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.

In some way you need to estimate $2\sqrt7$. First observe that $2\sqrt7=\sqrt{4\cdot 7}=\sqrt{28}$, thus $5=\sqrt{25}<\sqrt{28}<\sqrt{36}=6$, so
$$4=\left\lfloor\frac{7+5}3\right\rfloor\le\left\lfloor\frac{7+2\sqrt7}3\right\rfloor\le\left\lfloor\frac{7+6}3\right\rfloor=4$$
