6 men and 9 women can do a piece of work in 4 days.4 men and 3 women can do it in 8 days. In how many days can 20 men and 6 women do the same work ?

Options

A. 2
B. 3
C. 1
D. 4

Explanation

To solve the problem step by step, we will first determine the work done by men and women in terms of a common unit of work, and then use that to find out how many days 20 men and 6 women will take to complete the same work.

### Step 1: Determine the total work done by 6 men and 9 women in 4 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:
- 6 men and 9 women can complete the work in 4 days.
- Total work done = \( (6M + 9W) \times 4 \)

### Step 2: Determine the total work done by 4 men and 3 women in 8 days.
From the second scenario:
- 4 men and 3 women can complete the work in 8 days.
- Total work done = \( (4M + 3W) \times 8 \)

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

### Step 4: Simplify the equation.
Expanding both sides:
\[ 24M + 36W = 32M + 24W \]

### Step 5: Rearranging the equation.
Rearranging gives:
\[ 24M - 32M = 24W - 36W \]
\[ -8M = -12W \]
Dividing both sides by -4:
\[ 2M = 3W \]
Thus, we can express \( M \) in terms of \( W \):
\[ M = \frac{3}{2}W \]

### Step 6: Substitute \( M \) in terms of \( W \) into one of the work equations.
We can substitute \( M \) into the first work equation:
\[ (6 \times \frac{3}{2}W + 9W) \times 4 = Total Work \]
Calculating:
\[ (9W + 9W) \times 4 = 18W \times 4 = 72W \]
So, the total work is \( 72W \).

### Step 7: Determine the work done by 20 men and 6 women in a day.
Now, we need to find out how many days it will take for 20 men and 6 women to complete the same work.
The work done by 20 men and 6 women in one day is:
\[ (20M + 6W) = (20 \times \frac{3}{2}W + 6W) = (30W + 6W) = 36W \]

### Step 8: Calculate the number of days required.
To find the number of days \( D \) required to complete the total work of \( 72W \):
\[ D = \frac{Total Work}{Work \, done \, in \, one \, day} = \frac{72W}{36W} = 2 \, days \]

### Final Answer:
Thus, 20 men and 6 women can complete the work in **2 days**.

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