The Potato Paradox

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The Potato Paradox

Postby SciameriKen on January 17th, 2018, 3:40 pm 

Can someone explain the potato paradox to me?

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.

Wikipedia explains this with two methods:
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.

What I don't get is if 100 lbs is 99% water - then solids = 1 lb and water = 99 lbs -- if it now consists of 98% water - then why wouldn't the answer simply be 99 lbs? (1 lb solids, 98 lbs water?)
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Re: The Potato Paradox

Postby Braininvat on January 17th, 2018, 4:33 pm 

All the total weight number doesn't matter. Just say one tater is one pound. It's 99% water. 1% solids. Loses 1% of its water overnight, ergo .99 lb. next morning. (.01 lb of solid, plus .98 lb water)

No real paradox, but maybe people get lost in the wording?
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Re: The Potato Paradox

Postby doogles on January 17th, 2018, 6:19 pm 

Braininvat's method of calculation is good, but I was seeking more of an EXPLANATION for the apparent paradox.

I see where this is called a VIRIDICAL PARADOX. It’s the sort of thing where our brains get conned by an elementary error in thinking. Recently I posted a similar question on the painting of a shed wherein if one bloke could do it in 3 hours and another in 5, how long would it take if both painted it together? My mental arithmetic said at first that the average time was 4 hours and therefore it would take 2 h.

My brain did the same thing when I first saw your post. We reduced the water percentage here from 99 to 98% and my reflex elementary thinking said that we’d reduced the water content by 1%. Therefore, we’d reduced the total weight by 1 lb to give the answer 99 lbs. Like you I was initially surprised that the answer was 50 lbs. The question used the expression 'they consist of 98 percent water'.

If the question had been asked the other way around, our elementary thinking would have been totally different – How much water would have to be lost from the 100 lbs of potatoes for the solids to increase from 1% to 2%? - (50lbs)

Or even better - How much water would I need to dissolve 1 lb of solids in to get a 98% solution by weight? (49 lbs)

Rightly or wrongly, Sciameriken, that’s the way I see this apparent paradox.
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Re: The Potato Paradox

Postby Dave_C on January 17th, 2018, 7:34 pm 

I got tripped up too. Consider that if we remove just 1% of the water from the original 100 pounds, we have 1 pound of potatoes and 98 pounds of water. That's a ratio of 1/98 = .0102 or 1.02% potato by mass and 98.98% water. So removing 1% of the water doesn't change things much....

For it to be 98% water, we need 2% potato. If the potato is 1 pound, then water is 49 pounds. Total is 50 pounds.

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