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 counter-intuitive result. 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 (being purely mathematical water). He then leaves them outside overnight so that they consist of 98% water. What is their new weight?

Then reveals the answer:

The surprising answer is 50 kg.

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 proofs

Let x be the new total mass of the potatoes (dry + water).

Let d be the dry mass of the potatoes and w, the mass of water within the potatoes.

Recall w is 98% of the total mass, that is, w = 0.98x.

Therefore, x = d + w = d + 0.98x, i.e., x = d / 0.02 = 50 kg.

In our case, d = 1 kg so the new mass of the potatoes will indeed be 50 kg.

Let X be the mass lost. Since the solid (non-water) mass remains constant, then

X = initial water content – final water content
X = 99% 100 kg – 98% (100 kg – X)
X = 99 kg – 98 kg + 0.98X
0.02X = 1 kg
X = 1 kg / 0.02 = 50 kg

The potato paradox was a "Puzzler" on the Car Talk radio show.[2]

References

  1. ^ Darling, David J. (2004). The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. John Wiley & Sons. p. 253. ISBN 0-471-27047-4.
  2. ^ "Porch Potatoes", Car Talk, August 19, 2017.