A can finish a task in $20$ days and B can finish the same task in half the time taken by A. Working together, what part of the task can they finish in a day?

Given:Time taken by A = $20$ daysWork rate of A per day = $\frac{1}{20}$ Time taken by B = half of A's time = $10$ daysWork rate of B per day = $\frac{1}{10}$ Combined work rate per day when they work together:
$$\text{Work done in 1 day}=\frac{1}{20}+\frac{1}{10}=\frac{1+2}{20}=\frac{3}{20}$$