Potato Paradox: A Simple Question with an Impossible Answer

Potato Paradox, Can you Solve it
Potato Paradox, Can you Solve it?

The potato paradox presents a mathematical puzzle as follows: Fred purchases 100 kg of potatoes containing 99% water. After allowing them to dry, they now contain 98% water. The question arises: What is their revised weight?

This may seem like a simple question but the question itself is called potato paradox for a reason. The answer is contrary to what many people believe. It is a veridical paradox which means a scenario where a seemingly impossible or contradictory situation turns out to be true.

The potato paradox presents a mathematical conundrum with a counterintuitive outcome. It is described in The Universal Book of Mathematics


The surprising answer is 50 kg.

Now, if this answer has you scratching your head, you may want to look at how the answer is arrived at.

Explanation of the Potato Paradox

If the potatoes are composed of 99% water, then the remaining 1% constitutes the dry mass, which remains constant at 1 kg as only the water evaporates.

To attain a composition of 98% water in the potatoes, the dry mass must represent 2% of the total weight, effectively doubling its previous proportion. Given that the dry mass remains at 1 kg, the only feasible adjustment is to reduce the total mass of the potatoes.

Consequently, in order to double the proportion of dry mass, the total mass of the potatoes must be halved, resulting in a final weight of 50 kg.

The mathematical solution to the Potato Paradox

Let’s break it down:

  • Let ( x ) be the new total mass of the potatoes (dry + water).
  • Let ( d ) be the dry mass of the potatoes and ( w ) be the mass of water within the potatoes.
  • Given that the water content is 98% of the total mass, we have ( w = 0.98x ).
  • Therefore, the total mass ( x ) equals the sum of the dry mass ( d ) and the mass of water ( w ), which leads to ( x = d + w = d + 0.98x ).
  • Solving for ( x ), we find ( x = d / 0.02 = 50 ) kg.

Since ( d = 1 ) kg in our case, the new mass of the potatoes indeed amounts to 50 kg. This aligns with the conclusion drawn earlier.

The potato paradox has gained notable recognition in popular culture. It was notably featured as the “Puzzler” on the Car Talk radio show. Subsequently, it garnered attention on Neatorama and was also acknowledged as one of the “Five Famous Paradoxes.”

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