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I have a Betty Crocker white box cake mix and the box says to use either three egg whites or three whole eggs. This doesn't make sense to me: if I used whole eggs wouldn't I use less, more like one and a half?

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Think of the recipe as needing 3 egg whites, and optionally 3 egg yolks. Often if you have a cake mix that calls for oil, somewhere on the package will be a "light version" which is the very same additions and quantities, just not the oil. No extra water or milk or whatnot to make up for it.

The liquid volumes for most cake mixes are not as precise as they seem. Just leaving out the egg yolks, or just leaving out the oil, doesn't require adding some other liquid to make up for it. The cake may end up with a slightly different texture, but it will still be cake.

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Well, as it turns out, no...

I don't know where you are from in the world or what eggs you are using, but I found a scientific paper that studied the ratios of egg components over age in Leghorn chickens (the type Foghorn Leghorn is named for) in the USA (Ahn et al., 1997 you can read the abstract for free at the link, I can access the full paper). It seems that:

When calculated on the basis of interior contents, 65% of egg content is white and 35% is yolk.

But, what you are getting from the egg in this case, is not the water/volume that matters, but instead it is the protein content that is critical. Eggs overall are about 88% water:

The major constituent of egg white is water, approximately 88% of total weight.

However, protein makes up:

The total solids content of white is approximately 12%, and protein is the major component of the solids at 11%... the major constituents of yolk are protein (16%) and lipid (32%).

So, by adding the whole egg you are getting a little extra protein (16% of 35%) relative to using just the white alone. You also get a bit of extra fat (lipid), but in such small amounts it won't matter for the recipe and calorifically is pretty insignificant compared to the cake mix itself.

Edited to add: You might also be concerned about the water content of the eggs, though you are most likely also adding either milk or water to the boxed mix. I had a look at a couple of recipes from the Betty Crocker site and it looks like somewhere between 1 and 1.5 cups (USA measurements as Betty Crocker is a USA company) of water/milk per box, depending on the recipe. This is equivalent to 8-12 fluid ounces (240 - 360 ml). A large egg is about 46 ml (3.25 Tbsp), which is a reasonable amount of extra liquid, but the yolks being about 50% water and only 30% of the volume only adds about 15% extra (20 ml or 1.3 Tbsp). The fat content of the eggs is also insignificant compared to the oil added in the recipes.

Ref:

  1. Ahn DU, Kim SM, Shu H. Effect of egg size and strain and age of hens on the solids content of chicken eggs. Poult Sci. 1997 Jun;76(6):914-9. doi: 10.1093/ps/76.6.914. PMID: 9181628.

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