Acar traveled from city a to city b at an average speed of 3x miles per hour, where x > 0. the car then immediately returned along the same route at an average speed of 4x miles per hour. in terms of x, what was the car's average speed, in miles per hour, for the round trip

This is a classic question on Speed, time, & distance.

1. When time traveled in each segment is constant, then average speed is simple mean of speeds.

2. When distance traveled in each segment is constant, then average speed is reciprocal of simple mean of reciprocal of speeds. It is basically called Harmonic mean.

So this question falls in the category of 2.

=> So, average speed = Reciprocal of mean of reciprocals of 40 & 50.

=> Average speed = Reciprocal of mean of 1/40 & 1/50.

=> Average speed = Reciprocal of (1/40 + 1/50)/2 = Reciprocal of (5+4)/400 = 44 4/9 .
Hope this helps