Cake math

Jordan and I are in the midst of baking his birthday cake (even as I write, the layers are in the oven) and we encountered some interesting math situations in the process. He requested a red, white, and blue cake, one layer of each color, with white frosting, and raspberries and blueberries on top. Very seasonal. So he and I made up a big batch of cake batter. He was entertained to learn that it is a 1-2-3-4 cake: 1 cup butter, 2 cups sugar, 3 cups flour, and 4 eggs. We needed three layers and I only have two nine-inch pans, but we can handle that. When the batter was prepared, though, we found ourselves with a puzzle. How would we make three different-colored layers all the same size?

Jordan: I know! Split the batter into three halves. Put one half in one bowl and make it red, the other half in the other bowl and and make it blue, and leave the third half where it is.
Rachel: Great idea. Now, this is going to be a little tricky, because see my three bowls? They are not the same size. So how will we make sure each one gets the same amount?
Jordan: Fill them up to the same height.
Rachel: Hmm. But if I do that, this bowl is bigger so it will get more. See?
Jordan: (blank look)
Rachel: Here, I'll show you. See this little bowl and this big bowl? If I fill the little bowl up to the top, and the big bowl up to the height of the little bowl, the big bowl has more.
Jordan: OH! Because it's wider.
Rachel: Right.  So what should we do?
Jordan: Put less in the bigger bowl so that it comes out the same.  Not as high.
Rachel: That would be good.  That's what I was thinking I should do.  But it would be hard to know just how much less, you know?  So I had another idea:
(I show him my kitchen scale.)
Jordan:  A waiter!!  (Weighter, I guess.)
Rachel: I suggest that we put the same amount of batter in each bowl by weighing it.  Now, here is a trick: I just want to weigh the batter, not the bowl.  So I put the bowl on the scale and press this little button to make it be zero with the bowl on.  Now, I don't know how much to put in, so I will just guess. Here, can you remember this number?
Jordan: Twenty!
Rachel: Twenty ounces in bowl number one.  Twenty ounces in bowl number two.  What will it be in bowl number three?
Jordan:  Twenty!
Rachel: Well, I want that, but I don't know if I did it right.  Maybe twenty was too much in the other bowls and there's not enough left.  And look, it's not enough, this is only twelve.  What should we do?
Jordan: Take some from the other bowls to make twelve, so that it will be twelve, twelve, twelve.
Rachel: That would make it even.  But then we would have leftover batter, and I want to use all the batter.  Hmm.  Okay, here's my idea.  How much batter is there all together?  Twenty plus twenty plus twelve, can you do that?
Jordan: No.
Rachel: Twenty plus twenty?
Jordan: Forty!
Rachel: And then forty plus twelve?
Jordan (intense look):  FIFTY-TWO.
Rachel:  Wow, awesome!
Jordan: Mommy I thought about the forty and the twelve and the ten part of the twelve goes with the forty and that makes it fifty and then there was the two part of the twelve and I remembered it was fifty and two more after fifty is fifty-two!
Rachel: Fantastic.  That is some great addition.  Now, we have fifty-two ounces, and we want to put the same amount into each of the three bowls.  How much should we put?  Can you figure that out?
Jordan: No.
Rachel: I don't blame you.  That is a tough one.  I can figure it out and it's between seventeen and eighteen.  So we will take about two ounces out of each of the bowls of twenty, and put those in the bowl of twelve, and that will make them just about the same.

That's a lot of math.