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The potato paradox is a mathematical calculation that has 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.
Contents
Description
The paradox has been described as:
You have 100 lbs of 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.
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 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.
Method 1
After the evaporating of the water, the remaining total quantity, x, contains 1 lbs pure potatoes and (98/100)x water. The equation becomes:
resulting in x = 50 lbs.
Method 2
The weight of water in the fresh potatoes is
If
Expanding brackets and simplifying
Subtracting the smaller
And solving:
Which gives the lost water as:
And the dehydrated weight of the potatoes as:
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.