At a certain university, 60% of the professors are women, and 70% of the professors are tenured. If 90% of the professors are women, tenured, or both, then what percent of the men are tenured?

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Accepted Answer

সঠিক উত্তর: । ব্যাখ্যা: Let $T$ be the total number of professors (or 100%). Women ($W$) = $60\%$ of $T$. Men ($M$) = $100\% - 60\% = 40\%$ of $T$. Tenured ($A$) = $70\%$ of $T$. Women or Tenured or Both ($W \cup A$) = $90\%$ of $T$. The Inclusion-Exclusion Principle for two sets is: $P(W \cup A) = P(W) + P(A) - P(W \cap A)$. $90\% = 60\% + 70\% - P(W \cap A)$. $90\% = 130\% - P(W \cap A)$. $P(W \cap A) = 130\% - 90\% = 40\%$. The percentage of professors who are **Women and Tenured** ($W \cap A$) is **40\%**. We need to find the percentage of professors who are **Men and Tenured** ($M \cap A$). Total Tenured = (Women and Tenured) + (Men and Tenured). $P(A) = P(W \cap A) + P(M \cap A)$. $70\% = 40\% + P(M \cap A)$. $P(M \cap A) = 70\% - 40\% = 30\%$. The percentage of professors who are Men and Tenured is 30\%. The question asks: **What percent of the men are tenured?** This is $\frac{\text{Men and Tenured}}{\text{Total Men}}$. $\frac{30\%}{40\%} = \frac{3}{4} = 75\%$. (The solution uses this logic: $60\% + \text{Men Tenured} = 90\%$, leading to Men Tenured = $30\%$, then $\frac{30\%}{40\%} = 75\%$).