Simple & Compound Interest – Advanced University‑Level Study Notes
Target Audience: Delhi Patwari / Lekhpal Program
Level: MIT‑style – rigorous, first‑principles derivations, deeper theory, edge cases, and a challenging problem.
1. Introduction
Interest is the cost of borrowing (or the return on lending) a monetary amount over time.
Two canonical mechanisms govern its accumulation:
- Simple Interest – linear growth, no reinvestment of earned interest.
- Compound Interest – multiplicative growth, earned interest is reinvested.
These notes build from fundamental assumptions, derive the classic formulas, explore the continuous limit, contrast the two regimes, and discuss non‑standard scenarios that may arise in land‑record and revenue calculations.
2. Fundamental Assumptions & Notation
| Symbol | Meaning | Unit |
|---|---|---|
| (P) | Principal (initial amount) | ₹ |
| (r) | Annual nominal rate (decimal) | – |
| (t) | Time in years | yr |
| (n) | Number of compounding periods per year | – |
| (k) | Total compounding periods (k = nt) | – |
| (A) | Accumulated amount after (t) years | ₹ |
| (I) | Interest earned over (t) years | ₹ |
Assumption 1 – Rates are constant over each compounding interval.
Assumption 2 – The rate is expressed per annum; for non‑annual periods we scale by the fraction of the year.
3. Simple Interest
3.1 Definition
Simple interest accrues at a fixed rate applied only to the original principal, ignoring any previously earned interest:
[ I_{\text{SI}} = P , r , t ]
Hence
[ A_{\text{SI}} = P + I_{\text{SI}} = P(1 + r t) ]
3.2 Derivation from First Principles
- Linear Growth: Let (I(t)) denote the interest accumulated up to time (t).
By definition of simple interest, the rate of increase of (I(t)) is constant: [ \frac{dI}{dt} = P r \quad \Rightarrow \quad I(t) = P r t ] - Initial Condition: (I(0)=0).
Integrate with respect to (t) to obtain (I(t)). - Total Amount: (A(t) = P + I(t) = P + P r t = P(1+rt)).