Potato paradox

The potato paradox is a mathematical calculation that results in an un-intuitive result. However it also is another name for the potato-effect, an economical paradox that was discovered in the potato-famine in Ireland in the 1840ies. It describes how the law of demand in economical recessions favors cheap products which in turn become more expensive than higher quality products.

Description

The paradox has been described as:

You have 100 pounds 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 pounds 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 pounds.

It is not really a paradox, but most people find the answer counter-intuitive.

Explanation

One explanation begins by saying that initially the non-water weigh is 1 pound, which is 1% of 100 pounds. The 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 dehdrated 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

$100-x=100-50=50$

See also

• List of paradoxes

External links

• Eric Weisstein