AFCAT Numerical Ability · Speed, Time and Distance

Circular Tracks: Meetings, Laps and the Starting Point

On a closed track, two runners meet whenever the distance one has gained on the other reaches a full lap, and all of them are back at the start together after the LCM of their lap times.

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Why this matters

A quarter of this page is HARD, the largest share in the chapter, though no AFCAT paper has set one here. The trap is treating the track as a straight road: runners going the same way meet only when the faster has gained a whole lap, and meeting somewhere is not meeting at the start.

Concept 1 of 3: The first meeting: opposite ways or the same way

Running opposite ways, the two together cover one lap before they meet. Running the same way, the faster must gain one whole lap on the slower.

Definition

Opposite directions: first meeting after $\dfrac{L}{a + b}$ .
Same direction: first meeting after $\dfrac{L}{a - b}$ .
Starting from diametrically opposite points, the gap to close is half a lap.
'Half a lap ahead' means a gain of $\tfrac L2$ .
'The faster laps the slower every $T$ ' means $a - b = \tfrac LT$ .

First meeting on a track

t_{\text{opp}} = \dfrac{L}{a + b}, \qquad t_{\text{same}} = \dfrac{L}{a - b}

Worked example

On a 600 m track two runners start together from one point at 5 m/s and 3 m/s. When do they first meet if they run (a) opposite ways, (b) the same way?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Speed, Time and Distance · Circular Tracks

Two runners A and B start simultaneously from the same point on a circular track of

400

m. A runs at

8

m/s and B runs at

12

m/s in the same direction. After how much time will they meet for the first time?

The same way needs a whole lap

Two runners going the same way from one point next meet when the faster has gained a full lap, not half a lap. Half a lap ahead is the moment they are furthest apart.

Drill 21 more on the first meeting: opposite ways or the same way

Concept 2 of 3: Counting meetings and laps

After the first meeting the pattern repeats: the runners meet again each time the distance gained, or covered together, grows by another lap.

Definition

Time between meetings $=$ the first-meeting time. In a time $T$ , count only complete meetings: the whole number part of $T$ divided by that time.
The $n$ th meeting comes at $n$ times the first.
Laps a runner has run at a meeting: its distance $\div L$ .
Lap times instead of speeds: work in laps per second, $\tfrac{1}{t_1}$ and $\tfrac{1}{t_2}$ .
Speeds in the ratio $p : q$ , same way: the faster has run $\tfrac{p}{p - q}$ laps when it first laps the slower.

Meetings in a time T

n = \dfrac{T\,v_{\text{rel}}}{L} \;\;(\text{whole meetings only})

Worked example

On a 300 m track two joggers start together and run opposite ways at 4 m/s and 6 m/s. How many times do they meet in 2 minutes?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Speed, Time and Distance · Circular Tracks

A and B start running simultaneously from the same point on a

600

m circular track in opposite directions at

5

m/s and

7

m/s respectively. How many times will they meet in

10

minutes?

Whole meetings only

If the time allows 3.6 laps of gain, A has lapped B 3 times. The fourth lapping has not happened yet.

Drill 22 more on counting meetings and laps

Concept 3 of 3: Back at the start together

A runner is back at the start after every whole lap. All the runners are there together at a common multiple of their lap times, and the first such moment is the LCM.

Definition

Lap time $= \dfrac{L}{\text{speed}}$ . Together at the start after the LCM of the lap times.
A fractional lap time such as $\tfrac{400}{3}$ s: the LCM of fractions is the LCM of the numerators over the HCF of the denominators.
Meeting anywhere on the track (the first concept) is different from meeting at the start.
Positions after a time: the distance run, less whole laps. The gap between two runners is the shorter way round.

Together at the start

T_{\text{start}} = \text{LCM}(t_1, t_2, t_3)

Worked example

Three runners lap a track in 40 s, 50 s and 75 s. When are they next together at the start?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Speed, Time and Distance · Circular Tracks

Three runners A, B, and C start simultaneously from the same point on a circular track. A completes the track in

60

seconds, B in

90

seconds, and C in

120

seconds. After how much time will all three meet again at the starting point?

Meeting is not meeting at the start

At 6 m/s and 4 m/s on a 300 m track, opposite runners first meet after 30 s, but they are both at the start only after the LCM of their lap times, 50 s and 75 s, which is 150 s.

Drill 8 more on back at the start together

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

The first meeting: opposite ways or the same way
First meeting on a track
$t_{\text{opp}} = \dfrac{L}{a + b}, \qquad t_{\text{same}} = \dfrac{L}{a - b}$
Counting meetings and laps
Meetings in a time T
$n = \dfrac{T\,v_{\text{rel}}}{L} \;\;(\text{whole meetings only})$
Back at the start together
Together at the start
$T_{\text{start}} = \text{LCM}(t_1, t_2, t_3)$

Watch out for (3)

The same way needs a whole lap
→ The first meeting: opposite ways or the same way
Whole meetings only
→ Counting meetings and laps
Meeting is not meeting at the start
→ Back at the start together

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