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 (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.

Explanation of the paradox

The misunderstanding that causes the paradox stems from the overlap between the amount of water in the potatoes, and the proportion of the potatoes that is water.

The initial 100kg of potatoes is described as 99% water. That means that the 100kg of potatoes are made up of 99kg of water and 1kg of dry mass. The percentage water and the mass of water are the same number, which can make it appear that removing 1kg of water is all that is necessary.

If the 1kg of water could be replaced with something else, so that the potatoes remained at a weight of 100kg, then replacing 1kg of water would indeed make them 98% water; but the problem does not allow this. If the water is not replaced with something else, then removing 1kg of water also reduces the total mass of the potatoes by 1kg, to 99kg. That means that the proportion of water is then 98kg out of 99kg, which is 98.98%. Removing 1kg mass of water reduces the proportion of water by only 0.02%.

Because removing water lowers the total mass of the potatoes as well as the mass of water, the proportion of water is reduced much more slowly than the mass of water.

The difficulty in mentally calculating 98/99 can also have an effect on the problem. 98/99 is 0.98989898.. (recurring), so a reader who estimates the result may wrongly believe that it is closer to 98%.

Calculating the correct answer

Method 1

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 proportion of the potatoes must be halved, giving the answer 50kg.

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 proportion 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.