This problem involves calculating the time taken to complete a task when two people, A and B, work on alternate days. We need to determine their individual work rates and how their combined effort progresses over the alternate days.

Understanding Work Rates

First, let's determine the amount of work each person completes in a single day.

• Person A can complete the work in 24 days. So, A's work rate is $\frac{1}{24}$ of the work per day.
• Person B can complete the work in 28 days. So, B's work rate is $\frac{1}{28}$ of the work per day.

Calculating Work Done in Cycles

They work on alternate days, with B starting first. This means a cycle of work consists of two days: Day 1 (B works) and Day 2 (A works).

1. Day 1 (B works): Work done = $\frac{1}{28}$
2. Day 2 (A works): Work done = $\frac{1}{24}$
3. Work done in 2 days (1 cycle): Sum of work done by B and A = $\frac{1}{28} + \frac{1}{24}$

To add these fractions, we find a common denominator. The least common multiple (LCM) of 28 and 24 is 168.

• Work done by B in 2 days = $\frac{168}{28} = 6$ parts
• Work done by A in 2 days = $\frac{168}{24} = 7$ parts
• Total work done in 2 days = $6 + 7 = 13$ parts.
• So, in every 2-day cycle, 13 parts of the total work (represented as 168 parts) are completed.

Determining the Number of Cycles

We need to find out how many such 2-day cycles are needed to complete the total work of 168 parts.

Divide the total work by the work done per cycle: $$ \frac{168 \text{ parts}}{13 \text{ parts/cycle}} = 12 \text{ cycles with a remainder} $$ Let's calculate the work done in exactly 12 cycles:

• Number of days = $12 \text{ cycles} \times 2 \text{ days/cycle} = 24$ days.
• Work completed in 24 days = $12 \text{ cycles} \times 13 \text{ parts/cycle} = 156$ parts.

Calculating Remaining Work and Time

After 24 days, there is still some work left.

• Remaining work = Total work - Work done = $168 - 156 = 12$ parts.
• The 24 days completed the 12th cycle, which ended with A working. The next day, Day 25, is B's turn.
• B's work rate is 6 parts per day.
• Time needed for B to complete the remaining 12 parts = $\frac{12 \text{ parts}}{6 \text{ parts/day}} = 2$ days.

Wait, let's recheck the calculation.

After 12 cycles (24 days), 156 parts are done. Remaining work is 12 parts.

Day 25: B works. B completes 6 parts. Remaining work = $12 - 6 = 6$ parts.

Day 26: A works. A's rate is 7 parts per day. Time A needs for the remaining 6 parts = $\frac{6 \text{ parts}}{7 \text{ parts/day}} = \frac{6}{7}$ days.

Total Time Calculation

The total time taken is the sum of the time for the full cycles and the time taken on the last day.

• Time for 12 cycles = 24 days.
• Time for the remaining work = 1 day (B on Day 25) + $\frac{6}{7}$ days (A on Day 26) = $1\frac{6}{7}$ days.
• Total time = $24 + 1\frac{6}{7} = 25\frac{6}{7}$ days.

Therefore, the work will be finished in $25\frac{6}{7}$ days.