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 ?

Answer: Option B

Solution

Answer: Option B

Solution:

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

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