Pounds in the newspaper
Eric Weisstein's MathWorld gives the puzzle in pounds: 100 pounds of potatoes, 99% water, then 98%. As its source it names John Allen Paulos's 1995 book "A Mathematician Reads the Newspaper", page 81.
A sack holds 100 kg of potatoes, and they are 99% water. The sack is left outside overnight, some water evaporates, and now the potatoes are 98% water. How much do they weigh? Lock in your answer, then check it: dry the sack yourself, count on a computer and read the explanation.
A sack holds 100 kg of potatoes, and they are 99% water. Overnight some water evaporates, and now they are 98% water. It feels as if the sack lost a kilo, since the water went down by "1%". In fact the potatoes weigh 50 kg: the dry matter, 1 kg, was 1% of the mass and is now 2%, so the whole mass had to halve.
There is no probability here and no advanced formulas. Just percentages, which everyone knows from primary school. But try to answer quickly, as you would at a market stall: it is more fun to find out whether your first thought matches the right answer.
A sack holds exactly 100 kg of potatoes. They are 99% water, and the rest, 1%, is dry matter: starch, fibre, everything that does not evaporate.
The sack is left outside overnight. Some of the water evaporates, while the dry matter stays where it is.
In the morning the potatoes are checked: water now makes up 98% of their mass.
How much do the potatoes weigh now?
"The water went down by 1%, so the potatoes weigh a kilo less, 99 kg"
In fact 50 kg. The "1%" here is not a kilo but a percentage point of a share that is measured against a mass that has itself changed. There is one solid anchor: the dry matter. There was 1 kg of it, and there is still 1 kg.
Pick an answer with a row under the slider, then fine-tune it or type a number in kilograms.
Your answer will be compared with the right one here.
Hard to decide? Open the hints one at a time: the easy one first, then stronger ones.
Don't watch the water: it changes, and the share of water is taken of a mass that changes too. Find what stayed the same overnight.
How many kilos of dry matter are in the sack in the evening? And in the morning? What share of the mass is it in the morning?
If 1 kg is 2% of the sack, how much does the whole sack weigh?
The next rounds weigh the potatoes: you dry the sack yourself, the computer dries it gram by gram, and then comes the explanation. Answer round 1 first, or go on?
You chose to always show all rounds.
The sack is made of 100 cells of 1 kg each: 99 blue cells of water and one brown cell of dry matter. The slider evaporates water kilo by kilo. The buttons set a target, the share of water you want, and the sack dries down to it by itself.
There is always just one brown cell. The only way to change the share of water is to take blue cells away, and each of them weighs a kilo. For the brown cell to be 2% of the sack, the sack must consist of 50 cells.
The computer takes 100,000 g of potatoes and evaporates water one gram at a time, weighing the sack each time and working out the share of water. When the share falls to the value it wants, it plots a dot. To the right of the starting share it does the opposite and adds water a gram at a time, as if the potatoes were soaked. No formulas here, just scales. The grey dashed line is the intuition "one percent less water, one kilo less weight".
Once you answer, the simulation code appears here so you can run it yourself.
The same simulation, short and with no libraries. Run the JavaScript in your browser console (F12) or in Node; the Python needs only the standard library.
// 100 kg of potatoes at 99% water. Evaporate water a gram at a time and weigh the sack until the water share reaches the target.
// Shares in tenths of a percent, masses in grams: whole numbers only, no rounding errors.
const START = 100000, DRY = 1000; // 1% of 100,000 g is dry matter; it does not evaporate
function weigh(target) { // target: the water share, e.g. 980 = 98%
let total = START;
if (1000 * (total - DRY) >= target * total) { while (1000 * (total - DRY) > target * total) total--; } // dry
else { while (1000 * (total - DRY) < target * total) total++; } // soak
return total;
}
for (const t of [995, 980, 970, 950, 900, 500]) {
const kg = weigh(t) / 1000, formula = (DRY / 1000) / (1 - t / 1000); // dry matter / dry share
console.log(t / 10 + '% water:', kg.toFixed(3), 'kg, formula', formula.toFixed(3), 'kg, intuition', (100 - (99 - t / 10)).toFixed(1));
}
# 100 kg of potatoes at 99% water. Evaporate water a gram at a time and weigh the sack until the water share reaches the target.
# Shares in tenths of a percent, masses in grams: whole numbers only, no rounding errors.
START, DRY = 100000, 1000 # 1% of 100,000 g is dry matter; it does not evaporate
def weigh(target): # target: the water share, e.g. 980 = 98%
total = START
if 1000 * (total - DRY) >= target * total:
while 1000 * (total - DRY) > target * total:
total -= 1 # dry
else:
while 1000 * (total - DRY) < target * total:
total += 1 # soak
return total
for t in [995, 980, 970, 950, 900, 500]:
kg = weigh(t) / 1000
formula = (DRY / 1000) / (1 - t / 1000) # dry matter / dry share
print(f"{t / 10:g}% water: {kg:.3f} kg, formula {formula:.3f} kg, intuition {100 - (99 - t / 10):.1f}")
Intuition watches the water, because the water is what changes. But the share of water is measured against the whole mass, and the mass changes too. There is only one solid anchor: the dry matter, which weighs 1 kg throughout.
“In plain words” is hidden on all puzzles.
In the evening, 1 kg of dry matter is 1% of the sack. If 1 kg is 1%, the whole sack is 100 kg. In the morning the same 1 kg is 2%. If 1 kg is 2%, the whole sack is 50 kg. Nothing else needs to be worked out.
This is the core of the puzzle, though not the whole story: why intuition goes so wrong, and where this trap turns up in real life, comes next.
A room holds 99 men and one woman. How many men must leave for the room to be 98% men? There is still one woman, and she has to be 2% of the room, one person in fifty. So the room holds 50 people, and 50 men have to leave. It is the same puzzle, with men for water and the woman for dry matter.
We hear "99% water became 98%" and see a small change: it was 99, it is now 98. But look at the other side of the same numbers. The dry part was 1% and is now 2%. That is not "a little more", it is twice as much. And the dry part's share can only double if everything else halves, because the dry part itself does not grow.
The asymmetry is strong only near 100%. If the potatoes were 50% water and became 49%, the dry part would go from 50% to 51%, and the mass would barely change, from 100 kg to 98 kg. The closer the water share is to 100%, the smaller the dry part, and the more each percent changes it.
“Rigorous” is hidden on all puzzles.
| evening | morning | change | |
|---|---|---|---|
| water share | 99% | 98% | −1 pt |
| dry share | 1% | 2% | +100% |
| water, kg | 99 | 49 | −50.5% |
| dry matter, kg | 1 | 1 | 0 |
| mass, kg | 100 | 50 | −50% |
The water share changed by one percentage point, or by 1/99 ≈ 1.01% relative to itself. Neither tells you how much water is gone. For that you multiply the share by the mass, and in the morning the mass is different. The water went from 99 kg to 49 kg, down 50 kg, more than half.
| water | dry | mass |
|---|---|---|
| 99.5% | 0.5% | 200 kg |
| 99% | 1% | 100 kg |
| 98% | 2% | 50 kg |
| 97% | 3% | 33⅓ kg |
| 95% | 5% | 20 kg |
| 90% | 10% | 10 kg |
The mass is inversely proportional to the dry share: twice the share, half the mass. That is why the chart in round 3 is a hyperbola, not a straight line. Note the top row: soak the potatoes up to 99.5% water and the mass doubles to 200 kg.
A mass m₀ with water share p₀ dries to a share p₁. The dry mass does not change.
The potatoes: 100 kg, 99% water, then 98%.
How much water must evaporate: the difference in mass.
The "one kilo less" intuition is a different formula: m₁ = m₀ − m₀(p₀ − p₁), as if a percentage point always weighed 1% of the starting mass. That would be true if the percentage were taken of a fixed 100 kg. But the 98% is taken of the new mass, and the denominator 1 − p₁ turns the relationship into a hyperbola.
“Deeper” is hidden on all puzzles.
Near 100%, percentages squeeze huge differences into a few decimal places. Write the water share as a ratio of water to dry matter: 99% is 99 : 1, 98% is 49 : 1, and 99.9% is 999 : 1. Now you can see that "one percent less" halved the water per kilo of dry matter.
In the middle of the scale it is the other way round: 50% is 1 : 1, 49% is 0.96 : 1, a barely visible change. That is why statisticians often switch to ratios when shares are close to 0 or 100%, and why engineers compare not "99.9% and 99.8% uptime" but "one failed minute in 1,000 and one in 500".
The philosopher and logician Willard Van Orman Quine, in his essay "The Ways of Paradox", divided paradoxes into three kinds. Falsidical ones: the conclusion sounds absurd and really is false, because the reasoning hides an error, as in a "proof" that 1 = 2. Antinomies: reasoning that looks correct leads to a contradiction and forces a rethink of the rules of thought themselves, like the liar paradox.
The potatoes belong to the third kind, veridical paradoxes: the conclusion seems absurd, but it is true and the reasoning is sound. It is not the mathematics that is wrong but the expectation. The Monty Hall problem and the birthday paradox are of the same kind.
The formula works for any share, but near 100% the curve gets very steep. The trap itself is wider than potatoes: it springs whenever a percentage is taken of something that has changed too. And in real life you can check it in the kitchen, in a customer database and in a glass.
The general formula m₁ = m₀ · (1 − p₀)/(1 − p₁) says the mass depends only on how many times the dry share has changed. So the mass halves at 99% → 98%, at 90% → 80%, at 80% → 60% and at 50% → 0%. In the first case the water share fell by 1 point, in the last by 50, and the result is the same.
Near 100% this makes every tenth of a percent heavy. Potatoes at 99.9% water that dry to 99% lose 90% of their mass: 100 kg becomes 10 kg. And from 99.9% to 99.8% it is exactly half again. Try it with the sliders on the left.
A percentage is always a share of something. The potato trap springs when we quietly assume that "something" stays fixed. A price cut by 50% and then raised by 50% ends up at 75% of where it started, because the rise is taken of the already reduced price. Unemployment that rose from 1% to 2% rose "by 1 percentage point" and also "by 100%", and both headlines are true.
Before comparing two percentages, ask: what base is each of them taken from, and is it the same one? If the base has changed, look for a quantity that has not, like the dry matter in the potatoes, and count from that.
Each time we find what does not change and divide it by the new share. Water contents of foods are from the USDA FoodData Central database (SR Legacy).
| Situation | What stays fixed | What comes out |
|---|---|---|
| Grapes are dried into raisins: grapes are 80.54% water, dark seedless raisins 15.46% | the dry matter: 194.6 g per kilo of grapes | 1 kg of grapes gives 0.1946 / 0.8454 ≈ 230 g of raisins. A kilo of raisins takes ≈ 4.34 kg of grapes. |
| A mailing list: 10,000 contacts, 99% of whom have not opened an email in ages. Inactive ones are removed until they make up 98% | the 100 active contacts | They must become 2% of the list, so 5,000 contacts remain. "1% fewer" inactive contacts means deleting half the list. |
| 1 l of a drink at 40% alcohol is diluted with water down to 20% | the 0.4 l of alcohol | 0.4 / 0.2 = 2 l: you add 1 l of water. The real volume comes out slightly smaller, because a mix of alcohol and water shrinks a little. |
| An asset is 1% of a 100,000 portfolio. Its share is to be raised to 2% without buying more of it | the 1,000 in that asset | The whole portfolio has to be 1,000 / 0.02 = 50,000: the rest must shrink from 99,000 to 49,000, more than half. |
Eric Weisstein's MathWorld gives the puzzle in pounds: 100 pounds of potatoes, 99% water, then 98%. As its source it names John Allen Paulos's 1995 book "A Mathematician Reads the Newspaper", page 81.
In David Darling's "The Universal Book of Mathematics" (2004, p. 253), Fred brings home 100 kg of potatoes which, "being purely mathematical potatoes", are 99% water, and leaves them outside overnight. This is the wording Wikipedia quotes.
The American car radio show "Car Talk" set its listeners a puzzler every week. The potato puzzle was one of them, under the name "Porch Potatoes": a sack left out on the porch dries out.
Raw potatoes with skin are 79.25% water (USDA). That is 20.75 kg of dry matter per 100 kg, so if the water drops by one point, to 78.25%, the sack only slims down to 95.4 kg. The trap really works near 100%.
A cucumber with peel is 95.23% water, a watermelon 91.45% (USDA). Even the cucumber is far from 99%: 100 kg of cucumbers at 95.23% water that dry to 94.23% weigh 82.7 kg, not 50.
Grapes are 80.54% water, dark raisins 15.46%. The water share falls from 80% to 15%, and a kilo of raisins takes about 4.3 kg of grapes. That is one reason dried fruit costs more than fresh.
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