Vehicular trouble

This week’s Riddler Classic is about steady-state mixing of fluids. Here is the paraphrased problem.

Your old van holds 12 quarts of transmission 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. 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?

Here is my solution:
[Show Solution]

2 thoughts on “Vehicular trouble”

  1. It seems to me there’s a much simpler way to think about this, which is:

    The quart you just put in is fresh
    There is 11/12 left of the quart you put in last month, which is fresh
    There is (11/12)^2 left of the quart you put in two months ago, which is fresh

    There is (11/12)^11 left of the quart you put in eleven months ago, which is fresh
    The rest is old.

    So the fraction of the oil which is old is (12-(\sum_{i=0}^11 (11/12)^i))/12 = 0.352.

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