Technique

Counting overtakes on a circular track

Arithmetic • Time, Speed and Distance • Races

Statement

On a closed course of one lap length

L

, the faster runner passes the slower exactly once each time the lead grows by

L

, so over a race in which the total lead gained is

D

the number of passes is

⌊ \frac{D}{L} ⌋

How to use it

1Find the lead at the finish first: in equal time the distances are in the ratio of the speeds, so the lead is the difference of those distances.
2Divide the lead by the lap length and take the whole part — a partial lap of lead is not a pass.
3Equivalently, a pass occurs every $\frac{L s _{1}}{s _{1} - s _{2}}$ of distance run by the leader, which locates each pass along the race.

Two runners whose speeds are in the ratio

3 : 2

run a

6

km race on a

0.5

km circular track. How many times does the faster pass the slower?

Step-by-step

1
When the faster covers $6$ km the slower covers $\frac{2}{3} \times 6 = 4$ km
Why: In the same time the distances are in the ratio of the speeds.
2
$lead = 6 - 4 = 2 km = \frac{2}{0.5} = 4 laps$
Why: Convert the lead into laps of the circuit.
3
$⌊ 4 ⌋ = 4$