Percentages – Advanced University‑Level Study Notes
Prepared for the Goa Police Constable Quantitative Aptitude Module
Goal – Provide a self‑contained, rigorous treatment of percentages that can be used to solve both textbook and contest‑style problems.
Audience – Students of MIT‑level mathematics, physics, engineering, or competitive programming.
1. From Ratios to Percentages – First Principles
| Symbol | Meaning | Explanation |
|---|---|---|
| (r) | Ratio of two numbers (a) and (b) | (r = \dfrac{a}{b}) |
| (p) | Percentage value | (p = r \times 100) |
| (x) | Base value (quantity to be scaled) | The number we are applying a percentage to |
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Percent as a scaled ratio
[ p% \text{ of } x ;=; \frac{p}{100},x ] This is the definition of a percentage. -
Inverse operation (recovering the base)
[ x ;=; \frac{p% \text{ of } x}{p/100} ] This shows that a percentage is a linear scaling. -
Linear map view
Define (T_p : \mathbb{R}\rightarrow \mathbb{R}) by (T_p(x)=\frac{p}{100}x).
Then (T_{p+q}=T_p+T_q) and (T_{pq}=T_p\circ T_q) (composition).
Hence percentages form a commutative monoid under addition and a semigroup under multiplication. -
Basis change
The map (x \mapsto \frac{x}{100}) is a change of basis from the base‑100 system to the unit system.
Percentages are simply coordinates in the base‑100 basis.
2. Fundamental Properties (with Proofs)
| Property | Formula | Proof |
|---|---|---|
| Additivity | ((p+q)% \text{ of } x = p% \text{ of } x + q% \text{ of } x) | (\frac{p+q}{100}x = \frac{p}{100}x + \frac{q}{100}x) (algebraic). |
| Multiplicativity | ((pq)% \text{ of } x = p% \text{ of } (q% |