A certain Math Teacher traveled from Winchester, Virginia to Harrisburg, Pennsylvania at a constant rate of 25 mph. How fast must she travel on the return trip so that she averages 50 mph over the entire trip?
(The distance from Harrisburg to Winchester is not relevant to the problem. You only need to know that the distance from Winchester to Harrisburg is the same as the distance from Harrisburg to Winchester.)
Solution to the Problem:
It is not possible to go fast enough so that the teacher will average 50 mph over the entire trip.
First, it is necessary to realize that "average speed" is defined to be total distance traveled divided by the total time it takes to get there.
To see that it is not possible to go fast enough to average 50 mph,
let the distance from Winchester to Harrisburg be
100 miles (any number will do, or you can use a variable for this).
Traveling at 25 mph, it would take 4 hours to get to Harrisburg.
In order to average 50 mph over the entire trip of 200 miles, you would
need to make the trip in 200 / 50 = 4 hours.
But that can't be done
unless you make the return trip in 0 seconds!!!
| Rate | Time | Distance | |
|---|---|---|---|
| Winchester to Harrisburg | 25 mph | 4 hours | 100 miles |
| Harrisburg to Winchester | --- mph | 0 hours | 100 miles |
| Entire Trip | 50 mph | 4 hours | 200 miles |
The most common wrong answer was 75 mph.
But if you returned at 75 mph,
you would average only 37.5 mph over the entire trip (NOT 50 mph)!
Remember that average speed equals the total distance traveled divided by the
total time. Look at the Rate-Time-Distance
table below to see why 75 mph is incorrect:
| Rate | Time | Distance | |
|---|---|---|---|
| Winchester to Harrisburg | 25 mph | 4 hours | 100 miles |
| Harrisburg to Winchester | 75 mph | 1 1/3 hours | 100 miles |
| Entire Trip | 37.5 mph | 5 1/3 hours | 200 miles |
Correctly solved by:
| 1. Walt Arrison | Philadelphia, Pennsylvania |
| 2. Rick Jones | Kennett Square, Pennsylvania |
| 3. Richard K. Johnson | La Jolla, California |
| 4. David and Judy Dixon | Bennettsville, South Carolina |
| 5. Kirstine Wynn | St. Olaf's College Northfield, Minnesota |
| 6. James Alarie | University of Michigan -- Flint
Flint, Michigan |
| 7. Izzy Kushner | Closter, New Jersey |
| 8. Tina Zahel | Winchester, Virginia |
| 9. Frank Maire | Norwalk, Ohio |
| 10. Tom Hanzl | Brno, Czech republic |
| 11. Jeff Gaither | Winchester, Virginia |
| 12. Geoff Keith | Santa Monica, California |
| 13. Kevin Thompson | Oregon |
| 14. Daniel Wilberger | Winchester, Virginia |
| 15. Peggah Sadeghzadeh | Winchester, Virginia |
| 16. Torbjorn Jansson | Garphyttan, Sweden |
| 17. Steven Strasburg | Memphis, Tennessee |
| 18. George Gaither | Winchester, Virginia |
| 19. Kathleen Altemose | Winchester, Virginia |
| 20. Matt Stewart | Columbus, Georgia |
| 21. Richard Mocarski | University of Michigan
Ann Arbor, Michigan |
| 22. Paul E. Haak | ---------- |