Problem: Cars A and B travel the same distance. Car A travels half that distance at u miles per hour and half at v miles per hour. Car B travels half the time at u miles per hour and half at v miles per hour. The average speed of Car A is x miles per hour and that of Car B is y miles per hour: Then we always have
Answer Choices:
A. x≤y
B. x≥y
C. x=y
D. x<y
E. x>y
Solution:
Let s denote the distance. The time t for car A is t=2us​+2vs​ and the average speed is
ts​=2us​+2vs​s​=u+v2uv​=x
For car B, let s1​,s2​ denote the diatance at speed u,v reapectively so that if the time for car B is T,=T/2s1​​=u and T/2s2​​=v. Adding these T/2s​=T/2s1​+s2​​=u+v and the average speed of car B is
s/T=2u+v​=y.
The fact that x≤y for positive u and v follows trom 4uv≤(u+v)2.