A mother along with her two sons is entrusted with the task of cooking Biryani for a family get- together. It takes 30 minutes for all three of them cooking together to complete 50 percent of the task. The cooking can also be completed if the two sons start cooking together and the elder son leaves after 1 hour and the younger son cooks for further 3 hours. If the mother needs 1 hour less than the elder son to complete the cooking, how much cooking does the mother complete in an hour?

Let the time taken by the Younger son to make the whole biryani be $$y$$ hours and the time taken by the Elder son to make the whole biryani be $$e$$ hours.
Thus, the time taken by the mother to make the biryani = $$e-1$$
Now we are given that, the biryani will be made in 4 hours if the elder son works for 1 hour and the younger son works for 4 hours.
Thus, in 1 hour the elder son will complete $$\dfrac{1}{e}$$ of the biryani and in 4 hours the younger son will complete $$\dfrac{4}{y}$$ of the biryani.
Thus, $$\dfrac{1}{e}$$+$$\dfrac{4}{y}$$ = $$1$$
=> $$y+4e = ey$$
=> $$e = \dfrac{y}{y-4}$$
Thus, the time taken by the mother = $$\dfrac{y}{y-4} - 1$$ = $$\dfrac{4}{y-4}$$
We are given that, It takes 30 minutes for all three of them cooking together to complete 50 percent of the task.
Thus, in 1 hour they'll complete the task.
Thus, $$\dfrac{1}{y}+\dfrac{y-4}{y}+\dfrac{y-4}{y} = 1$$
=> $$\dfrac{y-3}{y} + \dfrac{y-4}{4} = 1$$
=> $$4(y-3)+y(y-4) = 4y$$
=> $$4y-12+y^2-4y = 4y$$
=> $$y^2 - 4y - 12 = 0$$
Solving we get, $$y=6$$ hours and thus, $$e = \dfrac{6}{6-4} = 3$$ hours and thus, the time taken by the mother = $$2$$ hours
Thus, the mother will complete 50% of the biryani in 1 hour.
Hence, option B is the correct answer.