Linear & Circular Races · Topic 3 of 5

🔄 Circular Meetings

3 exam question types, fully solved

On a circular track the runners keep lapping, so “meeting” happens again and again. Two questions get asked: when do they first meet anywhere on the track, and when do they next meet at the starting point. They use different tools — don't mix them up.

Relative speed for the track, LCM for the start

• First meeting anywhere. One has to gain a whole lap on the other. Time — use when they run the same way, when they run opposite ways.

• Meeting at the starting point. Each returns to start every “lap time” and . They are both at the start together after the LCM of those two lap times.

• Number of distinct meeting points on the track (same direction) or (opposite), using the speeds in lowest ratio terms.

Take LCM/HCF of fractions with the rule LCM .

⭕Circular Race SimulatorWatch meeting points

Speed A (m/s): 8Speed B (m/s): 5Track length (m): 200

1st meeting: 66.7 s

Meetings so far: 0

Distinct points: 3

Sim time: 0.0 s

Same direction: Time to meet = L / |S_A − S_B| = 200 / 3 = 66.67 s

LCM of lap times: lap times are L/a = 25.0 s and L/b = 40.0 s → both meet again at the starting point after LCM(L/a, L/b) = 200.0 s

On a 400 m track A runs at 8 m/s and B at 6 m/s the same way. They first meet after s.

Q1On a 400 m loop, runners at 5 m/s and 3 m/s go opposite ways. First meeting time:

Q2On a 600 m loop, runners at 9 m/s and 6 m/s go the same way. First meeting time:

Q3Two runners lap a track in 24 s and 36 s. They meet at the START after:

📚

Real exam questions — Circular Meetings

3 question types · 9 solved examples from real SSC papers

On a loop, opposite runners close the gap at the SUM of speeds, same-way runners at the DIFFERENCE. "Meet anywhere" uses relative speed; "meet at the start" uses the LCM of lap times — keep the two apart.

How to solve this type

Running TOWARDS each other on a loop, the gap between them closes at the SUM of their speeds — the relative speed is

v_1+v_2

.

They meet for the first time after the pair together covers one whole lap

L

: time

=\dfrac{L}{v_1+v_2}

. After that they meet again every

\dfrac{L}{v_1+v_2}

seconds. So in a window of

T

seconds the number of meetings

=\dfrac{(v_1+v_2)\,T}{L}

.

If speeds are in km/h but the track is in metres, convert with

\times\dfrac{5}{18}

first. (Distinct meeting POINTS on the track: reduce the speed ratio to

a:b

in lowest terms; opposite ⇒

a+b

points, same direction ⇒

|a-b|

points.)

A circular track is 2500 m. A man runs at 37 km/h and a woman at 35 km/h in opposite directions from the same point. After how long do they first meet?

A2 min 40 secB2 min 5 secC2 min 30 secD2 min 20 sec

Ram at 4 m/s and Shyam at 6 m/s run in opposite directions on a 200 m track for 200 s. How many times do they meet?

A9B8C11D10

X runs at 4 m/s and Y at 6 m/s in opposite directions on a circular track. At how many distinct points do they keep meeting?

A5B10C4D8