The Potato Paradox

Key idea. Only the water evaporates, so the mass of dry matter does not change. Track that invariant rather than the water.

Initially the dry matter is $1\%$ of $100$ pounds, so there is $1$ pound of it. After drying the watermelon is $98\%$ water, so this same pound is $2\%$ of the new weight $W$. Hence

The watermelon has lost $49$ of its $99$ pounds of water and kept $49$ pounds. Check: $49/50 = 98\%$.

Generalisation. Suppose the water fraction falls from $p$ to $q$ (with $q<p$) and the dry matter is unchanged. Then the dry matter is $(1-p)W_0 = (1-q)W_1$, so $W_1 = W_0\,\frac{1-p}{1-q}$. The ratio depends on the dry fractions, not on the water fractions. Here the dry fraction doubles from $1\%$ to $2\%$, so the weight halves.