Step 1: Set up the total work equations

Let the daily efficiency of one man be $M$ and one woman be $W$. Equating the total work from both conditions: $(8M + 12W) \times 10 = (6M + 8W) \times 14$

$(8M + 12W) \times 10 = (6M + 8W) \times 14$

Step 2: Find the efficiency ratio between men and women

Expanding and simplifying the equation: $80M + 120W = 84M + 112W \implies 4M = 8W \implies M = 2W$ This shows a man is twice as efficient as a woman ($M/W = 2/1$).

$M = 2W$

Step 3: Calculate the total work in units

Substitute $M = 2W$ into the first equation to find total work: $\text{Total Work} = (8(2W) + 12W) \times 10 = (16W + 12W) \times 10 = 280W$

$\text{Total Work} = 280W$

Step 4: Calculate time taken by 1 man and 2 women

Combined efficiency of 1 man and 2 women is $1M + 2W = 1(2W) + 2W = 4W$. Time required is: $\text{Time} = \frac{280W}{4W} = 70 \text{ days}$

$\text{Time} = 70 \text{ days}$