# Potato paradox
The potato paradox is a mathematical puzzle in which 100 kg of potatoes that are 99% water by weight are dehydrated until they are 98% water, and the correct answer, 50 kg, contradicts the near-99 kg that intuition suggests.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/) It is classified as a veridical paradox: a result that is undoubtedly true but that many people find hard to believe.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/)
| Key fact | Detail |
|---|---|
| Setup | 100 kg of potatoes, 99% water by weight, dried until the water content is 98%[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/) |
| Intuitive answer | About 99 kg, from treating the 1% drop in water content as 1% of the total mass[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/) |
| Correct answer | 50 kg, half the original weight[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/) |
| Why | The 1 kg of dry matter is conserved and doubles its share from 1% to 2% of the total, so the total must halve[2](https://onlineminitools.com/potato-paradox-calculator) |
| General formula | New weight = initial weight × (1 − initial water fraction) ÷ (1 − final water fraction)[3](https://www.toolead.com/math/potato) |
| Realism | Real white potatoes contain about 79% water, not 99%[4](https://clever-geek.imtqy.com/articles/7566387/index.html) |
| Type | Veridical paradox, related in spirit to the bat and ball problem[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/) |
## The puzzle and the wrong answer
The standard statement asks: you buy 100 kg of potatoes containing 99% water; overnight, water evaporates until the potatoes are 98% water; how much do they now weigh?[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/) Many people guess 99 kg or a similar figure.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/)
That guess fails a quick consistency check. If 98 kg of water remained out of a total mass of 99 kg, the water percentage would still be very nearly 99%, not 98%.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/) The error comes from a flawed [mental model](https://www.edgepedia.org/mental-model): attention fixes on the change in the water proportion, from 99% to 98%, and jumps to the conclusion that the water lost equals 1% of the total mass. The problem seems simple enough that the brain glosses over the fact that this would require some water to turn into potato solids, which is impossible.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/)
## Why the answer is 50 kg
The resolution is to track the dry matter instead of the water. Initially, 1 kg of potato solids makes up 1% of the 100 kg total. Drying removes only water, so the 1 kg of solids is unchanged, but it now constitutes 2% of the remaining mass. For 1 kg to be 2% of the total, the total must be 1 ÷ 0.02 = 50 kg.[2](https://onlineminitools.com/potato-paradox-calculator)
The same result follows from algebra. If the dry mass is x and the water fraction is f, the total mass is x ÷ (1 − f). The dry mass x is the same before and after drying, so the ratio of final to initial mass is (1 − f₁) ÷ (1 − f₂). With f₁ = 0.99 and f₂ = 0.98, this ratio is 0.01 ÷ 0.02 = 1/2, proving the potatoes weigh half as much after drying.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/)
A square-diagram visualization shows the same thing: one dark square of solids among 99 light squares of water; when the single solid square must represent 2% of the total, the total shrinks to 50 squares.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/)
## The general formula
Because the dry mass never changes, the calculation holds for any starting weight and any pair of water percentages:[3](https://www.toolead.com/math/potato)
\[ M_{\text{final}} = M_0 \cdot \frac{1 - f_{\text{initial}}}{1 - f_{\text{final}}} \]
For example, 8 kg of potatoes at 98% water contain 0.16 kg of dry matter; dried to 95% water, they weigh 0.16 ÷ 0.05 = 3.2 kg.[3](https://www.toolead.com/math/potato) The same principle applies to any substance that loses water, such as grapes dried into raisins.[3](https://www.toolead.com/math/potato)
The effect does not depend on the numbers being 99 and 98. If a potato initially contains 99.999% water, a reduction to 99.998% still halves the weight, because what doubles is the dry-matter fraction, from 0.001% to 0.002%.[4](https://clever-geek.imtqy.com/articles/7566387/index.html)
## Relation to real potatoes
The setup is deliberately unrealistic. Real white potatoes contain about 79% water, not 99%; a 99% water content is closer to substances such as agar-agar.[4](https://clever-geek.imtqy.com/articles/7566387/index.html) Potatoes also do not shrivel to half their size overnight, so the framing encourages the assumption that little has changed.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/) The direction is still real: a potato that dries out in the back of a store cupboard is noticeably smaller than it was, and industrial drying for crisps or freeze-dried potato products reduces moisture to 2–10%, concentrating the remaining mass dramatically.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/) • [5](https://we-calculate.com/calculator/potato-calculator)
The paradox's lesson is about where to focus in percentage problems: the small change in the water percentage conceals a doubling of the dry-matter fraction, and tracking the conserved quantity, the dry mass, makes the result immediate.[2](https://onlineminitools.com/potato-paradox-calculator) • [5](https://we-calculate.com/calculator/potato-calculator) The bat and ball problem is another well-known puzzle that tricks readers in a similar way, by directing attention to the wrong quantity.[1](https://graphicmaths.com/recreational/paradoxes/potato-paradox/)
## References
1. [Potato paradox, Martin McBride, GraphicMaths (2023)](https://graphicmaths.com/recreational/paradoxes/potato-paradox/)
2. [Potato Paradox Calculator – Dehydration & Mass Loss, OnlineMiniTools](https://onlineminitools.com/potato-paradox-calculator)
3. [Free Potato Calculator, Toolead](https://www.toolead.com/math/potato)
4. [The Potato Paradox, clever-geek](https://clever-geek.imtqy.com/articles/7566387/index.html)
5. [Potato Paradox Calculator — Water Content & Weight Loss, we-Calculate](https://we-calculate.com/calculator/potato-calculator)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Conditional probability and independence › Conditioning paradoxes and pitfalls*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026; Nov 11, 2026 · Last review: Oct 15, 2026*
*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*
License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license