# Why does the minimum control group size decrease, when total sample size increases?

I'm gathering methods to calculate the minimum size of a control group, given the total sample size, power, significance level, minimum change I want to be able to detect, etc. I want to do this for different kinds of outcomes, such as proportions and continuous outcomes, for which I have found nice formulas.

I have tested these formulas and one thing I find, is the fact that in all cases the control group can be smaller when the total sample size increases (given that all other inputs stay the same). Now, for me, this does not sound like a weird result, but I also want to be able to explain this to people who are less skilled in statistics. So basically, I am looking for an intuitive way to explain why the control group size can be smaller, when the total sample size increases.

I was thinking of something like this: Because the treatment group size increases, you are more certain on the outcome within this group, and as such, you can be less certain about the results in the control group.

However, I am not completely satisfied with this explanation and think this will not be sufficient to convince someone who is less skilled in statistics, that this is a correct result (i.e., control groups getting smaller when total sample size increases).

Is there a better way of explaining this result in an intuitive manner?