Examkin

Question

Difficulty: HardRate, Time, and Distance Problems

An athlete completes a three-leg training course consisting of cycling, running, and swimming, covering a total distance of 6060 miles in a total time of 44 hours. The athlete cycles at a constant speed of 2424 miles per hour, runs at a constant speed of 1010 miles per hour, and swims at a constant speed of 22 miles per hour. If the time spent cycling is equal to the combined time spent running and swimming, how many miles did the athlete run?

Answer: 10 miles

Answer

The athlete ran 10 miles.
Because cycling time equals combined running and swimming time, the 4-hour total splits evenly into 2 hours for cycling and 2 hours for the remaining legs. Cycling at 24 mph covers 48 miles, leaving 12 miles and 2 hours for running and swimming. Writing the total time equation as (d_running / 10) + ((12 - d_running) / 2) = 2 and solving yields d_running = 10 miles.

Step-by-Step Solution

1
Determine the time spent cycling
2 hours
Since the total time is 4 hours and cycling time equals the sum of running and swimming times, 2 * t_cycling = 4.
2
Calculate the distance covered while cycling
48 miles
Distance = Speed * Time = 24 mph * 2 hours = 48 miles.
3
Determine the combined distance and time for running and swimming
Combined distance = 12 miles, Combined time = 2 hours
Total distance minus cycling distance is 60 - 48 = 12 miles; total time minus cycling time is 4 - 2 = 2 hours.
4
Set up and solve the system of equations for running distance
Running distance = 10 miles
Using time = distance / speed gives (d_running / 10) + ((12 - d_running) / 2) = 2. Multiplying by 10 yields d_running + 60 - 5 * d_running = 20, leading to d_running = 10 miles.

Key Concept

Multi-leg rate, time, and distance problems using systems of linear equations.
Rate this question

Topics

Rate, Time, and Distance Problems

More questions from this topic