SSC CGL Maths · Time, Speed and Distance

Circular Tracks

On a circular track two runners meet whenever the faster has gained a whole lap (same direction) or the two together have covered a whole lap (opposite directions), and they are back at the start together after the LCM of their lap times.

All Time, Speed and Distance notes

Why this matters

Twenty-seven questions in the bank, fourteen of them MODERATE and two HARD. Three questions get asked: when do they first meet, when are they together at the starting point, and at how many points do they meet. Each has its own rule, and mixing them up is how the printed wrong options are made.

Concept 1 of 3: The first meeting on a circular track

Going the same way, the faster runner meets the slower one when he has run exactly one lap more, so the lap is the gap and the difference of the speeds closes it. Going opposite ways, they meet when their two distances add up to one lap, so the sum of the speeds closes it.

Definition

Same direction: first meeting after $\frac{L}{s_1 - s_2}$ , and again after every such interval.
Opposite directions: first meeting after $\frac{L}{s_1 + s_2}$ .
Speeds in rounds an hour: the same way they meet $r_1 - r_2$ times an hour; opposite ways $r_1 + r_2$ times.
Not starting together: if one starts ahead on the track, the first catch closes only that start gap, not a full lap.
Meetings during a race: the second meeting comes at twice the first meeting time, and so on. Count only the meetings before the faster runner finishes the race.
Work in metres and seconds: change km/h with $\frac{5}{18}$ first.
Limit: the two lap formulas need both runners to start from the same point at the same moment and keep steady speeds. A start gap changes only the first meeting, as above.

First meeting on a circular track

t_{\text{same}} = \frac{L}{s_1 - s_2} \qquad t_{\text{opp}} = \frac{L}{s_1 + s_2}

$L$ length of one lap
$s_1, s_2$ the two speeds, the first one the faster

Worked example

Two cyclists start together from one point of a 1.8 km circular track at 18 km/h and 36 km/h. When do they first meet if they ride (a) the same way, (b) opposite ways?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 26 Sep 2024, Shift 3 · Q73Moderate

Example 1 · Time, Speed and Distance · Circular Tracks

In a circular race of 1600 m in length, Bhaskar and Vinay start with speeds of 27 km/h and 45 km/h starting at the same time from the same point. When will they meet for the first time on the track when running in the opposite directions and the same direction, respectively?

Using the difference of speeds for opposite directions

Running towards each other round the track, both close the lap, so the speeds add. The difference is only for runners going the same way.

Waiting for a full lap when one starts ahead

If one runner starts some way ahead on the track, the first overtaking closes only that start gap. A full lap is needed only when both start from the same point.

Counting a meeting after the race is over

In a race of fixed length, the meetings stop when the faster runner finishes. A meeting time later than his finishing time does not count.

Drill 17 more on the first meeting on a circular track

Concept 2 of 3: Back at the starting point together

A runner is at the starting point only after whole laps, that is at multiples of his lap time. Two runners are there together at a common multiple of both lap times, and the first such time is the LCM.

Definition

Lap time $= \frac{L}{s}$ for each runner.
Together at the start for the first time after the LCM of the lap times, whatever directions they run.
Lap times with decimals or fractions: scale them to whole numbers, take the LCM and scale back. The LCM of 2.5 hours and 3 hours is 15 hours.
Three or more runners meeting anywhere is a different question: the first time all of them meet is the LCM of the times between meetings of each pair, not of the lap times.
How many times at the start in a period: divide the period by the LCM. Say plainly whether the moment of starting counts.

Together at the starting point

t = \text{LCM}\left(\frac{L}{s_1}, \frac{L}{s_2}\right)

$\frac{L}{s}$ a runner's time for one lap

Worked example

On a 360 m track A runs at 9 m/s and B at 6 m/s, starting together from the start line. When are they first together at the start line again?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 17 Sep 2024, Shift 3 · Q55Easy

Example 2 · Time, Speed and Distance · Circular Tracks

Arvind and Chetan started running simultaneously from the same point in the same direction on a circular track of length 210 m. If Arvind takes 180 seconds to complete one round, and Chetan takes 420 seconds to complete one round, after how much time will they meet for the first time at the starting point on the track?

Taking the LCM of the speeds

The LCM is taken of the lap TIMES. For speeds of 4 m/s and 6 m/s on a 120 m track the lap times are 30 s and 20 s, so they are at the start together after 60 s.

Giving the first meeting when the start is asked

Runners can meet anywhere on the track long before both are back at the starting point. Read whether the question asks for any meeting or a meeting at the start.

Drill 4 more on back at the starting point together

Concept 3 of 3: Distinct meeting points on a circular track

Runners who keep going meet again and again, but only at a few fixed spots. Between two meetings each covers the same fraction of a lap as before, so the meeting points step round the track by equal amounts and come back to the start after a set number of steps.

Definition

Write the two speeds in their lowest ratio $a : b$ .
Same direction: they meet at $a - b$ distinct points. Opposite directions: at $a + b$ points.
The points are spaced evenly round the track, and the starting point is one of them.
How many meetings in a time: divide the time by the time between meetings and keep the whole number. Count a meeting that falls exactly at the end only if the question includes that moment.
Limit: the rule needs a start from the same point at the same moment and steady speeds.

Number of distinct meeting points

\text{same direction: } a - b \qquad \text{opposite directions: } a + b

$a : b$ the ratio of the speeds in lowest terms

Worked example

Two runners start together from one point of a circular track at 8 m/s and 6 m/s. At how many distinct points do they meet if they run (a) the same way, (b) opposite ways?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2023 · Tier I, 19 Jul 2023, Shift 3 · Q60Moderate

Example 3 · Time, Speed and Distance · Circular Tracks

Two persons started running on a circular track at a speed of 20 m/s and 30 m/s in opposite directions. If the circumference of the circular track is 100 m, find at how many distinct points they will cross each other?

Using the speeds before reducing the ratio

Speeds of 12 m/s and 18 m/s give 5 meeting points running opposite ways, from the lowest ratio 2 : 3. Adding 12 and 18 to get 30 points is wrong.

Rounding up a part of a meeting

10.5 meetings in a period means 10 meetings. The eleventh comes after the period is over.

Drill 3 more on distinct meeting points on a circular track

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 on a circular track
First meeting on a circular track
$t_{\text{same}} = \frac{L}{s_1 - s_2} \qquad t_{\text{opp}} = \frac{L}{s_1 + s_2}$
Back at the starting point together
Together at the starting point
$t = \text{LCM}\left(\frac{L}{s_1}, \frac{L}{s_2}\right)$
Distinct meeting points on a circular track
Number of distinct meeting points
$\text{same direction: } a - b \qquad \text{opposite directions: } a + b$

Watch out for (7)

Using the difference of speeds for opposite directions
→ The first meeting on a circular track
Waiting for a full lap when one starts ahead
→ The first meeting on a circular track
Counting a meeting after the race is over
→ The first meeting on a circular track
Taking the LCM of the speeds
→ Back at the starting point together
Giving the first meeting when the start is asked
→ Back at the starting point together
Using the speeds before reducing the ratio
→ Distinct meeting points on a circular track
Rounding up a part of a meeting
→ Distinct meeting points on a circular track

Test yourself on Time, Speed and Distance

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