Explanation:

- Let's break down the problem:

- In \( \frac{1}{6} \)th of the time B takes, A completes half the work.

- This means A can complete the entire work in \( \frac{1}{3} \) of B's time.

- If A does the work in \( \frac{1}{3} \) of B's time, their rates can be compared.

- Together, they complete the work in 16 days. Thus, their combined rate is \( \frac{1}{16} \) of the work per day.

- Let's assume B takes \( x \) days for the entire work:

- A will take \( \frac{x}{3} \) days alone to complete the work.

- A's rate alone is \( \frac{3}{x} \) of the work per day.

- B's rate alone is \( \frac{1}{x} \) of the work per day.

- Combined, their rate is:

$$

\frac{3}{x} + \frac{1}{x} = \frac{1}{16}

$$

$$

\frac{4}{x} = \frac{1}{16}

$$

$$

x = 64

$$

- The correct answer is Option:4 - 64 days.