Complicated Percentage Question

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.

Answer 1

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 $= 80\% + 4\% - 16\% = 68\%$

Answer 2

$\frac15$ (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.

Answer 3

$20\%$ 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.