Explanation: Let \(A, B, C\) be the amount of work done by each person in one day. \(A + B = \frac{1}{12}\) (work/day). \(B + C = \frac{1}{18}\) (work/day). \(A + C = \frac{1}{9}\) (work/day). Add the three equations: \(2(A + B + C) = \frac{1}{12} + \frac{1}{18} + \frac{1}{9}\). Find the LCM of 12, 18, 9, which is 36. \(2(A + B + C) = \frac{3}{36} + \frac{2}{36} + \frac{4}{36} = \frac{9}{36} = \frac{1}{4}\). \(A + B + C = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}\) (work/day). To find A's rate, subtract \(B + C\) from \(A + B + C\). \(A = (A + B + C) - (B + C) = \frac{1}{8} - \frac{1}{18}\). Find the LCM of 8 and 18, which is 72. \(A = \frac{9}{72} - \frac{4}{72} = \frac{5}{72}\) (work/day). Time taken by A alone = \(\frac{1}{A} = \frac{1}{\frac{5}{72}} = \frac{72}{5} = 14.4\) days. Since 14.4 is not an option, the correct answer is **None of these**. (The provided solution reaches $72/5$ days).
