The Master Formula Box
Net Change: $x + y + \frac{xy}{100}$Base Value: $\frac{\text{Final Value}}{\text{Multiplier}}$Product Stability: If $A \times B = \text{Constant}$, and $A$ increases by $x\%$, $B$ must decrease by $\frac{100x}{100+x}\%$.
Dimensional Analysis of Percentage
Percentages are Dimensionless. They are pure ratios $[M^0 L^0 T^0]$. This is why you can add a percentage error to a value without changing its unit. However, the absolute change carries the unit of the base variable.
Variations: Percentage to Fraction Conversion
Memorizing these common fractions will make your calculations $5\times$ faster:
- $12.5\% = 1/8$
- $16.66\% = 1/6$
- $33.33\% = 1/3$
- $37.5\% = 3/8$
- $66.66\% = 2/3$
Shortcuts & Mnemonics
- The "Split" Trick: To find $15\%$ of $400$, find $10\%$ ($40$) and half of that ($20$), then add them ($60$).
- AB Rule: If A is $25\%$ more than B, B is $20\%$ less than A. If A is $50\%$ more, B is $33.33\%$ less.
- Mnemonic: "Change over Original, keep it under the hundred-fold."
Edge Cases
- $100\%$ Increase: Means the value has doubled ($1 \to 2$).
- $100\%$ Decrease: Means the value is now zero.
- Infinite Increase: In percentages, this is simply a very large multiplier, but in a decrease, you can never go beyond $100\%$ (unless you are dealing with negative quantities like debt).
