X can do a piece of work in 40 days. He works at it for 8 days and then Y finished it in 16 days. How long will they together take to complete the work?

Explanation:-

$$\eqalign{ & {\text{Work}}\,{\text{done}}\,{\text{by}}\,{\text{X}}\,{\text{in}}\,{\text{8}}\,{\text{days}} \cr & = \left( {\frac{1}{{40}} \times 8} \right) = \frac{1}{5} \cr & {\text{Remaining}}\,{\text{work}} \cr & = \left( {1 - \frac{1}{5}} \right) = \frac{4}{5} \cr & {\text{Now}},\,\frac{4}{5}\,{\text{work}}\,{\text{is}}\,{\text{done}}\,{\text{by}}\,{\text{Y}}\,{\text{in}}\,{\text{16}}\,{\text{days}} \cr & {\text{Whole}}\,{\text{work}}\,{\text{will}}\,{\text{be}}\,{\text{done}}\,{\text{by}}\,{\text{Y}}\,{\text{in}}\, \cr & \left( {16 \times \frac{5}{4}} \right) = 20\,{\text{days}} \cr & \therefore {\text{X's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{40}} \cr & \therefore {\text{Y's}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} = \frac{1}{{20}} \cr & \left( {{\text{X + Y}}} \right){\text{'s}}\,{\text{1}}\,{\text{day's}}\,{\text{work}} \cr & = \left( {\frac{1}{{40}} + \frac{1}{{20}}} \right) = \frac{3}{{40}} \cr & {\text{Hence,}}\,{\text{X}}\,{\text{and}}\,{\text{Y}}\,{\text{will}}\,{\text{together}}\,{\text{complete}}\,{\text{the}}\,{\text{work}}\,{\text{in}} \cr & \left( {\frac{{40}}{3}} \right) = 13\frac{1}{3}\,{\text{days}} \cr}$$