Don’t trust your intuition, part 5: The Potato Paradox
The potato paradox is a simple question: You have 1 Kg of potato. 99% of the mass of the potato is water. You leave it out to dehydrate until it is 98% water. How much do the potatoes weigh now?
Make a guess and keep it in mind as we explore this question. The answer may surprise you.
What 1% looks like
The potatoes in this paradox are made up of 99% water and 1% “potato stuff”. This means that, if we have 100 grams of potato, 1g will be potato and 99g will be water, as show in Figure 1 below.
Figure 1: 100 blocks: 1 yellow (potato) and 99 blue (water)
Having 1kg of potato, or 10 rows of 100g, we can see that we will have 10g of actual “potato stuff”, in total, as shown in Figure 2 below.
Figure 2: 1% of 1kg is 10g
The key to the solution is the fact that these 10 grams of potato won’t go anywhere, no matter what happens to the water content.
Solving the potato paradox
The problem tells us that, after some evaporation, the water content now makes up 98% of the total mass of the potatoes. This leaves 2% for the potato stuff. Now, remember that this 2% is made up of 10g of potato, the part that does not change no matter what happens to the water content. The question then becomes:
If 2% of of the total weight of potatoes is 10g, what is the total weight?
Well, let’s keep drawing rows of 100 blocks, make 2 of them yellow, until we have 10 yellow blocks, representing the potato stuff. This is what Figure 3 does for us. We can see that, once we have drawn 5 rows, all 10 yellow blocks have been used. This approach leads us to the answer: 500g of potato!
Figure 3: 10g is 2% of 500g
The answer in summary
So, if 1Kg of potatoes is 99% water by weight, and if, after evaporation, the water makes up 98% of the new weight, then the total weight of the potatoes must now be 500 grams or half a kilogram.
Percentages are hard!
Most people guess a much larger number that 0.5 Kg because, they think, we have only lost 1% of the water content. A more helpful way of thinking about it is this: The significance of the potato stuff has doubled. This corresponds to a loss of around half of the water content.
If you have made the same error, do not despair! Percentages are hard and many questions involving percentages are counter-intuitive.
Please provide your feedback
This post was meant to be a personal challenge in explaining a tricky problem involving percentages without assuming knowledge of fractions or algebra. I hope I have succeeded but I want to know if I haven’t. Please provide me with feedback, especially if you are a “words person”.
If you want to test your new understanding, then see if you can work out the new weight of the potatoes if the water content became 50%. A more challenging version would be to bring the water content down to 40%.
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