#### Question

*A* and *B* can do a piece of work in 18 days; *B* and *C* in 24 days and *A* and *C* in 36 days. In what time can they do it, all working together?

#### Solution

\[\text{ Time taken by } \left( A + B \right) \text{ to do the work = 18 days } \]

\[\text{ Time taken by } \left( B + C \right) \text{ to do the work = 24 days } \]

\[\text{ Time taken by } \left( A + C \right) \text{ to do the work = 36 days } \]

\[\text{ Now } , \]

\[\text{ Work done by } \left( A + B \right) = \frac{1}{18}\]

\[ \text{ Work done by } \left( B + C \right) = \frac{1}{24}\]

\[ \text{ Work done by } \left( A + C \right) = \frac{1}{36}\]

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

\[ = \frac{1}{18} + \frac{1}{24} + \frac{1}{36}\]

\[ = \frac{4 + 3 + 2}{72} = \frac{9}{72}\]

\[ = \frac{1}{8}\]

\[ \therefore \text{ Work done together } = 2\left( A + B + C \right) = \frac{1}{8}\]

\[ \therefore \text{ Work done by } \left( A + B + C \right) = \frac{1}{16}\]

\[ \text{ Thus, together they can finish the work in 16 days } .\]