The potato paradox is a mathematical calculation that has a counter-intuitive result. The Universal Book of Mathematics states the problem as such:

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?

The surprising answer is 50 kg. 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 100kg of potatoes contains 1kg of dry mass. This mass will 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 - 1kg - cannot be changed, 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 50kg.

## Mathematical proofs

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

Let's call d, the dry mass of the potatoes and w the mass of the water inside the potatoes.

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

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

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

Let the weight lost be equal to X

As the solid weight remains constant, So

X = initial water content - final water content

X = 99% 100 - 98% (100 - X)

X = 99 - 98 + 0.98X

1 = 0.02X

X = 50