Here's how to solve the problem step-by-step:
1. Calculate Individual Work Rates
- A's rate: If A finishes the task in 12 days, A completes $\frac{1}{12}$ of the task per day.
- B's rate: If B finishes the task in 16 days, B completes $\frac{1}{16}$ of the task per day.
2. Calculate Work Done in a 2-Day Cycle
Since they work alternatively and A starts:
- Day 1: A works, completes $\frac{1}{12}$.
- Day 2: B works, completes $\frac{1}{16}$.
- Work done in 2 days = Work by A + Work by B
- Work = $\frac{1}{12} + \frac{1}{16}$
- To add these fractions, find a common denominator (LCM of 12 and 16 is 48):
- Work = $\frac{4}{48} + \frac{3}{48} = \frac{7}{48}$ of the task.
3. Determine Full Cycles Completed
We need to find how many 2-day cycles are needed to complete the task. Calculate the number of cycles ($k$) so that the work done is close to 1 (the whole task).
- Work per cycle = $\frac{7}{48}$.
- Let's estimate $k$. Total work needed is 1. $1 \div \frac{7}{48} = \frac{48}{7} \approx 6.85$. This suggests around 6 full cycles.
- Work done in 6 cycles = $6 \times \frac{7}{48} = \frac{42}{48} = \frac{7}{8}$ of the task.
- Number of days passed = $6 \text{ cycles} \times 2 \text{ days/cycle} = 12$ days.
4. Calculate Remaining Work
After 12 days, the remaining work is:
- Remaining Work = $1 - \frac{7}{8} = \frac{1}{8}$ of the task.
5. Calculate Time for Remaining Work
The 13th day begins, and A starts working:
- Work done by A on Day 13 = $\frac{1}{12}$.
- Total work completed after 13 days = Work after 12 days + Work on Day 13
- Total Work = $\frac{7}{8} + \frac{1}{12} = \frac{21}{24} + \frac{2}{24} = \frac{23}{24}$ of the task.
- Remaining work after Day 13 = $1 - \frac{23}{24} = \frac{1}{24}$ of the task.
The 14th day begins, and it's B's turn:
- B's rate = $\frac{1}{16}$ per day.
- Time needed for B to complete the remaining $\frac{1}{24}$ work = $\frac{\text{Remaining Work}}{\text{B's Rate}}$
- Time = $\frac{1/24}{1/16} = \frac{1}{24} \times 16 = \frac{16}{24} = \frac{2}{3}$ days.
6. Calculate Total Time
Total time = Days from full cycles + Day 13 + Fractional day on Day 14
- Total Time = $12 \text{ days} + 1 \text{ day} + \frac{2}{3} \text{ days}$
- Total Time = $13 \frac{2}{3}$ days.
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