Three friends, X, Y, and Z, are cycling on a circular track with a circumference of 2 km. They all start from the same point at 9:00 a.m. and travel in the same direction with speeds of 10 km/h, 12 km/h, and 15 km/h respectively. How many times (after the start) will all three of them meet at the starting point if they continue cycling until 1:00 p.m.?
2 times
The correct answer is 2 times.
Step 1 – time to complete one lap (in minutes):
X: \(\dfrac{2\,\text{km}}{10\,\text{km/h}} = \dfrac{1}{5}\,\text{h} = 12\,\text{min}\)
Y: \(\dfrac{2}{12}\,\text{h} = 10\,\text{min}\)
Z: \(\dfrac{2}{15}\,\text{h} = 8\,\text{min}\)
Step 2 – LCM of lap times:
\(\text{LCM}(12,\,10,\,8) = 120\,\text{min} = 2\,\text{hours}\)
So all three meet at the starting point every 2 hours after the start.
Step 3 – count meetings after the start (9 a.m.) up to 1 p.m.:
They meet at the starting point at 11:00 a.m. and 1:00 p.m. — 2 times. The starting moment (9:00 a.m.) is not counted as a meeting.
Short trick: Number of meetings = total time ÷ LCM = 240 ÷ 120 = 2.
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