A can do a work in 8 hours & B can do the same work in 12 hours. If they workalternatively (one hour A works and one hour B works) and A starts the work at 9 AM, then the work will be completed at __O'clock.

Options

A. `7:00` PM
B. `5:30` PM
C. `6:30` AM
D. `6:30` PM

Answer

To solve the problem step by step, we will first determine the work done by A and B in one hour and then calculate how long it will take them to complete the work when they work alternatively.

### Step 1: Determine the work done by A and B in one hour.
- A can complete the work in 8 hours, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{8} \text{ of the work per hour} \]
- B can complete the work in 12 hours, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{12} \text{ of the work per hour} \]

### Step 2: Find the combined work done in 2 hours (1 hour by A and 1 hour by B).
- In the first hour, A works:
\[ \text{Work done by A in 1 hour} = \frac{1}{8} \]
- In the second hour, B works:
\[ \text{Work done by B in 1 hour} = \frac{1}{12} \]
- Therefore, the total work done in 2 hours is:
\[ \text{Total work in 2 hours} = \frac{1}{8} + \frac{1}{12} \]
- To add these fractions, we need a common denominator. The least common multiple of 8 and 12 is 24.
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
- So,
\[ \text{Total work in 2 hours} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24} \]

### Step 3: Calculate how many 2-hour cycles are needed to complete the work.
- The total work to be done is 1 (the whole work).
- In 2 hours, they complete \(\frac{5}{24}\) of the work.
- To find out how many such cycles are needed to complete 1 unit of work:
\[ \text{Number of cycles} = \frac{1}{\frac{5}{24}} = \frac{24}{5} = 4.8 \text{ cycles} \]

### Step 4: Calculate the total time taken to complete the work.
- Each cycle takes 2 hours, so:
\[ \text{Time for 4 complete cycles} = 4 \times 2 = 8 \text{ hours} \]
- After 4 cycles (8 hours), they will have completed:
\[ 4 \times \frac{5}{24} = \frac{20}{24} = \frac{5}{6} \text{ of the work} \]
- Remaining work:
\[ 1 - \frac{5}{6} = \frac{1}{6} \]

### Step 5: Determine who will work next and how long it will take to finish the remaining work.
- A will work next (since they alternate and A started first).
- A's work rate is \(\frac{1}{8}\) of the work per hour.
- To complete \(\frac{1}{6}\) of the work:
\[ \text{Time taken by A} = \frac{\frac{1}{6}}{\frac{1}{8}} = \frac{8}{6} = \frac{4}{3} \text{ hours} \approx 1 \text{ hour and } 20 \text{ minutes} \]

### Step 6: Calculate the total time from the start.
- Total time taken = 8 hours (for 4 cycles) + 1 hour and 20 minutes = 9 hours and 20 minutes.
- If they started at 9 AM, we add 9 hours and 20 minutes:
\[ 9 \text{ AM} + 9 \text{ hours} + 20 \text{ minutes} = 6:20 \text{ PM} \]

### Final Answer:
The work will be completed at **6:20 PM**.

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