Discussion
In a series of 100 houses actually invaded by smallpox, 70% of the inhabitants are attacked and 85% have been vaccinated. What is the lowest percentage of the vaccinated that must have been attacked ?
Answer: A
At least 64.7% of the vaccinated must have been attacked.
Of the inhabitants, 70% were attacked and 85% vaccinated. The overlap is smallest when the two groups overlap as little as possible:
\[ \begin{aligned} \text{Attacked and vaccinated} &\ge 70 + 85 - 100 \\ &= 55\% \end{aligned} \]
As a percentage of the vaccinated:
\[ \frac{55}{85} \times 100 \approx 64.7\% \]
63.7%, 62.7% and 61.7% are lower than this minimum, so they are not possible for the data given.
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