# Help me to understand the potato paradox

From wiki - Fred brings home $$100$$ kg of potatoes, which (being purely mathematical potatoes) consist of $$99\%$$ water. He then leaves them outside overnight so that they consist of $$98\%$$ water. What is their new weight? The surprising answer is $$50$$ kg.

When I look at the problem, I assume that $$99\%$$ water is equal to $$99$$ kg and the rest is $$1$$ kg. When the percentage of the water is now $$98\%$$ the weight is now $$98$$ kg and the rest is $$2$$ kg.

I don't understand how the total weight can be $$50$$ kg at the end, maybe the water turns into a solid?

• The "rest" are dry materials of the potatos, i.e. starch, protein, minerals, etc. How can these increase from 1kg to 2kg after being left outside overnight? The point is that these things don't increase and remain 1kg. The only change is that water decreases. Apr 13, 2021 at 13:26
• Where is Fred living, that a potato would lose half its weight in water after just one night left outside? :P Apr 14, 2021 at 0:07

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.

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.

We start with $$1$$ kg solid potato. Next morning there are still $$1$$ kg solid potato, which allegedly make up $$2\%$$ of the total rest weight. The latter then has to be $$50$$ kg.

This is a picture of the potatoes before dehydrating.

About 99% water (not to scale). This is a picture of the potatoes after dehydrating:

About twice the amount of potato. Oh, and I should mention that I zoomed in on the second diagram so that it's the same width as the first. The second bar is actually half as wide as the first one, since it represents 50kg instead of 100kg. In fact, all that changed between the two bars is that the blue bar got cut roughly in half (50kg of water were removed from 99kg of water). Since the bar overall gets cut in half and the amount of potato doesn't change, the potato looks twice as big once you zoom in. In other words, its size relative to the total amount of mass is twice as much, or it constitutes double the percentage that it originally did.