6 men and 4 women can do a piece of work in 32 days. 7 men and 12 women can do it in 18 days. In how many days can 18 men and 8 women do the same work, working together ?

Options

A. 10
B. 12
C. 14
D. 16

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Define Variables
Let the efficiency of one man be \( x \) and the efficiency of one woman be \( y \).

### Step 2: Calculate Total Work from the First Scenario
From the first scenario, we have:
- 6 men and 4 women can complete the work in 32 days.

The total work done can be expressed as:
\[ \text{Total Work} = \text{Efficiency} \times \text{Days} \]
Thus, the total work done by 6 men and 4 women in 32 days is:
\[ 6x + 4y \text{ (efficiency per day)} \times 32 \text{ (days)} = 32(6x + 4y) \]

### Step 3: Calculate Total Work from the Second Scenario
From the second scenario, we have:
- 7 men and 12 women can complete the work in 18 days.

The total work done in this case is:
\[ 7x + 12y \text{ (efficiency per day)} \times 18 \text{ (days)} = 18(7x + 12y) \]

### Step 4: Set the Equations Equal
Since both expressions represent the total work, we can set them equal to each other:
\[ 32(6x + 4y) = 18(7x + 12y) \]

### Step 5: Expand and Simplify
Expanding both sides:
\[ 192x + 128y = 126x + 216y \]

Rearranging gives:
\[ 192x - 126x = 216y - 128y \]
\[ 66x = 88y \]

### Step 6: Solve for the Ratio of Efficiencies
Dividing both sides by 22:
\[ 3x = 4y \quad \Rightarrow \quad \frac{x}{y} = \frac{4}{3} \]

### Step 7: Express Efficiencies in Terms of a Common Variable
Let \( x = 4k \) and \( y = 3k \) for some constant \( k \).

### Step 8: Calculate Total Work Using Either Scenario
Using the first scenario to find total work:
\[ \text{Total Work} = 32(6x + 4y) = 32(6(4k) + 4(3k)) = 32(24k + 12k) = 32(36k) = 1152k \]

### Step 9: Calculate Work Done by 18 Men and 8 Women
Now, we need to find out how many days \( t \) it takes for 18 men and 8 women to complete the same work:
\[ \text{Efficiency of 18 men and 8 women} = 18x + 8y = 18(4k) + 8(3k) = 72k + 24k = 96k \]

### Step 10: Set Up the Equation for Total Work
Setting up the equation:
\[ 96k \cdot t = 1152k \]

### Step 11: Solve for \( t \)
Dividing both sides by \( 96k \):
\[ t = \frac{1152k}{96k} = 12 \]

### Conclusion
Thus, 18 men and 8 women can complete the work together in **12 days**.

---