Two trains $A$ and $B$ are entering a railway platform from opposite direction. The length of the platform is 300 m. The speed of $B$ is two times the speed of $A$. The time taken by $B$ to cross the platform is one-third of the time taken by $A$ to cross the platform. If the sum of the lengths of $A$ and $B$ is 500 m, what is the difference in their lengths?

To cross a platform, a train covers a distance equal to its length plus the platform's length (L + 300). Train B has twice the speed but takes one-third the time of Train A, meaning it covers two-thirds the distance of A. This gives the equation (LB + 300) = 2/3(LA + 300), which simplifies to 2LA - 3LB = 300. Given the sum LA + LB = 500, we solve the system to find LA = 360 m and LB = 140 m. The difference is 360 - 140 = 220 m.