Let's break down the problem and solve it step by step.

Step 1: Determine the rate of work for A, B, and C.

• A can finish the task in 42 days, so A's rate of work is $\frac{1}{42}$ of the task per day.
• B can finish the task in 84 days, so B's rate of work is $\frac{1}{84}$ of the task per day.
• C can finish the task in 28 days, so C's rate of work is $\frac{1}{28}$ of the task per day.

Step 2: Analyze the work phases.

• First 3 days: Only A is working. In 3 days, A completes: $3 \times \frac{1}{42} = \frac{3}{42} = \frac{1}{14} \text{ of the task}.$

• From day 4 to day 5 (2 days): A and B are working together. Their combined rate of work is: $\frac{1}{42} + \frac{1}{84} = \frac{2}{84} + \frac{1}{84} = \frac{3}{84} = \frac{1}{28}.$ In 2 days, they complete: $2 \times \frac{1}{28} = \frac{2}{28} = \frac{1}{14} \text{ of the task}.$

• From day 6 onward: A, B, and C are all working together. Their combined rate of work is: $\frac{1}{42} + \frac{1}{84} + \frac{1}{28}.$ To add these fractions, we need a common denominator, which is 168: $\frac{1}{42} = \frac{4}{168}, \quad \frac{1}{84} = \frac{2}{168}, \quad \frac{1}{28} = \frac{6}{168}.$ Their combined rate is: $\frac{4}{168} + \frac{2}{168} + \frac{6}{168} = \frac{12}{168} = \frac{1}{14}.$ So, when A, B, and C work together, they complete $\frac{1}{14}$ of the task per day.

Step 3: Calculate the total work done.

• In the first 3 days, A completed $\frac{1}{14}$ of the task.

• In the next 2 days, A and B completed another $\frac{1}{14}$ of the task.

• The total completed after 5 days is $\frac{1}{14} + \frac{1}{14} = \frac{2}{14} = \frac{1}{7}$ of the task.

• The remaining work after 5 days is $1 - \frac{1}{7} = \frac{6}{7}$.

Since A, B, and C together complete $\frac{1}{14}$ of the task per day, it will take: $\frac{\frac{6}{7}}{\frac{1}{14}} = 12 \text{ days}$ for them to complete the remaining work.

Step 4: Calculate how many days A worked.

• A worked alone for 3 days.
• A worked with B for 2 days.
• A worked with B and C for the remaining 12 days.

Thus, A worked a total of: $3 + 2 + 12 = 17 \text{ days}.$

Final Answer:

A worked for 17 days until the completion of the task.

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Would you like any further details or clarifications?

Here are 5 related questions to practice:

1. If B started working after 4 days instead of 3, how would it affect the total days A worked?
2. What if C joined after 6 days instead of 5? How many days would A have worked?
3. If A could finish the task in 35 days instead of 42, how many days would A have worked?
4. How would the solution change if B could complete the work in 60 days instead of 84 days?
5. What if A, B, and C all started together from day 1? How many days would the task take?

Tip: Breaking down a problem into phases and analyzing each phase separately is a powerful strategy in solving time and work problems.