To complete a task, A takes 15 days whereas B takes 21 days. If they work alone on alternate days by turn, B starting on the first day, the number of days required to complete the task is

Correct option is D

Given:
A completes the task in 15 days → A's 1 day work = 1/15
B completes the task in 21 days → B's 1 day work = 1/21
They work on alternate days, starting with B

Solution:

​$$\begin{aligned}& \quad \text{A completes in 15 days } \Rightarrow \frac{1}{15}, \quad \text{B completes in 21 days } \Rightarrow \frac{1}{21} \\[6pt]&\textbf{Work in 2 days (B + A): } \frac{1}{21} + \frac{1}{15} = \frac{5 + 7}{105} = \frac{12}{105} = \frac{4}{35} \\[6pt]&\textbf{In 8 such cycles (16 days): } \frac{4}{35} \times 8 = \frac{32}{35} \\[6pt]&\textbf{Remaining work: } 1 - \frac{32}{35} = \frac{3}{35} \\[6pt]&\text{On 17th day, B works. Let } x \text{ be fraction of day: } \frac{x}{21} = \frac{3}{35} \\[6pt]&x = \frac{3 \times 21}{35} = \frac{63}{35} = \frac{9}{5} = 1.8 \text{ days} \Rightarrow 0.8 \text{ of 1 day} \\[6pt]&\text{Total days = } 16 + 0.8 = 16 + \frac{28}{35} = \boxed{17 \frac{4}{7}}\end{aligned}$$

Final Answer: (D)​​