A mathematical conundrum!

This morning I baked snickerdoodles, the fourth batch of this holiday season. Each cookie has two components, which I make separately: the dough and the coating. The dough is a blob, and the coating is a mixture of sugar, cinnamon, and coloured sprinkles.

As written, the recipe that I use can make 24 cookies. I’ve found that that’s too large of a batch size for me, so I’ve cut all of the ingredient amounts in half to make 12 cookies instead. However, I’ve since experimented with making more cookies by making them smaller, so now I can make 24 half-size cookies.

When I roll the balls of dough in the coating, I find that I have made too much coating. I end up with some coating to spare after all of the dough-balls have been duly rolled, which I sprinkle indiscriminately with my spoon over the baking sheet. At first I figured this was by the recipe’s design, but then a consideration occurred to me.

Does the amount of coating I need to make scale linearly with the amount of dough I need to make? If I cut the amount of dough I make in half, should I cut the amount of coating I make in half, too?

The answer might be no! The amount of coating I need to make scales with the surface area of the dough, but not with the volume.

So how much should I adjust the amounts of the ingredients in the recipe that are used for the coating?