A lily pad doubles in area every second. After one minute, it fills the pond. How long would it take to quarter fill the pond ?

I think it's easiest to work backwards: if the area doubles every second and the pond is totally covered at time $t=60$, then it must be half covered at $t=59$, and therefore one quarter covered at $t=58$.

Alternately, let $f(t)$ be the fraction of the pond's area covered at time $t\leq 60$. Then $f(t)=f(0)2^t$ since the area doubles every second, and since $f(60)=1$ we get $f(0)=2^{-60}$. Therefore $f(t)=2^{-60}2^t=2^{t-60}$. Then setting $ 2^{t-60}=\frac{1}{4}$ and solving for $t$ yields $t=58$.

Forget formulas for this one!

If going forward 1 second the area gets doubled, then going back 1 second the area gets halved.

So, 1 second before the pond was filled the pond must have been half filled, and 1 second before that it must have been quarter filled.

This is exponential rather than linear. If $A$ is the initially covered area, then after one second the covered area will be $2A$, after two second $2\cdot 2A=4A$, after three seconds $2\cdot 4A=8A$. And so on: after $t$ seconds the covered area will be $2^tA$.

After $60$ seconds it will be $2^{60}A$, by assumption this is the whole pond. A quarter of this is $$ \frac{2^{60}A}{4}=2^{58}A $$

_{Of course Carmichael’s answer is slicker.}