Finding an Equation for a Fish Word Problem

Date: 12/8/95 at 14:16:55
From: Anonymous
Subject: MATH QUESTION

A boy has the hobby of breeding goldfish.
He decides to sell all his fish.
He does this in five steps.
1.  He sells one half of his fish plus a fish.
2.  He sells a third of what remains, plus one third of a fish.
3.  He sells a fourth of what remains, plus one fourth of a fish.
4.  He sells a fifth of what remains, plus one fifth of a fish.
He now has 19 goldfish left.
Of course, no fish is divided or injured in any way.
How many did he start with?

Date: 3/22/96 at 0:39:19
From: Doctor Ken
Subject: Re: MATH QUESTION

Hello!

I think the best way to solve a problem like this is to find an 
equation that fits the problem, then see if you can solve the 
equation.  So let's first get ourselves a variable: let f 
represent the number of fish that  the boy starts out with.

Then here's how to get an equation that tells the story of what 
happened:  First, he sold 1/2 of his fish, plus 1. So what he had 
left was f - (f/2 + 1). But there's a better way to write that.  
Instead of thinking about the half that he sold, let's think about 
the half that he kept. Then we can use the expression f/2 + 1.

The advantage of this method comes in the next steps: if he sold 
1/3 of his fish plus a fish, then he kept 2/3 of them, so he's 
left with (f/2 + 2)*2/3 + 1/3

Then if you keep going like this to the end of the problem, you'll 
end up with an expression for the number of fish that he has left, 
and you can set that equal to 19, and BINGO!  You've got your 
equation.

-Doctor Ken,  The Math Forum