Simple & Compound Interest – Study Notes
(For Odisha Patwari / Lekhpal & IIT‑JEE level exams)
1. Definitions
| Concept | Meaning | Formula |
|---|---|---|
| Principal (P) | Initial amount of money invested or borrowed. | – |
| Rate of Interest (r) | Annual rate expressed in % per annum. | – |
| Time (t) | Duration of investment/loan in years. | – |
| Simple Interest (SI) | Interest earned only on the principal. | (SI = \dfrac{P,r,t}{100}) |
| Maturity Value (A) | Total amount after interest is added. | (A = P + SI) |
| Compound Interest (CI) | Interest earned on both principal and previously earned interest. | (A = P\left(1+\dfrac{r}{100}\right)^{t}) |
| Compound Interest (on interest) | Interest earned on the interest part of an amount. | (CI = A - P) |
Tip – In competitive exams, the compounding period (annually, quarterly, monthly…) is often implicit; always look for the phrase “compounded n times a year”.
2. Derivations
2.1 Simple Interest
Interest is linear in time:
[ SI = \frac{P,r}{100}\times t ]
Maturity value:
[ A = P + SI = P\left(1+\frac{r,t}{100}\right) ]
2.2 Compound Interest
Assume compounding m times per year.
The effective rate per compounding period is ( \dfrac{r}{100m} ).
After one period:
[ P_1 = P\left(1+\frac{r}{100m}\right) ]
After m periods (one year):
[ P_m = P\left(1+\frac{r}{100m}\right)^m ]
After t years (i.e. (mt) periods):
[ A = P\left(1+\frac{r}{100m}\right)^{mt} ]
If (m=1) (annual compounding), this reduces to
[ A = P\left(1+\frac{r}{100}\right)^t ]
3. Equivalent Rates
| Scenario | Formula |
|---|---|
| Nominal rate (R_N) compounded m times a year → Effective annual rate (R_E) | (1+\frac{R_E}{100} = \left(1+\frac{R_N}{100m}\right)^m) |
| Effective rate → Nominal rate | (1+\frac{R_N}{100m} = \left(1+\frac{R_E}{100}\right)^{1/m}) |
Tricky point – When the exam says “rate of 10% per annum compounded quarterly”, the nominal rate is 10%, effective rate is
( (1+0.10/4)^4-1 \approx 10.38% ).
4. Continuous Compounding
Let (e) denote the base of natural logarithms.
[ A = P,e^{rt/100} ]
When the rate is given per annum and compounding is continuous, use the natural log to solve for unknowns.
5. Interest on Interest
For compound interest, the interest earned during the first year is (P,r/100).
During the second year, interest is earned on the new principal (P(1+r/100)), and so on.
Practice – The “interest on interest” after t years is simply (A - P).
6. Solving for Unknowns – Quick Tricks
| Unknown | Quick Method |
|---|---|
| Principal (P) | (P = \dfrac{A}{(1+r/100)^t}) |
| Rate (r) | Take logarith |