Cake mathematics

We ate warm carrot cake with melted cream cheese frosting for my birthday. When Naomi asked what kind of cake I would like, I didn't consider mud cake, though Naomi concocted such a confection this summer for her play bakery (photo above). 

When I was younger I reasoned that smaller families ate more birthday cake. One cake mix makes the same size cake no matter the size of your family. Therefore I concluded that children in smaller families ate more cake because they had more leftovers.

Dad set up a math problem for me in which we hypothetically examined the cake consumption of two families--a family with four members and a larger family with eight members--through a year. A cake was baked for each member's birthday. We assumed that the cakes were the same size and divided evening among family members.

Family A (four member)

At each birthday each member receives a quarter of the cake. At the end of the year, each member has consumed the equivalent of an entire cake. 

Family B (eight members)

At each birthday each member receives an eighth of the cake. At the end of the year each member consumed the equivalent of an entire cake. 

Conclusion: Each person received the same amount of cake, no matter how it was initially divided.

Of course, life isn't hypothetical. From my more mature view, I can see advantages of large families (in our problem we assume all cakes were the same size, but last night my cake was 11x14". That's bigger than the typical 9x13" cake). The advantages aren't limited to cake consumption though. Nothing beats younger siblings who are more excited about your birthday than you are. Logan was so pleased with the present he bought me that I had to open it before lunch. Naomi created a birthday card for me, complete with a sketch of me wearing the hat she hates.