Mathilda and Matt Share a Pie....

Date: 01/20/2010 at 18:54:58
From: Jitka
Subject: a fraction word problem eating pie, how much eaten

Say Mathilda and Matt are going to eat a pie.  Mathilda starts out 
with 3/7 of the pie, and Matt gets 4/7.

But then Matt eats 4/7 of Mathilda's portion, and Mathilda eats 3/7 
of Matt's portion.  Finally, each of them eats what is left on their 
own plate.

How much pie does Matt eat in total?

I am thinking that, after the first split, Mathilda's 3/7 becomes 1 
whole, and now somehow I have to figure out what 4/7 of that is; and 
similarly for Matt's pie portion -- 4/7 now becomes one whole.

Most confusing is how to find out what 4/7 of that "new whole" really 
is; and then to add what Matt eats from Mathilda's plate, and what is 
left on his that he will be eating.

I have no idea if I need to find a common denominator or what I need 
to do to work through this problem.  I have no idea how to even get 
started and would love to know how this is done and why it is done 
this way.  I have always been bad at math.  No one I know knows how 
to approach this problem, either.

Date: 01/21/2010 at 09:49:51
From: Doctor Ian
Subject: Re: a fraction word problem eating pie, how much eaten

Hi Jitka,

Let's try this with simpler numbers.

Suppose Mathilda (M) gets 1/3 of the pie, and Matt (m) gets 2/3:

   +---+---+---+
   |   |   |   |
   |   |   |   |
   |   |   |   |
   | M | m | m |
   |   |   |   |
   |   |   |   |
   |   |   |   |
   +---+---+---+

(I made a rectangular "pie" because that is easier to draw with a 
keyboard!)

Now Mathilda eats 2/3 of Matt's portion, 

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

and Matt eats 1/3 of Mathilda's portion,

   +---+---+---+
   | m |   |   |
   |   |   |   |
   +---+---+---+
   |   | M | M |
   |   |   |   |
   +---+---+---+
   |   | M | M |
   |   |   |   |
   +---+---+---+

and then each of them eats what's left on his or her own plate:

   +---+---+---+
   | m | m | m |
   |   |   |   |
   +---+---+---+
   | M | M | M |
   |   |   |   |
   +---+---+---+
   | M | M | M |
   |   |   |   |
   +---+---+---+

This way, Mathilda ends up with 2/3 of the pie, and Matt ends up with 
1/3... even though the portions were reversed at first.  Which is 
kind of interesting!

Do the pictures make sense?  If so, let's think about how we might do 
that WITHOUT pictures.

We start with this situation:

  Mathilda:  1/3 of a pie

  Matt:      2/3 of a pie

Mathilda eats 2/3 of Matt's portion -- which means she gets 2/3 of 
2/3 of a pie, and he loses that much:

  Mathilda:  1/3 of a pie + (2/3)(2/3) of a pie

  Matt:      2/3 of a pie - (2/3)(2/3) of a pie

Matt eats 1/3 of Mathilda's portion -- which means she gets 1/3 of 
1/3 of a pie, and she loses that much:

  Mathilda:  1/3 of a pie + (2/3)(2/3) of a pie - (1/3)(1/3) of a pie

  Matt:      2/3 of a pie - (2/3)(2/3) of a pie + (1/3)(1/3) of a pie

And now we know how much each of them ate, right?  We just have to 
simplify:

  Mathilda:    1/3 + (2/3)(2/3) - (1/3)(1/3) 

             = 1/3 + 4/9 - 1/9

             = 3/9 + 4/9 - 1/9

             = 6/9

             = 2/3

  Matt:        2/3 - (2/3)(2/3) + (1/3)(1/3)

             = 2/3 - 4/9 + 1/9

             = 6/9 - 4/9 + 1/9

             = 3/9

             = 1/3

So we don't really need the picture... although it did show us what 
was going on, which is always helpful!

Anyway, does this make sense?  Can you follow the same reasoning with 
your fractions?  Try, and let me know what you find out.

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

Date: 01/24/2010 at 23:14:54
From: Jitka
Subject: Thank you (a fraction word problem eating pie, how much 
eaten)

Thanks so much for answering my pie question.  I ran it by my friend, 
who looked at the length of the process toward the answer, and he 
thinks there has got to be a shorter, less complicated way to do this 
math problem.  If he figures it out, I will let you know.

Have a great day.

Thanks much.

P.S.  I have always been terrible at math and even if I had all the 
formulas memorized I could not quite use them properly lol

Date: 01/25/2010 at 10:05:41
From: Doctor Ian
Subject: Re: Thank you (a fraction word problem eating pie, how much 
eaten)

Hi Jitka,

Well, you could do it algebraically, assuming that you start with one 
guy taking a/b of the pie, and the other taking (b - a)/b of the 
pie.  Then you might find a way to generalize it, and get the answer.

And there might, in fact, be some way to "see" the answer directly.  
This kind of reminds me of couple of other classic problems,

  http://mathforum.org/library/drmath/view/56822.html
    
  http://mathforum.org/library/drmath/view/56735.html

where there is a hard way, and an easy way, to figure out the answer.

I hope your friend is able to come up with an insightful solution.  
It would make a great addition to our archives!

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