"Because it’s not just the half life, caffeine has a quarter life, meaning it’s still has its hooks in my patient long after that morning sip. According to sleep scientist Matthew Walker, author of Why We Sleep, caffeine’s quarter life is about double its half life. This means that slow metabolizers like my patient may still be feeling the effects of their morning coffee well into the night. No wonder caffeine has been shown to be so disruptive of sleep. 4 After all, it binds to and blocks our adenosine receptors, the signaling pathways the body uses to fall asleep each night."
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.
"Because it’s not just the half life, caffeine has a quarter life, meaning it’s still has its hooks in my patient long after that morning sip. According to sleep scientist Matthew Walker, author of Why We Sleep, caffeine’s quarter life is about double its half life. This means that slow metabolizers like my patient may still be feeling the effects of their morning coffee well into the night. No wonder caffeine has been shown to be so disruptive of sleep. 4 After all, it binds to and blocks our adenosine receptors, the signaling pathways the body uses to fall asleep each night."