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.