Mathnasium Weekly Problem: The Olympics!

We're right in the thick of Olympics fever! As you root for your favorites, take a quick math minute to solve this week's installment of Word Problem Wednesday.

It takes an Olympic canoe paddler 1 minute and 20 seconds to paddle a 300 meter course in the direction of the current. If the current moves at a rate of 1.5 meters per second, then how fast can the paddler paddle in still water?

This question brings back a memory. In the early 1980s, both my sister and I were electrical engineering student. We were in an engineering honor society and quite active, being elected to officer positions. Each year, we’d traditionally organize an end-of-school event. One event was a society outing on the Delaware River. We brought food for a picnic and rented canoes. It was exciting.

The day arrived and we assembled at the canoe rental site; then were hauled several miles up river. We were paired two to a canoe. There were an odd number of students and an elderly faculty chaperone. That meant someone had to be paired with the teacher. Everyone quickly paired with their favored friend, myself with my sister. I think we felt a little sorry for that odd student with the teacher.

Off we launched. It was a beautiful day. We enjoyed the tranquil water and wide vista of the river. However, after the adrenalin wore off, the beautiful day became a warm day, then a hot day. The sun beat down relentessly. It was our first time paddling a canoe and it grew into a chore. The goal now became completing the journey back to the rental site and the picnic awaiting us! Geometrically, the shortest distance downstream is to take the inside corner of every turn in the river. The shortest distance should be the fastest approach. And that is what we did.

After struggling, and cursing (mildly), we managed some semblance of coordination and steered as straight a course as we could. It was hard work, constantly paddling to keep the boat moving efficiently without too much wiggling. Our group fragmented into small clumps. Every now and then, we’d catch our breath and watch the other clumps meander about, but everyone seemingly attempting the same strategy of getting downstream as fast as possible. Everyone except a lonely canoe in the far outer banks with that odd student and elderly teacher. They were moving slowly. We smirked when we heard the teacher, lolling comfortably umbrella aloft, instructing that student how to steer.

We eventually arrived, exhausted at the rental dock. One by one, we unloaded with sunburns and blistered hands. And very shortly, that lonely canoe arrived. We watched the student and old teacher unload, very calmly, and very relaxed, and not a single blister! We were incredulous. Acting nonchalant, we asked that very lucky student “How did you paddle so well?”

Let’s answer the Olympic question first. “It takes an Olympic canoe paddler 1 minute and 20 seconds to paddle a 300 meter course in the direction of the current. If the current moves at a rate of 1.5 meters per second, then how fast can the paddler paddle in still water?”

Speed is the unit rate of distance covered, that is, how much distance traveled for each time unit. We’re given the rate of the current in meters (distance) per second (time unit). So the time unit is seconds. 1 minute and 20 seconds = 80 seconds. 300 meters per 80 seconds = 300 / 80 meters per second = 3¾ meters per second. However, the paddler is moving in the direction of the current of 1½ meters per second which means the current is adding to the total speed of the padder. Hence, without the current, in still water, the paddler is actually moving at 3¾ - 1½ = 2¼ meters per second. The paddler is moving not too much faster compared to the current at 1½ meters; 2¼ / 1½ = 9 / 6 = 1.5 times faster (notice how we eliminated pesky fractions by multiplying with 4 -- a technique we teach).

That also means that without paddling effort, the canoe will be moved by the current the entire 300 meters in 300 / 1½ seconds = 600 / 3 = 200 seconds = 3 minutes 20 seconds.

And… thus the answer to our second question. Since the water in a river moves downstream enmass somewhat in unison, the outer banks is the farthest the water must travel and so water travels faster at the outer banks than the inner banks. So the very experienced teacher and student only had to guide the canoe in the faster current and let the water do the rest.

Many, many lessons never forgotten.

Contact:
Ruby Yao and Benedict Zoe
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