Two friends, Ravi and Sara, agree to meet at a cafe some time between 5 pm and 6 pm…

Step 1. The square. Let $x$ be Ravi's arrival and $y$ be Sara's, in minutes after $5$ pm. Every point of the square $0 \le x, y \le 60$ is equally likely. The area is $3600$.

Step 2. Both wait 20 minutes. They fail to meet when $|x - y| > 20$. That is two triangles with legs of $40$ minutes, each of area $\dfrac{40^2}{2} = 800$. So $P(\text{meet}) = 1 - \dfrac{1600}{3600} = \dfrac{5}{9} \approx 0.5556$.

Step 3. Waits of 20 and 10 minutes. They fail to meet when $y - x > 20$ (Sara arrives too late for Ravi), which is a triangle with legs $40$ and area $800$. They also fail when $x - y > 10$ (Ravi arrives too late for Sara), which is a triangle with legs $50$ and area $1250$. So $P(\text{meet}) = 1 - \dfrac{800 + 1250}{3600} = \dfrac{1550}{3600} = \dfrac{31}{72} \approx 0.4306$.