X can do $\frac{1}{4}$ of a work in 10 days, Y can do 40% of the work in 40 days and Z can do $\frac{1}{3}$ of the work in 13 days. Who will complete the work first ?

### Concept & Strategy To find out who completes the work first, calculate the total time each person takes to complete 100% (or $1$ whole) of the work. The person who takes the minimum total time will complete the work first. ### Step-by-Step Solution * **Time taken by X:** X completes $\frac{1}{4}$ of the work in $10$ days. Time taken to complete the whole work = $10 \times 4 = 40$ days. * **Time taken by Y:** Y completes $40\%$ ($\frac{40}{100}$ or $\frac{2}{5}$) of the work in $40$ days. Time taken to complete the whole work = $40 \times \frac{5}{2} = 100$ days. * **Time taken by Z:** Z completes $\frac{1}{3}$ of the work in $13$ days. Time taken to complete the whole work = $13 \times 3 = 39$ days. * **Comparison:** Comparing the total times ($40$ days, $100$ days, $39$ days), Z takes the least number of days ($39$ days). ### Exam Strategy & Shortcut Rather than finding full work times, you can quickly estimate their pace to find the fastest worker: X does $25\%$ in $10$ days -> $100\%$ in $40$ days. Y does $40\%$ in $40$ days -> very slow. Z does $33.33\%$ in $13$ days -> $100\%$ in $39$ days. Clearly, Z is the fastest. ### Common Pitfall A common mistake is directly comparing the given days ($10$, $40$, $13$) without adjusting for the fraction of work completed, which leads to incorrectly choosing X. ### Final Answer Therefore, the correct answer is **Z**.