Visualisation of the potato paradox: Blue boxes represent 99 water and orange represents 1 non-water parts (left figure). To double the ratio of non-water to water to 1:49, the amount of water is reduced to 49 to retain same amount of non-water part (middle figure). This is equivalent to doubling the concentration of the non-water part (right figure).
White potatoes are actually around 79% water;[1] agar is 99% water.[2]

The potato paradox is a mathematical calculation that has a counter-intuitive result. The so-called paradox involves dehydrating potatoes by a seemingly minuscule amount, and then calculating a change in mass which is larger than expected.

## Description

The paradox has been stated as:

You have 100 lb of potatoes, which are 99% water by weight. You let them dehydrate until they're 98% water. How much do they weigh now?

The Universal Book of Mathematics states the problem as follows:

Fred brings home 100 pounds 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? The surprising answer is 50 pounds.[3]

## Simple explanations

### Method 1

One explanation begins by saying that initially the non-water weight is 1 pound, which is 1% of 100 pounds. Then one asks: 1 pound is 2% of how many pounds? In order for that percentage to be twice as big, the total weight must be half as big.

### Method 2

100 lb of potatoes, 99% water (by weight), means that there's 99 lb of water, and 1 lb of solids. It's a 1:99 ratio.

If the water decreases to 98%, then the solids account for 2% of the weight. The 2:98 ratio reduces to 1:49. Since the solids still weigh 1 lb, the water must weigh 49 lb for a total of 50 lbs for the answer.

## Explanations using algebra

### Method 1

After the evaporating of the water, the remaining total quantity, ${\displaystyle x}$, contains 1 lb 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 lb.

### Method 2

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:

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

## Implication

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.