Potato paradox

Initially, if the potatoes are 99% water, the non-water mass is 1%. The potatoes mass 100 kg; 1% of 100 kg is 1 kg. This mass is static; it will not change, as only the water evaporates.

If after leaving them overnight, the water mass shrinks to 98%, now the potatoes' non-water mass is 2% of the total mass - but that's still just 1 kg. For non-water percentage to be twice as big, the total mass must be half as big. In other words, 1 kg is 2% of 50 kg.

Another way to word it starting from the beginning:

1% of 100 kg is 1 kg of solid potato. This amount does not change. So we have 1 kg of solid potato comprising 1% of the total mass.

When water evaporates so that the water is 98% of the total mass, meaning that 2% of the total mass is the unchanged 1 kg of solid.

Simple algebra: 1 kg is 2% of what? The answer is 50 kg.

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).

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).

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.

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.

After the evaporating of the water, the remaining total quantity, ${\displaystyle x}$, contains 1 kg pure potatoes and (98/100)x water. The equation becomes:

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

resulting in ${\displaystyle x}$ = 50 kg.

The weight of water in the fresh potatoes is ${\displaystyle 0.99\cdot 100}$.

If ${\displaystyle x}$ is the weight of water lost from the potatoes when they dehydrate then ${\displaystyle 0.98(100-x)}$ is the weight of water in the dehydrated potatoes. Therefore:

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

Expanding brackets and simplifying

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

Subtracting the smaller ${\displaystyle x}$ term from each side

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

Which gives the lost water as:

${\displaystyle 50=x}$

And the dehydrated weight of the potatoes as:

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

After the potatoes are dehydrated, the potatoes are 98% water.

This implies that the proportion of non-water weight of the potatoes is ${\displaystyle (1-.98)}$.

If x is the weight of the potatoes after dehydration, then:

${\displaystyle {\begin{aligned}(1-.98)x&=1\\.02x&=1\\x&={\frac {1}{.02}}\\x&=50\end{aligned}}}$