One hour after a hiker left a camp, a cyclist set out to catch up. If the hiker traveled 3 miles per hour and the cyclist traveled 6 miles per hour faster, how long will it take the cyclist to catch up with the hiker
The Correct Answer and Explanation is:
Correct Answer:
The cyclist will catch up with the hiker in 0.5 hours (or 30 minutes) after starting.
Explanation
To solve this problem, we need to understand how the relative motion between the hiker and the cyclist works.
Let’s break it down step by step.
Step 1: Define Speeds
- The hiker’s speed = 3 miles per hour (mph)
- The cyclist’s speed = 6 mph faster than the hiker = 3 + 6 = 9 mph
Step 2: Understand the Head Start
Since the hiker left one hour earlier, we need to calculate how far the hiker has traveled before the cyclist starts.
- Distance hiker travels in 1 hour = speed × time = 3 mph × 1 hour = 3 miles
So, when the cyclist starts, the hiker is already 3 miles ahead.
Step 3: Set Up a Catch-Up Equation
Now we want to find how much time it will take the cyclist to catch up with the hiker after starting.
Let ttt be the time in hours the cyclist travels before catching up.
In that time:
- The cyclist will travel: 9t9t9t miles
- The hiker, continuing at 3 mph, will travel: 3t3t3t miles
Since the hiker already had a 3-mile head start, the cyclist’s distance must be equal to the hiker’s total distance:Cyclist’s distance=Hiker’s head start+Hiker’s additional distance\text{Cyclist’s distance} = \text{Hiker’s head start} + \text{Hiker’s additional distance}Cyclist’s distance=Hiker’s head start+Hiker’s additional distance9t=3+3t9t = 3 + 3t9t=3+3t
Step 4: Solve the Equation
9t=3+3t9t−3t=36t=3t=36=0.5 hours9t = 3 + 3t \\ 9t – 3t = 3 \\ 6t = 3 \\ t = \frac{3}{6} = 0.5 \text{ hours}9t=3+3t9t−3t=36t=3t=63=0.5 hours
Final Answer
It will take the cyclist 0.5 hours, or 30 minutes, to catch up with the hiker after starting
