7 men and 6 women together can complete a piece of work in 8 days and work done by a woman in one day is half the work done by a man in one day. If 8 men and 4 women started working and after 3 days 4 men left the work and 4 new women joined then, in how many more days will the work be completed.

Options

A. A)7 days
B. B)6 days
C. C)5.25 days
D. D)6.25 days

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Determine the Work Done by Men and Women
Let the work done by one man in one day be \( m \) and the work done by one woman in one day be \( w \). According to the problem, the work done by a woman in one day is half the work done by a man in one day. Therefore, we can express this as:
\[ w = \frac{1}{2}m \]

### Step 2: Calculate Total Work
We know that 7 men and 6 women can complete the work in 8 days. The total work can be calculated as:
\[ \text{Total Work} = \text{(Number of Workers)} \times \text{(Efficiency)} \times \text{(Time)} \]
Thus,
\[ \text{Total Work} = (7m + 6w) \times 8 \]
Substituting \( w \) in terms of \( m \):
\[ \text{Total Work} = (7m + 6 \cdot \frac{1}{2}m) \times 8 = (7m + 3m) \times 8 = 10m \times 8 = 80m \]
So, the total work is \( 80m \).

### Step 3: Work Done by 8 Men and 4 Women in 3 Days
Next, we calculate how much work is done by 8 men and 4 women in 3 days:
\[ \text{Work Done} = (8m + 4w) \times 3 \]
Substituting \( w \):
\[ \text{Work Done} = (8m + 4 \cdot \frac{1}{2}m) \times 3 = (8m + 2m) \times 3 = 10m \times 3 = 30m \]

### Step 4: Remaining Work After 3 Days
Now, we find the remaining work after 3 days:
\[ \text{Remaining Work} = \text{Total Work} - \text{Work Done} = 80m - 30m = 50m \]

### Step 5: Work Done After 4 Men Leave and 4 Women Join
After 3 days, 4 men leave and 4 new women join. The new workforce consists of 4 men and 8 women. We calculate their combined work in one day:
\[ \text{New Work Rate} = (4m + 8w) = (4m + 8 \cdot \frac{1}{2}m) = (4m + 4m) = 8m \]

### Step 6: Calculate Days to Complete Remaining Work
Let \( t \) be the number of days required to complete the remaining work of \( 50m \):
\[ \text{Remaining Work} = \text{New Work Rate} \times t \]
Thus,
\[ 50m = 8m \times t \]
Solving for \( t \):
\[ t = \frac{50m}{8m} = \frac{50}{8} = 6.25 \text{ days} \]

### Final Answer
The work will be completed in **6.25 days** after the changes in workforce.

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