If 2 men and 7 women can do a piece of work in 4 days, 4 men and 2 women can do the same work in 5 days, then in how many days 6 men and 27 women can do the same work ?

Options

A. `7(1)/(2)` days
B. `2(1)/(3)` days
C. `1(1)/(9)` days
D. None of these

Answer

To solve the problem step by step, we will first establish the work done by men and women based on the information given.

### Step 1: Establish the equations based on the work done
From the problem, we have two scenarios:
1. **2 men and 7 women can complete the work in 4 days.**
2. **4 men and 2 women can complete the work in 5 days.**

Let the work done by 1 man in 1 day be \( M \) and the work done by 1 woman in 1 day be \( W \).

From the first scenario:
\[ (2M + 7W) \times 4 = \text{Total Work} \]
This simplifies to:
\[ 2M + 7W = \frac{\text{Total Work}}{4} \quad \text{(Equation 1)} \]

From the second scenario:
\[ (4M + 2W) \times 5 = \text{Total Work} \]
This simplifies to:
\[ 4M + 2W = \frac{\text{Total Work}}{5} \quad \text{(Equation 2)} \]

### Step 2: Equate the total work from both equations
Since both equations represent the same total work, we can equate them:
\[ (2M + 7W) \times 4 = (4M + 2W) \times 5 \]

### Step 3: Expand and simplify the equation
Expanding both sides gives:
\[ 8M + 28W = 20M + 10W \]
Rearranging this gives:
\[ 8M - 20M + 28W - 10W = 0 \]
\[ -12M + 18W = 0 \]
\[ 12M = 18W \]
Dividing by 6:
\[ 2M = 3W \quad \Rightarrow \quad \frac{W}{M} = \frac{2}{3} \]

### Step 4: Substitute \( W \) in terms of \( M \)
From \( W = \frac{2}{3}M \), we can substitute \( W \) back into either equation. Let's use Equation 1:
\[ 2M + 7\left(\frac{2}{3}M\right) = \frac{\text{Total Work}}{4} \]
This simplifies to:
\[ 2M + \frac{14}{3}M = \frac{\text{Total Work}}{4} \]
Converting \( 2M \) to a fraction:
\[ \frac{6}{3}M + \frac{14}{3}M = \frac{\text{Total Work}}{4} \]
Combining the fractions:
\[ \frac{20}{3}M = \frac{\text{Total Work}}{4} \]
Cross-multiplying gives:
\[ \text{Total Work} = \frac{20}{3}M \times 4 = \frac{80}{3}M \]

### Step 5: Calculate the work done by 6 men and 27 women
Now, we need to find out how many days 6 men and 27 women can complete the work:
\[ 6M + 27W \]
Substituting \( W \):
\[ 6M + 27\left(\frac{2}{3}M\right) = 6M + 18M = 24M \]

### Step 6: Calculate the number of days
Now, we know the total work and the work done per day:
\[ \text{Total Work} = \frac{80}{3}M \]
The work done by 6 men and 27 women in one day is \( 24M \).

To find the number of days \( D \):
\[ D = \frac{\text{Total Work}}{\text{Work done per day}} = \frac{\frac{80}{3}M}{24M} \]
This simplifies to:
\[ D = \frac{80}{3 \times 24} = \frac{80}{72} = \frac{10}{9} \]

### Final Answer
Thus, the number of days 6 men and 27 women can complete the work is \( \frac{10}{9} \) days, or \( 1 \frac{1}{9} \) days.