# THE POTATO PARADOX

Imagine that a farmer has a sack containing 100 lbs of potatoes. The potatoes, he discovers, are comprised of 99% water and 1% solids, so he leaves them in the heat of the sun for a day to let the amount of water in them reduce to 98%. But when he returns to them the day after, he finds his 100 lb sack now weighs just 50 lbs. How can this be true? Well, if 99% of 100 lbs of potatoes is water then the water must weigh 99 lbs. The 1% of solids must ultimately weigh just 1 lb, giving a ratio of solids to liquids of 1:99. But if the potatoes are allowed to dehydrate to 98% water, the solids must now account for 2% of the weight—a ratio of 2:98, or 1:49—even though the solids must still only weigh 1lb. The water, ultimately, must now weigh 49lb, giving a total weight of 50lbs despite just a 1% reduction in water content. Or must it?

Although not a true paradox in the strictest sense, the counterintuitive Potato Paradox is a famous example of what is known as a veridical paradox, in which a basic theory is taken to a logical but apparently absurd conclusion.

There is no paradox here, or anything even close.

At 99 pounds water and 1 pound solids, the water is 99/(99+1) = 99% by weight of the potatoes.

Removing 1% of the water (0.99 pounds) makes the ratio of solids be 98.01/(98.01+1) = 98.99% by weight of the potatoes. (Or, removing water equivalent to 1% of the weight of the potatoes makes the ratio of solids be 98/(98+1) = 98.9899% by weight.) This amounts to reducing the quantity of water by 1% (or 1 pound), but reducing the ratio of the water by just 0.01% (or 0.010101…%).

The potential confusion results from linguistics, not math. It has to do with the two different uses of the word “percentage”.

Let’s look at another example. Let’s say that 99% of 100 people surveyed like ice cream, leaving one person (1%) of people that don’t like ice cream. To reduce the percentage of people that like ice cream to 98%, just 1% (one person) needs to join the group of people that don’t like ice cream, making 2% of people not like ice cream. However, if instead of an ice cream fan converting I simply eliminate one person that likes ice cream from the survey, instead of 2% of people not liking ice cream, there are just 1.01%.

### Your Answer

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