A particular work can be completed by 6 men and 6 women in 24 days, whereas the same work can be completed by 8 men and 12 women in 15 days. Find :
The time taken by 4 men and 6 women to complete the same work.

Answer

To solve the problem step by step, we will first analyze the information given and set up equations based on the efficiencies of men and women.

### Step 1: Establish the work done by men and women

We know from the problem that:
- 6 men and 6 women can complete the work in 24 days.
- 8 men and 12 women can complete the same work in 15 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: Set up the equations

From the first scenario:
- The total work done by 6 men and 6 women in 24 days can be expressed as:
\[ (6M + 6W) \times 24 = \text{Total Work} \quad \text{(1)} \]

From the second scenario:
- The total work done by 8 men and 12 women in 15 days can be expressed as:
\[ (8M + 12W) \times 15 = \text{Total Work} \quad \text{(2)} \]

Since both expressions equal the total work, we can set them equal to each other:
\[ (6M + 6W) \times 24 = (8M + 12W) \times 15 \]

### Step 3: Simplify the equation

Expanding both sides:
\[ 144M + 144W = 120M + 180W \]

Now, rearranging the equation:
\[ 144M - 120M = 180W - 144W \]
\[ 24M = 36W \]

### Step 4: Find the relationship between men and women

Dividing both sides by 12:
\[ 2M = 3W \]
This gives us the relationship:
\[ M = \frac{3}{2}W \quad \text{(3)} \]

### Step 5: Set up the equation for 4 men and 6 women

Now, we need to find the time taken by 4 men and 6 women to complete the same work. Let \( x \) be the number of days taken by 4 men and 6 women. The work done can be expressed as:
\[ (4M + 6W) \times x = \text{Total Work} \]

### Step 6: Substitute \( M \) in terms of \( W \)

Using equation (3):
\[ 4M + 6W = 4 \left(\frac{3}{2}W\right) + 6W = 6W + 6W = 12W \]

Thus, the equation becomes:
\[ 12W \times x = \text{Total Work} \quad \text{(4)} \]

### Step 7: Use the total work from either scenario

Now, we can use the total work from either scenario. Let's use the first scenario:
\[ \text{Total Work} = (6M + 6W) \times 24 = (6 \times \frac{3}{2}W + 6W) \times 24 = (9W + 6W) \times 24 = 15W \times 24 = 360W \]

### Step 8: Set the equations equal

Now we can set equation (4) equal to the total work:
\[ 12W \times x = 360W \]

### Step 9: Solve for \( x \)

Dividing both sides by \( 12W \):
\[ x = \frac{360W}{12W} = 30 \]

### Final Answer

The time taken by 4 men and 6 women to complete the same work is **30 days**.

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