Four men and six women can complete 150% of the work in 12 days and three men and 7 women can complete half of the work in 5 days. Find the ratio of efficiency of man to woman.

The correct option is 11: 1.

Let the efficiency (work done per day) of one man be represented by $M$ and that of one woman by $W$. Let the total work be represented by $T$.

From the first statement, four men and six women can complete 150% of the work (which is $1.5T$ or $\frac{3}{2}T$) in 12 days. Therefore, their combined rate of work per day is given by:

$4M+6W=\frac{1.5T}{12}$

Simplifying the fraction on the right side:

$4M+6W=\frac{3T}{2\times 12}=\frac{T}{8}$

Multiplying by 8, we get our first equation for the total work $T$:

$T=8(4M+6W)$

$T=32M+48W$ (Equation 1)

From the second statement, three men and seven women can complete half of the work (which is $0.5T$ or $\frac{T}{2}$) in 5 days. Therefore, their combined rate of work per day is given by:

$3M+7W=\frac{0.5T}{5}$

Simplifying this fraction:

$3M+7W=\frac{T}{10}$

Multiplying by 10, we get our second equation for the total work $T$:

$T=10(3M+7W)$

$T=30M+70W$ (Equation 2)

Since both Equation 1 and Equation 2 are expressions for the same total work $T$, we can equate them:

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

Rearranging the terms to group $M$ on one side and $W$ on the other:

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

$2M=22W$

Solving for the ratio of the efficiency of a man ($M$) to a woman ($W$):

$\frac{M}{W}=\frac{22}{2}=\frac{11}{1}$

Thus, the ratio of the efficiency of a man to that of a woman is 11:1.