Two particles move at different but constant speeds along a circle of circumference 276 ft. starting at the same instant and from the same place, when they move in opposite directions, they pass each other every 6 seconds and when they move in the same direction they pass each other every 23 seconds. determine their rates.

This is the concept of relative speed, given that the particles are moving at a circumference of 276 ft and they have the speed x and y, this rate can be calculated as follows;
Relative speed of the particles are moving towards each other will be:
(x+y) ft/s
time taken for them to meet will be:
time=distance/speed
time=276/(x+y)=6
thus;
276=6x+6y
this can be simplified as
46=x+yi
Time taken for them to meet when they are moving away from each other will be:
relative speed=x-y
thus;
time=276/(x-y)=23
this can be written as:
276=23x-23y
when we simplify we get
12=x-yii
from ii,
x=12+yiii
substituting iii in i we get
46=x+y
46=12+y+y
46-12=2y
34=2y
y=17
x=12+y
x=12+17
x=29
therefore we conclude that particle A is moving at 29 ft/s and B at 17 ft/s