9 men and 12 women can complete a work in 4 days, whereas 3 men and 6 women can complete it in 10 days. The numberof days in which 15 women will complete the work is:

Options

A. 9
B. 10
C. 12
D. 8

Answer

To solve the problem step by step, we need to determine how many days it will take for 15 women to complete the work based on the information given about men and women working together.

### Step 1: Establish the work done by men and women
We know from the problem that:
- 9 men and 12 women can complete the work in 4 days.
- 3 men and 6 women can complete the work in 10 days.

Let's denote the work done by one man in one day as \( M \) and the work done by one woman in one day as \( W \).

### Step 2: Calculate the total work done in terms of \( M \) and \( W \)
From the first scenario:
- Total work done by 9 men and 12 women in 4 days:
\[ \text{Total Work} = (9M + 12W) \times 4 \]

From the second scenario:
- Total work done by 3 men and 6 women in 10 days:
\[ \text{Total Work} = (3M + 6W) \times 10 \]

Since both expressions represent the same total work, we can set them equal to each other:
\[ (9M + 12W) \times 4 = (3M + 6W) \times 10 \]

### Step 3: Simplify the equation
Expanding both sides:
\[ 36M + 48W = 30M + 60W \]

Now, rearranging the equation:
\[ 36M - 30M = 60W - 48W \]
\[ 6M = 12W \]

Dividing both sides by 6:
\[ M = 2W \]

### Step 4: Substitute \( M \) back into the work equation
Now we can substitute \( M \) in terms of \( W \) back into one of our work equations. Let's use the first equation:
\[ \text{Total Work} = (9M + 12W) \times 4 \]
Substituting \( M = 2W \):
\[ \text{Total Work} = (9(2W) + 12W) \times 4 \]
\[ = (18W + 12W) \times 4 \]
\[ = 30W \times 4 = 120W \]

### Step 5: Determine how many days 15 women will take to complete the work
Now we need to find out how many days it will take for 15 women to complete the total work of \( 120W \):
\[ \text{Work done by 15 women in one day} = 15W \]

Let \( D \) be the number of days required for 15 women to complete the work:
\[ 15W \times D = 120W \]

Dividing both sides by \( 15W \):
\[ D = \frac{120W}{15W} = 8 \]

### Final Answer
Thus, 15 women will complete the work in **8 days**.