Section 32.3

Percentages, Mixtures and Growth

The potato paradox, repeated dilution, lily-pad doubling, compound-growth estimates.

Toolkit

These problems hinge on tracking the right invariant or multiplicative factor through repeated operations, rather than re-deriving each step from scratch.

Mixture/dilution: if a fraction $r$ of a mixture is replaced by pure solvent each time, the concentration of the original substance after $n$ repetitions is $C_0(1-r)^n$ ; solve for $n$ or $r$ via logarithms.
Percentage changes compound multiplicatively, not additively: a sequence of changes $p_1,\dots,p_n$ (as fractions) gives final value $V_0\prod_i(1+p_i)$ . A rise of $a\%$ then fall of $a\%$ gives factor $1-(a/100)^2<1$ : symmetric percentage moves never cancel.
Invariants: in “two quantities, one changes while a combination (sum, product, difference) stays fixed” problems, identify the conserved quantity first (e.g. dry mass $=$ total mass $\times$ (1 $-$ water fraction)).
Doubling/exponential growth: if a quantity doubles every $T$ days, multiplying the start amount by $2^k$ shifts every milestone earlier by exactly $kT$ days — independent of the target level.
Compound growth estimates: $(1+1/n)^n = e\left(1-\frac{1}{2n}+O(n^{-2})\right)$ for large $n$ ; for small $n$ , expand directly via the binomial theorem.

Problems