Help me to understand the potato paradox

Solution 1:

The water doesn't turn into a potato; it evaporates.

You assumed that the mass of the potato-water system remains the same. But the water can't become potato overnight.

Initially, we had $99$ kg water and $1$ kg potato.

Later, let there be $x$ kg water and $1$ kg potato.

Then $$\frac{x}{x+1}=\frac{98}{100}$$ $$100x=98x+98$$ $$x=49$$ $$x+1=50$$

Thus the total weight (potato$+$water) is $x+1=50$ kg.

Solution 2:

The potatoes will be $1kg$ of solid part and $99kg$ of water. the ration will be $1/(99+1)=1\%$ and the water will constitute $99\%$. the ratio of $1kg$ solid and $49kg$ water will be $1/(1+49)=2\%$ and the water will thus constitute $98\%$ of the potatoes in this second case. The intuition is that to go from being $1\%$ of the mass to $2\%$ of the mass the solid part has to double relative to the total mass. But since the solid part does not change the total mass has to half.

The apparent paradox lies in showing the change in percentage of the water which is relatively small but this hides the large change in the quantity of solid part.