The error in your reasoning is that you're replacing "quarter life" and "half life" with "1/4" and "1/2" (scalars), when the definition is actually about the number of hours the caffeine stays in your system. You are also putting the (2 *) on the wrong side. It should instead be:
half_life_hrs * 2 = quarter_life_hrs
Think of it this way: if caffeine's half life is 6 hours, and you drink 100mg, it means that after 6 hours you still have 50mg in your system.
Given that, you could try to extrapolate and say that after another 6 hours, you would have 0mg left in your system.
However, since "caffeine's quarter life is about double its half life", that means that:
6hrs * 2 = quarter_life_hrs
Therefore, after 12 hours (6hrs * 2), you would still have 25mg (a quarter of 100mg) of caffeine in your system.
half_life_hrs * 2 = quarter_life_hrs
Think of it this way: if caffeine's half life is 6 hours, and you drink 100mg, it means that after 6 hours you still have 50mg in your system. Given that, you could try to extrapolate and say that after another 6 hours, you would have 0mg left in your system.
However, since "caffeine's quarter life is about double its half life", that means that:
6hrs * 2 = quarter_life_hrs
Therefore, after 12 hours (6hrs * 2), you would still have 25mg (a quarter of 100mg) of caffeine in your system.