The Potato Paradox, math that surprises you lesson

A batch of potatoes weighs 100 kilograms and is 99 percent water by mass. It dries until it is 98 percent water. The water share has dropped by only one point, so a weight of about 99 kilograms sounds right, yet the batch now weighs 50 kilograms. This page explains the answer by following the one amount that never changes, the dry material, and it separates percentage points from changes in mass.

Key terms

Potato paradox

Definition The potato paradox is an idealized puzzle about drying. A batch of potatoes has a total mass of 100 kilograms, and 99 percent of that mass is water, while the rest is dry material. The batch dries, and only water leaves, so the dry material keeps its mass. Drying stops when water makes up 98 percent of the new total mass. The puzzle asks for the new total mass, and the answer is 50 kilograms, half the original. More generally, if a batch contains a fixed dry mass $D > 0$ and ends with water fraction $p$ , where $0 \le p < 1$ , its final total mass is

M = \frac{D}{1 - p}.

The rule that only water leaves matters. If some dry material were lost as well, the final mass would be smaller than 50 kilograms and could not be found without knowing how much was lost. The numbers form a model chosen for clear arithmetic, not a claim about the water content of real potatoes.

In plain words The water percentage looks almost unchanged, but the dry part tells a different story. It was 1 percent of the batch and becomes 2 percent, while its mass stays the same, so the whole batch must have shrunk to half its size.

Example At the start the batch holds 99 kilograms of water and 1 kilogram of dry material. At the end water is 98 percent of the total, so the dry material is the other 2 percent, and the total mass is

M = \frac{1 \text{ kg}}{0.02} = 50 \text{ kg}.

The final batch holds $50 - 1 = 49$ kilograms of water, and $49/50 = 0.98$ , so water is 98 percent of it, as required. The batch lost $99 - 49 = 50$ kilograms of water. The tempting answer of 99 kilograms fails the same check. Losing only 1 kilogram of water leaves 98 kilograms of water in a total of 99 kilograms, a water share of

\frac{98}{99} \approx 98.99\%,

almost a full percentage point above the required 98 percent.

Percentage

Definition A percentage is a ratio written as a number of hundredths. A part is $p$ percent of a whole when

\frac{\text{part}}{\text{whole}} = \frac{p}{100},

and $p$ percent of a quantity $Q$ means $\tfrac{p}{100}\, Q$ . Every percentage refers to a particular whole, also called its base. When the whole changes, the same part becomes a different percentage, and the same percentage describes a different amount.

In plain words Percent means per hundred. A percentage says nothing about how much of something there is until you know what it is a percentage of.

Example The 1 kilogram of dry material is 1 percent of the original 100 kilograms and 2 percent of the final 50 kilograms. The phrase 98 percent water refers to the new total, so it means $0.98 \times 50 = 49$ kilograms of water, not $0.98 \times 100 = 98$ kilograms. The tempting answer of 99 kilograms takes one percent of the old total away, as if the new percentage still had the old 100 kilograms as its base.

Percentage point

Definition A percentage point is the unit for the difference between two percentages. A change from $a$ percent to $b$ percent is a change of $b - a$ percentage points. It differs from the relative change of a quantity, which divides the change by the starting value. For a quantity that goes from a value $x_0 \ne 0$ to a value $x_1$ , the relative change is

\frac{x_1 - x_0}{x_0},

usually written as a percentage and then called a percent change. Neither a change in percentage points nor a percent change of a share gives the change in the underlying amounts unless the wholes are known.

In plain words Percentage points say how far apart two percentages are. A percent change says how big a change is compared with where it started. Neither tells you the change in kilograms, which here is large even though the water share moves by only one point.

Example In the potato paradox the water share falls from 99 percent to 98 percent, a drop of 1 percentage point and a relative change of $-1/99 \approx -1.01\%$ . The dry share rises from 1 percent to 2 percent, also 1 percentage point, but its relative change is $+100\%$ . Meanwhile the total mass falls from 100 to 50 kilograms, by 50 percent, and the water mass falls from 99 to 49 kilograms, by about 50.5 percent. One drying process gives all of these numbers, and the doubling of the dry share is the one that matches the halving of the total.

Ratio

Definition The ratio of a quantity $a$ to a quantity $b$ , with $b \ne 0$ , compares them by division, as the quotient $a/b$ . A fraction of a whole is the ratio of a part to the whole. If a part $D$ stays fixed and makes up the fraction $f$ of the whole, where $0 < f \le 1$ , then the whole is

M = \frac{D}{f}.

So the whole is inversely proportional to the fraction, and doubling the fraction halves the whole.

In plain words When the part does not change, a larger share can only come from a smaller whole. If one kilogram is a fiftieth of the batch, the batch weighs fifty kilograms.

Example With 1 kilogram of dry material and water fraction $p$ , the dry fraction $f = 1 - p$ fixes the total mass at $M = 1/f$ kilograms.

Water share	Dry fraction	Total mass	Water
99 percent	$1/100$	100 kg	99 kg
98 percent	$1/50$	50 kg	49 kg
96 percent	$1/25$	25 kg	24 kg
90 percent	$1/10$	10 kg	9 kg
50 percent	$1/2$	2 kg	1 kg

The ratio of water to dry material tells the same story. It is 99 to 1 at the start and 49 to 1 at 98 percent water, so each kilogram of dry material carries 49 kilograms of water instead of 99.

Check yourself

The batch from the potato paradox keeps drying until it is 95 percent water. What does it weigh then, and how much water does it hold?

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The dry material is now 5 percent of the total, which is one twentieth, and it still weighs 1 kilogram. So the total mass is

M = \frac{1 \text{ kg}}{0.05} = 20 \text{ kg},

of which $20 - 1 = 19$ kilograms are water. The check $19/20 = 0.95$ confirms the 95 percent water share.

A different batch weighs 100 kilograms and is 60 percent water. It dries, again losing only water, until it is 59 percent water. What does it weigh now? Why is the change so much smaller than in the potato paradox?

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The dry material weighs 40 kilograms and now makes up 41 percent of the total, so

\begin{aligned} M &= \frac{40 \text{ kg}}{0.41} = \frac{4000}{41} \text{ kg} \\[0.4em] &\approx 97.56 \text{ kg}. \end{aligned}

The water is $4000/41 - 40 = 2360/41 \approx 57.56$ kilograms, and $2360/4000 = 0.59$ confirms the 59 percent share. The batch lost only about 2.44 kilograms. A drop of one percentage point in the water share always means a rise of one percentage point in the dry share, but here the dry share grows from 40 to 41 percent, a relative increase of only 2.5 percent, so the total shrinks only by the factor $40/41$ . In general, a dry share that grows from $d$ to $d + 1$ percent shrinks the total by the factor $d/(d + 1)$ . That factor is one half only for $d = 1$ , a batch that starts at 99 percent water, where one more point doubles the dry share.

Run the process backward. Water is added to the dried 50 kilogram batch, which is 98 percent water, until it is 99 percent water again. How much water is added? Describe the change in the water share in percentage points and as a percent change, and compare it with the change in total mass.

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The 1 kilogram of dry material must become 1 percent of the total, so the total must be $1/0.01 = 100$ kilograms, and 50 kilograms of water are added. The water share rises from 98 to 99 percent, by 1 percentage point, which is a relative change of

\frac{99 - 98}{98} = \frac{1}{98} \approx 1.02\%.

The total mass doubles from 50 to 100 kilograms, a change of 100 percent. A small change in a share can again go with a large change in mass.

The Potato Paradox

Video 1, chapter 5

Key terms

Potato paradox

Percentage

Percentage point

Ratio

Check yourself