A, B and C start together from the same point to go round a circular island of circumference 30 km. A and B go in the same direction and C goes in the opposite direction. If A, B and C move at 5 km/h, 7 km/h and 8 km/h respectively, after how many hours will all three first be together again?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 30 h
Two people moving in the same direction are together each time the faster one gains one full round; two people moving in opposite directions are together each time they cover one full round between them. All three are together when A meets B and A meets C at the same moment.
A and B: B gains \(7-5=2\) km every hour, so they are together every \(\frac{30}{2}=15\) h
A and C: they close \(5+8=13\) km every hour, so they are together every \(\frac{30}{13}\) h
All three together: LCM of 15 and \(\frac{30}{13}\) \(\displaystyle =\frac{\text{LCM of }15\text{ and }30}{\text{HCF of }1\text{ and }13}=\frac{30}{1}=30\) h
At 15 h, A and B are together after 75 km and 105 km (half a round past the start), but C has gone 120 km, 4 full rounds, and is at the start
At 30 h, A, B and C have gone 150, 210 and 240 km, that is 5, 7 and 8 full rounds, so all three are at the start
All three are first together again after 30 hours.