Percentage Formulas and Concepts in Depth

Vocabulary flashcards covering core formulas, rules, methods, and dimensional relations in percentage arithmetic.

Per Cent

Literally meaning "per hundred," it defines a ratio relative to a reference base of $100$, where every quantity represents $100\%$ of itself.

Percent to Fraction Conversion

The algebraic conversion of a percentage into an irreducible fraction by dividing by $100$, represented by the formula $x\% = \frac{x}{100}$.

Fraction to Percent Conversion

The arithmetic conversion of a fraction into a percentage by multiplying the ratio by $100$, represented by the formula $\frac{a}{b} = \left(\frac{a}{b} \times 100\right)\%$.

Commutative Property of Percentage

The operational identity stating that $x\% \text{ of } y = \frac{xy}{100} = y\% \text{ of } x$, allowing factors to be swapped to simplify calculations.

Mirror Identity of Percentage

The calculation identity stating that $a\% \text{ of } b + b\% \text{ of } a = \frac{2ab}{100} = 2a\% \text{ of } b$.

Percentage Comparison Base Rule

The structural rule specifying that the quantity directly following the words "of" or "than" is placed in the denominator as the reference base.

Same-Unit Rule

The mathematical rule requiring both compared quantities to be converted into identical units of measurement before calculating percentage ratios.

Equal-Percent Relation

The proportionality identity stating that if $x\% \text{ of } A = y\% \text{ of } B$, then the ratio of the quantities inverts to $A : B = y : x$.

Transfer of a Percent

The substitution rule stating that if $x\% \text{ of } A = y\% \text{ of } B$, then $z\% \text{ of } B = \left(\frac{zx}{y}\right)\% \text{ of } A$.

Parts Method

A problem-solving framework that converts percentage changes into irreducible fractions $\frac{a}{b}$, representing an increase as the ratio $b : (b + a)$ and a decrease as $b : (b - a)$.

The AB Rule

The reciprocal relationship establishing that if quantity $A$ is more than $B$ by $+\frac{1}{n}$, then $B$ is less than $A$ by $-\frac{1}{n + 1}$; conversely, if $B$ is less than $A$ by $-\frac{1}{n}$, $A$ is more than $B$ by $+\frac{1}{n - 1}$.

Constant Product Rule

An application of inverse variation where if the product of two quantities is held constant ($\text{Price} \times \text{Consumption} = \text{Expenditure}$), an increase of $+\frac{1}{n}$ in one factor requires a reduction of $-\frac{1}{n + 1}$ in the other factor.

Successive Percentage Change Formula (Two Terms)

The formula for combining two consecutive percentage changes $a\%$ and $b\%$, expressed as $\left(a + b + \frac{ab}{100}\right)\%$, where negative signs represent decreases.

Same Rate Twice Shortcut

The abbreviated successive change formula $2a + \frac{a^2}{100}\%$ applied when a single percentage rate $a\%$ increases consecutively over two cycles.

Equal Increase and Decrease Rule

The net result principle stating that when an identical rate $x\%$ is applied first as an increase and then as a decrease, the final result is always a net decrease of $-\frac{x^2}{100}\%$.

Successive Percentage Change Formula (Three Terms)

The net change equation for three consecutive percentages $a$, $b$, and $c$, defined as $(a + b + c) + \frac{ab + bc + ca}{100} + \frac{abc}{10000}\%$.

Unknown-Factor Formula

A product formula establishing that if factor $x$ changes by $p\%$ and the product $z$ changes by $e\%$, the second factor $y$ changes by $\left[\frac{100 + e}{100 + p} - 1\right] \times 100\%$.

100-Vala Method

A calculation method that assumes an initial base of $100$ for known terms to determine an unknown factor when $a \times b = c$, restricted strictly to two-factor problems.

Linear Dimension Scaling (1D)

The dimensional principle stating that one-dimensional measures (length, radius, diagonal, perimeter, circumference) scale by factor $k$ and share the exact same percentage change as the side or radius.

Area Dimension Scaling (2D)

The dimensional principle stating that two-dimensional measures (area, Curved Surface Area, Total Surface Area) scale by factor $k^2$, calculated using a two-term successive formula.

Volume Dimension Scaling (3D)

The dimensional principle stating that three-dimensional solid measures scale by factor $k^3$, evaluated using a three-term successive change calculation.

2D to 1D Reverse Scaling

The technique of establishing an old-to-new ratio for an area and extracting its square root to determine the corresponding linear percentage change of the side or radius.

3D to 1D Reverse Scaling

The technique of constructing the old-to-new ratio of a solid's volume and extracting its cube root to deduce the percentage change of its linear measurements.

Series Growth and Depreciation Formula

The compounded formulation $\text{Final} = \text{Initial} \times \left(1 \pm \frac{r}{100}\right)^n$, where growth factors are multiplied successively over $n$ intervals.

Remaining Concept (Series Method)

An operational method applied when percentage deductions are taken exclusively from the remaining balance, evaluated via $\text{Original} \times \prod \left(1 - \frac{\text{percent taken}}{100}\right) = \text{Final}$.

Remaining Concept (Tree Method)

A diagrammatic procedure setting total initial value to $100x$, utilized when deductions refer to the total initial value or when fixed numerical amounts are subtracted mid-sequence.

Weighted Mean Formula

The formulation $\text{Mean} = \frac{n_1 m_1 + n_2 m_2}{n_1 + n_2}$ applied to calculate the combined percentage from two distinct groups of sizes $n_1, n_2$ and percentage rates $m_1, m_2$.

Alligation Ratio Formula

A cross-subtraction method determining the ratio of two group quantities based on their individual rates and the overall mean: $n_1 : n_2 = (m_2 - \text{Mean}) : (\text{Mean} - m_1)$.

Income-Expenditure-Savings Relation

The financial identity $\text{Income} = \text{Expenditure} + \text{Savings}$ ($I = E + S$), where the net percentage change in income is the weighted average $\Delta I\% = \frac{E \cdot e\% + S \cdot s\%}{E + S}$.

Two-Set Venn Diagram Formula

The set identity $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ used to remove double-counted overlaps inside a $100\%$ universal set.

Election Margin of Victory Formula

The voting formula calculating the victory margin between two candidates based on valid votes: $\text{Margin} = (2W - 100)\% \text{ of valid votes}$, where $W\%$ represents the winner's share.

Pass-Fail Connecting Rule

The examination marks rule relating a failing shortfall $x$ and a passing excess $y$ across two percentage scores $a\%$ and $b\%$ to find maximum marks: $M = \frac{(x + y) \times 100}{b - a}$.

Add-Back Trick

A calculation shortcut for tiered sales commissions that treats bonus percentages as applying from zero sales, structured as $\text{Earning} + b\% \times L = (c + b)\% \times \text{Total}$.

Pulp Constant Principle

The mass-balance principle for fruit evaporation stating that solid pulp remains unchanged: $\text{Fresh} \times (100 - \text{water}\%) = \text{Dry} \times (100 - \text{water}\%)$.

Wrong-Operation Error Percentage

The formula measuring arithmetic discrepancy resulting from an incorrect operator or fraction: $\text{Error } \% = \frac{|\text{Correct} - \text{Wrong}|}{\text{Correct}} \times 100$.