A long-distance runner started on a course at an average speed of 7 mph. Half an hour later, a second runner began the same course at an average speed of 9 mph. How long after the second runner starts will the second runner overtake the first runner
The Correct Answer and Explanation is:
Correct Answer:
1.75 hours (or 1 hour and 45 minutes) after the second runner starts.
Explanation
To solve this problem, we need to determine how long it will take the second runner to catch up with the first runner. Since both are running the same course, we can use the concept of relative speed and distance.
Let’s break it down step by step.
Step 1: Find the Head Start
The first runner starts half an hour before the second runner and runs at 7 miles per hour. In half an hour, the first runner covers: Distance=Speed×Time=7 mph×0.5 hours=3.5 miles\text{Distance} = \text{Speed} \times \text{Time} = 7 \, \text{mph} \times 0.5 \, \text{hours} = 3.5 \, \text{miles}Distance=Speed×Time=7mph×0.5hours=3.5miles
So, the first runner has a 3.5-mile head start when the second runner begins.
Step 2: Determine Relative Speed
The second runner is faster and runs at 9 miles per hour. The first runner runs at 7 miles per hour. The relative speed of the second runner compared to the first is: 9 mph−7 mph=2 mph9 \, \text{mph} – 7 \, \text{mph} = 2 \, \text{mph}9mph−7mph=2mph
This means the second runner reduces the gap by 2 miles every hour.
Step 3: Calculate Time to Close the Gap
To find out how long it takes the second runner to cover the 3.5-mile head start with a relative speed of 2 mph, we use: Time=DistanceSpeed=3.5 miles2 mph=1.75 hours\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{3.5 \, \text{miles}}{2 \, \text{mph}} = 1.75 \, \text{hours}Time=SpeedDistance=2mph3.5miles=1.75hours
Final Answer
The second runner will overtake the first runner 1.75 hours after starting. Converting 0.75 hours into minutes (0.75 × 60), we get 45 minutes, so the second runner overtakes the first 1 hour and 45 minutes after starting.
