If 3 men and 7 women can complete a work in 10 days and 4 men and 6 women need 8 days to complete the same work. In how many days will 10 women complete the same work?

Options

A. 50
B. 40
C. 30
D. 20

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the work done by 3 men and 7 women in one day.
Given that 3 men and 7 women can complete the work in 10 days, the work done in one day by them is:
\[ \text{Work done in one day} = \frac{1}{10} \]

### Step 2: Determine the work done by 4 men and 6 women in one day.
Similarly, for 4 men and 6 women who can complete the work in 8 days, the work done in one day is:
\[ \text{Work done in one day} = \frac{1}{8} \]

### Step 3: Set up equations based on the work done.
Let the work done by one man in one day be \( m \) and the work done by one woman in one day be \( w \). We can set up the following equations based on the information given:
1. \( 3m + 7w = \frac{1}{10} \) (Equation 1)
2. \( 4m + 6w = \frac{1}{8} \) (Equation 2)

### Step 4: Eliminate one variable.
To eliminate \( m \), we can multiply Equation 1 by 4 and Equation 2 by 3:
- From Equation 1:
\[ 4(3m + 7w) = 4 \cdot \frac{1}{10} \implies 12m + 28w = \frac{4}{10} \implies 12m + 28w = \frac{2}{5} \quad (Equation 3) \]
- From Equation 2:
\[ 3(4m + 6w) = 3 \cdot \frac{1}{8} \implies 12m + 18w = \frac{3}{8} \quad (Equation 4) \]

### Step 5: Subtract the equations to find \( w \).
Now, subtract Equation 4 from Equation 3:
\[ (12m + 28w) - (12m + 18w) = \frac{2}{5} - \frac{3}{8} \]
This simplifies to:
\[ 10w = \frac{2}{5} - \frac{3}{8} \]

### Step 6: Find a common denominator and solve for \( w \).
The common denominator for 5 and 8 is 40. Thus, we convert:
\[ \frac{2}{5} = \frac{16}{40}, \quad \frac{3}{8} = \frac{15}{40} \]
Now substituting back:
\[ 10w = \frac{16}{40} - \frac{15}{40} = \frac{1}{40} \]
So,
\[ w = \frac{1}{400} \]

### Step 7: Substitute \( w \) back to find \( m \).
Now substitute \( w \) back into either Equation 1 or 2 to find \( m \). Using Equation 1:
\[ 3m + 7\left(\frac{1}{400}\right) = \frac{1}{10} \]
This simplifies to:
\[ 3m + \frac{7}{400} = \frac{40}{400} \implies 3m = \frac{40}{400} - \frac{7}{400} = \frac{33}{400} \]
Thus,
\[ m = \frac{11}{400} \]

### Step 8: Calculate the work done by 10 women in one day.
Now, the work done by 10 women in one day is:
\[ 10w = 10 \cdot \frac{1}{400} = \frac{10}{400} = \frac{1}{40} \]

### Step 9: Determine the number of days 10 women will take to complete the work.
If 10 women can do \(\frac{1}{40}\) of the work in one day, then the number of days required to complete the entire work is:
\[ \text{Days} = \frac{1}{\frac{1}{40}} = 40 \text{ days} \]

### Final Answer:
10 women will complete the work in **40 days**.

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