Sharing the Cost of the Weekend Trip

Date: 07/24/2003 at 21:58:21
From: Jill
Subject: Algebra word problem

A group of people planned to rent a large beach house for a weekend 
trip. They were to share the $800 cost equally. However, two people 
were unable to go and this increased the cost for each person by $20.  
How many persons were in the original group?

I know the answer is 10, but how do you figure it out algebraically 
and not just guess and check?  I tried a system of two equations 
with two unknowns, but that wouldn't do it. 

I'm actually a math teacher (fairly new), but I think maybe I'm making 
it too complicated. Please help. Thanks.

x is number in original group
y is cost per person
x-2 is original group less the two that dropped out

800/x = y
800/(x-2) = y+20

Use elimination of y, but this is not a linear system, so it can't 
be solved that way.

Date: 07/24/2003 at 23:27:36
From: Doctor Ian
Subject: Re: Algebra word problem

Hi Jill, 

Don't knock guess-and-check. If you do it the right way, it's often
faster than using algebra. 

Also, in all seriousness, guess-and-check is often a good first step
toward figuring out what equations you need. But if it's algebra you
want, it's algebra you'll get.  

You don't necessarily need two variables. If all N people go, the cost 
per person is $800/N, right? And if two fewer people go, then the cost 
per person is $800/(N-2). And we know that the latter is $20 higher 
than the former, so

  $800/(N-2) = $800/N + 20

So now we have one equation with one variable. It's ugly, but 
solvable. 

Now, how did I know to do this? What the problem is really telling me 
is 

  the price per person
  with N-2 people       = $20 + the price per person 
                                with N people

This is a clue that I should look for a way to express the 'price per
person' in both cases. Does that make sense?

Or we could try to be clever by distorting the problem a little bit
- enough to let us see it in a different way, but not enough to change
the meaning of the problem. Suppose that instead of just skipping out
on their part of the bill, the two people who can't come distribute
their share among the remaining people. Their share of the total is 

  (2/N) * $800

and since they're handing $20 to everyone else, their share must be
(N-2) times $20.  So 

   their share = their share

   (2/N) * 800 = (N - 2) * 20

This is a little less messy, and if we multiply both sides by N, we'll
have a well-behaved (i.e., factorable) quadratic equation. 

Does this help? 

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