A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
- A)
- B)
- C)
- D)
- E)
Answer
D
Key insight
Average speed is total distance over total time; with each leg 60 km the times are 20, 3 and 6 hours, giving 180/29, about 6.2.
Solution
The answer does not depend on the length of a leg, so choose a length divisible by , and : let each segment be km.
Time spent on each leg:
Total distance km, total time h, so the average speed is
The closest choice is .
Why this works
Average speed is a time-weighted average of the speeds, never the plain mean. Equal distances at different speeds means the slow leg dominates the time. Picking a common distance equal to the lcm of the speeds turns the problem into integer arithmetic; algebraically the answer is the harmonic mean .
The trap
Averaging the three speeds to get 11, ignoring that the slow swim takes far more time than the other legs.
Common mistakes
- Averaging the three speeds to get 11, ignoring that the slow swim takes far more time than the other legs.
- Computing total time correctly but dividing by (one leg) instead of the full km, which gives about .
Techniques
Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer