The questions given below contain two statements numbered I and II giving certain data. You have to decide whether the data given in the statements are sufficient to answer the question. Give answer
In how many days can 18 men and 14 women complete a piece of work?
I 9 men and 7 women together can complete the same piece of work in 6 days. II. 2 men and 4 women together can complete the same piece of work in 16 days.

Options

A. if statement I alone is sufficient but statement II alone is not sufficient.
B. if statement II alone is sufficient but statement I alone is not sufficient.
C. if either statement I or II is sufficient.
D. if both statement I and II together are not suficient.

Answer

To determine how many days 18 men and 14 women can complete a piece of work, we will analyze the two statements provided.

### Step 1: Analyze Statement I
Statement I states that 9 men and 7 women can complete the work in 6 days.

1. **Calculate the total work done in terms of man-days:**
- If 9 men and 7 women complete the work in 6 days, the total work can be expressed as:
\[ \text{Total Work} = \text{Number of Workers} \times \text{Days} = (9 + 7) \text{ workers} \times 6 \text{ days} = 96 \text{ worker-days} \]

2. **Determine the work rate of men and women:**
- Let the work done by one man in one day be \( m \) and by one woman be \( w \).
- The equation based on Statement I becomes:
\[ 6(9m + 7w) = 96 \implies 9m + 7w = 16 \quad \text{(1)} \]

### Step 2: Analyze Statement II
Statement II states that 2 men and 4 women can complete the work in 16 days.

1. **Calculate the total work done in terms of man-days for Statement II:**
- If 2 men and 4 women complete the work in 16 days, the total work can be expressed as:
\[ \text{Total Work} = (2 + 4) \text{ workers} \times 16 \text{ days} = 48 \text{ worker-days} \]

2. **Determine the work rate of men and women:**
- The equation based on Statement II becomes:
\[ 16(2m + 4w) = 48 \implies 2m + 4w = 3 \quad \text{(2)} \]

### Step 3: Solve the Equations
Now we have two equations:
1. \( 9m + 7w = 16 \) (from Statement I)
2. \( 2m + 4w = 3 \) (from Statement II)

We can solve these equations to find the values of \( m \) and \( w \).

1. From equation (2), we can express \( m \) in terms of \( w \):
\[ 2m + 4w = 3 \implies m = \frac{3 - 4w}{2} \quad \text{(3)} \]

2. Substitute equation (3) into equation (1):
\[ 9\left(\frac{3 - 4w}{2}\right) + 7w = 16 \]
\[ \frac{27 - 36w}{2} + 7w = 16 \]
\[ 27 - 36w + 14w = 32 \implies -22w = 5 \implies w = -\frac{5}{22} \quad \text{(not valid)} \]

### Step 4: Use the Validity of Statements
Since we cannot find valid values for \( m \) and \( w \) from both statements combined, we need to check if either statement alone is sufficient.

- **Statement I alone is sufficient** because it provides a direct relationship between men and women.
- **Statement II alone is not sufficient** as it does not provide enough information to determine the work rate of men and women.

### Conclusion
Thus, we can conclude that:
- **Only Statement I is sufficient to answer the question.**

### Final Answer
The answer is that Statement I alone is sufficient to determine how many days 18 men and 14 women can complete the work.