A certain work was started by 4 men and 10 women who completed 50% of the work in 6 days, if another 2 men and 2 women joined them and they could complete `(2)/(3)`rd of the remaining work in three days. How many men along with 6 women are required to complete the remaining work in two more days?

Options

A. 3
B. 5
C. 6
D. 9

Answer

To solve the problem step by step, we will break down the information given and calculate the required number of men along with 6 women to complete the remaining work in two days.

### Step 1: Calculate the total work done by 4 men and 10 women in 6 days.

Let the work done by 1 man in 1 day be \( M \) and the work done by 1 woman in 1 day be \( W \).

The total work done by 4 men and 10 women in 1 day is:
\[ \text{Work in 1 day} = 4M + 10W \]

In 6 days, the work done is:
\[ \text{Work in 6 days} = 6 \times (4M + 10W) = 24M + 60W \]

Since this work is 50% of the total work, we can say:
\[ 24M + 60W = \frac{1}{2} \text{Total Work} \]
Thus, the total work \( T \) is:
\[ T = 2(24M + 60W) = 48M + 120W \]

### Step 2: Calculate the remaining work after the first 6 days.

After completing 50% of the work, the remaining work is:
\[ \text{Remaining Work} = T - \frac{T}{2} = \frac{T}{2} = 24M + 60W \]

### Step 3: Calculate the work done by 6 men and 12 women in 3 days.

Now, 2 more men and 2 more women join, making it 6 men and 12 women. The work done by them in 1 day is:
\[ \text{Work in 1 day} = 6M + 12W \]

In 3 days, the work done is:
\[ \text{Work in 3 days} = 3 \times (6M + 12W) = 18M + 36W \]

According to the problem, they complete \( \frac{2}{3} \) of the remaining work in these 3 days:
\[ 18M + 36W = \frac{2}{3} \times (24M + 60W) \]

### Step 4: Set up the equation to find the relationship between \( M \) and \( W \).

Calculating \( \frac{2}{3} \) of the remaining work:
\[ \frac{2}{3} \times (24M + 60W) = 16M + 40W \]

Setting the two expressions equal gives:
\[ 18M + 36W = 16M + 40W \]

Rearranging:
\[ 2M = 4W \implies M = 2W \]

### Step 5: Calculate the total work in terms of \( W \).

Substituting \( M = 2W \) into the total work:
\[ T = 48(2W) + 120W = 96W + 120W = 216W \]

### Step 6: Calculate the remaining work.

The remaining work is:
\[ \text{Remaining Work} = 24M + 60W = 24(2W) + 60W = 48W + 60W = 108W \]

### Step 7: Determine the work left to be completed in 2 days.

Let \( X \) be the number of men required along with 6 women to complete the remaining work in 2 days. The total work done by \( X \) men and 6 women in 2 days is:
\[ \text{Work in 2 days} = 2(XM + 6W) = 2(X(2W) + 6W) = 4XW + 12W \]

Setting this equal to the remaining work:
\[ 4XW + 12W = 108W \]

### Step 8: Solve for \( X \).

Dividing the entire equation by \( W \) (assuming \( W \neq 0 \)):
\[ 4X + 12 = 108 \]
\[ 4X = 108 - 12 \]
\[ 4X = 96 \]
\[ X = \frac{96}{4} = 24 \]

### Conclusion

Thus, the number of men required along with 6 women to complete the remaining work in 2 days is:
\[ \text{Number of men} = 24 \]