Potato paradox

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

White potatoes are actually around 79% water, agar is 99% water.

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. He then leaves them outside overnight so that they consist of 98% water. What is their new weight? The surprising answer is 50 kg.

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

Simple explanations

Method 1

Initially, the non-water weight is 1 kg, which is 1% of 100 kg. If after leaving them overnight, the non-water weight shrinks to 98%, then 1 kg is 2% of how many kg? For non-water percentage to be twice as big, the total weight must be half as big. The non-water weight cannot grow or shrink in this question, therefore it will always be 1 kg, and the water weight must change around it.

Method 2

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).

In the beginning (left figure), there is 1 part non-water and 99 parts water. This is 99% water, or a non-water to water ratio of 1:99. To double the ratio of non-water to water to 1:49, while keeping the one part of non-water, the amount of water must be reduced to 49 parts (middle figure). This is equivalent to 2 parts non-water to 98 parts water (98% water) (right figure).

In 100 kg of potatoes, 99% water (by weight) means that there is 99 kg of water, and 1 kg of non-water. This is a 1:99 ratio.

If the percentage decreases to 98%, then the non-water part must now account for 2% of the weight: a ratio of 2:98, or 1:49. Since the non-water part still weighs 1 kg, the water must weigh 49 kg to produce a total of 50 kg.

Explanations using algebra

Method 1

After the evaporating of the water, the remaining total quantity, $x$, contains 1 kg pure potatoes and (98/100)x water. The equation becomes:

${\begin{aligned}1+{\frac {98}{100}}x&=x\\\Longrightarrow 1&={\frac {1}{50}}x\end{aligned}}$

resulting in $x$ = 50 kg.

Method 2

The weight of water in the fresh potatoes is $0.99\cdot 100$.

If $x$ is the weight of water lost from the potatoes when they dehydrate then $0.98(100-x)$ is the weight of water in the dehydrated potatoes. Therefore:

$0.99\cdot 100-0.98(100-x)=x$

Expanding brackets and simplifying

${\begin{aligned}99-(98-0.98x)&=x\\99-98+0.98x&=x\\1+0.98x&=x\end{aligned}}$

Subtracting the smaller $x$ term from each side

${\begin{aligned}1+0.98x-0.98x&=x-0.98x\\1&=0.02x\end{aligned}}$

Which gives the lost water as:

$50=x$

And the dehydrated weight of the potatoes as:

$100-x=100-50=50$

Method 3

After the potatoes are dehydrated, the potatoes are 98% water.

This implies that the percentage of non-water weight of the potatoes is $(1-.98)$.

If x is the weight of the potatoes after dehydration, then:

${\begin{aligned}(1-.98)x&=1\\.02x&=1\\x&={\frac {1}{.02}}\\x&=50\end{aligned}}$

Implication

The answer is the same as long as the concentration of the non-water part is doubled. For example, if the potatoes were originally 99.999% water, reducing the percentage to 99.998% still requires halving the weight.

The Language Paradox

After the first reading, one might wrongly assume that by reducing the water percentage by 1% you reduce its weight by 1 kg. But when the water percentage is reduced by 1%, what this actually means is that the non-water percentage is doubled while its weight stays constant, meaning that 50 kg of water evaporated.

Another way to interpret the initial query, is that the 99% water refers to the volume and not the weight of the potatoes. Though the volume of the potatoes would still be halved, the answer would be unknowable, as we do not know the weight of the potato solids. For example, the potato solids might weigh 75kg on their own, in which case the answer can never be 50kg, no matter how much the water is reduced. But since logic dictates the paradox must have a valid answer, we must assume the water makes up 99% of the weight. The paradox is, then not mathematical, but more so about our understanding of the language and logic used to define the question. Careful wording must be used to ensure that the "paradox" is correct.

External links

• Weisstein, Eric W. "Potato Paradox". MathWorld.