B is twice as fast as A and C is three times as fast as A. If B alone can complete a job in 12 days, how long will A, B and C take to complete the same job together?

Options

A. 3 days
B. 4 days
C. 6 days
D. None of these

Answer

To solve the problem, we need to determine how long A, B, and C will take to complete the job together, given their individual work rates.

### Step-by-step Solution:

1. **Determine B's Work Rate:**
- B can complete the job in 12 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{12} \text{ of the job} \]

2. **Determine A's Work Rate:**
- Since B is twice as fast as A, we can express A's work rate in terms of B's:
\[ \text{Work done by A in one day} = \frac{1}{2} \times \text{Work done by B in one day} = \frac{1}{2} \times \frac{1}{12} = \frac{1}{24} \text{ of the job} \]

3. **Determine C's Work Rate:**
- C is three times as fast as A, so we can express C's work rate in terms of A's:
\[ \text{Work done by C in one day} = 3 \times \text{Work done by A in one day} = 3 \times \frac{1}{24} = \frac{3}{24} = \frac{1}{8} \text{ of the job} \]

4. **Combine the Work Rates of A, B, and C:**
- Now, we can find the total work done by A, B, and C together in one day:
\[ \text{Total work done in one day} = \text{Work done by A} + \text{Work done by B} + \text{Work done by C} \]
\[ = \frac{1}{24} + \frac{1}{12} + \frac{1}{8} \]

5. **Find a Common Denominator:**
- The least common multiple of 24, 12, and 8 is 24. We can express each fraction with a denominator of 24:
\[ \frac{1}{12} = \frac{2}{24}, \quad \frac{1}{8} = \frac{3}{24} \]
- Therefore:
\[ \text{Total work done in one day} = \frac{1}{24} + \frac{2}{24} + \frac{3}{24} = \frac{6}{24} = \frac{1}{4} \text{ of the job} \]

6. **Calculate the Time Taken by A, B, and C Together:**
- If A, B, and C together can complete \(\frac{1}{4}\) of the job in one day, then the time taken to complete the entire job is:
\[ \text{Time taken} = \frac{1}{\text{Total work done in one day}} = \frac{1}{\frac{1}{4}} = 4 \text{ days} \]

### Final Answer:
A, B, and C together will take **4 days** to complete the job.