7 men and 6 women together can complete a piece of work in 8 days and work done by a women 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 them, in how many more days will the remaining work be completed?

Options

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

Answer

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

### Step 1: Determine the work done by men and women
Given:
- 7 men and 6 women can complete the work in 8 days.
- The work done by a woman in one day is half the work done by a man in one day.

Let the work done by one man in one day be \(2x\) (units of work).
Then, the work done by one woman in one day is \(x\).

### Step 2: Calculate total work
The total work done by 7 men and 6 women in one day is:
\[ \text{Work per day} = 7 \times 2x + 6 \times x = 14x + 6x = 20x \]
Since they complete the work in 8 days, the total work \(W\) is:
\[ W = \text{Work per day} \times \text{Number of days} = 20x \times 8 = 160x \]

### Step 3: Work done in the first 3 days
Now, 8 men and 4 women start working. The work done by them in one day is:
\[ \text{Work per day} = 8 \times 2x + 4 \times x = 16x + 4x = 20x \]
In 3 days, the total work done is:
\[ \text{Total work done in 3 days} = 20x \times 3 = 60x \]

### Step 4: Calculate remaining work
The remaining work after 3 days is:
\[ \text{Remaining work} = \text{Total work} - \text{Work done in 3 days} = 160x - 60x = 100x \]

### Step 5: Work after 3 days
After 3 days, 4 men leave, and 4 new women join. Now, the workforce consists of:
- Remaining men: \(8 - 4 = 4\)
- New women: \(4 + 4 = 8\)

The work done by the remaining workers in one day is:
\[ \text{Work per day} = 4 \times 2x + 8 \times x = 8x + 8x = 16x \]

### Step 6: Calculate days required to finish remaining work
To find the number of days required to complete the remaining work of \(100x\):
\[ \text{Days required} = \frac{\text{Remaining work}}{\text{Work per day}} = \frac{100x}{16x} = \frac{100}{16} = 6.25 \text{ days} \]

### Final Answer
The remaining work will be completed in **6.25 days**.

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