100 Birds for 100 Dollars

Date: 03/19/98 at 18:23:14
From: Lynn Cocchiola
Subject: Word problem

Dr. Math,

A bird collector wants to buy 100 birds and to spend exactly $100. 
Blue birds cost 10 dollars, green birds cost 3 dollars, and yellow 
birds cost 50 cents. The collector wants to purchase at least one bird 
of each color. How many blue, green, and yellow birds can he buy?

Date: 03/21/98 at 23:14:28
From: Doctor Mike
Subject: Re: Word problem

Dear Lynn, 

Here are a few things to help get you started. I don't want to spoil
the fun by doing it all.  

1. He must buy an even number of yellow birds. Why? Because if he
   buys an odd number, like 21 for $10.50, then the total cost cannot
   be "exactly" 100 dollars - so he will have to buy the yellow
   birds in pairs.  

2. You may be a little young yet to have encountered much algebra.
   That's fine, because this problem can be solved without algebra.
   But one really good idea from algebra is to give letter names to
   things you don't know yet. Let's use the letter P to stand for
   how many Pairs of yellow birds he buys. Also, use G for the number
   of Green birds, and B for the number of the expensive Blue birds.
   (There are other ways to do this; we could use Y yellow birds.)

3. We can use these letters to help write down the given facts in the
   form of equations. Remember that P is the number of Pairs of the
   cheap yellow birds, so 2*P will be the actual number of Yellows.
   The problem says that the total number of birds is 100, so: 
   
             B + G + 2*P = 100

4. We can make another equation out of the given fact that the total
   cost is $100. Blue birds are $10 apiece so the total cost of all
   the blue birds is 10*B. Green birds are $3 apiece so the total
   cost of all the green birds is 3*G. Yellow birds are 1/2 dollar,
   so each pair costs one dollar. Adding up all the costs gives : 
   
             10*B + 3*G + P = 100

5. The problem tells you that AT LEAST one of each color must be 
   bought. We already know more than that, because since the yellow
   ones must be bought in pairs, at least two of them must be bought. 
   But how many of the blue ones? I have figured out that six or
   more blue birds is impossible. Here's how I know that. If you get
   six or more blue birds, that will cost at least $60. We know that
   we have to buy at least one green bird, so the cost goes up to $63.
   Even if the remaining cost of $37 is all from the yellow birds, 
   that will only be 37*2 + 6 + 1 = 81 birds, which is not enough. 

6. Now you know a really important fact that you did not know at the
   beginning. The number B of blue birds is either 1, or 2, or 3, 
   or 4, or 5. There are only these five possibilities, so that 
   means a lot fewer different ways to check. Remember that the total
   number of birds has to be "exactly" 100, and the total cost has to
   be "exactly" 100 dollars. It cannot cost ANY less, or ANY more.
   
Good luck and have fun.  Really work hard on this and I think you will
figure it out, but if you get totally stuck you can write back. 
  
By the way, there is one and only one correct answer.    

-Doctor Mike,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/