If A and B together can complete a work in 18 days, A and C together in 12 days, and B and C together in 9 days, then B alone can do the work in:

Explanation:-

$$\eqalign{ & {\text{One}}\,{\text{day's}}\,{\text{work}}\,{\text{of}} \cr & \left( {A + B} \right) = \frac{1}{{18}}\,.......(1) \cr & {\text{One}}\,{\text{day's}}\,{\text{work}}\,{\text{of}} \cr & \left( {A + C} \right) = \frac{1}{{12}}\,.......\left( 2 \right) \cr & {\text{One}}\,{\text{day's}}\,{\text{work}}\,{\text{of}} \cr & \left( {B + C} \right) = \frac{1}{9}\,.......\left( 3 \right) \cr & {\text{Adding}}\,\left( {\text{1}} \right){\text{,}}\,\left( {\text{2}} \right)\,{\text{and}}\,\left( {\text{3}} \right) \cr & 2 \times \left( {A + B + C} \right) \cr & = \left\{ {\left( {\frac{1}{{18}}} \right) + \left( {\frac{1}{{12}}} \right) + \left( {\frac{1}{9}} \right)} \right\} \cr & = \frac{1}{4} \cr & {\text{One day's work of}} \cr & \left( {A + B + C} \right) = \frac{1}{8} \cr & B = \left( {\frac{1}{8}} \right) - \left( {A + C} \right) \cr & B = \left( {\frac{1}{8}} \right) - \left( {\frac{1}{{12}}} \right) \cr & {\text{One day's work of}} \cr & B = \frac{{\left( {3 - 2} \right)}}{{24}} = \frac{1}{{24}} \cr & B\,{\text{need}}\,24\,{\text{days}} \cr}$$