How much caffeine is left in your system? The half-life of caffeine in your system is 6 hours. If you drink 100mg of caffeine every day, what percentage is left in your system after a week?
If you drink 100mg caffeine every day for a month (30d), how long does it take for the level of caffeine in your system to drop below 1mg?

The answer to both depends on when (and how, since one rarely drinks the coffee instantaneously) the coffee is consumed during the day.
For the purposes of this problem, however, I will assume that all the coffee is consumed at exactly $t_{\text{coffee}}$ every day and that we are looking for the quantity remaining at the end of the $7$th day.
At the end of a day, the contribution from that day is $100 ( \frac{1}{2} )^{ \frac{24-t_{\text{coffee}}}{6}}$, after $n$ days, this contribution  decreases to 
$100 ( \frac{1}{2} )^{ \frac{24-t_{\text{coffee}}}{6}} ( \frac{1}{2^{4} } )^n$ (since $4 = \frac{24}{6}$). Summing up, and dividing by $700$ to get the ratio gives:
$$\frac{100}{700}  (\frac{1}{2} )^{ \frac{24-t_{\text{coffee}}}{6}} 
\left( ( \frac{1}{2^{4} } )^6 + ... + ( \frac{1}{2^{4} } )^1 + 1\right)  = 
\frac{1}{7}  (\frac{1}{2} )^{ \frac{24-t_{\text{coffee}}}{6}} 
\left( \frac{1-(\frac{1}{2^{4}})^7 }{1-\frac{1}{2^{4} }} \right)$$
So, doing the computation shows that the percentage left varies between $0.95 \% -15.24 \%$ (taking $t_{\text{coffee}} \in [0,24)$). Note that this is the percentage of the total caffeine consumed during the week, which is a bit misleading, since if you took the average over $n$ days, and let $n \to \infty$, then the percentage would decrease to zero.
The other computation follows similar lines (but is less misleading):
I am assuming that you drink coffee every day for 30 days and then go cold turkey.
Then you want to find the minimum $n$ (whole days) such that
$$100  (\frac{1}{2} )^{ \frac{24-t_{\text{coffee}}}{6}} 
\left( ( \frac{1}{2^{4} } )^{29} + ... + ( \frac{1}{2^{4} } )^1 + 1\right) (\frac{1}{2^4} )^{ n}  \le 1$$
A quick calculation shows that the answer is $n=1$ or $n=2$ (depending on $t_{\text{coffee}}$).
