Compute a division with integer and fractional part I have a problem that I don't know how to solve:
Compute $[\frac{\sqrt{7}}{frac(\sqrt{7})}]$ 
Here's what I've tried: $[\sqrt{7}]=2 \rightarrow frac(\sqrt{7}) = \sqrt{7}-2 \rightarrow [\frac{\sqrt{7}}{frac(\sqrt{7})}] =[\frac{\sqrt{7}}{\sqrt{7}-2}] = [\frac{\sqrt{7}(\sqrt{7}+2)}{7-4}] = [\frac{7+2\sqrt{7}}{3}]$
But from here I don't know what to do anymore.

It is immediate that $2<\sqrt7<3$, which yields
$$\frac{11}3<\frac{7+2\sqrt7}3<\frac{13}3$$ and the answer is one of $3$ or $4$.
So let us evaluate
$$\frac{7+2\sqrt7}3-4=\frac{2\sqrt7-5}3.$$
It is clear that the numerator is positive, by
$$2\sqrt7>5\iff 28>25.$$
Hence, $4$.

You can also start with a tighter bracketing,
$$2.5=\frac52<\sqrt7<3,$$ justified by $$\frac{25}4<7.$$
Then 
$$\frac{7+2\cdot\dfrac52}3=4<\frac{7+2\sqrt7}3<\frac{7+2\cdot3}3<5.$$
