P, Q and R start running at the same time from the same point on a circular track 42 km long, in the same direction, at speeds of 2 km/h, \(3\frac{1}{2}\) km/h and \(4\frac{1}{2}\) km/h respectively. After how many hours will they first meet again at the starting point?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 84 hours
Each runner is back at the starting point after every complete lap, so all three are there together after the LCM of their lap times. For fractions, LCM \(=\) LCM of the numerators \(\div\) HCF of the denominators.
Lap time of P \(\displaystyle=\frac{42}{2}=21\) hours
Lap time of Q \(\displaystyle=42\div\frac{7}{2}=12\) hours
Lap time of R \(\displaystyle=42\div\frac{9}{2}=\frac{84}{9}=\frac{28}{3}\) hours
LCM of 21, 12 and \(\frac{28}{3}\) \(\displaystyle=\frac{\text{LCM of }21, 12, 28}{\text{HCF of }1, 1, 3}=\frac{84}{1}=84\)
Check: in 84 hours P runs 4 laps, Q runs 7 laps and R runs 9 laps
They first meet again at the starting point after 84 hours.