Cake Recipe Ratio

Date: 03/05/2003 at 21:36:28
From: Lavos
Subject: This is a math problem I really need help with...

Cake Recipe

A recipe for four-egg cake calls for the ingredients listed below and 
is supposed to be baked in three 9-inch-diameter round cake pans.  
Unfortunately, you have three 10-inch-diameter cake pans. 

You want to make your cake about the same height as it would have been 
in a 9-inch-diameter cake pan. Determine the amounts needed for each 
ingredient. (Hint: do the eggs first.)
 
Ingredients:
    2 2/3 cups of cake flour 
    2 1/4 teaspoons of baking powder
    1 cup butter
    2 cups sugar
    4 eggs
    1 cup of milk

One of the methods I tried was to find the volume of both cake pans 
(the 9-inch and the 10-inch) and divide the smaller one by the bigger 
one.  This gave me approximately 0.81 or 81%. I then subtracted 1 by 
the 81% and got 0.19.  I multiplied 4 (because there are four eggs) by 
0.19 and got 0.79 and then added that to the original four. The answer 
was 4.79, making the total eggs needed 4.79.  I don't know if this is 
the solution though. It's a hard problem and I was hoping you could 
help me.

Date: 03/05/2003 at 23:04:06
From: Doctor Peterson
Subject: Re: This is a math problem I really need help with...

Hi, Lavos.

You used some good ideas, but I'm not sure you put them together 
right. I would approach it a little differently, so as to be sure of 
getting the right ratios.

You found that the ratio of base areas, and therefore of volumes at 
the same depth, is 81:100. That's a great start. 

Now, that means that you want to multiply every ingredient by 100/81. 
Do you see that? 

They tell you to start with the eggs; you will need 100/81 * 4 = 
almost 5 eggs. Since recipes are generally a little flexible, I would 
just go with 5 eggs and keep multiplying everything by 100/81 (so 
that there will be just a slight increase in the proportion of egg - 
after all, eggs vary in size anyway). They might expect you instead 
to adjust your ratio from 100/81 to 5/4 so that everything agrees 
with the number of eggs (and then check that your pan won't be too 
full); but I don't think that is necessary in practice. In my mind, 
I'm just making sure that I don't have to round too far in doing 
this; if it had turned out to need 4.5 eggs, I might have wanted to 
check how sensitive the recipe is to the number of eggs before going 
ahead with the rounding.

Your method was essentially multiplying everything by (1 - 81/100)+1, 
which is equal to 1.19, rather than by 100/81 = 1.23. You would be 
close, but the method is not correct.

If you have any further questions, feel free to write back.

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

Date: 03/05/2003 at 23:27:28
From: Lavos
Subject: Thank you (This is a math problem I really need help with...)

You helped me a lot with this problem and got back to me very fast. I 
see where my mistake was and I fixed it. Now that I see how to do it I 
can fix my problem.  Thank you VERY much for all the help.

-Lavos