Explanations:In work and time problems, the key concept is the work rate, which is the fraction of the work that can be completed in a unit of time. If A can complete a work in 30 days, A's work rate is \( \frac{1}{30} \) of the work per day. This means that in one day, A can complete \( \frac{1}{30} \) of the total work.
Let's break down the problem step by step:
1. **Work Done by A Alone:**
- A works alone for 6 days.
- The work done by A in 6 days is \( 6 \times \frac{1}{30} = \frac{6}{30} = \frac{1}{5} \) of the total work.
- Therefore, the remaining work after A has worked for 6 days is \( 1 - \frac{1}{5} = \frac{4}{5} \) of the total work.
2. **Combined Work Rate of A and B:**
- A and B together complete the remaining \( \frac{4}{5} \) of the work in 12 days.
- Let B's work rate be \( \frac{1}{x} \) of the work per day, where \( x \) is the number of days B would take to complete the work alone.
- The combined work rate of A and B is \( \frac{1}{30} + \frac{1}{x} \).
- Since they complete \( \frac{4}{5} \) of the work in 12 days, their combined work rate is \( \frac{4}{5} \div 12 = \frac{4}{60} = \frac{1}{15} \) of the work per day.
3. **Setting Up the Equation:**
- We have the equation \( \frac{1}{30} + \frac{1}{x} = \frac{1}{15} \).
- To solve for \( x \), we first isolate \( \frac{1}{x} \):
\[
\frac{1}{x} = \frac{1}{15} - \frac{1}{30}
\]
- Finding a common denominator (30):
\[
\frac{1}{x} = \frac{2}{30} - \frac{1}{30} = \frac{1}{30}
\]
- Therefore, \( x = 30 \).
Thus, B can complete the work alone in 30 days. This matches the claimed correct answer.
The key to solving such problems is to understand the work rates and how they combine when multiple individuals work together. By breaking down the work done by each individual and the combined work rate, we can accurately determine the time required for each person to complete the work alone.