Let the athlete take x minutes to cycle one mile.
Therefore he takes 3x/2 minutes to run one mile and
3x minutes to walk one mile.
Therefore 3x +3x/2 +x = 3x + 10 i.e. x = 4
The cyclist takes 12 minutes to walk the first mile, 6 minutes to run the second mile and 4 minutes to cycle the third mile, a total time of 22 minutes.
This problem is taken from the UKMT Mathematical Challenges.