If 5 men or 9 women can finish a piece of work in 19 days, then the number of days for 3 men and 6 women to finish the same work is

Let the total work = W

Number of days required for 9 women to finish the work = 19 days

$$\Rightarrow$$  Work done by 9 women in 1 day = $$\frac{W}{19}$$

$$\Rightarrow$$  Work done by 1 woman in 1 day = $$\frac{W}{19\times9}$$

5 men or 9 women can finish the work in 19 days

$$\Rightarrow$$ 5 men = 9 women

$$\Rightarrow$$ 1 men = $$\frac{9}{5}$$ women

3 men and 6 women = 3($$\frac{9}{5}$$women) + 6 women = $$\frac{27}{5}$$ women + 6 women = $$\frac{57}{5}$$ women

Work done by $$\frac{57}{5}$$ women in 1 day = $$\frac{57}{5}\times\frac{W}{19\times9}$$ = $$\frac{W}{15}$$

Number of days required for $$\frac{57}{5}$$ women to complete the work = 15 days

$$\therefore\ $$Number of days required for 3 men and 6 women = 15 days