Question

an athlete began a 110 - mile bicycle trip to build up stamina for a triathlon competition. unfortunately, the bicycle chain broke, so the athlete finished the trip walking. the whole trip took 3 hours. if the athlete walks at a rate of 5 miles per hour and rides at 43 miles per hour, find the amount of time spent on the bicycle. what amount of time did the athlete spend on the bicycle? the athlete rode for \\(\square\\) hours. mph. miles.

Explanation:

Step1: Define variables

Let \( t \) be the time spent riding the bicycle (in hours). Then the time spent walking is \( 3 - t \) hours (since the total time is 3 hours).

Step2: Set up distance equation

The distance ridden on the bicycle is \( 43t \) miles (using \( \text{distance} = \text{rate} \times \text{time} \), rate is 43 mph). The distance walked is \( 5(3 - t) \) miles (rate is 5 mph, time is \( 3 - t \) hours). The total distance is 110 miles, so:
\[
43t + 5(3 - t) = 110
\]

Step3: Solve the equation

First, expand the left side:
\[
43t + 15 - 5t = 110
\]
Combine like terms:
\[
38t + 15 = 110
\]
Subtract 15 from both sides:
\[
38t = 110 - 15
\]
\[
38t = 95
\]
Divide both sides by 38:
\[
t = \frac{95}{38} = 2.5
\]

Answer:

The athlete rode for 2.5 hours.