2 men and 3 women can finish a piece of work in 10 days, while 4 men can do it in 10 days. In how many days will 3 men and 3 women finish it ?

Options

A. 6 days
B. `5 2/3`days
C. 8 days
D. `8 1/3` days

Answer

To solve the problem step by step, we will first establish the work done by men and women and then calculate how long it will take for 3 men and 3 women to finish the work.

### Step 1: Determine the total work done by men and women.
We know that:
- 2 men and 3 women can finish the work in 10 days.
- 4 men can also finish the work in 10 days.

Let’s denote:
- The work done by 1 man in 1 day as \( m \).
- The work done by 1 woman in 1 day as \( w \).

From the first statement:
\[ (2m + 3w) \times 10 = \text{Total Work} \]
This simplifies to:
\[ 2m + 3w = \frac{\text{Total Work}}{10} \]

From the second statement:
\[ 4m \times 10 = \text{Total Work} \]
This simplifies to:
\[ 4m = \frac{\text{Total Work}}{10} \]

### Step 2: Set the equations equal to each other.
Since both expressions equal the total work, we can set them equal:
\[ 2m + 3w = 4m \]

### Step 3: Rearrange the equation to find the relationship between men and women.
Rearranging gives us:
\[ 3w = 4m - 2m \]
\[ 3w = 2m \]
Thus, we can express \( w \) in terms of \( m \):
\[ w = \frac{2}{3}m \]

### Step 4: Calculate the total work.
Now, we can substitute \( w \) back into one of our total work equations. Using \( 4m \):
\[ \text{Total Work} = 4m \times 10 = 40m \]

### Step 5: Calculate the efficiency of 3 men and 3 women.
Now, we need to find the efficiency of 3 men and 3 women:
\[ \text{Efficiency} = 3m + 3w \]
Substituting \( w \):
\[ \text{Efficiency} = 3m + 3\left(\frac{2}{3}m\right) \]
This simplifies to:
\[ \text{Efficiency} = 3m + 2m = 5m \]

### Step 6: Calculate the number of days for 3 men and 3 women to finish the work.
Now we can find the number of days it will take for 3 men and 3 women to finish the total work:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Efficiency}} \]
Substituting the values we have:
\[ \text{Days} = \frac{40m}{5m} = 8 \text{ days} \]

### Final Answer:
Thus, 3 men and 3 women will finish the work in **8 days**.

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