Alvin, Simon, and Theodore are running around a 1000 meter circular track starting at different positions. Alvin is running in the opposite direction of Simon and Theodore. He is also the fastest, running twice as fast as Simon and three times as fast as Theodore. If Alvin meets Simon for the first time after running 312 meters, and Simon meets Theodore for the first time after running 2526 meters, how far apart along the track (shorter distance) did Alvin and Theodore meet?
Say Alvin's speed is A and he travels CCW. Alvin and Simon head for
each other at speeds A and 1/2 A. When Alvin has run 312 m to reach
their rendezvous, Simon has run 156. So, they started
312 + 156 = 468 m apart. If the meter marks on the track run CW, let's
start Alvin at 468 and Simon at 0. They will meet at meter 156.
Simon travels at 1 1/2 the speed of Theo, so he not only keeps up,
but spends the additional 1/3 of his speed closing the gap between
them. So, Theo starts at meter mark 842 CW = (1/3) 2526.
At the start, Theo is 1000 - 842 = 158 m before completing the circle CW,
while Alvin is approaching CCW from mile mark 468.
There is 158 + 468 = 626 meters between them. Alvin is running 3 times
faster than Theo, so he will travel (3/4) 626 = 469.5 m and Theo will
travel (1/4) 626 = 156.5 m to where they meet. They will meet on meter mark
998.5
Edited on November 8, 2024, 5:55 pm
soln
