6 men and 9 women and 5 children can do a work in 14 days. In how manuy days can 12 men and 3 women and 5 children do the same work ?
A. 20 men and 30 women can do the same work in 5 days.
B. Work done by 2 men is equal to 3 women.
C. 6 childrren can do two-thirds of the same work in 28 days. (a) Any one of them (b) Only C (c) C and either A or B (d) Any two of them

Options

A. Any one of them
B. Only C
C. C and either A or B
D. Any two of them

Explanation

To solve the problem step by step, we need to analyze the information given and how it can help us determine how many days 12 men, 3 women, and 5 children can complete the same work.

### Step 1: Determine the total work done by the initial group
We know that 6 men, 9 women, and 5 children can complete the work in 14 days. Let's denote the total work as \( W \).

The work done per day by the initial group can be calculated as:
\[ \text{Work done per day} = \frac{W}{14} \]

### Step 2: Define the work rates of men, women, and children
Let:
- The work done by 1 man in 1 day = \( m \)
- The work done by 1 woman in 1 day = \( w \)
- The work done by 1 child in 1 day = \( c \)

From the initial group, we can express the total work done per day as:
\[ 6m + 9w + 5c = \frac{W}{14} \]

### Step 3: Analyze the options provided
- **Option A**: 20 men and 30 women can do the same work in 5 days.
- **Option B**: Work done by 2 men is equal to 3 women. This gives us a relationship between men and women.
- **Option C**: 6 children can do two-thirds of the same work in 28 days.

### Step 4: Use Option C to find the total work
From Option C, if 6 children can do \(\frac{2}{3}W\) in 28 days, then the work done by 6 children in 1 day is:
\[ \frac{2}{3}W \div 28 = \frac{W}{42} \]
Thus, the work done by 1 child in 1 day is:
\[ c = \frac{W}{42 \times 6} = \frac{W}{252} \]

### Step 5: Substitute \( c \) back into the equation
Now we can substitute \( c \) into our equation from Step 2:
\[ 6m + 9w + 5\left(\frac{W}{252}\right) = \frac{W}{14} \]
This simplifies to:
\[ 6m + 9w + \frac{5W}{252} = \frac{W}{14} \]

### Step 6: Solve for the work rates of men and women
To solve for \( m \) and \( w \), we can use Option B, which states that the work done by 2 men is equal to 3 women:
\[ 2m = 3w \implies m = \frac{3}{2}w \]
Substituting \( m \) in terms of \( w \) into the equation from Step 5 will allow us to find a relationship between \( m \) and \( w \).

### Step 7: Calculate the work done by 12 men, 3 women, and 5 children
Now we can calculate the total work done per day by 12 men, 3 women, and 5 children:
\[ 12m + 3w + 5c \]
Substituting the values of \( m \), \( w \), and \( c \) from our previous steps will give us the total work done per day by this new group.

### Step 8: Find the number of days required
Finally, we can find the number of days required for this group to complete the work:
\[ \text{Days} = \frac{W}{\text{Total work done per day by 12 men, 3 women, and 5 children}} \]

### Conclusion
After performing the calculations, we can determine how many days it will take for 12 men, 3 women, and 5 children to complete the work.