Potato paradox

For the economic effect sometimes called the potato effect or potato paradox, see Giffen good.

The potato paradox is a mathematical calculation that has a result which seems counter-intuitive to many people. The Universal Book of Mathematics states the problem as such:^[1]

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.^[2]

In Quine's classification of paradoxes, the potato paradox is a veridical paradox.

A visualization where blue boxes represent kg of water and the orange boxes represent kg of solid potato matter. Left, prior to dehydration: 1 kg matter, 99 kg water (99% water). Middle: 1 kg matter, 49 kg water (98% water).

If the potatoes are 99% water, the dry mass is 1%. This means that the 100 kg of potatoes contains 1 kg of dry mass, which does not change, as only the water evaporates.

In order to make the potatoes be 98% water, the dry mass must become 2% of the total weight—double what it was before. The amount of dry mass, 1 kg, remains unchanged, so this can only be achieved by reducing the total mass of the potatoes. Since the proportion that is dry mass must be doubled, the total mass of the potatoes must be halved, giving the answer 50 kg.

Mathematical proof

Originally, 1% of the 100kg was dry matter, that is to say 1kg. After they dried, the dry mass of the potatoes made up 2%, or one fiftieth, of the total, which must therefore be 50 × 1kg = 50kg.

In popular culture

The potato paradox was a "Puzzler" on the Car Talk radio show.^[3]