n this striking article [1], they analyzed why time appears to speed up with age mathematically. They deal with some additional complications. While those with a sound understanding of first year calc can still understand it, there is no reason to set the bar so high for such a simple idea. So let me rephrase this striking idea in middle school math.

The idea: Your life is twice as long at 2 y.o than at 1 y.o., but only 110% as long at 11 y.o. compared to 10y.o.

So the question is, if we place 80 tick marks from when you are 1 y.o. to when you are 80, such that a constant percentage of life pass with each tick mark, where should these tick marks be?

So, let x_n be the location of the nth tick mark, we have the relation that f(n)= f(n-1) * (1+x), so that f(1) = 1, and f(80)=80. So all you have to do is solve for x. 80=f(80)=(1+x)^79, sovling and we get that x is about 5.7%.

So if starting at age 1, we place a tick mark everytime when we pass 5.7% of the life that we've already passed. We are at age 80 at the 80th tickmark. The graph looks like this:

So in a way, the tick mark is another kind of age, call it effective age. When your effective age increase by 1, you have lived an additional 5.7% of your current physical age. Effective age might correspond well to our perception of how fast time passes. After 40 of these tick marks, we are only 10 years old. It takes the rest of 70 years for the other 40 tick marks to pass.

An example:

My first summer break in kindergarten seemed forever, because that was probably 3month /36month=9% of my life. The previous summer break past very quickly, because that was only 3months/283months=1% of my life.

So these exponential functions merely captures the above notion concretely. The results are striking nevertheless. In terms of how much money you make, the notion of physical age is better. But in terms of how much you might enjoy the next fancy dinner, or the next road trip, perhaps the effective age is better.