Simple & Compound Interest – Study Notes
(Tamil Nadu Forest Guard – Quantitative Aptitude)
1. Simple Explanation
| Concept | What it means | When it is used |
|---|---|---|
| Simple Interest (SI) | Interest earned only on the principal amount. | Short‑term loans, deposits, or any situation where the interest rate is fixed for the whole period. |
| Compound Interest (CI) | Interest earned on the principal and on the interest that has accumulated previously. | Scholarships, long‑term deposits, loans with periodic interest credit. |
Key idea – In CI the money “compounds” because the interest is added to the principal at regular intervals, so the next interest calculation is on a larger base.
2. Key Points & Formulas
| Symbol | Meaning | Formula |
|---|---|---|
| (P) | Principal (initial amount) | – |
| (R) | Annual rate of interest (in %) | – |
| (T) | Time (in years) | – |
| (n) | Number of times interest is compounded per year (SI has (n=1) by default) | – |
| (C) | Amount (principal + interest) | – |
| (I) | Interest earned | – |
2.1 Simple Interest
[ I_{SI} = \frac{P \times R \times T}{100} ] [ C_{SI} = P + I_{SI} = P\left(1 + \frac{R \times T}{100}\right) ]
- SI is linear: doubling the time doubles the interest.
2.2 Compound Interest
[ C_{CI} = P\left(1 + \frac{R}{100 \times n}\right)^{nT} ] [ I_{CI} = C_{CI} - P ]
- Compound factor: (\left(1 + \frac{R}{100n}\right)^{nT}).
- Common compounding frequencies: yearly ((n=1)), half‑yearly ((n=2)), quarterly ((n=4)), monthly ((n=12)), daily ((n=365)).
2.3 Effective Annual Rate (EAR)
When interest is compounded more frequently than once a year, the EAR is the equivalent annual rate that gives the same amount of CI:
[ EAR = \left(1 + \frac{R}{100n}\right)^{n} - 1 ]
2.4 Conversion between Rates
- From nominal to effective: Use EAR formula above.
- From effective to nominal (for a given compounding frequency):
[ R_{\text{nominal}} = n \times \left[(1 + EAR)^{1/n} - 1\right] \times 100 ]
2.5 Quick Rules for Exam
| Task | Quick Calculation |
|---|---|
| CI for 1 year, compounding yearly | (C = P(1 + R/100)) |
| CI for 1 year, compounding monthly | (C = P\left(1 + \frac{R}{1200}\right)^{12}) |
| Find rate when amount known | (R = \frac{100}{T}\left(\frac{C}{P}^{1/(nT)} - 1\right) \times n) |
| Find time when rate known | (T = \frac{\ln(C/P)}{n \ln\left(1 + \frac{R}{100n}\right)}) |
| Comparing two rates | Use EAR to decide which is better. |
3. Worked Example
Problem
A forest guard invests ₹10,000 in a deposit that yields 8 % per annum compounded quarterly. What will be the amount after 3 years?
Solution
-
Identify variables
- (P = ₹10,000)
- (R = 8%)
- (T = 3) years
- (n = 4) (quarterly)
-
Apply the CI formula
[ C = P\left(1 + \frac{R}{100 \times n}\right)^{nT} ]
[ C = 10,000 \left(1 + \frac{8}{100 \times 4}\right)^{4 \times 3} ]
[ C = 10,000 \left(1 + \frac{8}{400}\right)^{12} = 10,000 \left(1 + 0.02\right)^{12} = 10,000 \times (1.02)^{12} ]
- Compute ((1.02)^{12})
Using a calculator or successive squaring:
- ((1.02)^2 = 1.0404)
- ((1.02)^4 = (1.0404)^2 ≈ 1.0824)
- ((1.02)^8 = (1.0824)^2 ≈ 1.1719)
- ((1.02)^{12} = (1.02)^8 \times (1.02)^4 ≈ 1.1719 \times 1.0824 ≈ 1.2682)
- Final amount
[ C ≈ 10,000 \times 1.2682 = ₹12,682 ]
Answer: ₹12,682 (rounded to the nearest rupee).
Interest earned
(I = C - P = ₹12,682 - ₹10,000 = ₹2,682).
4. Quick Revision Summary
-
Simple Interest:
[ I = \frac{PRT}{100} ] Linear with time. Use when interest is calculated only once. -
Compound Interest:
[ C = P\left(1 + \frac{R}{100n}\