How Many People?

Date: 06/11/2001 at 22:26:47
From: Sheri
Subject: How many people?

Clancy, Nancy, and Mary Ellen are at the supermarket together.  Each 
of the girls has part of the list of things they need for the cookout. 
As they go the checkout stands, they decide to each get in a different 
line to save time. 

As Clancy gets in line, she notices that there are three more people 
in front of her than are in front of Nancy, and there are two times as 
many people in front of Mary Ellen as there are in front of Nancy.  
The total number of people in front of Clancy, Nancy, and Mary Ellen 
is 11. How many people are in front of each of the girls?

I am new at this problem solving, and my son gave me this problem but 
I need help so I can show him. Please give me ideas for easy ways to 
problem solve. I just don't get it.

Thank you for being here.

Date: 06/11/2001 at 23:21:09
From: Doctor Peterson
Subject: Re: How many people?

Hi, Sheri.

There are all sorts of ways to organize your thinking for this kind of 
question, depending on how your mind works. In this case, it would 
also help if I knew your son's age, so I'd know whether to use algebra 
or pictures. I'll try a hybrid that should be understandable at 
different levels, since I presume he's fairly young.

Let's say Clancy has C people in front of her, Nancy N people, and 
Mary Ellen M people. I'm just using letters to identify the different 
numbers and save writing.

We're told that

    C is 3 more than N
    M is 2 times as many as N
    C + N + M is 11

In algebra, I would write these as equations and solve them. But I 
don't need such a powerful tool, because I see that C and M are given 
in terms of N. If I can just find N, I can get everything. So let's 
just count all the people:

      C   +   M   + N

is   ___     ___
    /   \   /   \
    N + 3 + N + N + N

This is 4N + 3.

If you prefer, we can make this a picture:

        +---+
    C = |   | * * *  where the box is as many people as there are 
        +---+        in Nancy's line, and each star is another

        +---+---+
    M = |   |   |
        +---+---+

so altogether we have

     __C__   __M__   N
    /     \ /     \ / \
       +---+---+---+---+
    ***|   |   |   |   |
       +---+---+---+---+

However you do it, this has to be equal to 11. So if we take off the 
three extras, we'll have 11-3 = 8:

    +---+---+---+---+
    |   |   |   |   | = 8
    +---+---+---+---+

Okay: four of what number gives 8? The problem suddenly becomes easy.

Algebraically, we subtract 3 from each side of

    4N + 3 = 11

to get

    4N = 8

and then divide by 4:

    N = 2.

I'll let you finish the work, by finding all three numbers and 
checking.

What have we learned? First, that knowing algebra can get in my way, 
because I want to use that when a child doesn't have the background to 
use it; yet I can also use insights from algebra and translate them 
into a child's level. And if you don't remember any algebra, that's 
fine: my non-algebraic method is actually easier, if you're willing to 
think. (Algebra is just a way to do math without thinking - much.)

But mostly: anything you can do to visualize the problem and see the 
whole thing at once will help. I first had to see that knowing "N" was 
the key, so that I could then make my picture using boxes for the 
number of people in Nancy's line. Once I had that clue, a picture made 
it easy to manipulate numbers and unknowns as if I were telling people 
to switch lines. (Well, actually easier!)

I hope this gives you some good ideas. I sometimes hesitate to go too 
deep into this sort of problem solving, because each problem is 
different, and one technique won't solve them all. But these steps are 
crucial:

1. Do something to see what the problem is all about - read through 
   it, visualize it, act it out, whatever it takes.

2. Pay special attention to what you are asked to find. Sometimes I go 
   all through a problem and then realize that I was finding the wrong 
   thing.

3. Translate the problem into something symbolic, whether it's 
   variables, pictures, or objects you can move around, and then PLAY 
   with it. You'd be surprised how often mathematicians use that word 
   "play"! You need to get a feel for a problem in order to solve it, 
   and you may have to try a lot of things that don't work before you 
   find one that does.

4. Once you've found a way to solve it, go back and make sure you've 
   solved the right problem and that it makes sense.

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