Circles, Percents, and Butter Cake

I was reminded of my grandmother's butter cake recipe today after seeing a math-related scenario (sort of a prompt) on the Math Forum, posted by Annie (whom I think I remember from college decades ago, but that's another story).

I love Gram's butter cake recipe (posted below) for several reasons. The recipe style reminds me of her (I can just hear her voice saying "Don't DUMP it in"); it's absolutely delicious; it's fun to make; and then there is this lovely warning: "This is too much batter for 2 9" pans." If you're a member of my family, that is code for "You probably want to make a 9", two-layer cake, so you're just going to have to eat some of the batter so it doesn't go to waste. Oh, DARN." (I should add here that you're eating raw eggs if you choose to do this, which is not advised, not to mention the gazillion calories. So let's consider this a hypothetical scenario that you're too smart to follow.)

So the question that always occurred to me, and which I have solved several times in my life, is: just how much excess batter are we talking about? If we fill two 9" round pans to the same depth to which we would have filled three 8" pans, what percentage of the batter do the cooks need to... um... dispose of?

I suppose if I pose this problem to students, I could come up with how many cupcakes a full recipe would make (probably about 30), then ask how many cupcakes you should make with the excess batter.

I just remembered: the frosting I always use for it also has interesting math. This was a recipe of my other grandmother's. It says to bring 2 tablespoons milk, 3 tablespoons butter, and 4 tablespoons of brown sugar to a boil, then stir in 1 1/2-2 cups of confectioner's sugar and a pinch of salt. But there's also a parenthetical note "or 3-4-5 proportions" under the 3 ingredients you boil. I used to entertain myself figuring out how much the options shifted the share of each ingredient.

By the way, although my frosting grandmother died about 20 years ago, Gram-of-the-butter-cake is still in fine form at age 93. If you live in Greensboro, NC, you've probably met her; it seems like everybody has!