A can do a piece of work in 6 days working 8 hours a day while B can do the same work in 4 days working 10 hours a day. If the work has to be completed in 5 days, so how many hours do they need to work together in a days?

Options

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

Answer

To solve the problem step by step, we need to determine how many hours A and B need to work together each day to complete the work in 5 days.

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

- **A's Work Calculation:**
A can complete the work in 6 days, working 8 hours a day.
\[ \text{Total work by A} = 6 \text{ days} \times 8 \text{ hours/day} = 48 \text{ hours} \]

- **B's Work Calculation:**
B can complete the work in 4 days, working 10 hours a day.
\[ \text{Total work by B} = 4 \text{ days} \times 10 \text{ hours/day} = 40 \text{ hours} \]

### Step 2: Determine the total amount of work required.

The total work required to complete the task is the same as the work done by A or B, which is the maximum of the two:
\[ \text{Total Work} = 48 \text{ hours} \text{ (A's work)} \]

### Step 3: Calculate the work done per hour by A and B.

- **A's Work Rate:**
\[ \text{Work rate of A} = \frac{1 \text{ work}}{48 \text{ hours}} = \frac{1}{48} \text{ work/hour} \]

- **B's Work Rate:**
\[ \text{Work rate of B} = \frac{1 \text{ work}}{40 \text{ hours}} = \frac{1}{40} \text{ work/hour} \]

### Step 4: Combine their work rates.

To find the combined work rate of A and B:
\[ \text{Combined work rate} = \frac{1}{48} + \frac{1}{40} \]

To add these fractions, we need a common denominator. The least common multiple of 48 and 40 is 240.

Converting the fractions:
\[ \frac{1}{48} = \frac{5}{240}, \quad \frac{1}{40} = \frac{6}{240} \]
So,
\[ \text{Combined work rate} = \frac{5}{240} + \frac{6}{240} = \frac{11}{240} \text{ work/hour} \]

### Step 5: Calculate the total work needed in 5 days.

If they need to complete the work in 5 days, the total work done in 5 days at their combined work rate is:
\[ \text{Total work in 5 days} = 5 \text{ days} \times H \text{ hours/day} \times \frac{11}{240} \text{ work/hour} \]
Setting this equal to 1 (the total work):
\[ 5H \cdot \frac{11}{240} = 1 \]

### Step 6: Solve for H.

Rearranging the equation:
\[ H = \frac{240}{5 \times 11} = \frac{240}{55} = \frac{48}{11} \]

### Step 7: Convert hours into a mixed number.

\[ H = 4 \frac{4}{11} \text{ hours} \]

### Final Answer:
A and B need to work together for approximately \(4 \frac{4}{11}\) hours each day to complete the work in 5 days.

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