To solve this problem, we need to calculate the total work and how much each person contributes. Let's break down the work:

Step 1: Determine the work rates

• A's rate: A can complete the work in 10 days, so A's work rate is $\frac{1}{10}$ of the work per day.
• B's rate: B can complete the work in 20 days, so B's work rate is $\frac{1}{20}$ of the work per day.
• C's rate: C's efficiency is twice that of A. So, C can do the work in $\frac{10}{2} = 5$ days, which means C's work rate is $\frac{1}{5}$ of the work per day.

Step 2: Calculate the combined work rates when all three are working together

When A, B, and C work together: $\text{Combined work rate} = \frac{1}{10} + \frac{1}{20} + \frac{1}{5}$ Let's find the common denominator (which is 20): $\text{Combined work rate} = \frac{2}{20} + \frac{1}{20} + \frac{4}{20} = \frac{7}{20}$ So, together they complete $\frac{7}{20}$ of the work in one day.

Step 3: Work completed before C and A leave

Let the total time for the work completion be $t$ days. The events to consider:

1. For the first $t-4$ days: All three are working together, so they complete $(t-4) \times \frac{7}{20}$ of the work.
2. For the next day $t-3$: Only A and B are working, so their combined work rate is $\frac{1}{10} + \frac{1}{20} = \frac{3}{20}$.

Step 4: Calculate total work done

The total work done is 1 (which is the whole work): $\frac{7}{20}(t - 4) + \frac{3}{20} = 1$

Step 5: Solve for $t$

$\frac{7}{20}t - \frac{28}{20} + \frac{3}{20} = 1$ $\frac{7}{20}t - \frac{25}{20} = 1$ Multiply through by 20 to clear the fraction: $7t - 25 = 20$ $7t = 45$ $t = \frac{45}{7} \approx 6.43 \text{ days}$

So, the work is completed in approximately 6.43 days.

Conclusion:

The total work is completed in about 6.43 days.

Would you like a detailed explanation of any step or further questions? Here are some related questions:

1. How do we calculate work done when people leave at different times?
2. What happens if C leaves earlier than 4 days before completion?
3. How would the solution change if C's efficiency was different?
4. What if B had left earlier instead of A?
5. How to solve similar problems with varying work rates?

Tip: Always remember to break down complex problems into smaller, more manageable parts for easier analysis and solution.