X can do a piece of work in 24 days. When he had worked for 4 days, Y joined him. If complete work was finished in 16 days, Y can alone finish that work in ?

Solution(By Examveda Team)

$$\eqalign{ & {\text{According to the question,}} \cr & {\text{X}} \to {\text{24 days}} \cr & \Rightarrow {\text{Work done X in 4 days alone}} \cr & = 4 \times 1 \cr & = 4{\text{ units}} \cr & \Rightarrow {\text{Remaining work}} \cr & = 24 - 4 = 20{\text{ units}} \cr & \Rightarrow {\text{20 units done by both together in}} \cr & = \left( {16 - 4{\text{ days}}} \right) \cr & = 12{\text{ days}} \cr & {\text{Then efficiencies of }}\left( {{\text{X}} + {\text{Y}}} \right) \cr & = \frac{{{\text{Work}}}}{{{\text{Days}}}} \cr & = \frac{{20}}{{12}} \cr & = \frac{5}{3} \cr & = 1 + \frac{2}{3} \cr & \Rightarrow {\text{Efficiency of Y}} = \frac{2}{3} \cr} $$
Time taken by Y alone to complete the total work
$$\eqalign{ & = \frac{{24}}{{\frac{2}{3}}} \cr & = 36{\text{ days}} \cr} $$

Alternate :
$$\eqalign{ & {\text{X}} \times {\text{20}} = \left( {{\text{X}} + {\text{Y}}} \right) \times {\text{12}} \cr & \frac{{\text{X}}}{{{\text{X + Y}}}} \cr & = \frac{{12}}{{15}} \cr & = \frac{{3 \to {\text{Efficiency of X}}}}{{5 \to {\text{Efficiency of }}\left( {{\text{X}} + {\text{Y}}} \right)}} \cr & {\text{Efficiency of Y}} \cr & = 5 - 3 \cr & = 2{\text{units/day}} \cr & {\text{Total work}} \cr & = 24 \times 3 \cr & = 72{\text{units}} \cr & {\text{Total time taken by Y}} \cr & = \frac{{72}}{2} \cr & = {\text{36 days}} \cr} $$