A can do a piece of work in 48 days and B can do the same work in 64 days, while C take as half time as A and B take together. If they start working alternatively, starting by C, followed by B and then A respectively, then find in how many days work will be completed?

Options

A. A)`26(3)/(7)`days
B. B)`28(5)/(7)`days
C. C)`26(1)/(14)`days
D. D)`27(3)/(14)`days

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the work rates of A, B, and C
- A can complete the work in 48 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{48} \text{ work/day} \]
- B can complete the work in 64 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{64} \text{ work/day} \]

### Step 2: Calculate the combined work rate of A and B
- The combined work rate of A and B is:
\[ \text{Combined work rate of A and B} = \frac{1}{48} + \frac{1}{64} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 48 and 64 is 192.
\[ \frac{1}{48} = \frac{4}{192}, \quad \frac{1}{64} = \frac{3}{192} \]
- Therefore,
\[ \text{Combined work rate of A and B} = \frac{4 + 3}{192} = \frac{7}{192} \text{ work/day} \]

### Step 3: Determine C's work rate
- C takes half the time of A and B working together. The time taken by A and B together to complete the work is:
\[ \text{Time taken by A and B together} = \frac{192}{7} \text{ days} \]
- Therefore, C's time to complete the work is:
\[ \text{Time taken by C} = \frac{1}{2} \times \frac{192}{7} = \frac{96}{7} \text{ days} \]
- C's work rate is:
\[ \text{Work rate of C} = \frac{1}{\frac{96}{7}} = \frac{7}{96} \text{ work/day} \]

### Step 4: Calculate the total work done in one complete cycle of C, B, and A
- In one cycle (3 days), the work done is:
- Day 1 (C): C does \(\frac{7}{96}\)
- Day 2 (B): B does \(\frac{1}{64}\)
- Day 3 (A): A does \(\frac{1}{48}\)

- Total work done in one cycle:
\[ \text{Total work in 3 days} = \frac{7}{96} + \frac{1}{64} + \frac{1}{48} \]

### Step 5: Find a common denominator and calculate the total work done in one cycle
- The common denominator for 96, 64, and 48 is 192.
\[ \frac{7}{96} = \frac{14}{192}, \quad \frac{1}{64} = \frac{3}{192}, \quad \frac{1}{48} = \frac{4}{192} \]
- Therefore,
\[ \text{Total work in 3 days} = \frac{14 + 3 + 4}{192} = \frac{21}{192} \text{ work} \]

### Step 6: Calculate how many complete cycles are needed to finish the work
- The total work is 1 (or 192/192). The number of complete cycles needed:
\[ \text{Number of cycles} = \frac{192}{21} \approx 9.14 \text{ cycles} \]
- This means 9 complete cycles can be done, which takes:
\[ 9 \times 3 = 27 \text{ days} \]

### Step 7: Calculate the remaining work after 9 cycles
- Work done in 9 cycles:
\[ 9 \times \frac{21}{192} = \frac{189}{192} \]
- Remaining work:
\[ 1 - \frac{189}{192} = \frac{3}{192} = \frac{1}{64} \]

### Step 8: Determine how long it will take to finish the remaining work
- C starts the next cycle and can complete \(\frac{7}{96}\) work in one day. To finish \(\frac{1}{64}\):
- Time taken by C to finish \(\frac{1}{64}\):
\[ \text{Time} = \frac{\frac{1}{64}}{\frac{7}{96}} = \frac{96}{448} = \frac{3}{14} \text{ days} \]

### Final Calculation
- Total time taken to complete the work:
\[ 27 + \frac{3}{14} = 27 \frac{3}{14} \text{ days} \]

### Conclusion
The total time taken to complete the work is **27 whole 3/14 days**.