The Problem: While A–a and Bea are competing for first place in the triathlon, Les and Pat are competing for last place. Coming into the 10,000 meter run that ends it, Les is 800 meters behind Pat. Les can run (on average) 250 meters per minute, but Pat can run (on average) only 225 meters per minute. Who wins, Pat or Les? At the finish line, how far ahead of the loser is the winner? How long does it take her to drag herself over the finish line?
Comments This problem is one that I adapted from elsewhere in the book, and the adaptation produced some ambiguities. A number of groups got hung up, so it seems worth discussing. First off, although I actually didnÕt intend to produce them, I would like to point out that if learning mathematics is to have any impact outside of the classroom, then it has to include dealing with situations where you have to figure out some things about the information Ð real life rarely hands us cut and dried information that is exactly and only what we need to have. More specifically, though, each of the two major issues highlights a point worth noting.
1) Coming into the 10,000 meter run that ends it, Les is 800 meters behind Pat. From a grammatical point of view, as one of you pointed out, this one is not in fact ambiguous. A phrase like Òcoming intoÉÓ has no choice but to modify the noun that immediately follows it, in this case, Les. On the other hand, if grammatical cues donÕt work for you, consider the fact that this problem obviously expects to have an answer. If you assume that the phrase refers to the time when the whole group began coming into the last part, then you know absolutely nothing about Les or Pat, who are trailing the whole durned field. If you assume that it refers to the moment when Pat crosses the line, then Les could be either madly splashing out of the swim or whirling her bike pedals, and you have no clue at what speed she is doing so. So the only workable reading is that Les is starting her run with Pat already 800 meters down the track.
If you are worried when you make an assumption like that, it is fine to start with ÒWe are interpreting this to mean ÉÓ
2) The question of whether the winner of a competition for last place is the first or second one over the line is very much muddled in the problem wording (blush!). On the other hand, you have only to make a definition and stick with it and itÕs all workable. Instead of saying ÒwinnerÓ you say Òfirst over the lineÓ (or alternatively Òlast over the lineÓ) and stick with it.
The solution Note that this solution requires absolutely no algebra. Algebra can be used, but the hardware is likely to obscure the meaning of the quantities.
Quantities:
1. Total length of running portion of race 10,000 meters
2. Total distance Les must run 10,000 meters
3. Distance Pat has run as of the beginning of the problem 800 meters
5. Number of meters Pat can run in a minute 225 meters
6. Number of meters Les can run in a minute 250 meters
Values directly computable from those:
4. Distance Pat must still run 10,000 Ð 800 = 9200
7. Fraction of a minute it takes Pat to run a meter 1/225 minute
8. Fraction of a minute it takes Les to run a meter 1/250 minute
9. Time it takes Pat to run her total remaining distance
(1/225 minutes/meter)(9200 meters) = 40 8/9 minutes
10. Time it takes Les to run her distance
(1/250 minutes/meter)(10000) meters) = 40 minutes
So Les is the first one over the line.
11. Difference in those times
40 8/9 minutes Ð 40 minutes = 8/9 of a minute
12. Distance second one in still must run when first crosses finish line
In 40 minutes, Pat has gone
(225 meters/minute)(40 minutes) = 9000 meters. She was already at the 800 meter mark, so when Les finishes, Pat has run 9800 meters and has 200 meters to go.
Alternatively, she is going to run
(225 meters/minute)(8/9min) = 200 meters.