How can multiplication rule in sigfigs make sense? I've been going through significant figures video on khan academy and it says the product of two numbers cannot have more significant digits than the significant digits in any of the inputs.
Example:
length = $301 m$
width = $2 m$
area = $301*2 = 602 m^2$
But since the width has only $1$ sigfig, we must round area to $600 m^2$
This seems to convey that area can range between $550$ and $649$.
How can the error in area be as large as $50$ while using a $1m$  precision ruler?
NOTE
The extreme case area:
MIN:
length = $300.5$
width = $1.5$
area = $450.75$
MAX:
length = $301.5$
width = $2.5$
area = $753.75$
How do these min,max areas relate to $600m^2$ ?
 A: 
How can the error in area be as large as $50$ while using a $1\: \rm m$ precision ruler?

Well, because you are multiplying the reading of that "precise" ruler with a large number ($\approx 300$ in your case). So even an error of $0.5 \: \rm m$, could scale itself up and manifest itself as a $\approx 150 \:\rm m^2$ error in the final result. And this is the reason why the maximum and the minimum areas differ from the calculated area by approximately $150 \: \rm m^2$. If you also take into account the variation in the value of length, you could exactly predict the deviation of the minimum and the maximum areas from the calculated value.
So, it's quite logical to have a large deviation in such a quantity, and thus to reduce the sigfigs to show the amount of uncertainty. In fact, strictly speaking, even rounding off to one significant digit isn't enough, since the maximum and the minimum areas differ by more than $200 \:\rm m^2$. However, rounding off to the lowest number of sigfigs in either of the initial quantities, works as nice tool/trick to give a vague, albeit not precise, idea of uncertainty.
