Advanced Study Notes on Percentages
(MIT‑Level Quantitative Aptitude – Insurance & Finance Context)
1. Introduction
Percentages are ubiquitous in insurance (LIC, GIC) and finance:
- premium calculations,
- return on investment,
- discounting,
- risk‑adjusted rates.
In these notes we
- derive the notion of a percentage from first principles,
- explore algebraic properties and edge cases,
- connect percentages to linear transformations and logarithms,
- discuss applications in insurance, and
- present a challenging problem that blends multiple concepts.
Notation
- (P) : base quantity (e.g., premium).
- (p%) : percentage value expressed as a decimal (p/100).
- (r%) : rate of change or interest rate.
2. From Ratios to Percentages
2.1 Definition
A percentage is a ratio expressed per hundred.
If (x) is a fraction of (y), then
[ x = \frac{p}{100},y \quad \Longleftrightarrow \quad p = 100\cdot\frac{x}{y} ]
Thus a percentage is simply a dimensionless scalar that scales a base quantity.
2.2 Linear Transformation View
Given a base (y), the mapping
[ T_p : y \mapsto y + \frac{p}{100},y = \Bigl(1+\frac{p}{100}\Bigr)y ]
is a linear operator on (\mathbb{R}).
- Multiplicative factor (1+\frac{p}{100}) is the growth factor.
- Composition of percentages corresponds to multiplication of growth factors:
[ T_{p_1}\circ T_{p_2} = T_{p_1+p_2+\frac{p_1p_2}{100}} ]
This shows that percent changes do not commute additively unless they are small (so (p_1p_2/100 \approx 0)).
3. Fundamental Properties
| Property | Formula | Intuition |
|---|---|---|
| Percent of a percent | ((p%)%) of (x) → (\frac{p}{100}\cdot\frac{1}{100}x = \frac{p}{10,000}x) | Each “%” introduces a division by 100 |
| Percent change | (\frac{y-x}{x}\times100% = \left(\frac{y}{x}-1\right)\times100%) | Ratio of difference to initial value |
| Relative vs absolute | Absolute change: (y-x). Relative change: (\frac{y-x}{x}). | Relative change is dimensionless |
| Compound percentage | After (n) periods: (x(1+\frac{p}{100})^n). | Multiplicative growth across periods |
| Inverse operation | To undo a (p%) increase: divide by (1+p/100). | Equivalent to a (-p%) decrease only when (p) is small |
4. Edge Cases & Subtleties
| Edge Case | Explanation | Handling |
|---|---|---|
| Zero base | (\frac{0}{0}) undefined. | Percent change only defined when base ≠ 0. |
| Negative base | Sign matters: e.g., (p%) of (-100) is (-p). | Keep the sign of the base in calculations. |
| Percent > 100 | Implies multiplication factor > 2. | No mathematical restriction; represent as (>100%). |
| Negative percent | Decreases (discounts). | Treat (-p%) as a division by (1+ |
| Percent of percent | Each “%” multiplies by (1/100). | Many authors mistakenly treat ((p%)%) as (p%); correct is (p/100%). |
| Percent change with negative change | (\frac{y-x}{x}) can be negative. | Interpret as a percentage decrease. |
5. Percentages in Insurance & Finance
5.1 Premium & Return Calculations
Let (C) be the initial premium (cost).
If the policy yields a simple interest rate (r%) per annum for (t) years:
[ \text{Return} = C + C\cdot