If $9$ men and $6$ women working together can do six times as much work per hour as $1$ man and $2$ women working together, then the ratio of the work done by $1$ man to that of $1$ woman in a given time is:

Let the work efficiency (work done per hour) of $1$ man be $M$ and $1$ woman be $W$.According to the given condition:
$$\text{Work done by 9 men and 6 women per hour}=6\times (\text{Work done by 1 man and 2 women per hour})$$
Mathematically:
$$9M+6W=6(1M+2W)$$
Expand the right side:
$$9M+6W=6M+12W$$
Rearrange the terms to group $M$ and $W$:
$$9M-6M=12W-6W$$
$$3M=6W$$
Find the ratio of $M$ to $W$:
$$\frac{M}{W}=\frac{6}{3}=\frac{2}{1}$$
Thus, the ratio of work done by $1$ man to $1$ woman in a given time is $2:1$.