A and B together can finish a job in 24 days, while A, B and C together can finish the same job in 8 days. C alone will finish the job in

Options

A. 12 days
B. 14 days
C. 16 days
D. 24 days

Answer

To solve the problem step by step, we will break down the information given and use it to find out how long C will take to finish the job alone.

### Step-by-Step Solution:

1. **Understanding the Work Done by A and B Together:**
- A and B together can finish the job in 24 days.
- This means their combined work rate (efficiency) is \( \frac{1}{24} \) of the job per day.

2. **Understanding the Work Done by A, B, and C Together:**
- A, B, and C together can finish the job in 8 days.
- This means their combined work rate (efficiency) is \( \frac{1}{8} \) of the job per day.

3. **Calculating the Efficiency of A and B:**
- Let the efficiency of A + B be \( E_{AB} = \frac{1}{24} \).
- Let the efficiency of A + B + C be \( E_{ABC} = \frac{1}{8} \).

4. **Finding the Efficiency of C:**
- The efficiency of C can be found by subtracting the efficiency of A and B from the efficiency of A, B, and C:
\[ E_C = E_{ABC} - E_{AB} = \frac{1}{8} - \frac{1}{24} \]

5. **Finding a Common Denominator:**
- The least common multiple of 8 and 24 is 24. We convert \( \frac{1}{8} \) to have a denominator of 24:
\[ \frac{1}{8} = \frac{3}{24} \]
- Now, we can perform the subtraction:
\[ E_C = \frac{3}{24} - \frac{1}{24} = \frac{2}{24} = \frac{1}{12} \]

6. **Calculating the Time Taken by C to Finish the Job Alone:**
- If C's efficiency is \( \frac{1}{12} \), it means C can complete \( \frac{1}{12} \) of the job in one day.
- Therefore, the time taken by C to finish the entire job alone is:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of C}} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

### Conclusion:
C alone will finish the job in **12 days**.

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