Potato paradox

Potatoes

The Potato Paradox is a mathematical calculation that results in a counter-intuitive result. The "paradox" involves dehydrating potatoes by a seemingly minuscule amount, and then calculating a change in mass which is larger than expected. This is not to be confused for the Potato-Effect, which is sometimes known as the Potato Paradox, and is actually an economical paradox that was discovered in the potato-famine in Ireland in the 1840s.

Description

The paradox has been described as:

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

The Universal Book of Mathematics states the problem as follows:

Fred brings home 100 lbs of potatoes, which (being purely mathematical potatoes) consist of 99 percent water. He then leaves them outside overnight so that they consist of 98 percent water. What is their new weight? The surprising answer is 50 lbs.

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

Explanation

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 that that percentage be twice as big, the total weight must be half as big.

An explanation via algebra is as follows:

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

$99 - (98 - 0.98x) = x$

$99 - 98 + 0.98x = x$

$1 + 0.98x = x$

Subtracting the smaller $x$ term from each side

$1 + 0.98x - 0.98x = x - 0.98x$

$1 = 0.02x$

And solving:

$1 / 0.02 = 0.02x / 0.02$

Which gives the lost water as:

$50 = x$

And the dehydrated weight of the potatoes as:

$100 - x = 100 - 50 = 50$

A less rigorous explanation that may be more intuitive is as follows.

100 lbs of potatoes, 99% water (by weight), means that there's 99 lbs 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 lbs.

See also

• List of paradoxes

External links

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