Malia runs twice as fast as Eileen. If Malia and Eileen start from the same point and run around a circular track in opposite directions, how many times will they pass each other if they stop when Malia completes her sixth lap?
I figured 5 times. Is that correct?
To help you understand the solution, let's say the track is a typical 400 m track, Eillen can run 4 m/sec and Malia can run 8 m/sec
let t be the time when they first meet
Eileen has run 40t m, and Malia has run 80t m (distance = rate x time)
but when they meet:
4t + 8t = 400
12t = 400
t = 400/12 = 100/3 seconds or 33 1/3 sec
so they will meet every 100/3 seconds
after Malia's 6 laps she has gone 6(400)m or 2400 m
and at her speed of 8 m/s, it would take her
2400/8 = 300 seconds
she meets the other girl every 100/3 seconds, so number of times they meet = 300/(300/3) = 3 times
Now with general variables
let the distance around be k m
let Eileens speed be x m/sec
then Malia's speed is 2x m/sec
time of first meeting --- t sec
distance for Eileen = xt
distance for Malia = 2xt
2xt + xt = k
3xt = k
t = k/(3x) , the time it took for them to meet
total distance covered by Malia = 6k
time taken by Malia to cover the 6k = 6k/(2x) = 3k/x
time for them to meet = k/(3x)
number of meetings = (3k/x) / (k/3x) = (3k/x)(3x/k) = 3
they meet 3 times.