4 men and 6 women complete a work in 8 days. 2 men and 9 women also complete in 8 days in which. The number of days In which 18 women complete the work is :

Options

A. `4(1)/(3)` days
B. `5(1)/(3)` days
C. `4(2)/(3)` days
D. `5(2)/(3)` days

Answer

To solve the problem step by step, we will follow the logical deductions made in the video transcript.

### Step 1: Understand the given information
We have two scenarios:
1. 4 men and 6 women complete the work in 8 days.
2. 2 men and 9 women also complete the work in 8 days.

### Step 2: Set up the equations
From the information given, we can set up the following equations based on the amount of work done:

1. For the first scenario:
\[ (4 \text{ men} + 6 \text{ women}) \times 8 \text{ days} = \text{ Total Work} \]

2. For the second scenario:
\[ (2 \text{ men} + 9 \text{ women}) \times 8 \text{ days} = \text{ Total Work} \]

Since both scenarios complete the same total work, we can equate them:
\[ (4M + 6W) \times 8 = (2M + 9W) \times 8 \]

### Step 3: Simplify the equations
We can simplify the equation by dividing both sides by 8:
\[ 4M + 6W = 2M + 9W \]

### Step 4: Rearranging the equation
Rearranging the equation gives us:
\[ 4M - 2M = 9W - 6W \]
\[ 2M = 3W \]

### Step 5: Express men in terms of women
From the equation \(2M = 3W\), we can express men in terms of women:
\[ M = \frac{3}{2}W \]

### Step 6: Substitute into the first scenario
Now, we can substitute \(M\) back into the first scenario to find the equivalent number of women:
\[ 4M + 6W = 4 \left(\frac{3}{2}W\right) + 6W = 6W + 6W = 12W \]
Thus, \(4M + 6W\) is equivalent to \(12W\).

### Step 7: Use the second scenario
Now we can use the second scenario:
\[ 2M + 9W = 2 \left(\frac{3}{2}W\right) + 9W = 3W + 9W = 12W \]
This confirms that both scenarios equate to \(12W\).

### Step 8: Find the work done by 18 women
Now we need to find how many days it takes for 18 women to complete the same work:
Let \(D\) be the number of days for 18 women to complete the work:
\[ 18W \times D = 12W \times 8 \]
Dividing both sides by \(W\) (since \(W\) is not zero):
\[ 18D = 12 \times 8 \]
\[ 18D = 96 \]
Now, solving for \(D\):
\[ D = \frac{96}{18} = \frac{16}{3} \]

### Step 9: Convert to mixed fraction
Converting \(\frac{16}{3}\) to a mixed fraction gives:
\[ D = 5 \frac{1}{3} \]

### Final Answer
Thus, the number of days in which 18 women complete the work is:
\[ \text{5 } \frac{1}{3} \text{ days} \]