3 men and 5 women together can complete a work in 6 days, whereas 4 men and 9 women together can doit in 4 days. How many women are required to do the same work in 7 days?

Options

A. 15
B. 12
C. 14
D. 10

Answer

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

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

### Step 2: Set Up Equations Based on Given Information
From the problem, we know:
- 3 men and 5 women can complete the work in 6 days.
- 4 men and 9 women can complete the same work in 4 days.

Using the formula for total work:
\[ \text{Total Work} = \text{Efficiency} \times \text{Number of Days} \]

For the first group (3 men and 5 women):
\[ (3m + 5w) \times 6 = \text{Total Work} \quad \text{(1)} \]

For the second group (4 men and 9 women):
\[ (4m + 9w) \times 4 = \text{Total Work} \quad \text{(2)} \]

### Step 3: Equate the Total Work from Both Groups
Since both groups complete the same work, we can set the two equations equal to each other:
\[ (3m + 5w) \times 6 = (4m + 9w) \times 4 \]

### Step 4: Expand Both Sides
Expanding both sides gives:
\[ 18m + 30w = 16m + 36w \]

### Step 5: Rearrange to Solve for m and w
Rearranging the equation:
\[ 18m - 16m = 36w - 30w \]
\[ 2m = 6w \]
\[ m = 3w \quad \text{(3)} \]

### Step 6: Calculate Total Work
Now, we can substitute \( m \) in terms of \( w \) back into one of the original equations to find the total work. Using equation (1):
\[ (3(3w) + 5w) \times 6 = \text{Total Work} \]
\[ (9w + 5w) \times 6 = \text{Total Work} \]
\[ 14w \times 6 = \text{Total Work} \]
\[ \text{Total Work} = 84w \]

### Step 7: Find Number of Women Required to Complete Work in 7 Days
To find how many women are required to complete the work in 7 days, we use the formula:
\[ \text{Total Work} = \text{Efficiency} \times \text{Number of Days} \]
Let \( x \) be the number of women required:
\[ 84w = x \times 7 \]
\[ x = \frac{84w}{7} = 12w \]

### Step 8: Determine the Number of Women
Since we need to find the number of women, we set \( w = 1 \) (the efficiency of one woman):
\[ x = 12 \times 1 = 12 \]

### Final Answer
Thus, **12 women** are required to complete the same work in 7 days.

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