A, B and C can finish a task in 42 days, 84 days and
28 days, respectively. A started the work. B joined him
after 3 days. If C Joined them after 5 days from the
beginning, then for many days did A work till the
completion of the task?

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.

---

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.

A, B and C can finish a task in 42 days, 84 days and 28 days, respectively. A started the work. B joined him after 3 days. If C joined them after 5 days from the beginning, then for how many days did A work till the completion of the task?

A, B, and C can complete a task in 42, 84, and 28 days, respectively. A started the work alone, with B joining after 3 days and C after 5 days. This step-by-step solution explains how long A worked until the task's completion. Learn key concepts in time and work problems with this practical example.

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