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.

To find the ratio of the efficiency of a man to a woman, we can set up equations based on the work done by them.

Let the daily efficiency of a man be represented by $M$ and the daily efficiency of a woman be represented by $W$.
Let the total work be represented by 1 unit.

Step 1: Analyze the first case
Four men and six women can complete 150% (or 1.5 times) of the work in 12 days.
Therefore, the work completed in 1 day by 4 men and 6 women is:

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

We can simplify the right-hand side of the equation:

$\frac{1.5}{12}=\frac{3}{24}=\frac{1}{8}$

Thus, we have:

$4M+6W=\frac{1}{8}$

Multiplying both sides by 8 to express it in terms of the total work (1 unit):

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

Step 2: Analyze the second case
Three men and seven women can complete half (or 0.5 times) of the work in 5 days.
Therefore, the work completed in 1 day by 3 men and 7 women is:

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

Simplifying the fraction:

$\frac{0.5}{5}=\frac{1}{10}$

Thus, we have:

$3M+7W=\frac{1}{10}$

Multiplying both sides by 10 to express it in terms of the total work (1 unit):

$30M+70W=1$ (Equation 2)

Step 3: Solve for the ratio of efficiencies
Since both Equation 1 and Equation 2 are equal to 1 unit of total work, we can equate them:

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

Subtract $30M$ and $48W$ from both sides to group like terms:

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

$2M=22W$

Divide both sides by 2:

$M=11W$

To find the ratio of efficiency of a man to a woman ($\frac{M}{W}$):

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

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