Potato paradox

[Gavran]

100 kg of potatoes consists of 99% water. After some water evaporates, the potatoes consist of 98% water. What is the new weight of the potatoes? The answer is 50 kg.

The explanation is simple. 100 kg of potatoes consists of 99% water and 1% dry matter, so the weight of the dry matter is 1 kg. After some water evaporates, the potatoes consist of 98% water and 2% dry matter. The weight of the dry matter is still 1 kg, so 2% of the potatoes equals 1 kg. This means that 100% of the potatoes equals 50 kg.

Some people think the problem is not paradoxical and the answer is intuitive, but in my opinion, there is a paradox because a small 1% decrease in water content (from 99% to 98%) causes a large 50% decrease in the total weight of the potatoes.

 

[anuttarasammyak]

Salt matter condense up from 1% to 2% (doubles) by more than 50% water evaporation seems to be reasonable.

 

[Hill]

It seems paradoxical when one erroneously translates the "1% decrease in water content" as "1% of water evaporates."
If 100 kg potatoes consisted of 100% of water, it would stay 100% regardless of evaporation.

 

[PeroK]

Some people think the problem is not paradoxical and the answer is intuitive, but in my opinion, there is a paradox because a small 1% decrease in water content (from 99% to 98%) causes a large 50% decrease in the total weight of the potatoes.

It's only paradoxical if you insist on manipulating numbers without thining about what the numbers mean.

There was a viral problem in a Chinese exam that exploited this.

 

[Hill]

The guy in the video also makes logical mistakes, it seems. For example,

I doubt that the problem in China and in France has been asked in English. In this case, what does "in ALMOST EXACTLY the same wording" mean?
Both wordings are translations, not how the problem has been asked.

 

[DaveC426913]

The guy in the video also makes logical mistakes, it seems. For example,

View attachment 369849

I would be interested the methodology of the experiment.

I suspect many of the subjects are not literally failing to understand the meaning of the numbers. I suspect they are, instead self-imposing an external constraint on the problem. To-wit:

"I've been given this problem and asked to solve it. It makes no sense to me as written, but the instructor has given it to me for a reason, which means there is an answer I'm supposed to find. Perhaps there's a metaphor in there I don't know or care about. Anyway, I'll just take the numbers and produce an answer, and if it's right, all the better."

In other words: performance pressure.

 

[jedishrfu]

I was made aware of this by my doctor when we talked about whole milk vs 2% milk. Whole milk is made mostly of water, its ingredients are what determines its caloric content.

Marketing makes the consumer believe that 2% milk has only 2% fat ss compered to whole milk.

But the reality is 2% has 40% less fat than whole milk which calorically is about 20 cals less the whole milk.

 

[Hill]

I would be interested the methodology of the experiment.

I suspect many of the subjects are not literally failing to understand the meaning of the numbers. I suspect they are, instead self-imposing an external constraint on the problem. To-wit:

"I've been given this problem and asked to solve it. It makes no sense to me as written, but the instructor has given it to me for a reason, which means there is an answer I'm supposed to find. Perhaps there's a metaphor in there I don't know or care about. Anyway, I'll just take the numbers and produce an answer, and if it's right, all the better."

In other words: performance pressure.

Indeed. What the results would've been if it were a multiple-choice question with the choices, say, a) 25 yo, b) 36 yo, c) not enough data, d) are you kidding, e) none of the above ?

 

[Dale]

100 kg of potatoes consists of 99% water. After some water evaporates, the potatoes consist of 98% water. What is the new weight of the potatoes? The answer is 50 kg.

The explanation is simple. 100 kg of potatoes consists of 99% water and 1% dry matter, so the weight of the dry matter is 1 kg. After some water evaporates, the potatoes consist of 98% water and 2% dry matter. The weight of the dry matter is still 1 kg, so 2% of the potatoes equals 1 kg. This means that 100% of the potatoes equals 50 kg.

Some people think the problem is not paradoxical and the answer is intuitive, but in my opinion, there is a paradox because a small 1% decrease in water content (from 99% to 98%) causes a large 50% decrease in the total weight of the potatoes.

One of the reasons that this seems surprising is that a 1% change seems like a small change. I think the same thing happens with probabilities. As probabilities get close to 0 or 1, the changes in the probabilities don’t feel right. A change in odds or log odds becomes more intuitive.

 

[PAllen]

I would be interested the methodology of the experiment.

I suspect many of the subjects are not literally failing to understand the meaning of the numbers. I suspect they are, instead self-imposing an external constraint on the problem. To-wit:

"I've been given this problem and asked to solve it. It makes no sense to me as written, but the instructor has given it to me for a reason, which means there is an answer I'm supposed to find. Perhaps there's a metaphor in there I don't know or care about. Anyway, I'll just take the numbers and produce an answer, and if it's right, all the better."

In other words: performance pressure.

I have bumped into problems that are trivial given the assumption that there is a well defined answer, but more tedious if you don't assume this and must establish that there is enough information. My favorite is a cylindrical hole cut out of the center of a ball. It turns out the volume of what is left of the ball depends only on the height of resulting figure. This can be established with a straightforward computation with calculus. But if the problem is presented with a value for the height (and nothing else), and multiple choice numerical answers, you can instead reason: it must be true that the result is independent of anything except the height, so let me take the limit of the hole vanishing. Then I simply have a ball of radius half the height, so this must be volume.

Note, if the problem is presented open ended (i.e., what is the volume, with no specific answer given), the quick response of 'not enough information' would, of course be flat out wrong.

 

[anuttarasammyak]

There was a viral problem in a Chinese exam that exploited this.

We may take the problem as a study subject of statistics, sending a questionnaire to the captains:
a. How old are you?
b. Have you loaded 26 sheep and 10 goats?
c. If yes, how old were you then?
I often see this method in Medecin in order to find suspected correlation between the two events.

 

[QuarkyMeson]

I was made aware of this by my doctor when we talked about whole milk vs 2% milk. Whole milk is made mostly of water, its ingredients are what determines its caloric content.

Marketing makes the consumer believe that 2% milk has only 2% fat ss compered to whole milk.

But the reality is 2% has 40% less fat than whole milk which calorically is about 20 cals less the whole milk.

It would all make more sense if whole milk was just called 3% milk or something.

 

[A.T.]

100 kg of potatoes consists of 99% water.

Here it should be specified that it's 99% by mass (not by volume, or by particles, etc.).

Some people think the problem is not paradoxical and the answer is intuitive, but in my opinion, there is a paradox because a small 1% decrease in water content (from 99% to 98%) causes a large 50% decrease in the total weight of the potatoes.

I think about it geometrically, in terms of linear scaling. And of course you have to keep in mind what is kept constant:
1cm is 1% of a 100cm stick
To make 1cm be 2% of the stick, you have to cut the stick down to 50cm

 

[Herman Trivilino]

but in my opinion, there is a paradox because a small 1% decrease in water content (from 99% to 98%) causes a large 50% decrease in the total weight of the potatoes.

The potatoes are almost all water. Cutting the amount of water in half cuts the amount of potatoes in half.

What's "paradoxical" is that the potatoes would have to rot and be inedible

 

[Trysse]

The question is designed to suprise you.

To regain a feeling of intuition you can take the numbers and apply them to da different scenario.

Imagine you have a box with 10×10=100 compartments. In each compartment you put a white ping-pong ball (100% white ping-pong balls).

Next you replace one white ball with a yellow ball (99% white and 1% yellow).

If you now want to increase the percentage of yellow balls from 1% to 2% you have two options:

First option: replace a second white ball with a yellow ball (98 white vs 2 yellow).

Second option: Remove as many white balls as necessary so one yellow ball makes up 2% of the remaining balls. You must remove 50 so you have 49 white vs 1 yellow.

I think this scenario should give you a feeling, that 50% loss of the total quantity is logical and no way paradoxical.

Option 1 is not available for the potato scenario but it shows why you instinctively think only a small change is required to change the percentage from 1% to 2%.

In addition the word "some" as used in everday language is associated with a small amount.

I think the psychological effects at work here is what is discussed in Daniel Kahneman's book "Thinking fast and slow".

So if you not immediately thought "the weight loss must be small" and you were not suprised to learn that 50% of total weight loss this means you are not suspectible to fall for trick questions.

I think considering why the question tricked you is as important as thinking about the correct solution.

By the way, I was surprised by the 50kg answer.

 

[DaveC426913]

1cm is 1% of a 100cm stick
To make 1cm be 2% of the stick, you have to cut the stick down to 50cm

I would say this is the simplest, most eloquent deconstruction of the apparent paradox.

Simple, linear measurement is supremely intuitive. No boxes, no ping pong balls, just a yardmetrestick.

 

[OmCheeto]

I can see why this is considered paradoxical.
It took me an hour to build a spreadsheet yesterday to do just the basics of the maths.
From there I discovered that the solution to the problem was y = a/x. WTB?

Actually, it was my spreadsheet 'trendlines' feature that told me that.
I had to reduce the initial weight to 1 lb to see it in action.

		1%
100	initial total weight	% water	water weight	% potato dry matter	potato dry matter weight	final total weight
0.01	1	99%	0.99	1%	0.01	1
		98%	0.49	2%	0.01	1/2
		97%	0.323333333333333	3%	0.01	1/3
		96%	0.24	4%	0.01	1/4
		95%	0.19	5%	0.01	1/5
		94%	0.156666666666667	6%	0.01	1/6
		93%	0.132857142857143	7%	0.01	1/7
		92%	0.115	8%	0.01	1/8
		91%	0.101111111111111	9%	0.01	1/9
		90%	0.09	10%	0.01	1/10

And here I am 24 hours later, still questioning how I came up with 'y = a/x', and how to explain it to someone who is slightly worse at maths than I am.

 

[gmax137]

It isn't paradoxical if you have ever boiled sap to make maple syrup. The raw sap is 2 or 3 percent sugar, depending on the tree. If you start with 40 gallons of sap, and boil 20 gallons off, the sugar content doubles (!) to a whopping 5 or 6 percent. The final syrup is near 70 percent sugar. The magic doesn't happen until you boil off all but a pound or two of the original water.

 

[haruspex]

I am reminded of a bogus calculation by Australian politician Greg Hunt.
https://www.abc.net.au/news/2015-10-28/fact-check-direct-action-vs-carbon-tax/6847234
He claimed the previous government's carbon tax had cost $1300 per tonne of reduction. His method was to divide the money companies had had to pay for exceeding their caps by the reduction that was achieved. On that method, had they spent enough on reducing their emissions that the caps were not exceeded then it would have achieved a huge reduction for no cost!
Of course, none of the journalists who interviewed him understood the illogicality.

 

[.Scott]

There was a viral problem in a Chinese exam that exploited this.

The video mentions one answer of "at least 30" based on China boat licensing practice.
But, the voiceover claims a few times that there is not enough information.
Whether there is enough information depends on what is "enough" precision - and if your going to ask such an off-the-wall question, you're certainly due an off-the-wall precision.

In any of the countries listed (China, France, Germany, etc.) both occupations (boat captain and the shepherd) have always been filled by humans of an age where they are potentially fit for the task. So my answer would be an estimate - about 6 to 95 years.