📚 Study Notes – Simple & Compound Interest
(For Andhra Pradesh Police Sub‑Inspector (SI) – Quantitative Aptitude)
Exam Tip – All SI questions on interest involve either
“rate, time, principal (P), amount (A)” or “rate, time, amount, interest (I)”.
Memorise the short‑hand formulas and practice converting one unknown to the other with logarithms or algebraic manipulation.
1. Definitions
| Term | Meaning |
|---|---|
| Principal (P) | Initial sum of money. |
| Rate (r) | Annual interest rate (in % or decimal). |
| Time (t) | Duration of the investment (in years). |
| Simple Interest (SI) | Interest calculated only on the principal. |
| Compound Interest (CI) | Interest calculated on the principal plus any previously earned interest. |
| Amount (A) | Final sum after interest is added. |
| Effective Annual Rate (EAR) | Annual rate after accounting for compounding frequency. |
| Nominal Annual Rate | Stated rate before accounting for compounding frequency. |
2. Simple Interest (SI)
2.1 Basic Formula
[ \boxed{SI = P \times r \times t} ]
[ \boxed{A = P + SI = P(1 + rt)} ]
- r is expressed in decimal form (e.g., 10 % → 0.10).
- If interest is paid annually, use the above directly.
- For intermittent compounding (e.g., monthly, quarterly), treat each period separately and add the accrued SI for each period.
2.2 Derivation (Straightforward)
- Interest for 1 year: (I_1 = P \times r).
- For t years: (I = t \times I_1 = P \times r \times t).
- Add to principal: (A = P + I).
3. Compound Interest (CI)
3.1 Basic Formula
For n compounding periods per year (e.g., quarterly: n = 4):
[ \boxed{A = P\Bigl(1 + \frac{r}{n}\Bigr)^{nt}} ]
- r: nominal annual rate (decimal).
- n: compounding frequency.
- t: time in years.
3.2 Derivation
- First period (1/n year):
[ P_1 = P\Bigl(1 + \frac{r}{n}\Bigr) ] - Second period:
[ P_2 = P_1\Bigl(1 + \frac{r}{n}\Bigr) = P\Bigl(1 + \frac{r}{n}\Bigr)^2 ] - k‑th period:
[ P_k = P\Bigl(1 + \frac{r}{n}\Bigr)^k ] - After (nt) periods (i.e., t years):
[ A = P\Bigl(1 + \frac{r}{n}\Bigr)^{nt} ]
4. Continuous Compounding
When compounding is continuous (theoretically infinite), the limit gives
[ \boxed{A = P,e^{rt}} ]
- e ≈ 2.71828 (Euler’s number).
- Derivation: ( \displaystyle \lim_{n\to\infty}\Bigl(1+\frac{r}{n}\Bigr)^{nt} = e^{rt}).
5. Effective Annual Rate (EAR)
Given a nominal rate ( r_{\text{nom}}) compounded n times per year, the EAR is
[ \boxed{EAR = \Bigl(1 + \frac{r_{\text{nom}}}{n}\Bigr)^{n} - 1} ]
- Use: Convert any compounding scheme to an equivalent annual rate.
- Tip: For quick mental checks, remember
- (4%) nominal quarterly → ( (1+0.01)^4-1 ≈ 4.04% ).
- (12%) nominal monthly → ( (1+0.01)^12-1 ≈ 12.68% ).
6. Converting Between Variables
| Variable | Formula | Notes |
|---|---|---|
| Principal (P) | (P = \frac{A}{(1+rt)}) (SI) | |
| (P = \frac{A}{(1 + \frac{r}{n})^{nt}}) (CI) | ||
| Rate (r) | (r = \frac |