Finding the Number of Fishbowls Sold

Date: 3/15/96 at 12:34:32
From: Ellen Borger
Subject: fishbowls

Dear Dr. Math,

My son was given the following problem as homework. He is an 8th 
grade algebra student. Unfortunately, neither my husband or I were 
able to help him.  Could you give us some help figuring this one 
out?

Thanks for your assistance.

Ellen Borger

Problem:
A young entrepreneur named Trivia took some fishbowls that she had 
bought cheaply to the flea market. In the first hour she sold 
one-third of them and a third more. In the second hour, she sold 
half of them, and a half more.  In her third hour there, she sold 
one third of them and a third of one more.  The next hour, she 
sold half of them and half of one more.  Finally, she sold the 
last two and went home to Capitola.  

How many fishbowls did Trivia sell?

Date: 4/3/96 at 11:49:27
From: Doctor Ken
Subject: Re: fishbowls

Hello!

I'm sorry we've taken a little while to get back to you.  It took 
some time before the doctors could agree on what was meant by the 
wording of the problem (specifically, the parts like "one third of 
them and a third more").  But I think we've got it now. Here are a 
couple of hints about how to do this problem.  I'll try to lead 
you through the first steps, and then see if you can finish it on 
your own.  

Let's say that Trivia starts out with a certain number of bowls, 
and call that number x.  How many does she have after the first 
hour?  Well, she sold a third of them plus a third of a bowl (this 
is what you meant, right?).  So the way we can write that in 
algebraic notation is:

 Number of bowls left after 1 hour  =  
   x - x/3 - 1/3  =  2x/3 - 1/3

Does that make sense?  Essentially, what you have to do in the 
rest of the problem is just keep doing that kind of step over and 
over.  So let's figure out how many bowls she has left after the 
second hour:

 Number of bowls left after 2 hours =  (2x/3 - 1/3) - 
   (2x/3 - 1/3)/2 - 1/2
                                    =  2x/6 - 1/6 - 1/2
                                    =  2x/6 - 4/6
                                    =   x/3 - 2/3

Did that make sense to you?  To get that equation on top, I just 
took the amount we had left (2x/3 - 1/3), subtracted half of it, 
and then subtracted 1/2.  Then I just simplified to get x/3 - 2/3.

So you'd keep doing this kind of thing, taking the result from the 
previous hour each time and figuring out how many you'd have the 
next hour.  At the end of the last hour, you'll have 0 bowls, so 
you have an equation: set your last expression in x equal to 0, 
and then solve for x.

Good luck!

-Doctor Ken,  The Math Forum

Date: 4/3/96 at 12:13:6
From: Ellen Borger
Subject: Re: fishbowls

Thanks for your response. I was able to get some assistance from 
one of our economics faculty here at UC Santa Cruz.  We finally 
came up with the answer of 162 bowls.

Thanks again for your help.

Ellen Borger