Two circles, one of radius 5 inches, the other of radius 2 inches, are tangent at point P. Two bugs start crawling at the same time from point P, one crawling along the larger circle at $3\pi$ inches per minute, the other crawling along the smaller circle at $2.5\pi$ inches per minute. How many minutes is it before their next meeting at point P?

Answer:   40 minutesStep-by-step explanation:The circumference of the larger circle is ...   C = 2πr = 2π(5 in) = 10π inThe bug navigates the circumference at 3π in/min, so will take   time = distance/speed = (10π in)/(3π in/min) = 10/3 minto travel once around.__The circumference of the smaller circle is ...   C = 2πr = 2π(2 in) = 4π inThe bug navigates this circumference at 2.5π in/min, so will take   (4π in)/(2.5π in/min) = 8/5 minto travel once around.__The bugs will meet at a time that is the least common multiple of these times. Both can be expressed in 15ths of a minute as ...   {50/15, 24/15}Then the LCM of these will be ...   (1/15)LCM(50, 24) = (1/15)(50×24)/GCD(50, 24) = 1200/30 = 40It will be 40 minutes before the bugs next meet at point P.___A graphing calculator can make use of the mod function to show when the bugs meet at point P (total is displacement of the two bugs from P is zero). It shows the meeting occurs after 40 minutes.