Percentage Increase
Percentage increase measures how much a quantity has grown relative to its original value, expressed as a percent.
Formula
\text{percentage increase} = \left(\frac{\text{new} - \text{original}}{\text{original}}\right) \times 100\%
Definition
Percentage increase tells you how much something went up compared to where it started, as a percent: find the amount it increased, divide by the original amount, then multiply by $100$. Formally, percentage increase $= ((\text{new value} - \text{original value})/\text{original value}) \times 100\%$, which can also be written as new value $= \text{original} \times (1 + p/100)$ where $p$ is the percentage increase. In calculus, the instantaneous rate of relative change is the logarithmic derivative $\frac{d}{dt}[\ln f(t)] = f'(t)/f(t)$; for exponential growth $f(t) = P e^{rt}$, this relative rate is constant, and in finance the continuously compounded rate of return is $r = \ln(V_T/V_0)/T$.
Example
A price that goes from $\$50$ to $\$60$ increased by $\$10$, and $10/50 \times 100 = 20\%$. A town's population growing from $25{,}000$ to $31{,}000$ is a $24\%$ increase ($6{,}000/25{,}000 \times 100$), and the multiplier form confirms it: $25{,}000 \times 1.24 = 31{,}000$. Chained increases do not add simply: a $10\%$ increase followed by another $10\%$ increase gives a multiplier of $1.10 \times 1.10 = 1.21$, a $21\%$ total increase, not $20\%$; the equivalent log return, $\ln(1.21) = 0.1906 \approx 19.06\%$, is the one measure that does add linearly, since $2\ln(1.10) = \ln(1.21)$.
Key Insight
Percentage increase is always compared to the ORIGINAL amount, not the new amount: a jump from $\$50$ to $\$60$ is a $20\%$ increase, but the same $\$10$ increase from $\$100$ would only be a $10\%$ increase. The multiplier $(1 + p/100)$ is the key to chained percent increases, since successive percent increases do not add. Log returns (continuously compounded) have the additivity property that simple percent increases lack, which is why finance and information theory favor logarithmic scales: Shannon entropy, decibels, and Richter magnitude are all examples where logarithmic percent change gives a more natural, additive measure of relative magnitude.