From a point on a circular track 42 km long, P, Q, and R started running at the same time with the speed of 2 km/h, 3\(\frac{1}{2}\) km/h and 4\(\frac{1}{2}\) km/h respectively.  After what time will they meet again?

84 hrs

Time = \(\frac{Distance}{Speed}\)

Time taken by P, Q and R to cover the track once = \(\frac{42}{2}\), \(\frac{42}{3.5}\) and \(\frac{42}{4.5}\) (hours)

= \(\frac{21}{1}\), \(\frac{12}{1}\), \(\frac{28}{3}\)

LCM of time = \(\frac{LCM\;of\;Numerator}{HCF\;of\;Denominator}\) = \(\frac{84}{1}\) = 84