Challenge question #21 (25/07/2016)
A certain army performed badly on the battlefield. $70\%$ of the people injured an eye, $75\%$ injured an ear, $80\%$ injured an arm, and $85\%$ injured a leg.
What percentage, at least, must have had all four injuries?
At least $10\%$ must have had all four injuries.Worked Solution
Suppose that there are $100$ people in the army.
In total there are $70+75+80+85=310$ injuries.
The least number of people with all four injuries occurs when the injuries are as evenly distributed amongst people in the army as possible.
Now, $300\div100=3.1$, so each person would need to have at least $3$ injuries.
Since this would result in $300$ injuries in total, $10$ people must have $3+1=4$ injuries in this scenario.
$\therefore\enspace$at least $10\%$ must have had all four injuries.