Explanation: To solve this problem, we need to understand the concept of work and time, specifically how to calculate the efficiency of individuals and their combined work rate.
First, let's determine the efficiency of A. Since A can complete the work in 30 days, his efficiency is:
\[ \text{Efficiency of A} = \frac{1}{30} \text{ work per day} \]
A worked alone for 6 days, so the work done by A in 6 days is:
\[ \text{Work done by A in 6 days} = 6 \times \frac{1}{30} = \frac{6}{30} = \frac{1}{5} \]
The remaining work after A worked for 6 days is:
\[ \text{Remaining work} = 1 - \frac{1}{5} = \frac{4}{5} \]
Now, A and B together completed the remaining \(\frac{4}{5}\) of the work in 12 days. Let's denote the efficiency of B as \( \frac{1}{x} \) work per day. The combined efficiency of A and B is:
\[ \text{Combined efficiency of A and B} = \frac{1}{30} + \frac{1}{x} \]
Since they completed \(\frac{4}{5}\) of the work in 12 days, we can write:
\[ \left( \frac{1}{30} + \frac{1}{x} \right) \times 12 = \frac{4}{5} \]
Solving for \(x\):
\[ \frac{12}{30} + \frac{12}{x} = \frac{4}{5} \]
\[ \frac{2}{5} + \frac{12}{x} = \frac{4}{5} \]
\[ \frac{12}{x} = \frac{4}{5} - \frac{2}{5} \]
\[ \frac{12}{x} = \frac{2}{5} \]
\[ x = \frac{12 \times 5}{2} \]
\[ x = 30 \]
Thus, B can complete the work alone in 30 days. This matches the claimed correct answer, and the detailed calculation confirms that B's efficiency is indeed \(\frac{1}{30}\) work per day.
This problem demonstrates the importance of understanding individual and combined efficiencies in work and time problems. It also highlights the need to carefully calculate the remaining work and the combined work rate when multiple individuals are involved.