Why can the Potato Paradox be quantified as $1 + \dfrac{98}{100}{x} = x$?

Question

I understand the two Simple Explanations, but not the Algebraic Explanation Method 1.

After the evaporating of the water, the remaining total quantity, $x$ , contains 1 lb pure potatoes and $(98/100)x$ water. The equation becomes:

$1 + \dfrac{98}{100}{x} = \color{red}{x}$

Why do we make the LHS equal to $\color{red}{x}$? In other words, I know how to devise or derive the LHS, but how can you expect to equate the LHS with $\color{red}{x}$? It just feels bizarre for $x$ to appear on both sides.

Answer 1

"After the evaporating of the water, the remaining total quantity x..."

This tells us that there is $x$ lb left.

"... contains 1 lb pure potatoes and (98/100)x water..."

So the remaining total quantity (as we recall, x) is $1$ lb of potato (which we can deduce to be $2\%$ of $x$) and $98\%$ water. This is the same as saying that of the remaining $x$ lb, $2\%$ of it is the $1$ lb of potato and $98\%$ of it is water.

So,

$2\%\text{ of }x + 98\%\text{ of }x = 100\%\text{ of }x$

Which is the same as saying,

$1 \text{ lb} + \frac{98}{100}x = x$ (Recall that $2\%$ of $x$ is the $1$ lb of potato)

Answer 2

The total weight of the potatoes after evaporating the water is $x$. This is how we are defining $x$.

Another way to express the total weight of the potatoes is the weight of the potato flesh plus the weight of the water, both after evaporating the water. The potato flesh comprised $1\%$ of the original $100$ pounds of potatoes, so there is $1$ pound of potato flesh (this is true both before and after the potatoes are dried out). After drying the potatoes out, we know they are $98\%$ water, so the amount of water is $98\%$ of $x$, or $\frac{98}{100}x$. Therefore if we add the amount of potato flesh and the amount of water, we get $1+\frac{98}{100}x$.

Since we calculated the total weight of the potatoes in two different ways, these expressions must be equal. Hence $1+\frac{98}{100}x=x$.

Answer 3

I like letters. We start with $p$ weight of pure potato, $w$ of water, later we are down to $d$ of water.

$$ \frac{w}{p+w} = \frac{99}{100}, $$ $$ 100 w = 99 p + 99 w $$ $$ w = 99p $$ $$ w + p = 100p $$ $$ $$

=============================================================

$$ $$ $$ \frac{d}{p+d} = \frac{98}{100}, $$ $$ 100 d = 98 p + 98 d $$ $$ 2d = 98p $$ $$ d = 49 p $$ $$ d + p = 50p $$

=============================================================

I was curious what happens if even more water evaporates away; here the letter $a$ stands for arid. I am putting in a letter $k$ which does not really need to be an integer, but in any case $1 \leq k \leq 100.$

$$ $$ $$ \frac{a}{p+a} = \frac{100-k}{100}, $$ $$ 100 a = (100-k) p + (100-k) a $$ $$ ka = (100-k)p $$ $$ ka+kp = 100 p $$ $$ a + p = \left( \frac{100}{k} \right) p $$

This says that there is a hyperbola involved. If i take $k=4,$ this tells us that, to get 96%, we decrease to $25p$