2013 AIME I Problem 1

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?

Answer

Answer: 150

Concepts:distance rate and time ratio and proportion unit conversion

Difficulty rating: 1810

Small hint:

Tom runs

5

times as fast as he swims and bicycles

10

times as fast, so write each leg’s time using only his swimming speed

Big hint:

The total time is

\frac{0.5}{s} + \frac{30}{10s} + \frac{8}{5s} = 4.25

hours; solve for the swimming speed

s

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

\begin{aligned} &\frac{0.5}{s} + \frac{30}{10s} + \frac{8}{5s} \\ &= \frac{0.5 + 3 + 1.6}{s} \\ &= \frac{5.1}{s} = 4.25, \end{aligned}

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.