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

### Concept & Logic To find the time taken by an individual working alone, determine what fraction of the total work they completed and how long they worked on it. If someone works from the start to the end, calculate their total contribution first. $$ \text{Total Work} = \text{Work done by X} + \text{Work done by Y} $$ ### Step-by-Step Solution 1. **Analyze X's Work:** X can finish the whole work in 24 days, meaning X's 1-day work is $\frac{1}{24}$. The complete work finished in 16 days. Since X worked from the beginning to the end, X worked for all 16 days. Work done by X in 16 days = $16 \times \frac{1}{24} = \frac{16}{24} = \frac{2}{3}$. 2. **Analyze Y's Work:** Remaining work = $1 - \frac{2}{3} = \frac{1}{3}$. This remaining $\frac{1}{3}$ of the work was done by Y. Y joined after 4 days and worked until the end (day 16). Therefore, Y worked for $16 - 4 = 12$ days. 3. **Calculate Y's Total Time:** If Y can complete $\frac{1}{3}$ of the work in 12 days, then the time Y will take to complete the whole work alone is $12 \times 3 = 36$ days. ### Exam Strategy & Shortcut Instead of tracking work day-by-day (before and after Y joined), just look at the total duration each person worked. X worked for 16 days ($\frac{2}{3}$ of X's total required time), leaving exactly $\frac{1}{3}$ for Y. Since Y worked for 12 days to do that $\frac{1}{3}$, Y's total time is simply $12 \times 3$. ### Common Pitfall Assuming Y worked for 16 days alongside X. Pay close attention to the fact that Y joined *after* 4 days, meaning Y only worked for 12 days. ### Final Answer Therefore, the correct answer is **36 days**.