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.
The population consists of those who are sick and those who are healthy.
First month:
$20\%$ of the population are sick
$80\%$ of the population are healthy
Second month:
$20\%$ of the sick are now healthy
$\implies$ $20\%$ of $20\%$ of the population
$\quad\quad$ are added to the healthy group
$\quad\quad$and removed from the sick group
$\implies$ $4\%$ of the population has shifted from sick to healthy
$20\%$ of the healthy are now sick
$\implies 20\%$ of $80\%$ of the population
$\quad\quad$are added to the sick group
$\quad\quad$and removed from the healthy group
$\implies 16\%$ of the population has shifted from healthy to sick.
$\therefore$ Healthy Population $= 80\% + 4\% - 16\% = 68\%$
$20\%$ 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.
