2 men and 3 women can do a piece of work in 16 days . In how many days can 4 men and 6 women do the same work?

Answer

To solve the problem, we will follow these steps:

### Step 1: Define 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 \( W \).

### Step 2: Calculate the total work done by 2 men and 3 women in one day
The work done by 2 men in one day is:
\[ 2M \]
The work done by 3 women in one day is:
\[ 3W \]
Thus, the total work done by 2 men and 3 women in one day is:
\[ 2M + 3W \]

### Step 3: Calculate the total work done in 16 days
Since 2 men and 3 women can complete the work in 16 days, the total work \( T \) can be expressed as:
\[ T = (2M + 3W) \times 16 \]

### Step 4: Calculate the work done by 4 men and 6 women in one day
Now, we need to find out how much work is done by 4 men and 6 women in one day. The work done by 4 men in one day is:
\[ 4M \]
The work done by 6 women in one day is:
\[ 6W \]
Thus, the total work done by 4 men and 6 women in one day is:
\[ 4M + 6W \]

### Step 5: Set up the equation for the total work
Let \( X \) be the number of days required for 4 men and 6 women to complete the work. The total work can also be expressed as:
\[ T = (4M + 6W) \times X \]

### Step 6: Equate the two expressions for total work
Now we can equate the two expressions for total work:
\[ (2M + 3W) \times 16 = (4M + 6W) \times X \]

### Step 7: Simplify the equation
We can simplify the equation:
\[ 16(2M + 3W) = X(4M + 6W) \]
Dividing both sides by 2:
\[ 8(2M + 3W) = X(4M + 6W) \]

### Step 8: Cancel out common terms
Now we can cancel \( 2M + 3W \) from both sides (assuming it's not zero):
\[ 8 = X \cdot 2 \]

### Step 9: Solve for \( X \)
Now, divide both sides by 2:
\[ X = 4 \]

### Conclusion
Thus, the number of days required for 4 men and 6 women to complete the work is:
\[ \boxed{8} \]