Potato paradox: Difference between revisions
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{{short description| |
{{short description|Mathematical calculation with a counter-intuitive result}} |
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{{about||the economic effect sometimes called the ''potato effect'' or ''potato paradox''|Giffen good}} |
{{about||the economic effect sometimes called the ''potato effect'' or ''potato paradox''|Giffen good}} |
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{{Multiple issues| |
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{{Incomprehensible|date=January 2021}} |
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{{More citations needed|date=January 2021}} |
{{More citations needed|date=January 2021}} |
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}} |
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[[File:Hillview Farms white potatoes.jpg|thumb|150px|White potatoes are actually around 79% water,<ref>{{cite web|title=''Water Content of Fruits and Vegetables'', Cooperative Extension Service, University of Kentucky|url=https://www2.ca.uky.edu/enri/pubs/enri129.pdf|access-date=11 January 2016}}</ref> |
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[[agar]] is 99% water.<ref>{{cite web|title=''Agar production methods – Food grade agar'', UN Food and Agriculture Organization|url=http://www.fao.org/docrep/006/y4765e/y4765e06.htm#bm06.1.1|access-date=11 January 2016}}</ref>]] |
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The '''potato paradox''' is a mathematical calculation that has a counter-intuitive result. ''[[The Universal Book of Mathematics]]'' states the problem as such: |
The '''potato paradox''' is a mathematical calculation that has a counter-intuitive result. ''[[The Universal Book of Mathematics]]'' states the problem as such:<ref>{{cite book |
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| last = Darling | first = David J. | authorlink = David J. Darling |
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| title = The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes |
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| year = 2004 |
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| publisher = [[John Wiley & Sons]] |
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| url = https://archive.org/details/universalbookofm0000darl/page/253/mode/1up?view=theater |
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| page = 253 |
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| isbn = 0-471-27047-4 |
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}}</ref> |
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{{quote|Fred brings home 100 kg of potatoes, which (being purely mathematical potatoes) consist of 99% water. He then leaves them outside overnight so that they consist of 98% water. What is their new weight? |
{{quote|Fred brings home 100 kg of potatoes, which (being purely mathematical potatoes) consist of 99% water (being purely mathematical water). He then leaves them outside overnight so that they consist of 98% water. What is their new weight?}} |
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Then reveals the answer: |
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{{quote|The surprising answer is 50 kg.}} |
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In [[Paradox#Quine's classification|Quine's classification of paradoxes]], the potato paradox is a [[Paradox#Quine's classification|veridical]] paradox. |
In [[Paradox#Quine's classification|Quine's classification of paradoxes]], the potato paradox is a [[Paradox#Quine's classification|veridical]] paradox. |
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== Simple explanations == |
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=== Method 1 === |
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Initially, the non-water weight is 1 kg, which is 1% of 100 kg. If after leaving them overnight, the water weight shrinks to 98%, then 1 kg is 2% of how many kg? For non-water percentage to be twice as big, the total weight must be half as big. The non-water weight cannot grow or shrink in this question, therefore it will always be 1 kg, and the water weight must change around it. |
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=== Method 2 === |
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[[File:potato paradox.svg|thumb|A visualization where blue boxes represent kg of water and the orange boxes represent kg of solid potato matter. Left, prior to dehydration: 1 kg matter, 99 kg water (99% water). Middle: 1 kg matter, 49 kg water (98% water).]] |
[[File:potato paradox.svg|thumb|A visualization where blue boxes represent kg of water and the orange boxes represent kg of solid potato matter. Left, prior to dehydration: 1 kg matter, 99 kg water (99% water). Middle: 1 kg matter, 49 kg water (98% water).]] |
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If the potatoes are 99% water, the dry mass is 1%. This means that the 100 kg of potatoes contains 1 kg of dry mass, which does not change, as only the water evaporates. |
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In the beginning (left figure), there is 1 part non-water and 99 parts water. This is 99% water, or a non-water to water ratio of 1:99. To double the ratio of non-water to water to 1:49, while keeping the one part of non-water, the amount of water must be reduced to 49 parts (middle figure). This is equivalent to 2 parts non-water to 98 parts water (98% water) (right figure). |
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In 100 kg of potatoes, 99% water (by weight) means that there is 99 kg of water, and 1 kg of non-water. This is a 1:99 ratio. |
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If the percentage decreases to 98%, then the non-water part must now account for 2% of the weight: a ratio of 2:98, or 1:49. Since the non-water part still weighs 1 kg, the water must weigh 49 kg to produce a total of 50 kg. |
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== Explanations using algebra == |
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=== Method 1 === |
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After the evaporating of the water, the remaining total quantity, <math>x</math>, contains 1 kg pure potatoes and (98/100)x water. The equation becomes: |
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: <math>\begin{align} |
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1 + \frac{98}{100}x &= x \\ |
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\Longrightarrow 1 &= \frac{1}{50}x |
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\end{align}</math> |
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resulting in <math>x</math> = 50 kg. |
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=== Method 2 === |
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The weight of water in the fresh potatoes is <math>0.99 \cdot 100</math>. |
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If <math>x</math> is the weight of water lost from the potatoes when they dehydrate then <math>0.98(100 - x)</math> is the weight of water in the dehydrated potatoes. Therefore: |
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: <math>0.99 \cdot 100 - 0.98(100 - x) = x</math> |
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Expanding brackets and simplifying |
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: <math>\begin{align} |
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99 - (98 - 0.98x) &= x \\ |
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99 - 98 + 0.98x &= x \\ |
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1 + 0.98x &= x |
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\end{align}</math> |
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Subtracting the smaller <math>x</math> term from each side |
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: <math>\begin{align} |
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1 + 0.98x - 0.98x &= x - 0.98x \\ |
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1 &= 0.02x |
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\end{align}</math> |
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In order to make the potatoes be 98% water, the dry mass must become 2% of the total weight—double what it was before. The amount of dry mass, 1 kg, remains unchanged, so this can only be achieved by reducing the total mass of the potatoes. Since the proportion that is dry mass must be doubled, the total mass of the potatoes must be halved, giving the answer 50 kg. |
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Which gives the lost water as: |
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== Mathematical proofs == |
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: <math>50 = x</math> |
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Let ''x'' be the new total mass of the potatoes (dry + water). |
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Let ''d'' be the dry mass of the potatoes and ''w'', the mass of water within the potatoes. |
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: <math>100 - x = 100 - 50 = 50 </math> |
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Recall ''w'' is 98% of the total mass, that is, ''w'' = 0.98''x''. |
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=== Method 3 === |
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After the potatoes are dehydrated, the potatoes are 98% water. |
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Therefore, ''x'' = ''d'' + ''w'' = ''d'' + 0.98''x'', i.e., ''x'' = ''d'' / 0.02 = 50 kg. |
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This implies that the percentage of non-water weight of the potatoes is <math>(1 - .98)</math>. |
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In our case, ''d'' = 1 kg so the new mass of the potatoes will indeed be 50 kg. |
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: <math>\begin{align} |
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(1-.98)x &= 1 \\ |
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.02x &= 1 \\ |
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x &= \frac{1}{.02} \\ |
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x &= 50 |
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\end{align}</math> |
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---- |
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== Implication == |
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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. |
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Let ''X'' be the mass lost. Since the solid (non-water) mass remains constant, then |
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== The Language Paradox == |
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{| |
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After the first reading, one might wrongly assume that by reducing the water percentage by 1% you reduce its weight by 1 kg. But when the water percentage is reduced by 1%, what this actually means is that the non-water percentage is doubled while its weight stays constant, meaning that 50 kg of water evaporated. |
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|align="right"|''X''|| = initial water content – final water content |
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|- |
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|align="right"|''X''|| = 99% 100 kg – 98% (100 kg – ''X'') |
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|- |
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|align="right"|''X''|| = 99 kg – 98 kg + 0.98''X'' |
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|- |
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|0.02''X''|| = 1 kg |
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|- |
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|align="right"|''X''|| = 1 kg / 0.02 = 50 kg |
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|} |
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==In popular culture== |
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Another way to interpret the initial query, is that the 99% water refers to the volume and not the weight of the potatoes. Though the volume of the potatoes would still be halved, the answer would be unknowable, as we do not know the weight of the potato solids. For example, the potato solids might weigh 75kg on their own, in which case the answer can never be 50kg, no matter how much the water is reduced. |
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The potato paradox was a "Puzzler" on the ''[[Car Talk]]'' radio show.<ref>"[https://www.cartalk.com/radio/puzzler/porch-potatoes Porch Potatoes]", ''Car Talk'', August 19, 2017.</ref> |
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But since logic dictates the paradox must have a valid answer, we must assume the water makes up 99% of the weight. |
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The paradox is, then not mathematical, but more so about our understanding of the language and logic used to define the question. Careful wording must be used to ensure that the "paradox" is correct. |
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== References == |
== References == |
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Latest revision as of 14:15, 21 May 2024
 | This article needs additional citations for verification. (January 2021) |
The potato paradox is a mathematical calculation that has a counter-intuitive result. The Universal Book of Mathematics states the problem as such:[1]
Fred brings home 100 kg of potatoes, which (being purely mathematical potatoes) consist of 99% water (being purely mathematical water). He then leaves them outside overnight so that they consist of 98% water. What is their new weight?
Then reveals the answer:
The surprising answer is 50 kg.
In Quine's classification of paradoxes, the potato paradox is a veridical paradox.
If the potatoes are 99% water, the dry mass is 1%. This means that the 100 kg of potatoes contains 1 kg of dry mass, which does not change, as only the water evaporates.
In order to make the potatoes be 98% water, the dry mass must become 2% of the total weight—double what it was before. The amount of dry mass, 1 kg, remains unchanged, so this can only be achieved by reducing the total mass of the potatoes. Since the proportion that is dry mass must be doubled, the total mass of the potatoes must be halved, giving the answer 50 kg.
Mathematical proofs
[edit]Let x be the new total mass of the potatoes (dry + water).
Let d be the dry mass of the potatoes and w, the mass of water within the potatoes.
Recall w is 98% of the total mass, that is, w = 0.98x.
Therefore, x = d + w = d + 0.98x, i.e., x = d / 0.02 = 50 kg.
In our case, d = 1 kg so the new mass of the potatoes will indeed be 50 kg.
Let X be the mass lost. Since the solid (non-water) mass remains constant, then
| X | = initial water content – final water content |
| X | = 99% 100 kg – 98% (100 kg – X) |
| X | = 99 kg – 98 kg + 0.98X |
| 0.02X | = 1 kg |
| X | = 1 kg / 0.02 = 50 kg |
In popular culture
[edit]The potato paradox was a "Puzzler" on the Car Talk radio show.[2]
References
[edit]- ^ Darling, David J. (2004). The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. John Wiley & Sons. p. 253. ISBN 0-471-27047-4.
- ^ "Porch Potatoes", Car Talk, August 19, 2017.