Three friends, X, Y, and Z, can paint a room in 180/17 minutes together. If X and Y work together, they can complete the same job in 12 minutes. How long would it take for Z alone to paint the room?

Correct option is D

​$$\textbf{Let the total work be } 1 \text{ unit.} \\\text{Time taken by } X + Y + Z = \frac{180}{17} \text{ minutes} \\\Rightarrow \text{Work rate of } X + Y + Z = \frac{1}{\frac{180}{17}} = \frac{17}{180} \text{ work per minute} \\\text{Time taken by } X + Y = 12 \text{ minutes} \\\Rightarrow \text{Work rate of } X + Y = \frac{1}{12} \text{ work per minute} \\\text{Therefore, Z's rate:} \\\text{Rate of Z} = \text{Rate of } (X + Y + Z) - \text{Rate of } (X + Y) \\= \frac{17}{180} - \frac{1}{12} \\\text{Take LCM of 180 and 12 = 180:} \\\frac{17}{180} - \frac{15}{180} = \frac{2}{180} = \frac{1}{90} \\\text{So, Z alone can complete the work in } \\\text{Time} = \frac{1}{\text{Rate}} = \frac{1}{\frac{1}{90}} = 90 \text{ minutes}$$​​