Potato paradox

Visualisation of the potato paradox: Blue boxes represent 99 water and orange represents 1 non-water parts (left figure). To double the ratio of non-water to water to 1:49, the amount of water is reduced to 49 to retain same amount of non-water part (middle figure). This is equivalent to doubling the concentration of the non-water part (right figure).

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

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

The potato paradox is a mathematical calculation that has a counter-intuitive result. The so-called paradox involves dehydrating potatoes by a seemingly minuscule amount, and then calculating a change in mass which is larger than expected.

Description

The paradox has been stated as:

You have 100 lb of potatoes, which are 99% water by weight. You let them dehydrate until they're 98% water. How much do they weigh now?

The Universal Book of Mathematics states the problem as follows:

Fred brings home 100 pounds 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 pounds.

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

Simple explanations

Method 1

One explanation begins by saying that initially the non-water weight is 1 pound, which is 1% of 100 pounds. Then one asks: 1 pound is 2% of how many pounds? In order for that percentage to be twice as big, the total weight must be half as big.

Method 2

100 lb of potatoes, 99% water (by weight), means that there's 99 lb of water, and 1 lb of solids. It's a 1:99 ratio.

If the water decreases to 98%, then the solids account for 2% of the weight. The 2:98 ratio reduces to 1:49. Since the solids still weigh 1 lb, the water must weigh 49 lb for a total of 50 lbs for the answer.

Explanations using algebra

Method 1

After the evaporating of the water, the remaining total quantity, $x$, contains 1 lb 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 lb.

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:

$100-x=100-50=50$

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

We imagine, from the initial query, that we are reducing water by 1%. This is not the case. If it were, we would be right that we would then have 99 pounds of potatoes (99.01 pounds if you reduce 1% of the 99 pounds of water; 99 pounds if you reduce the total weight by 1% by removing a pound of water - how it is worded makes all the difference!). But reducing water so that it is then 98% of the whole new weight of the potatoes means that we are eliminating half (actually a bit over half - 50/99) of the water so that the potato solids are then 2% of the total weight. The paradox, then, is not in the math, but in our understanding of the language used to define the question. Careful wording must be used to ensure that the "paradox" is correct.

Note that the introductory phrase "by a seemingly minuscule amount" is misleading, and so is part of this Language Paradox. Minuscule would be the 1% reduction that we expect. The more than 50% actually required will not seem minuscule to anyone.

External links

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