If 4 men or 6 women can do a piece of work in 24 days, then how many men should join 3 women to complete the work in 16 days

Options

A. 6
B. 5
C. 4
D. 2

Explanation

To solve the problem, we need to determine how many men should join 3 women to complete the work in 16 days, given that 4 men or 6 women can complete the work in 24 days.

### Step 1: Determine the total work in terms of man-days.
Since 4 men can complete the work in 24 days, the total work can be calculated as:
\[ \text{Total Work} = \text{Number of Men} \times \text{Days} = 4 \text{ men} \times 24 \text{ days} = 96 \text{ man-days} \]

### Step 2: Determine the work done by 3 women in 16 days.
First, we need to find out how much work one woman can do in one day. Since 6 women can complete the work in 24 days, we can calculate the work done by one woman in one day:
\[ \text{Work done by 1 woman in 1 day} = \frac{96 \text{ man-days}}{6 \text{ women} \times 24 \text{ days}} = \frac{96}{144} = \frac{2}{3} \text{ man-days} \]

Now, the work done by 3 women in one day is:
\[ \text{Work done by 3 women in 1 day} = 3 \times \frac{2}{3} = 2 \text{ man-days} \]

In 16 days, the total work done by 3 women is:
\[ \text{Total work done by 3 women in 16 days} = 2 \text{ man-days/day} \times 16 \text{ days} = 32 \text{ man-days} \]

### Step 3: Calculate the remaining work to be done.
The total work is 96 man-days, and the work done by 3 women in 16 days is 32 man-days. Therefore, the remaining work is:
\[ \text{Remaining Work} = 96 \text{ man-days} - 32 \text{ man-days} = 64 \text{ man-days} \]

### Step 4: Determine how many men are needed to complete the remaining work in 16 days.
Let \( x \) be the number of men required. The total work done by \( x \) men in 16 days is:
\[ \text{Work done by } x \text{ men in 16 days} = x \times 16 \text{ man-days} \]
We set this equal to the remaining work:
\[ x \times 16 = 64 \]
Solving for \( x \):
\[ x = \frac{64}{16} = 4 \]

### Conclusion:
Therefore, **4 men** should join the 3 women to complete the work in 16 days.

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