Purple Milk
(page 76)

Here is a solution from Maribeth Dann's class (March 2005). This is an eighth-grade group from Hermann Middle School in Hermann, MO.

Thank you for giving us the opportunity to solve your United We Solve Math Problems.

They were very difficult but we managed to solve them. We solved the Purple Milk problem. The solution we came up with was:

         Blue = 350ml
         Red = 450ml
         White = 550ml
         Yellow = 400ml

We came up with this by using the following mathematical problem solving methods. We started by writing all the combinations and the amount of milk they held. Then we just plugged the numbers into different colors and saw if they added up to what the clues stated. After we found the amount of blue and yellow we were able to solve the other colors and their amounts fairly easily.
 
Sincerely,
The One Eyed, One Horned, Flying Purple People Eaters

Great, I told them, but there was something missing: One clue even tells you that there are many solutions. I asked if they could find another solution? How many solutions are there?

They wrote back:

We are the 8th Graders from Hermann Middle School and we have accepted your challenge for the Purple Milk Math Problem. We think we have found all of the solutions and hope that you are pleased with our work. It was difficult until we found a pattern. Our Solutions to your problem, Purple Milk, are:

                        Blue     Red     Yellow     White               
---------------------------------------------------------------
Milliliters             350      450      400        550
                        351      449      399        551
                        352      448      398        552
                        353      447      397        553 
                        354      446      396        554
                        355      445      395        555
                        356      444      394        556
                        357      443      393        557
                        358      442      392        558
                        359      441      391        559
                        360      440      390        560
                       -----------------------------------
through numbers         400      400      350        600

After we found these numbers, we noticed a pattern. For Blue and White, you add 1 each time to white, you subtract 1 from yellow, you subtract 1 from red, and add 1 to blue. each time you just keep adding and subtracting from each specific color until you you exceeded the max for each color where the numbers dont add up to the clues. we also knew that we could'nt exceed 350 for yellow. When we got to 350 for yellow we knew to stop.

Very interesting! What do you think? First, are they right? Are their solutions really solutions? Did they get them all?

Second, it's not obvious how you should explain a problem with multiple solutions -- especially a LOT of solutions. How did they do? Were they clear in how they explained it?

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