Question

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 will all three of them meet at the starting point if they continue cycling until 1:00 p.m.?

• A. 1 time
• B. 2 times
• C. 3 times
• D. 4 times

Answer 1

Answer:

C. 3 times

We find how often each returns to the starting point.

Step 1: Time period for one full round

Track circumference = 2 km

• X speed = 10 km/h → time per lap:
  $\frac{2}{10} = 0.2 \text{ h}$

• Y speed = 12 km/h:
  $\frac{2}{12} = \frac{1}{6} \approx 0.1667 \text{ h}$

• Z speed = 15 km/h:
  $\frac{2}{15} \approx 0.1333 \text{ h}$

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Step 2: When do all meet at start together?

They meet at the starting point together when all complete whole number of laps simultaneously.

Find LCM of lap times:

• $0.2 = ( \frac{1}{5} )$

• $( \frac{1}{6} )$

• $( \frac{2}{15} )$

Convert to common denominator 30:

$\frac{1}{5}=\frac{6}{30},\quad \frac{1}{6}=\frac{5}{30},\quad \frac{2}{15}=\frac{4}{30}$

So we need LCM of 6/30, 5/30, 4/30 → LCM of numerators 6,5,4 is 60.

$\text{Meeting interval} = \frac{60}{30} = 2 \text{ hours}$

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Step 3: Time duration

From 9:00 a.m. to 1:00 p.m. = 4 hours

Meetings at:

• 9:00 a.m. (start)

• 11:00 a.m.

• 1:00 p.m.

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Step 4: Count meetings

Including the start time:

Total meetings = 3 times