A can finish a work in 14 days at 6 hrs a day. B can finish it in 8 days at 2 hrs a day. Find in how many days they can finish it working together 4 hrs a day.

Answer

To solve the problem, we need to calculate the total work done by A and B, their individual efficiencies, and then find out how long it will take them to complete the work together when working 4 hours a day.

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

1. **Work done by A**:
- A can finish the work in 14 days at 6 hours a day.
- Total hours worked by A = 14 days × 6 hours/day = 84 hours.
- Therefore, the total work (in terms of hours) = 1 work unit = 84 hours.

2. **Work done by B**:
- B can finish the work in 8 days at 2 hours a day.
- Total hours worked by B = 8 days × 2 hours/day = 16 hours.
- Therefore, the total work (in terms of hours) = 1 work unit = 16 hours.

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

1. **Efficiency of A**:
- Efficiency of A = Total work / Total time = 1 work unit / 84 hours = 1/84 work units per hour.

2. **Efficiency of B**:
- Efficiency of B = Total work / Total time = 1 work unit / 16 hours = 1/16 work units per hour.

### Step 3: Calculate the combined efficiency of A and B.

- Combined efficiency = Efficiency of A + Efficiency of B
- Combined efficiency = (1/84 + 1/16)

To add these fractions, we need a common denominator:
- The least common multiple of 84 and 16 is 336.

Converting the fractions:
- (1/84) = (4/336)
- (1/16) = (21/336)

Now, adding them:
- Combined efficiency = (4/336 + 21/336) = 25/336 work units per hour.

### Step 4: Calculate the total work done when working together for 4 hours a day.

- In 4 hours, the work done together = 4 hours × (25/336) work units/hour = 100/336 work units.

### Step 5: Calculate the total days required to finish the work.

- Since the total work is 1 work unit, we need to find out how many days it will take to complete the work at the rate of 100/336 work units per day.
- Total days required = Total work / Work done per day = 1 / (100/336) = 336/100 = 3.36 days.

### Final Answer:
They can finish the work together in approximately **3.36 days**.

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