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# News from Mathnasium of Fort Lee

### Word Problem Wednesday: It's a race!

May 3, 2017

The internet is obsessed with cats. I think it's the way they move. It's pretty difficult to find a small cute warm fuzzy pet that moves like a cat. A ferret perhaps? Anyway, we're contributing to cat obsession with this puzzler.

A cat starts racing a mouse from 10 meters away. The cat runs at a rate of 14 meters per second, and the mouse scampers at a rate of 4 meters per second. How many seconds will it take for the cat to catch up to the mouse?

Thousands of years ago, a Greek philosopher named Zeno, puzzled about similar problems. Have a look!

Zeno hypothesized in a race between Achilles and a tortoise, that it was impossible for Achilles to reach the tortoise! Zeno split the time to cover the distance between Achilles and the tortoise into an infinite series of halve segments thusly: 1/1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + ... Each piece of time segment half being a half of the previous piece. Our students should appreciate this because we teach halving so much, starting as early as kindergarten. Zeno's reasoning was that an infinite amount of time segments summed up must be infinite; and hence he reasoned, no distance could ever be covered in a given time. So, why is it that in a race, it's always possible to catch up, and get ahead? By the way, that's like learning math at Mathnasium, we help students to catch up and get ahead, and fortunately it won't take infinite time!

So it means that it's possible for an sum of infinite terms to be finite! In fact, the infinite sum of halves of halves = 1/+ 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + ... = 1 as expected. So we've resolved Zeno's paradox; but Zeno contributed to the valuable exploration of infinity that still serves to amuse, befuddle, and intrigue us today! Watch on for a wonderful answer for that infinite sum of halves.

So, now that we know the faster cat is going to catch up to the slower mouse, when will that happen?

We define distance = speed × time  -- because speed is defined as the distance covered per time unit. We're happy to explain all these concepts at Mathnasium. Let "t" be the time that cat reaches mouse. In that time "t", they will both be the same distance from the starting line; hence cat's distance compared to mouse's distance is:

14t = 10 + 4t
Hence, t = 1 second.

And now you'll appreciate this brainy joke. An infinite many mathematicians walk into a bar. The first asks for a mug of beer. The second says, I'll have half of what he has; the third says, I'll have half of what the second has; and the fourth says "I'll ... " but the bar tender yells "HEY! I got this" and pours two mugs of beer.

Contact:

Ruby Yao and Benedict Zoe, Mathnasium of Fort Lee
201-969-6284 (WOW-MATH), fortlee@mathnasium.com
246 Main St. #A
Fort Lee, NJ 07024

Happily serving communities of Cliffside Park, Edgewater, Fort Lee, Leonia, Palisades Park, North Bergen, West New York, and Fairview.