Ayesha can complete a piece of work in 16 days. Amita can complete the same piece of work in 8 days. If both of them work together in how many days can they complete the same piece of work?

### Concept & Time and Work The rate of work is inversely proportional to the time taken. If a person can complete a work in $n$ days, their 1-day work is $\frac{1}{n}$. When working together, their combined 1-day work is the sum of their individual 1-day works. ### Step-by-Step Solution * **Given:** Ayesha takes 16 days to complete the work. Amita takes 8 days to complete the work. * **Calculation:** Ayesha's 1-day work = $\frac{1}{16}$ Amita's 1-day work = $\frac{1}{8}$ Combined 1-day work = $\frac{1}{16} + \frac{1}{8}$ To add the fractions, find a common denominator (which is 16): Combined 1-day work = $\frac{1}{16} + \frac{2}{16} = \frac{3}{16}$ * Therefore, the time taken to complete the work together is the reciprocal of their combined 1-day work, which is $\frac{16}{3}$ days. * Converting $\frac{16}{3}$ to a mixed fraction: $16 \div 3 = 5$ with a remainder of 1. So, $\frac{16}{3} = 5\frac{1}{3}$ days. ### Exam Strategy & Shortcut Use the LCM (Total Work) method. Let Total Work = LCM of (16, 8) = 16 units. Ayesha's efficiency = 16 / 16 = 1 unit/day. Amita's efficiency = 16 / 8 = 2 units/day. Combined efficiency = 1 + 2 = 3 units/day. Total time = Total Work / Combined efficiency = 16 / 3 = $5\frac{1}{3}$ days. ### Common Pitfall A common mistake is simply averaging the days (e.g., $(16+8)/2 = 12$ days). Remember that working together always takes less time than the fastest individual. ### Final Answer Therefore, the correct answer is **$5\frac{1}{3}$ days**.