Riddler Classic Puzzle 1-28-2022

From ​​Travis Henry comes some vehicular trouble:

You want to change the transmission fluid in your old van, which holds 12 quarts of fluid. At the moment, all 12 quarts are “old.” But changing all 12 quarts at once carries a risk of transmission failure.

Instead, you decide to replace the fluid a little bit at a time. Each month, you remove one quart of old fluid, add one quart of fresh fluid and then drive the van to thoroughly mix up the fluid. (I have no idea if this is mechanically sound, but I’ll take Travis’s word on this!) Unfortunately, after precisely one year of use, what was once fresh transmission fluid officially turns “old.”

You keep up this process for many, many years. One day, immediately after replacing a quart of fluid, you decide to check your transmission. What percent of the fluid is old?

FiveThirtyEight

SOLUTION

Generally, for any given mixture of Old and New oil in the tank:

If  p  if the fraction of oil mixture removed (and New oil added), then  1 - p  is the fraction of previous oil mixture in the tank:

So after the 1st oil change:

1 - p  =  fraction of Old oil in tank

And after the 2nd oil change:

(1 - p) * (1 - p)  =  fraction of Old oil in tank

After the Nth oil change:

(1  -  p) ^ N  =  fraction of Old oil in tank 

BUT

Since any New oil becomes Old oil after K ( = 12) oil changes, the fraction of Old oil doesn't change after K oil changes.

(Another way to see this is to look at the fraction of New oil in tank and notice a geometric sum with K terms)

So for  p  =  1/12  and  K  =  12:

(1 - p) ^ K  =  (1 - 1/12) ^ 12  =  (11/12) ^ 12  ≈  .352

So the percentage of Old oil in tank:

35.2%