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 first determine the work done by men and women in terms of their efficiencies and then calculate how many days 18 women will take to complete the work.

### Step 1: Determine the total work done by 4 men and 6 women in 8 days.
Let the work done by 1 man in 1 day be \( M \) and the work done by 1 woman in 1 day be \( W \).

The total work done by 4 men and 6 women in 8 days can be expressed as:
\[ \text{Total Work} = (4M + 6W) \times 8 \]

### Step 2: Determine the total work done by 2 men and 9 women in 8 days.
Similarly, the total work done by 2 men and 9 women in 8 days can be expressed as:
\[ \text{Total Work} = (2M + 9W) \times 8 \]

### Step 3: Set the two expressions for total work equal to each other.
Since both groups complete the same total work, we can set the two equations equal:
\[ (4M + 6W) \times 8 = (2M + 9W) \times 8 \]

### Step 4: Simplify the equation.
Dividing both sides by 8 gives us:
\[ 4M + 6W = 2M + 9W \]

### Step 5: Rearrange the equation to find the relationship between M and W.
Rearranging gives:
\[ 4M - 2M = 9W - 6W \]
\[ 2M = 3W \]
This means:
\[ M = \frac{3}{2}W \]

### Step 6: Substitute M back into the total work equation.
Now we can substitute \( M \) in terms of \( W \) back into either total work equation. Let's use the first one:
\[ \text{Total Work} = (4M + 6W) \times 8 \]
Substituting \( M = \frac{3}{2}W \):
\[ \text{Total Work} = \left(4 \times \frac{3}{2}W + 6W\right) \times 8 \]
\[ = \left(6W + 6W\right) \times 8 \]
\[ = 12W \times 8 = 96W \]

### Step 7: Calculate the number of days for 18 women to complete the work.
Now, we need to find out how many days it will take for 18 women to complete the total work of \( 96W \):
\[ \text{Days} = \frac{\text{Total Work}}{\text{Work done by 18 women in 1 day}} \]
The work done by 18 women in 1 day is:
\[ 18W \]
Thus,
\[ \text{Days} = \frac{96W}{18W} = \frac{96}{18} = \frac{16}{3} \text{ days} \]

### Step 8: Convert the answer to a mixed number.
\[ \frac{16}{3} = 5 \frac{1}{3} \text{ days} \]

### Final Answer:
The number of days in which 18 women complete the work is \( 5 \frac{1}{3} \) days.

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