I am hesitant whether this question is appropriate for this forum, but I will go ahead and ask it anyways; so please bear with me.
$6$.$\bar 6$ = $20\over 3$
So if you have them multiplied by 30 ; then you should get the same numbers as the answer but obviously you do not. Why is that?
Your question is really about why is $6.\overline{6}=\frac{20}{3}$, since if you know those numbers are equal it is obvious that they multiply equally by anything.
Now,
$$6.\overline{6}= 6+ \sum_{i=1}^{\infty}\frac{6}{10^i}.$$
Multiplying by $\frac{3}{20}$,
$$(\frac{3}{20})(6. \overline{6})=\frac{9}{10}+\sum_{i=1}^{\infty}\frac{9}{10^{i+1}}=\frac{9}{10}+\frac{1}{10}\sum_{i=1}^{\infty}\frac{9}{10^i}=\frac{9}{10}+\frac{1}{10}=1.$$
Hence, $6.\overline{6}=\frac{20}{3}$.
