4 men and 6 women can complete a job in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women take to complete the same job?

40 days

The correct answer is Option (3) → 40 days

Step 1: Establish the Efficiency Ratio

Let the work done by 1 man in one day be $M$ and by 1 woman be $W$. Since the total work remains the same, we can set up the following equation:

$(4M + 6W) \times 8 = (3M + 7W) \times 10$

Simplify the equation:

$32M + 48W = 30M + 70W$

$32M - 30M = 70W - 48W$

$2M = 22W$

$1M = 11W$

This tells us that 1 man does as much work as 11 women.

Step 2: Calculate Total Work in terms of Women

Now, substitute the value of $M$ into the first scenario to find the total work units:

$\text{Total Work} = (4M + 6W) \times 8$

$\text{Total Work} = [4(11W) + 6W] \times 8$

$\text{Total Work} = (44W + 6W) \times 8$

$\text{Total Work} = 50W \times 8 = \mathbf{400W \text{ units}}$

Step 3: Find Time Taken by 10 Women

Now, we find how many days 10 women will take to complete these 400 units of work:

$\text{Days} = \frac{\text{Total Work}}{\text{Daily Work of 10 Women}}$

$\text{Days} = \frac{400W}{10W}$

$\text{Days} = \mathbf{40 \text{ days}}$