The Potato Paradox Explained: Why 99% Water Becomes 50kg

The potato paradox is a famous mathematical puzzle involving 100 kg of potatoes that are initially 99% water. After drying overnight, their water content falls to 98%. Surprisingly, the potatoes now weigh just 50 kg – half their original weight!

Many people get this completely wrong. They guess 99 kg or similar.

How can losing just one percentage point of water cause such a dramatic weight change? The answer lies in the amount of potato solids, which remains unchanged. Let's work through the puzzle and discover why the answer is correct.

Why does the potato paradox have such a counterintuitive answer?

The key to solving the potato paradox is to concentrate on the part of the potato that isn't water.

Initially, we have:

Notice that the potato solids weigh exactly 1 kg.

When the potatoes dry out, some water evaporates, but the amount of potato solids doesn't change. They still weigh 1 kg.

However, they now represent 2% of the total weight.

If 1 kg represents 2% of the total, the total weight must be:

$$ \frac{1\,\text{kg}}{0.02} = {50\,\text{kg}} $$

Therefore, the potatoes now weigh 50 kg. Here is the situation after the potatoes have dried out overnight:

It seems strange, but the only way that 1 kg of potato solids can make up 2% of the total is if the total weight is 50 kg.

Veridical paradox

A common response is that this isn't a true paradox, it is simply a counterintuitive fact. But, actually, it is an example of a veridical paradox - a result that is undoubtedly true, but many people find it hard to believe.

The confusion usually arises because we look at the change in the proportion of water, from 99% to 98%, and jump to the conclusion that the amount of water lost equals 1% of the total mass. But that isn't correct. It isn't even close.

A moment's thought proves that the answer of 99 kg can't possibly be true. Initially, there is 99 kg of water out of a total mass of 100 kg, so the water percentage is:

$$ \frac{99\,\text{kg}}{100\,\text{kg}} = 99\% $$

If there were 98 kg of water out of a total mass of 99 kg, the water percentage would be:

$$ \frac{98\,\text{kg}}{99\,\text{kg}} = 98.\overline{98}\% $$

So the water percentage wouldn't be 98%. It would still be very nearly 99%.

Mental model

The essence of this paradox is that it misdirects you into a flawed mental model of what is going on.

The problem seems simple, so we might not think it requires much thought. We tend to assume the potato's total mass can't have changed much (because the water content, the main component of its mass, appears to have changed by only 1%). This offers an easy and intuitive solution, so our brain glosses over the fact that this would require some of the water to magically transform into potato solids (which is obviously impossible).

Some people complain that the scenario is unfair or unrealistic. For one thing, real potatoes aren't 99% water. That's true, but potatoes do contain quite a lot of water. If you have ever found an old potato that rolled to the back of the store cupboard and dried out, you will see it's genuinely much smaller than it was.

The other part of the problem that might be misleading is that potatoes don't generally shrivel to half their size overnight. So again, that part of the setup encourages you to imagine that the potato won't have changed very much, which could be seen as slightly underhand. Regardless, if you work through the basic maths, the correct answer is easy to derive.

The main misdirection is that it focuses your attention on the water. If we concentrate on the potato solids instead of the water, the result is clear.

A better way of viewing the problem

Let's reframe the problem and focus on the potato's dry content.

At the start, we have x amount of potato solids that make up 1% of the total mass. In the end, we still have x amount of potato solids, but it now makes up 2% of the total mass. How has the total mass changed?

Here, the top set of squares represents the 100 kg of potato of which 1% is solid potato (the single dark green square), and 99% (the 99 light blue squares) is water:

If we remove water until the single green square represents 2% of the total mass, then clearly we must halve the total mass, as the bottom set of squares illustrates.

Or just use algebra

This problem can also be solved with straightforward algebra. Let's say that the mass of the potato solids is x, the mass of the water is w, and the total mass is t. The fraction of the potato that is water is f (eg, if the potato is 99% water, f will be 0.99).

Looking at the initial state, we know that the total mass is the sum of the mass of the water and the mass of the potato solids:

$$ t_1 = w_1 + x $$

We've used the suffix 1 to indicate the initial state before drying out. We also know that the water mass is the total mass multiplied by the water fraction, f:

$$ x_1 = t_1 f_1 $$

Combining and rearranging these expressions gives us:

$$ x = t_1 - w_1 = t_1(1 - f_1) $$

So we can find the total mass in terms of the potato mass and the fraction:

$$ t_1 = \frac{x}{(1 - f_1)} $$

You can calculate the total mass after drying the same way using the second set of values (with suffix 2). Notice, of course, that the value x is unchanged:

$$ t_2 = \frac{x}{(1 - f_2)} $$

Now we can calculate the ratio of the total mass after drying versus the original total mass:

$$ \frac{t_2}{t_1} = \left ( \frac{x}{1 - f_2} \right ) \Big/ \left ( \frac{x}{1 - f_1}\right ) = \frac{x (1 - f_1)}{x (1 - f_2)} $$

The x terms cancel out. We also know that f1 is 0.99 and f2 is 0.98, so the mass ratio is:

$$ \frac{t_2}{t_1} = \frac{1 - 0.99}{1 - 0.98} = \frac{0.01}{0.02} = \frac{1}{2} \qquad \text{(1)} $$

This proves that, after drying, the potatoes have half their original weight.

What happens with real potatoes?

Real potatoes typically contain about 80% water, although that varies depending on the potato variety, growing conditions, and storage. But we will use it as a typical value. If the water content of those potatoes changes from 80% to 79%, what happens to the total weight?

We can use our previous formula (1), but using percentages of 80% and 79% (instead of 99% and 98%):

$$ \frac{t_2}{t_1} = \frac{1 - 0.80}{1 - 0.79} = \frac{0.20}{0.21} \approx 0.9524 $$

This means that the potatoes now weigh about 95% of their weight when the water content changes from 80% to 79%. This isn't as dramatic as the original paradox, but the change in weight (about 5%) is still much bigger than the change in water percentage (about 1%).

FAQ

What is the potato paradox?

The potato paradox is a mathematical puzzle in which 100 kg of potatoes containing 99% water are dried until they contain 98% water. Surprisingly, their weight falls to 50 kg.

Why do the potatoes lose half their weight?

The potatoes initially contain 1 kg of solid material. When the water content falls to 98%, that same 1 kg must represent 2% of the total weight. Therefore, the new total weight is 50 kg.

Is the potato paradox mathematically correct?

Yes. It follows directly from the fact that the amount of solid potato material remains unchanged while water evaporates.

Is the potato paradox a real paradox?

Some people say that it isn't really a paradox, it is just a surprising fact. But it is classed as a veridical paradox: something true but hard to believe.

How much water is in a real potato?

A typical raw potato contains approximately 80% water by weight.

A related problem

Another well-known problem that is tricky in a similar way is the bat-and-ball problem.

It goes like this. A toy bat and ball cost \$1.10 in total. If the bat costs \$1 more than the ball, how much does the ball cost?

A surprising number of people, including people who might generally be regarded as being quite good at maths, will immediately answer 10c even though it is obviously not correct (if the ball is 10c, the bat would be \$1.10, so the total would be \$1.20).

It works similarly to the potato paradox. The problem itself appears very simple, so you aren't encouraged to think about it too much. And a quite plausible value for the bat price, $1, appears right there in the question. So there is a good chance that someone might answer with that poorly thought-out guess.

Again, some simple algebra gives the correct answer. If the ball costs x and the bat costs t, then the total cost must add up to \$1.10:

$$ t + x = 1.1 $$

And the bat costs \$1 more than the ball:

$$ t = 1 + x $$

Substituting t into the first formula gives:

$$ 1 + x + x = 1.1 \implies x = 0.05 $$

So of course the ball costs \$0.05 and the bat costs \$1.05.

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