The following questions are accompanied by three statements A, B and C. You have to determine which statement(s) is/are necessary/sufficient to answer the question.
In how many days 3 men, 2 women and 3 children can complete the work while working together?
A. 3 men are as efficient as 4 women.
B. 2 women and 5 children can work in 12 days.
C. Ratio of the efficiencies of a man and a child is 2:1.

Options

A. All the three together are not sufficient
B. All the three together
C. Only B and C together
D. Only A and B together

Answer

To determine how many days 3 men, 2 women, and 3 children can complete the work while working together, we need to analyze the given statements A, B, and C.

### Step 1: Understand the Work Rates
First, we need to define the work rates based on the information provided in the statements.

1. **Statement A**: 3 men are as efficient as 4 women.
This implies that the work done by 1 man is equal to the work done by \( \frac{4}{3} \) women.

2. **Statement B**: 2 women and 5 children can work in 12 days.
This provides a direct relationship between the work done by women and children. We can calculate the total work done in terms of women and children.

3. **Statement C**: The ratio of the efficiencies of a man and a child is 2:1.
This means that 1 man can do the work of 2 children.

### Step 2: Express All Efficiencies in Terms of a Common Unit
Let's denote:
- Efficiency of 1 man = \( M \)
- Efficiency of 1 woman = \( W \)
- Efficiency of 1 child = \( C \)

From Statement A:
\[ 3M = 4W \Rightarrow M = \frac{4}{3}W \]

From Statement C:
\[ M = 2C \Rightarrow C = \frac{M}{2} = \frac{4}{6}W = \frac{2}{3}W \]

### Step 3: Calculate Total Efficiency of 3 Men, 2 Women, and 3 Children
Now, we can express the total efficiency of 3 men, 2 women, and 3 children in terms of women’s efficiency:
- Total efficiency of 3 men = \( 3M = 3 \times \frac{4}{3}W = 4W \)
- Total efficiency of 2 women = \( 2W \)
- Total efficiency of 3 children = \( 3C = 3 \times \frac{2}{3}W = 2W \)

Now, adding these efficiencies together:
\[ \text{Total Efficiency} = 4W + 2W + 2W = 8W \]

### Step 4: Determine Total Work and Days Required
From Statement B, we know that 2 women and 5 children can complete the work in 12 days. We can calculate the total work in terms of women’s efficiency:
- Total efficiency of 2 women and 5 children = \( 2W + 5C = 2W + 5 \times \frac{2}{3}W = 2W + \frac{10}{3}W = \frac{16}{3}W \)

The total work done in 12 days:
\[ \text{Total Work} = \text{Efficiency} \times \text{Time} = \frac{16}{3}W \times 12 = 64W \]

### Step 5: Calculate Days for 3 Men, 2 Women, and 3 Children
Now, we can find how many days it will take for 3 men, 2 women, and 3 children to complete the same total work:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{64W}{8W} = 8 \text{ days} \]

### Conclusion
Thus, 3 men, 2 women, and 3 children can complete the work in **8 days**.