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 ?

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?

To me this seems like we can set up a fraction-like equation:

$$\frac{60 \ \text{seconds}}{1} = \frac{x \ \text{seconds}}{1/4}$$ then $x = 15$ seconds. But the answer is $58$ seconds which really makes no sense to me. Any suggestions are greatly appreciated.

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.}