# Why can the Potato Paradox be quantified as $0.99⋅100−0.98(100−w)=w$?

I understand the two Simple Explanations, but not Algebraic Explanation Method 2. I substitute $$x$$ in Wikipedia with $$w$$, as $$x$$ is already used for another method.

The weight of water in the fresh potatoes is $$0.99 ⋅ 100.$$

If $$w$$ is the weight of water lost from the potatoes when they dehydrate then $$\color{green}{0.98 ( 100 − w )}$$ is the weight of water in the dehydrated potatoes. Therefore:

$$0.99 ⋅ 100 \color{darkorange}{−}\color{green}{0.98 ( 100 −w )} = \color{red}{w}. \tag{?}$$

1. Why $$\color{orange}{−}$$ here, when we added (and never subtracted) in the LHS in Algebraic Explanation Method 1?

2. How's the LHS devised? I understand the 0.98, as the problem statement requires 98% water after dehydration. But I would've never dreamed or excogitated of $$\color{green}{0.98 ( 100 −w )}$$?

3. Why do we make the LHS equal to $$\color{red}{w}$$? I would've never excogitated equating the LHS with $$\color{red}{w}$$?

I have a BA in Economics, and already know how $$99 − 0.98 ( 100 − w ) = w \iff 1 + 0.98x = x$$.

• (original weight due to water) - (weight due to water, after dehydration) = (weight lost) – Zubin Mukerjee Jan 23 at 6:49
• You really need to edit the question to include some missing details - otherwise it is simply unclear where the 0.98 figure comes from. You have potatoes which are 99% water and they dehydrate until they are 98% water. How much weight do they lose? The equations are simply computing the weight of water before and after using the proportions given. The difference is the change in weight. – Mark Bennet Jan 23 at 7:02

Perhaps $$0.99\cdot 100=w+0.98(100-w)$$ is more intuitive: The total amount of water is the same before and after the evaporation. Thus the weight of water in the potatoes before the evaporation is equal to the weight of evaporated water, $$w$$, plus the weight of the water that's left in the potatoes, $$0.98(100-w)$$.
You have the original water weight equal to $$0.99\cdot 100$$. The total weight of the potatoes before is $$100$$, after you lose $$w$$ water you have the total weight $$100-w$$. Out of that, $$98\%$$ is water, so the water left is $$0.98(100-w)$$. You can probably better understand if you switch the order of the terms, and you write "original-lost=final": $$0.99\cdot 100-w=0.98(100-w)$$ This is the same equation, I just moved some terms around.