A and B can do a work in 13 days and 26 days, respectively. If they work for a day alternately, starting with A, then in how many days will the work be completed?

The correct answer is Option 17 (or 17 days).

Step 1: Understand the individual work rates of A and B

Let the total work be 1 unit (or 100%).

A can complete the entire work in 13 days. Therefore, A's 1-day work is:

$\frac{1}{13}$

B can complete the entire work in 26 days. Therefore, B's 1-day work is:

$\frac{1}{26}$

Step 2: Calculate the total work done in one 2-day cycle

A and B work on alternate days, starting with A.

Day 1 (A's turn): Work done = $\frac{1}{13}$

Day 2 (B's turn): Work done = $\frac{1}{26}$

Total work completed in 2 days (1 cycle) = $\frac{1}{13}+\frac{1}{26}=\frac{2+1}{26}=\frac{3}{26}$

Step 3: Determine the number of complete 2-day cycles

We need to find how many such 2-day cycles can occur without exceeding the total work of 1.

Multiply the cycle work by 8:

Work done in 8 cycles (which equals 8 × 2 = 16 days) = $8\times \frac{3}{26}=\frac{24}{26}=\frac{12}{13}$

Step 4: Calculate the remaining work and the final day

Remaining work after 16 days = $1-\frac{24}{26}=\frac{2}{26}=\frac{1}{13}$

On Day 17, it is A's turn to work. A's 1-day work is exactly $\frac{1}{13}$.

Thus, A completes the remaining work in exactly 1 full day.

Conclusion:

Total time taken = 16 days + 1 day = 17 days.