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?

Explanation: 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 \(\frac{30\%}{40\%} = 75\%\)).