Percentage Decrease
Fractions & DecimalsPercentage decrease measures how much a quantity has fallen relative to its original value, expressed as a percent.
Formula
\text{percentage decrease} = \left(\frac{\text{original} - \text{new}}{\text{original}}\right) \times 100\%
Definition
Percentage decrease tells you how much something went down compared to where it started. Find the amount it decreased, divide by the original amount, and multiply by $100$.
Example
A jacket was $\$80$ and is now on sale for $\$60$. Decrease $= \$80 - \$60 = \$20$. Percentage decrease $= 20/80 \times 100 = 25\%$. The jacket decreased by $25\%$.
Key Insight
Just like percentage increase, percentage decrease is always compared to the original (starting) value. Losing $\$20$ from $\$80$ is a $25\%$ decrease; losing the same $\$20$ from $\$200$ would only be a $10\%$ decrease.
Definition
Percentage decrease $= ((\text{original} - \text{new})/\text{original}) \times 100\%$. New value $= \text{original} \times (1 - p/100)$, where $p$ is the percent decrease. A $100\%$ decrease means the value reaches zero. A decrease greater than $100\%$ is not defined in this context, as quantities cannot decrease below zero in most real-world settings.
Example
A car depreciates from $\$24{,}000$ to $\$18{,}000$. % decrease $= (6000/24000) \times 100 = 25\%$. Using the multiplier: $\$24{,}000 \times 0.75 = \$18{,}000$. Two successive $20\%$ decreases: $0.80 \times 0.80 = 0.64$, a total $36\%$ decrease, not $40\%$.
Key Insight
Percentage decrease and increase are not symmetrical. A $50\%$ decrease followed by a $50\%$ increase does not return to the original: $\$100 \times 0.5 = \$50$, then $\$50 \times 1.5 = \$75$. The decrease wins because the subsequent increase is applied to the smaller value.
Definition
Percentage decrease is a relative change in the negative direction: $(f(t_1) - f(t_2))/f(t_1) \times 100$. For exponential decay $f(t) = P e^{-rt}$, the percentage decrease over interval $T$ is $(1 - e^{-rT}) \times 100$. The half-life $T_{1/2} = \ln(2)/r$ is the time for a $50\%$ decrease, fundamental in radioactive decay and pharmacokinetics.
Example
Carbon-14 has a half-life of $5730$ years. After $11{,}460$ years ($2$ half-lives), the amount decreases by $75\%$ (to $25\%$ of original). Percent remaining $= (1/2)^{t/5730} \times 100\%$. At $t = 5730$: $50\%$ remaining ($50\%$ decrease).
Key Insight
The asymmetry between percent increase and decrease reflects the multiplicative structure of ratios: a $p\%$ decrease followed by a $p\%$ increase yields a factor of $(1-p/100)(1+p/100) = 1 - (p/100)^2 < 1$. This is related to the AM-GM inequality and has a surprising consequence: any sequence of percent ups and downs that includes even one round trip results in a net loss.