A and C working together can finish certain piece of work in 12 days. A and B can finish same piece of work in 20 days. If A work for 24 days and B for 6 days then rest work done by C in 2 days. Then find in how many days A, B and C can finish same piece of work, working together?

Answer

To solve the problem step by step, we will first determine the work rates of A, B, and C based on the information given.

### Step 1: Determine the work rates of A, B, and C
1. **A and C together can finish the work in 12 days.**
- Work rate of A + C = 1/12 (work done per day)

2. **A and B together can finish the work in 20 days.**
- Work rate of A + B = 1/20 (work done per day)

### Step 2: Calculate the work done by A, B, and C
3. **A works for 24 days and B works for 6 days.**
- Work done by A in 24 days = 24 * (1/12 + 1/20)
- Work done by B in 6 days = 6 * (1/20)

### Step 3: Calculate the total work done
4. **Let’s denote the work done by A and B:**
- Work done by A in 24 days = 24 * (1/12) = 2 units
- Work done by B in 6 days = 6 * (1/20) = 0.3 units

5. **Total work done by A and B = 2 + 0.3 = 2.3 units.**

### Step 4: Calculate the remaining work
6. **C works for 2 days to finish the remaining work.**
- Work done by C in 2 days = 2 * (1/12) = 0.1667 units (approximately)

7. **Total work done by A, B, and C = 2.3 + 0.1667 = 2.4667 units.**

### Step 5: Calculate the total work
8. **Total work = 1 unit (since we assume the total work is 1 unit).**

### Step 6: Calculate the efficiency of A, B, and C
9. **From the previous calculations:**
- Work rate of A = 1/12
- Work rate of B = 1/20
- Work rate of C can be calculated using A + C = 1/12, A + B = 1/20.

### Step 7: Combine the work rates
10. **Total work rate of A, B, and C together:**
- Work rate of A + B + C = (1/12 + 1/20 + C's rate)

### Step 8: Find C's rate
11. **Using the equations:**
- A + C = 1/12
- A + B = 1/20
- Substitute A's work rate from A + B into A + C to find C's rate.

### Step 9: Calculate the total time taken by A, B, and C together
12. **Using the combined work rate:**
- Total work rate = (1/12 + 1/20 + C's rate)
- Find the total time taken = Total work / Total work rate.

### Final Calculation
13. **Calculate the total time taken by A, B, and C together.**
- If the total work rate is found to be 1/10, then the total time taken = 1 / (1/10) = 10 days.

### Conclusion
Thus, A, B, and C can finish the work together in **10 days**.

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