#### Question

*A* and *B* can do a piece of work in 6 days and 4 days respectively. A started the work; worked at it for 2 days and then was joined by *B*. Find the total time taken to complete the work.

#### Solution

\[\text{ A can do a work in 6 days, and B can do the same work in 4 days } . \]

\[ \therefore \text{ Work done by A in 2 days } = \frac{2}{6} = \frac{1}{3}\]

\[\text{ Remaining work } = 1 - \frac{1}{3} = \frac{2}{3}\]

\[ \therefore \text{ Work done by } \left( A + B \right)\text{ in 1 day } = \left( \frac{1}{6} + \frac{1}{4} \right)\]

\[ = \frac{2 + 3}{12} = \frac{5}{12}\]

\[ \because \frac{5}{12}\text{ th work is done by A and B in 1 day } . \]

\[ \therefore \frac{2}{3}\text{ rd work will be done by A and B in } \left( \frac{12}{5} \times \frac{2}{3} \right) \text{ days or } \frac{8}{5} \text{ days } . \]

\[ \therefore \text{ Total time taken } = \left( \frac{8}{5} + 2 \right) \text{ days } = \frac{18}{5} \text{ days } = 3\frac{3}{5} \text{ days } \]