Potato paradox

Visualization of the potato paradox: blue squares represent 99 parts of water, and orange - 1 part of dry matter (left picture). To bring the ratio of dry matter to water to 1:49, the amount of water is reduced to 49 in order to maintain the same amount of dry part (middle figure). This is equivalent to doubling the concentration of the dry part (figure on the right)

In fact, white potatoes contain about 79% water ^[1] ; 99% of water contains, for example, agar-agar ^[2]

The potato paradox is an example of mathematical calculation, the result of which contradicts intuition. This paradox assumes that the drying of potatoes is not significant at first glance, however, the calculated change in mass is more intuitively expected.

Content

Description

The paradox is formulated as follows:

“There are 100 kg of potatoes that have 99 percent water by weight. Potatoes are dried to 98 percent water. What is the mass of potatoes now? ”

David Darling's The Universal Book of Mathematics textbook defines the problem as follows ^[3] :

“Fred brings home 100 pounds of potatoes, which (being perfect math potatoes) are 99 percent water. He leaves them to dry outside for the night, so that they contain 98 percent of the water. What is their new weight? The unexpected answer is 50 pounds. "

In the classification of Quine’s paradoxes, the potato paradox is “authentic”.

Simple explanations

Method 1

One of the explanations begins with the fact that initially the dry matter mass is 1 kg, which is 1% of 100 kg. Then the question is asked: 1 kg is 2% of how many kg? In order for this share to become twice as large, the total mass should be half as much.

Method 2

100 kg of potatoes, 99% of water (by weight), means 99 kg of water and 1 kg of dry residue. This ratio is 1:99.

If the amount of water is reduced to 98%, the dry matter is 2% by weight. The ratio of 2:98 decreases to 1:49. Since dry matter still weighs 1 kg, water should weigh 49 kg, which gives a total weight of 50 kg in response.

Algebra Explained

Method 1

After water evaporation, the remaining total mass $x$ ${\ displaystyle x}$ contains 1 kg of pure potatoes and $(98/100)x$ ${\ displaystyle (98/100) x}$ water. This is expressed in the equation:

{\begin{aligned}1+{\frac {98}{100}}x&=x\\\Longrightarrow 1&={\frac {1}{50}}x\end{aligned}}

{\ displaystyle {\ begin {aligned} 1 + {\ frac {98} {100}} x & = x \\\ Longrightarrow 1 & = {\ frac {1} {50}} x \ end {aligned}}}

whose decision gives $x$ ${\ displaystyle x}$ = 50 kg.

Method 2

The mass of water in fresh potatoes is $0,99\cdot 100$ ${\ displaystyle 0.99 \ cdot 100}$ .

If a $x$ ${\ displaystyle x}$ Is the mass of water lost when drying potatoes, then $0,98(100-x)$ ${\ displaystyle 0.98 (100-x)}$ Is the weight of water in dried potatoes. Therefore:

0,99\cdot 100-0,98(100-x)=x

{\ displaystyle 0.99 \ cdot 100-0.98 (100-x) = x}

Bracket expansion and simplification:

{\begin{aligned}99-(98-0,98x)&=x\\99-98+0,98x&=x\\1+0,98x&=x\end{aligned}}

{\ displaystyle {\ begin {aligned} 99- (98-0.98x) & = x \\ 99-98 + 0.98x & = x \\ 1 + 0.98x & = x \ end {aligned}}}

Subtract Multiplier $x$ ${\ displaystyle x}$ from each side:

{\begin{aligned}1+0,98x-0,98x&=x-0,98x\\1&=0,02x\end{aligned}}

{\ displaystyle {\ begin {aligned} 1 + 0.98x-0.98x & = x-0.98x \\ 1 & = 0.02x \ end {aligned}}}

And the solution:

{\frac {1}{0,02}}={\frac {0,02x}{0,02}}

{\ displaystyle {\ frac {1} {0.02}} = {\ frac {0.02x} {0.02}}}

What gives a lot of lost water:

50=x

{\ displaystyle 50 = x}

And a lot of dried potatoes:

100-x=100-50=50

{\ displaystyle 100-x = 100-50 = 50}

Corollary

The answer is maintained for any doubling of the dry matter fraction. For example, if a potato initially contains 99.999% water, a percentage reduction of 99.998% still requires a half weight reduction.

Links

↑ Water Content of Fruits and Vegetables , Cooperative Extension Service, University of Kentucky (neopr.) . Date of treatment January 11, 2016.
↑ Agar production methods - Food grade agar , UN Food and Agriculture Organization (neopr.) . Date of treatment January 11, 2016.
↑ potato paradox (neopr.) . Encyclopedia of Science . Archived February 2, 2014.

Links

Weisstein, Eric W. Potato Paradox . mathworld.wolfram.com. Date of treatment August 14, 2018.

[1] Water Content of Fruits and Vegetables , Cooperative Extension Service, University of Kentucky (neopr.) . Date of treatment January 11, 2016.

[2] Agar production methods - Food grade agar , UN Food and Agriculture Organization (neopr.) . Date of treatment January 11, 2016.

[3] tato paradox (neopr.) . Encyclopedia of Science . Archived February 2, 2014.

Potato paradox

Content

Description

Simple explanations

Method 1

Method 2

Algebra Explained

Method 1

Method 2

Corollary

Links

Links

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