Which Grasshopper Wins the Race?Date: 06/11/2003 at 11:43:56 From: Kathy Subject: Winning distance Two grasshoppers have a race. The course is 12 feet long one way, and they race down and back for a total of 24 feet. They both start at the signal. The larger grasshopper covers 10 inches per jump, while the smaller one covers 5 inches. However, the smaller grasshopper takes 2 jumps for every jump of the larger grasshopper. One of them manages to win. Which one and why? Date: 06/11/2003 at 13:56:02 From: Doctor Ian Subject: Re: Winning distance Hi Kathy, One thing that makes this confusing is that the jumps are measured in inches, while the course distance is measured in feet. Let's measure everything in inches: | A | ----------------------------------------| Turn around here B 144 inches | | Now, what's interesting here is that the course length isn't a multiple of 5 or 10, so it won't be an even number of hops from one end to the middle. Let's say they've hopped to the 140-inch line. Since they each jump 10 inches in the same amount of time, they get there together. | A | ...|--|--|--|--| B | | But there are only four inches to the turnaround point. The smaller grasshopper will jump once, turn around, and jump again: | <-------------- second jump A--------------> first jump ...|--|--|--|--|..| B | | In the same amount of time, the larger grasshopper will jump once: <-------------- A--------------> ...|--|--|--|--|..|..|..|..|..|..| B-----------------------------> | So now the smaller one is in the lead. So the question is, will he keep the lead all the way back to the finish line? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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