Complicated Percentage Question In a certain town some people were affected by a ’flu’ epidemic. In the first
month $20$% of the population contracted the flu whilst $80$% were healthy.
In the following month $20$% of the sick people recovered and $20$% of the
healthy people contracted the disease.
What fraction of the population is healthy at the end of the second month?

Can someone help me tackle this problem as I am struggling to proceed.
$\frac15$  (20%) of $\frac45$ (80% of affected are still affected) plus $\frac45$ (healthy after first month) of $\frac15 $ (20% became sick): 
$$\frac15\cdot \frac45+\frac45\cdot\frac15 = \frac8{25} = 32\%$$
 32% are affected;  68% are not.
$20\%$ of the $20\%$ of the population that was sick became healthy.

$80\%$ of the $80\%$ of the population that was healthy stayed healthy.

Multiply then add.  It's not that complicated.