Who Picked the Most?

Date: 09/12/98 at 00:27:23
From: Will
Subject: Summer Vacation

During summer vacation, Anita, Brent, Carlo, and Donna worked together 
picking peaches. At the end of the season, they compared the number of 
peaches that they each had picked. They made note of the following 
facts:

  1. Brent picked more than Anita and Carlos picked together.

  2. The total of what Anita and Brent picked just equaled the number 
     Carlos and Donna picked together.

  3. The total picked by Donna and Carlos was more than the total 
     picked by Brent and Carlos.

Arrange the names of the people in the order of the number of peaches 
that each picked, starting with the person who picked the most.

The answer I have is that Donna picked the most because of fact number 
3. Brent came in second because of fact number 1. But the part that 
I have trouble with is finding who came in 3rd and who picked the 
fewest peaches, Anita or Carlos.

Date: 09/12/98 at 10:34:04
From: Doctor Kate
Subject: Re: Summer Vacation

Will:

I would find this problem very difficult to solve in my head. Instead, 
I'm going to try a very useful translation technique. Let's call the 
amounts Anita, Brent, Carlos and Donna picked "A", "B", "C", and "D" 
for short. Now let's translate the clues.

1. Brent picked more than Anita and Carlos picked together.

   That is, the amount Brent picked, B, is more than what Anita and 
   Carlos picked together, A + C. (Do you see why I've added A and C?  
   If A is what Anita picked, and C is what Carlos picked, A + C is 
   what they've picked together.) So this clue really means:

      B > A + C

   Now that you see what I'm doing, let's translate the others. You try 
   it first, and then see if you get the same thing I get.

2. The total of what Anita and Brent picked just equaled the amount 
   Carlos and Donna picked together: 

      A + B = C + D

3. The total picked by Donna and Carlos was more than the total picked 
   by Brent and Carlos:  

      D + C > B + C

Okay, so what does that do for us? Well, it gives us three pieces of 
information in the form of equations and inequalities. We can combine 
these bits of information mathematically. Here's where all the algebra 
comes in.

First, notice that if D + C > B + C , we can remove C from both sides 
and find that D > B. In plain English, if Donna and Carlos picked more 
together than Brent and Carlos did, Donna must have picked more than 
Brent.

So Donna's looking good as a winner, but we don't know that quite yet.  
We don't know anything about Anita and we aren't sure where Carlos is.  
Let's do some more algebra.

We also know B > A + C. This means that B > A and B > C because we 
know that A and C are both zero or positive numbers. This one makes 
most sense in english. If Brent picked more than Anita and Carlos 
combined, surely he picked more than Anita and more than Carlos 
separately.

So we have D > B, B > A and B > C. So the only thing left to know is 
whether A > C, A = C or A < C. That's where you got stuck. This part 
is a little more tricky, and we have to rely more on the algebra 
instead of the mathematics.

A good idea to find out where to start is to ask "What information 
have I not used?" and the answer is that we haven't used the fact that 
A + B = C + D. So let's try to use this information.

Well, looking at it by itself, it doesn't tell us whether A or C is 
bigger, but we are forgetting what we already know. That is, that 
D > B. Now, in algebra, we say that:

   A + B = C + D   and   C + D > C + B

So we know A + B > C + B. We can use the same trick as before and get 
rid of the B, finding that A > C. And we're done.

What does it mean?  Well, this is a little trickier in English, but I 
could say it like this: We know that Anita and Brent picked the same 
as Carlos and Donna, but Donna picked more than Brent. So Anita must 
have picked more than Carlos in order to make the pairs equal.

It's hard to imagine in English, and it's worth the effort of trying, 
but later in mathematics you'll encounter all sorts of problems that 
no one can do in English, but the Mathematical methods will be able to 
solve. So being able to do this problem both ways is important.

So please let me know if you have any questions.

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