#### Question

*A* and *B* can do a piece of work in 12 days; *B* and *C* in 15 days; *C* and *A* in 20 days. How much time will *A* alone take to finish the work?

#### Solution

\[\text{ Time taken by } \left( A + B \right) \text{ to do the work = 12 days } \]

\[\text{ Time taken by } \left( B + C \right) \text{ to do the work = 15 days } \]

\[ \text{ Time taken by } \left( A + C \right) \text{ to do the work = 20 days } \]

\[ \text{ Now, } \]

\[\text{ Work done by } \left( A + B \right) = \frac{1}{12}\]

\[ \text{ Work done by } \left( B + C \right) = \frac{1}{15}\]

\[ \text{ Work done by } \left( A + C \right) = \frac{1}{20}\]

\[ \therefore \text{ Work done together } = \left( A + B \right) + \left( B + C \right) + \left( A + C \right)\]

\[ = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}\]

\[ = \frac{5 + 4 + 3}{60} = \frac{12}{60}\]

\[ = \frac{1}{5}\]

\[ \therefore \text{ Work done together } = 2\left( A + B + C \right) = \frac{1}{5}\]

\[ \therefore \text{ Work done by } \left( A + B + C \right) = \frac{1}{10}\]

\[ \therefore \text{ Work done by A alone } = \left( A + B + C \right) - \left( B + C \right)\]

\[ = \frac{1}{10} - \frac{1}{15} = \frac{3 - 2}{30} = \frac{1}{30}\]

\[\text{ Thus, A alone can do the work in 30 days } .\]