Question

The AIME Triathlon consists of a half-mile swim, a $30$ -mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

答案

Solution:

Let Tom's swimming speed be $s$ miles per hour. Then he runs at $5s$ and bicycles at $10s.$ The total time in hours is $\frac{0.5}{s} + \frac{30}{10s} + \frac{8}{5s} = \frac{0.5 + 3 + 1.6}{s} = \frac{5.1}{s} = 4.25,$ so $s = \frac{5.1}{4.25} = 1.2$ miles per hour.

He bicycles at $12$ miles per hour, so the ride takes $\frac{30}{12} = 2.5$ hours, which is $150$ minutes.

2013 AIME I Problem 1

Solution:

Problem 1 in Other Years