A can complete a work in 12 days with a working of 8 hours per day. B can complete the same work in 8 days when working 10 hours a day. If Both start the work together, working 8 hours a day, the work can be completed in __ days.

Options

A. `5 (5)/11`
B. `4 (5)/11`
C. `6 (4)/11`
D. `6 (5)/11`

Answer

To solve the problem step by step, we will first determine the total work done by A and B, their efficiencies, and then calculate how long it will take them to complete the work together.

### Step 1: Calculate the total work done by A and B

1. **Work done by A:**
- A can complete the work in 12 days, working 8 hours a day.
- Total hours worked by A = 12 days × 8 hours/day = 96 hours.
- Therefore, the total work (in units) can be defined as 1 unit of work per hour.
- Total work = 96 units.

2. **Work done by B:**
- B can complete the work in 8 days, working 10 hours a day.
- Total hours worked by B = 8 days × 10 hours/day = 80 hours.
- Therefore, the total work (in units) can be defined as 1 unit of work per hour.
- Total work = 80 units.

3. **Finding the LCM for total work:**
- The total work can be taken as the LCM of the total hours worked by A and B.
- LCM of 96 and 80 is 480 units.

### Step 2: Calculate the efficiency of A and B

1. **Efficiency of A:**
- A's efficiency = Total work / Total time = 480 units / 96 hours = 5 units/hour.

2. **Efficiency of B:**
- B's efficiency = Total work / Total time = 480 units / 80 hours = 6 units/hour.

### Step 3: Calculate combined efficiency when A and B work together

1. **Combined efficiency:**
- When A and B work together, their combined efficiency = Efficiency of A + Efficiency of B = 5 units/hour + 6 units/hour = 11 units/hour.

### Step 4: Calculate the total work done in one day

1. **Working hours per day:**
- Both A and B work together for 8 hours a day.
- Work done in one day = Combined efficiency × Working hours per day = 11 units/hour × 8 hours = 88 units/day.

### Step 5: Calculate the total time to complete the work

1. **Total work to be completed:**
- Total work = 480 units.

2. **Time taken to complete the work:**
- Time = Total work / Work done in one day = 480 units / 88 units/day = 5.45 days.

### Step 6: Convert to mixed fraction

1. **Convert 5.45 to mixed fraction:**
- 5.45 = 5 + 0.45 = 5 + 5/11.

Thus, the work can be completed in **5 5/11 days**.

### Final Answer:
The work can be completed in **5 5/11 days**.

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