Potato Paradox: a clever play on common confusion about percentages

Problem. Fred brings home 100 kg of potatoes, made up of 99% water. He leaves them outside, allowing for some water to evaporate, thus reducing the water content to 98%. How much do the potatoes weigh now?

Answer. The new weight is 50 kg.

Fig 1. Potatoes contain about 80% water on average.

Solution. At first glance, the answer might seem illogical. The water content dropped by only 1%, so the weight shouldn't change much. However, the solid content doubled from 1% to 2%. The solids didn't evaporate, meaning their weight stayed the same. This is the factor we should focus on because the new proportions must account for this constant. If the proportion of solids in the new weight has doubled, then the amount of water, and thus its weight, must be halved. The Potato Paradox skillfully diverts attention from the key point, playing on a common confusion about percentages. Once this confusion is cleared, the solution is simple: 1 kg (constant) + 49 kg (halved) = 50 kg.

Now, let's do it slowly, starting with defining percentages. Percentages represent the ratio of ingredients in a mix. The entire mix is considered to be 100%, meaning it is divided into 100 equal parts. The percentage of each ingredient indicates how many parts out of 100 it makes up. The initial figures, 1% and 99%, were calculated based on the weight of 100 kg. Dividing 100 kg into 100 parts, we estimate that each part weighs 1 kg:

1% (solids) + 99% (water) = 100% (initial weight)

1 part (solids) + 99 parts (water) = 100 parts (initial weight),

1 kg + 99 kg = 100 kg

The new figures, 2% and 98%, are based on the new weight, which we again take as 100%:

2% (solids) + 98% (water) = 100% (new weight)

2 parts (solids) + 98 parts (water) = 100 parts (new weight)

The new weight differs because the weight of the water has changed, while the solids weigh the same, 1 kg. Now, this 1kg accounts for 2 parts of the new weight, meaning each part weighs 1/2 kg.

2 parts = 1kg, so 1 parts = 1/2 kg

1/2 kg x 2 + 1/2 kg x 98 = 1 kg + 49 kg = 50 kg

The new weight is indeed 50 kg.