Six men and 4 women can do a piece of work in 32 days. Seven 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, we need to find out how many days 18 men and 8 women can complete the same work when working together. We will first establish the work rates of men and women based on the information given.

### Step 1: Establish the work done by men and women
Let the work done by one man in one day be \( m \) and the work done by one woman in one day be \( w \).

From the first piece of information:
- 6 men and 4 women can complete the work in 32 days.
This means the total work \( W \) can be expressed as:
\[ W = (6m + 4w) \times 32 \]

From the second piece of information:
- 7 men and 12 women can complete the work in 18 days.
This means the total work \( W \) can also be expressed as:
\[ W = (7m + 12w) \times 18 \]

### Step 2: Set the two equations for total work equal to each other
Since both expressions equal the total work \( W \), we can set them equal to each other:
\[ (6m + 4w) \times 32 = (7m + 12w) \times 18 \]

### Step 3: Simplify the equation
Expanding both sides gives:
\[ 192m + 128w = 126m + 216w \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation to isolate \( m \) and \( w \):
\[ 192m - 126m = 216w - 128w \]
\[ 66m = 88w \]

### Step 5: Find the relationship between men and women
Dividing both sides by 22 gives:
\[ 3m = 4w \quad \Rightarrow \quad m = \frac{4}{3}w \]

### Step 6: Substitute \( m \) back into one of the original equations
Now, we can substitute \( m \) into one of the total work equations. Let's use the first one:
\[ W = (6m + 4w) \times 32 \]
Substituting \( m \):
\[ W = \left(6 \times \frac{4}{3}w + 4w\right) \times 32 \]
\[ = \left(8w + 4w\right) \times 32 = 12w \times 32 = 384w \]

### Step 7: Find the work done by 18 men and 8 women
Now, we need to find how many days 18 men and 8 women can complete the work:
\[ W = (18m + 8w) \times D \]
Substituting \( m \):
\[ W = \left(18 \times \frac{4}{3}w + 8w\right) \times D \]
\[ = \left(24w + 8w\right) \times D = 32w \times D \]

### Step 8: Set the two expressions for work equal
Now we can set the two expressions for \( W \) equal to each other:
\[ 384w = 32w \times D \]

### Step 9: Solve for \( D \)
Dividing both sides by \( 32w \):
\[ D = \frac{384}{32} = 12 \]

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