The potato paradox

I came across this question last night on some trivia app I have on my phone:

When you let fruits consisting of 99% water by weight dry so that they become 98% water, what percentage of weight do they lose?

Stop scrolling (or looking) down right now, if you want to give it some thought first! I got the answer wrong and curious as I usually am, I pushed the "Learn" button in the app, which opened up the relevant Wikipedia article. In this case, it opened up the Potato Paradox page. I read through it and was still flummoxed. The question is worded slightly differently on Wikipedia:

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.

The explanation on Wikipedia is this:

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.

What confused me was how losing 1% of water can amount to half the weight. Is there some definition of dry mass I don't understand!? 99% is 99kgs. Losing 1% should be losing like 1kg!

I did the maths on the Wikipedia article and sure enough the answer was 50kg but I'm not sure I understand what the equations were representing.

Finally, this line of thinking and wording (by me) helped towards understanding the solution a bit better.

1. Losing 1% of water (to 98%) means that the dry mass now makes up 2% of the weight
2. The dry mass does not change, so 2% = 1kg or 1% = 0.5kg
3. Therefore 98% = 98 * 0.5 = 49kg
4. Therefore the total weight is 49 + 1 = 50kg

If we lost 9% of water (to 90%), it would mean 10% = 1kg, and therefore 100% = 10kg.

Let's graph this in R!

calc_weight <- function(w){
  stopifnot(w >= 0 & w <= 99)
  d <- 100 - w
  1 / d * 100
}

library(ggplot2)
x <- 99:70
data.frame(
  water = factor(x, levels = x),
  weight = calc_weight(x)
) |>
  ggplot(aes(water, weight)) +
  geom_point() +
  theme_minimal() +
  labs(x = "Water %", y = "Weight", title = "Total weight of potatoes as a function of water percent")

So what did I learn from this? Don't expect percentages to follow some linear relationship!

Ask a friend

I asked some friends the same trivia question and one of them replied me with output from a LLM, which was wrong. However that gave me the idea of trying a LLM myself! (My friends all got the answer wrong too, guess that's why we're friends!)

I input the following to AnonChatGPT:

When you let fruits consisting of 99% water by weight dry so that they become 98% water, what percentage of weight do they lose?

and the output was:

To determine the percentage of weight lost, we need to calculate the difference in the total weight of the fruit before and after drying.

If the fruit consists of 99% water by weight, then the remaining 1% is non-water content. When the fruit is dried and becomes 98% water, then the non-water content must increase to 2%.

Let's assume the original weight of the fruit is 100 grams.

Before drying: Water content = 99% of 100g = 99g Non-water content = 1% of 100g = 1g

After drying: Water content = 98% of x = 0.98x grams (where x is the new weight after drying) Non-water content = 2% of x = 0.02x grams

Since the total weight remains the same before and after drying: 99g + 1g = 0.98x + 0.02x 100g = 1x x = 100g

Therefore, after drying, the weight of the fruit is 100 grams.

Weight lost = Original weight - Final weight = 100g - 100g = 0g

Percentage weight loss = (Weight lost / Original weight) x 100% = (0g / 100g) x 100% = 0%

Therefore, the fruit does not lose any weight in this scenario.

It was going well until the model couldn't capture the fact that it is the "Non-water content" weight that remains the same and not "the total weight".