$2$ men and $1$ woman together can complete a piece of work in $14$ days. $4$ women and $2$ men together can do it in $8$ days. If a man gets $\text{₹}600$ per day, how much should a woman get per day?

Let the daily efficiency (work done per day) of a man be $M$ and a woman be $W$.
According to the question, total work is equal in both cases:

$$(2M+1W)\times 14=(2M+4W)\times 8$$

Divide both sides by $2$:

$$(2M+1W)\times 7=(2M+4W)\times 4$$

$$14M+7W=8M+16W$$

$$14M-8M=16W-7W$$

$$6M=9W$$

$$\frac{M}{W}=\frac{9}{6}=\frac{3}{2}$$

The ratio of efficiency of a man to a woman is $3:2$. Wages are distributed in proportion to their work efficiency.
Given, daily wage of a man = $\text{₹}600$.

$$\text{Daily wage of a woman}=\frac{2}{3}\times 600=\text{₹}400$$