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### 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

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
```
Associated Topics:
Middle School Ratio and Proportion

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