Percentages – Advanced University‑Level Study Notes
(MIT‑style treatment for the JNU Entrance Quantitative Aptitude Section)
1. Introduction
Percentages are ubiquitous in quantitative reasoning: they quantify relative change, compare proportions, and encode error estimates. At the university level, we treat percentages as ratios and study them through the lens of algebra, calculus, and statistics. These notes begin with first‑principles definitions, progress to rigorous derivations, explore edge cases, and culminate in a challenging problem that synthesises multiple concepts.
2. The Percent as a Ratio
| Symbol | Meaning | Conversion |
|---|---|---|
| (p) | Percentage value | (p% = \dfrac{p}{100}) |
| (\displaystyle \frac{a}{b}) | Ratio | (\displaystyle 100,\frac{a}{b}% = \frac{100a}{b}% ) |
Definition 1 (Percent).
A percentage (p%) is the number (p) per hundred. In other words, it is the ratio (p/100).
First‑principle derivation.
Let a quantity (Q) be partitioned into two parts (Q_1) and (Q_2) such that (Q_1 = k,Q) for some real (k). Then the percentage of (Q) represented by