December Solution

Solution to December’s Challenge

The Challenge: Dr. Myles McNally, chair of the Department of Mathematics and Computer Science at Alma College, is an avid bicyclist. During the past summer he decided to bike to one of his favorite destinations in Michigan. He got up early one morning and was able to make it to his destination at an average speed of 20 m.p.h. Later that day he returned home riding into a stiff headwind. As a result, he averaged only 10 m.p.h. on the return trip. What was his average speed for the round trip? It’s not 15 m.p.h.

Solution 1 by David Zhang, Ann Arbor Huron High School

Using rate = distance / time, and give r₁ = 20 mph, r₂ = 10 mph, and d = distance to destination, we can obtain and . Therefore, the average rate is .

Solution 2 by David Kye, Troy Athens High School

Let x be the number of hours to get to the destination. Then the distance to the destination will be
20x miles. On the way back, Dr. McNally rides his bike at the rate of 10 mph. However, we do
not know the time to come back. But, we do know the rate and the distance, so we can find the time
by using the distance formula d = rt. The time to come back is . Now that we know the round trip distance and the total time, the average speed is

Also solved by: Jacob Arnett (C.M.U. Accelerated Math Program.); Matt Lang (Coldwater H.S.); Peter Denten, Nicolas Timkovich (East Grand Rapids H.S.); Rebecca Baker, Dan Comperchio, Katelyn Evans, Becky Fauble, Alicia Hamilton, Shane Hanson, Dana Harrison, Danielle Hockstra, Kelsey Johnson, Steve Kilpatrick, B.J. Klotz, Kristen Klotz, Ashley Kohlbeck, Sarah Lewis, Wen Fei Liu, Rachel Loh, Elise Miller, Adam Rosenblum, Josh Samp, Chase Schuler, Becky Striker, Courtney Sundstrom, Katelyn Van Slyke, Jack Ventimiglia, Emily West (Grand Rapids Forest Hills Northern H.S.); Andrew Kraus, Stephen Mancewicz, David Pangle (Grand Rapids Catholic Central); Eric Schmidt (Grosse Ile H.S.); Matt Monahan, Joe Pawlowski (Grosse Pointe South H.S.); Joe King, Alexander Nezich (Marquette H.S.); Amy Pavlov, Lindsay Walker (Marysville H.S.); Saranapoom Klomjit (Grand Rapids Ottawa Hills H.S.); Myron Chang, Ray Eisbrenner, Vikram Raghunathan, Lisa Tian, Shelly Vermani, Teresa Wang, Don Zhang (Plymouth-Canton Educational Park); Rachelle Grandia (Rockford H.S.); Bethany Alger, Jeff Beltnick, Kimberly Curtiss, Alex Dawe, Noel Dominick, Justin Dunevant, Beke Hollenbach, Bret Krabill, Michelle Leisenring, Laura Merritt, Mike Nelson, Travis Rhynard, Laura Schmidt (Shepherd H.S.); Siyuan Liu (Troy H.S.); Kelly Keeler (Vestaburg H.S.); Elliot Weingarten, Ranjit Sood (Andover H.S.).