Buying and Selling Horses

Date: 10/22/95 at 20:18:52
From: Anonymous
Subject: Word Problems/Problem Solving

This is a problem that you might know.  My students and I have come up 
with the answer but we want to know why a certain way of solving the 
problem does not work.

It is a horse problem.
A man buys a horse for $60.
Sells the horse for $70.
Rebuys the horse for $80.
Sells the horse for $90.

The problem to solve is then: Did the man break even, lose money or gain 
money?  And if he lost or gained money, how much?

The answer is that he gained $20.

But when we approach it in a different way it seems like he should only 
gain $10.

       He spent $60.
       He gains $70.
       He thus has gotten $10 extra dollars.  

But then in order to buy the horse at $80 he must take $10 from someone 
else so he no longer is $10 up - he is even.

       So then he sells the horse at $90 and is thus $10 up.

Again when we do this problem with play money we see that it is $20 that 
we gain. But when we sit back and try and figure out why this other 
answer is incorrect we run into problems.  We can't seems to explain why 
this solution does not work without just saying it doesn't work because 
the other one does. Where are we not seeing our mistake?

Date: 11/10/95 at 11:5:59
From: Doctor Ken
Subject: Re: Word Problems/Problem Solving

Hello!

Here's where the mistake lies. Let's assume that Bill (the guy buying 
and selling the horses) has $60 at the beginning of this little horse-
trading. So he spends it on the horse. Then he gets $70 for it. If he 
wants to buy it again for $80, you're right that he has to borrow $10 
from someone else, but we have to keep track of this debt.  See, he's 
not _even_ now, he actually has -10 dollars.  

So then he gets $90 back for the horse. But since he had -$10, the net 
effect of getting that $90 is to make him have _80_ dollars (you might
think about that as paying back the $10 he owed). So since he started 
out with $60, and he has $80 at the end, the overall effect was to gain 
$20.

Make sense?

-Doctor Ken,  The Geometry Forum