A can do a piece of work in 10 days and B in 20 days. Another friend C whose efficiency is twice of A joined them. All three started working together but C left 4 days before the completion and A left 3 days before the completion. In how many days work get completed?

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.

Discover how to calculate the completion time for a work problem involving workers A, B, and C with varying efficiencies. Solve equations with combined work rates and fractions effectively. Suitable for high school students studying work rate problems.

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