A can complete $\frac{1}{3}$ of a work in 5 days and B, $\frac{2}{5}$ of the work in 10 days. In how many days both A and B together can complete the work?

### Concept & Extrapolating Total Time When a fraction of the work and its completion time is given, the time required to complete the *entire* work is calculated by dividing the time taken by the fraction of the work completed. Once individual total times are found, use standard time and work formulas. ### Step-by-Step Solution * **Given:** A completes $\frac{1}{3}$ of the work in 5 days. B completes $\frac{2}{5}$ of the work in 10 days. * **Extrapolating Total Times:** Time for A to complete the whole work = $5 \times \frac{3}{1} = 15$ days. Time for B to complete the whole work = $10 \times \frac{5}{2} = 25$ days. * **Calculation:** A's 1-day work = $\frac{1}{15}$ B's 1-day work = $\frac{1}{25}$ Combined 1-day work = $\frac{1}{15} + \frac{1}{25}$ * Find the LCM of 15 and 25, which is 75. Combined 1-day work = $\frac{5}{75} + \frac{3}{75} = \frac{8}{75}$ * Total time for both A and B together = Reciprocal of their combined 1-day work. Total Time = $\frac{75}{8}$ days. * Convert to a mixed fraction: $75 \div 8 = 9$ with a remainder of 3. Total Time = $9\frac{3}{8}$ days. ### Exam Strategy & Shortcut First, quickly establish total days: $A \rightarrow 5 \times 3 = 15$, $B \rightarrow 10 \times (5/2) = 25$. Use the formula for two people working together: $T = \frac{xy}{x + y}$. $T = \frac{15 \times 25}{15 + 25} = \frac{375}{40}$. Simplify by dividing numerator and denominator by 5: $\frac{75}{8} = 9\frac{3}{8}$ days. ### Common Pitfall A frequent error is directly using the given partial days (5 and 10) in the standard formula without first converting them into the time required for 100% of the work. ### Final Answer Therefore, the correct answer is **$9\frac{3}{8}$**.