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, we will follow these steps:

### Step 1: Determine the work done by men and women
We know that:
- 6 men and 9 women can complete the work in 4 days.
- 4 men and 3 women can complete the work in 8 days.

Let's denote the work done by one man in one day as \(M\) and the work done by one woman in one day as \(W\).

From the first scenario:
\[ \text{Total work} = \text{Work done in 4 days} = 6M + 9W \text{ (for 4 days)} \]
Thus,
\[ 1 \text{ unit of work} = (6M + 9W) \times 4 \]
This simplifies to:
\[ 1 = 24M + 36W \quad \text{(Equation 1)} \]

From the second scenario:
\[ \text{Total work} = \text{Work done in 8 days} = 4M + 3W \text{ (for 8 days)} \]
Thus,
\[ 1 \text{ unit of work} = (4M + 3W) \times 8 \]
This simplifies to:
\[ 1 = 32M + 24W \quad \text{(Equation 2)} \]

### Step 2: Set up the equations
Now we have two equations:
1. \(24M + 36W = 1\)
2. \(32M + 24W = 1\)

### Step 3: Solve the equations
We can solve these equations simultaneously.

From Equation 1:
\[ 24M + 36W = 1 \quad \text{(Multiply by 2)} \Rightarrow 48M + 72W = 2 \quad \text{(Equation 3)} \]

From Equation 2:
\[ 32M + 24W = 1 \quad \text{(Multiply by 3)} \Rightarrow 96M + 72W = 3 \quad \text{(Equation 4)} \]

Now, we can subtract Equation 3 from Equation 4:
\[ (96M + 72W) - (48M + 72W) = 3 - 2 \]
This simplifies to:
\[ 48M = 1 \Rightarrow M = \frac{1}{48} \]

Substituting \(M\) back into Equation 1 to find \(W\):
\[ 24\left(\frac{1}{48}\right) + 36W = 1 \]
\[ \frac{1}{2} + 36W = 1 \Rightarrow 36W = \frac{1}{2} \Rightarrow W = \frac{1}{72} \]

### Step 4: Calculate the work done by 20 men and 6 women
Now we know:
- Work done by 1 man in 1 day = \(M = \frac{1}{48}\)
- Work done by 1 woman in 1 day = \(W = \frac{1}{72}\)

Now, calculate the total work done by 20 men and 6 women in one day:
\[ \text{Work done by 20 men} = 20M = 20 \times \frac{1}{48} = \frac{20}{48} = \frac{5}{12} \]
\[ \text{Work done by 6 women} = 6W = 6 \times \frac{1}{72} = \frac{6}{72} = \frac{1}{12} \]

Total work done by 20 men and 6 women in one day:
\[ \text{Total work} = \frac{5}{12} + \frac{1}{12} = \frac{6}{12} = \frac{1}{2} \]

### Step 5: Calculate the number of days to complete the work
If 20 men and 6 women can do \(\frac{1}{2}\) of the work in one day, then to complete 1 unit of work, they will take:
\[ \text{Days} = \frac{1}{\frac{1}{2}} = 2 \text{ days} \]

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

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