Percentage Increase
Fractions & DecimalsPercentage 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 $100$.
Example
A price went from $\$50$ to $\$60$. Increase $= \$60 - \$50 = \$10$. Percentage increase $= 10/50 \times 100 = 20\%$. The price increased by $20\%$.
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.
Definition
Percentage increase $= ((\text{new value} - \text{original value})/\text{original value}) \times 100\%$. This can also be expressed as: new value $= \text{original value} \times (1 + r)$, where $r = \text{percentage increase}/100$. If the percentage increase is known, the new value $= \text{original} \times (1 + p/100)$.
Example
A town's population grew from $25{,}000$ to $31{,}000$. Increase $= 6{,}000$. % increase $= (6000/25000) \times 100 = 24\%$. New value formula check: $25{,}000 \times 1.24 = 31{,}000$.
Key Insight
The multiplier $(1 + p/100)$ is the key to chained percent increases. A $10\%$ increase followed by another $10\%$ increase gives a multiplier of $1.10 \times 1.10 = 1.21$, which is a $21\%$ total increase - not $20\%$. Successive percent increases do not add simply.
Definition
Percentage increase is the relative change: $(f(t_2) - f(t_1))/f(t_1) \times 100$. 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 $f(t) = P e^{rt}$, the percentage increase over any fixed interval is constant. In finance, the continuously compounded rate of return is $r = \ln(V_T/V_0)/T$.
Example
A stock price grows from $\$100$ to $\$110$ ($10\%$ increase), then from $\$110$ to $\$121$ ($10\%$ again). Total: $\$21$ increase on $\$100$ original $= 21\%$ increase. Log return: $\ln(121/100) = \ln(1.21) = 0.1906 \approx 19.06\%$ - the continuously compounded equivalent, which adds linearly: $2\ln(1.10) = \ln(1.21)$.
Key Insight
Log returns (continuously compounded) have the additivity property that simple percent increases lack. This is why finance and information theory use logarithmic scales: Shannon entropy, decibels, and Richter magnitude are all examples where logarithmic (not linear) percent change gives additivity and a more natural measure of relative magnitude.