Two Steps Forward, One Step Back

Date: 10/14/2003 at 00:06:52
From: Manda
Subject: A fish

There once was a salmon who tried to swim upstream to the place where
she was born.  She had to swim 1,010 feet upstream to reach the shady
pool of water where she once hatched from her egg.  Every five minutes
she swam 30 feet upstream, but then in the next two minutes, while she
rested, then rushing current would push her 10 feet back downstream. 
She continued this way, swimming for five minutes then resting for two
minutes, until she reached her destination.  How long did it take her
to swim the entire 1010 feet?

I made a chart... but is there a faster way to solve this?

Date: 10/14/2003 at 09:47:06
From: Doctor Ian
Subject: Re: A fish

Hi Manda,

There _is_ a faster way, but you need to be careful with it.  Let's
look at a smaller version of the same problem.

Suppose a snail tries to climb out of a well 12 feet deep.  He climbs
2 feet each day, and slides back 1 foot each night.  When will he
finally get out? 

A quick way to do this is to say:  Each day he makes progress of 2
feet minus 1 foot, or 1 foot.  So it should take him 12 days to get out. 

This is close!  But if you actually work it out, you'll find that
after 10 days he only has 2 feet left to go... so he makes it out on
the 11th day.  

So the quick way misses the fact that the snail might make it all the
way out _before_ he slides back.  Your problem is basically the same
as this one, but with different numbers. 

Is this enough to get started? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/14/2003 at 09:47:27
From: Doctor Edwin
Subject: Re: A fish

Hi, Manda. 

These "two steps forward, one step back" problems are pretty easy to
solve, but they have one tricky part.

It's tempting to say, "Okay, the fish swims forward 30 feet in 5
minutes and then back 10 feet in 2 minutes, so she swims 20 feet every
7 minutes.  So all I have to do is divide 1,010 by 20 and multiply by
7 to get my answer."

But that won't work out.  Why?  Because once she reaches her goal (at 
the end of swimming forward) she stops.  She doesn't drift backward 
any more.  So if we call each swim-forward-drift-back cycle a "step", 
each one is 7 minutes long and moves her 20 feet, except for her last 
one, which is 5 minutes long and moves her 30 feet.

Given that, can you solve the problem with an equation, rather than a 
chart?  Write back if you need more help.

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/15/2003 at 12:11:17
From: Manda
Subject: Thank you (A fish)

Thank you very much for helping me with the fish problem... I think
after your second explanation I figured it out!