leq \left\lfloor\frac{\frac{4}{5}+2j+5}{8}\right\rfloor & \Leftrightarrow j-1 \leq \frac{\frac{4}{5}+2j+5}{8}\\ 
& \Leftrightarrow 8j-8\leq \frac{4}{5}+2j+5\\ 
& \Leftrightarrow 10j\leq 23\\ 
& \Leftrightarrow j\leq 2. \end{split} 
$$
Therefore, we get the maximal solution for the equation by substituting $j=2$ in $x=\frac{2+5j}{15}$, i.e. $x=\frac{4}{5}$. This is indeed a solution to the equality. As we have noted before, $\frac{4}{5}$ is not necessarily the only solution. Indeed, it is easy to check that $j=1$ also gives a solution to the equation. But for $j\leq 0$ the inequality is strict, so the only solutions to the equation are $x=\frac{4}{5}$ and $x=\frac{7}{15}$.

Here is a link to the answer to another question, where the OP had to deal with the floor function. Maybe you'll find it useful.
A: 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.
A: Let $\mathbb{Z} \ni k = \lfloor \frac{6 x+ 5}{8} \rfloor$. Then the equation says $x = \frac{k}{3} + \frac{7}{15}$. Now we write the equation for $k$:
$$
    k = \left\lfloor \frac{k}{4} + \frac{39}{40} \right\rfloor
$$
It now remains to test few small values of $k$. Values $k=0$ and $k=1$ work out. 

These correspond to $x = \frac{7}{15}$ and $x=\frac{12}{15} = \frac{4}{5}$.
A: When you have to chose between different answers, you don't need to solve the equation, you just need to test the proposed answers.
