2 men and 3 women can do a piece of work in 10 days, while 3 men nd 2 women can do the same work in 8 days. Then, 2 men and 1 woman can do the same work in

Options

A. 12 days
B. `12frac(1)(2)days`
C. 13 days
D. `13frac(1)(2) days`

Answer

To solve the problem, we need to establish a relationship between the work done by men and women based on the information provided.

### Step 1: Set up equations based on the given information

Let the work done by one man in one day be \( M \) and the work done by one woman in one day be \( W \).

From the first statement:
- 2 men and 3 women can complete the work in 10 days.
- Therefore, in one day, they complete \( \frac{1}{10} \) of the work.

This gives us the equation:
\[ 2M + 3W = \frac{1}{10} \]

From the second statement:
- 3 men and 2 women can complete the work in 8 days.
- Therefore, in one day, they complete \( \frac{1}{8} \) of the work.

This gives us the equation:
\[ 3M + 2W = \frac{1}{8} \]

### Step 2: Solve the equations simultaneously

We have the two equations:
1. \( 2M + 3W = \frac{1}{10} \) (Equation 1)
2. \( 3M + 2W = \frac{1}{8} \) (Equation 2)

To eliminate one variable, we can multiply Equation 1 by 3 and Equation 2 by 2:
\[ 6M + 9W = \frac{3}{10} \quad (3 \times \text{Equation 1}) \]
\[ 6M + 4W = \frac{1}{4} \quad (2 \times \text{Equation 2}) \]

Now, subtract the second modified equation from the first:
\[ (6M + 9W) - (6M + 4W) = \frac{3}{10} - \frac{1}{4} \]
This simplifies to:
\[ 5W = \frac{3}{10} - \frac{2.5}{10} = \frac{0.5}{10} = \frac{1}{20} \]
Thus, we find:
\[ W = \frac{1}{100} \]

### Step 3: Substitute \( W \) back to find \( M \)

Now that we have \( W \), we can substitute it back into one of the original equations to find \( M \). Let's use Equation 1:
\[ 2M + 3\left(\frac{1}{100}\right) = \frac{1}{10} \]
This simplifies to:
\[ 2M + \frac{3}{100} = \frac{10}{100} \]
\[ 2M = \frac{10}{100} - \frac{3}{100} = \frac{7}{100} \]
Thus:
\[ M = \frac{7}{200} \]

### Step 4: Calculate the work done by 2 men and 1 woman

Now we need to find out how long it takes for 2 men and 1 woman to complete the work:
\[ 2M + 1W = 2\left(\frac{7}{200}\right) + \frac{1}{100} \]
Calculating this:
\[ = \frac{14}{200} + \frac{2}{200} = \frac{16}{200} = \frac{2}{25} \]

This means that in one day, 2 men and 1 woman can complete \( \frac{2}{25} \) of the work.

### Step 5: Find the number of days to complete the work

To find the number of days required to complete the entire work, we take the reciprocal of the work done in one day:
\[ \text{Days} = \frac{1}{\frac{2}{25}} = \frac{25}{2} = 12.5 \text{ days} \]

### Final Answer
Thus, 2 men and 1 woman can complete the work in **12.5 days**.

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