Henry Ernest Dudeney/Puzzles and Curious Problems/79 - A Cycle Race/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $79$

A Cycle Race

Two cyclists race on a circular track.

Brown can ride once round the track in six minutes,

and Robinson in four minutes.

In how many minutes will Robinson overtake Brown?

Solution

$12$ minutes.

Proof

After $4$ minutes, Robinson has completed one circuit, while Brown is $\dfrac 1 3$ the way round his first circuit.

After $8$ minutes, Robinson has completed two circuits, while Brown has completed $1 \tfrac 1 3$ circuits, but Robinson has not yet caught up with Brown.

After $12$ minutes, Robinson has completed $3$ circuits, while Brown has just about completed $2$ circuits, which is the point Robinson overtakes Brown.

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $79$. -- A Cycle Race
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $93$. A Bicycle Race