4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women complete it?

Options

A. 50
B. 45
C. 40
D. 35

Answer

To solve the problem step by step, we will use the information given about the work done by men and women.

### Step 1: Determine the Work Done by Men and Women
Let the efficiency of one man be \( m \) and the efficiency of one woman be \( w \).

From the problem:
- **4 men and 6 women can complete the work in 8 days.**
\[ \text{Total work} = \text{(Number of workers)} \times \text{(Efficiency)} \times \text{(Days)} \]
\[ \text{Total work} = (4m + 6w) \times 8 \]

- **3 men and 7 women can complete the work in 10 days.**
\[ \text{Total work} = (3m + 7w) \times 10 \]

Since both expressions represent the same total work, we can equate them:
\[ (4m + 6w) \times 8 = (3m + 7w) \times 10 \]

### Step 2: Expand and Simplify the Equation
Expanding both sides:
\[ 32m + 48w = 30m + 70w \]

Now, rearranging the equation:
\[ 32m - 30m = 70w - 48w \]
\[ 2m = 22w \]
\[ m = 11w \]

### Step 3: Find the Efficiency of Men and Women
From the equation \( m = 11w \):
- The efficiency of one man is \( 11 \) units.
- The efficiency of one woman is \( 1 \) unit.

### Step 4: Calculate the Total Work
Now, we can substitute \( m \) back into the total work equation. We can use either of the original work equations. Let's use the first one:
\[ \text{Total work} = (4m + 6w) \times 8 \]
Substituting \( m \) and \( w \):
\[ \text{Total work} = (4 \times 11 + 6 \times 1) \times 8 \]
\[ = (44 + 6) \times 8 \]
\[ = 50 \times 8 = 400 \text{ units} \]

### Step 5: Calculate the Time Taken by 10 Women
Now, we need to find out how many days will 10 women take to complete the same work:
\[ \text{Total work} = \text{(Number of workers)} \times \text{(Efficiency)} \times \text{(Days)} \]
Let \( d \) be the number of days taken by 10 women:
\[ 400 = (10 \times 1) \times d \]
\[ 400 = 10d \]
\[ d = \frac{400}{10} = 40 \text{ days} \]

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

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