16 men and 12 women together complete a work in 20 days. If 18 women complete the same work in 40 days. Then how many days will be taken by 12 men and 27 women together to complete the same work ?

Options

A. 18 days
B. 15 days
C. 14 days
D. 16 days

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Determine the total work done in terms of women’s work
Given that 16 men and 12 women complete a work in 20 days, we can express the total work \( W \) as:
\[ W = (16M + 12W) \times 20 \]
Where \( M \) is the work done by one man in one day and \( W \) is the work done by one woman in one day.

### Step 2: Determine the work done by women alone
We also know that 18 women complete the same work in 40 days:
\[ W = 18W \times 40 \]

### Step 3: Set the two expressions for work equal to each other
From the two equations for \( W \):
\[ (16M + 12W) \times 20 = 18W \times 40 \]

### Step 4: Simplify the equation
Expanding both sides gives:
\[ 320M + 240W = 720W \]
Now, rearranging gives:
\[ 320M = 720W - 240W \]
\[ 320M = 480W \]
Dividing both sides by 80 gives:
\[ 4M = 6W \]
Thus, the ratio of men to women is:
\[ \frac{M}{W} = \frac{6}{4} = \frac{3}{2} \]

### Step 5: Substitute the ratio into the work equation
Now, we can express the work done by 12 men and 27 women:
\[ W = (12M + 27W) \times X \]
Substituting \( M = \frac{3}{2}W \):
\[ W = \left(12 \times \frac{3}{2}W + 27W\right) \times X \]
\[ W = (18W + 27W) \times X \]
\[ W = 45W \times X \]

### Step 6: Set the two expressions for work equal again
Now, we can equate this to the total work calculated earlier:
\[ 720W = 45W \times X \]

### Step 7: Solve for \( X \)
Dividing both sides by \( 45W \):
\[ X = \frac{720}{45} \]
Calculating this gives:
\[ X = 16 \]

### Conclusion
Thus, 12 men and 27 women will take **16 days** to complete the work.

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