Karnataka PSC – Quantitative Aptitude
Topic: Simple & Compound Interest
Quick Tip:
Simple Interest (SI) is charged only on the principal.
Compound Interest (CI) is charged on the principal plus all previously earned interest.
1. Simple Explanation
| Concept | What it is | Formula | When to use |
|---|---|---|---|
| Simple Interest (SI) | Interest earned only on the original amount (principal) for the entire period. | (\displaystyle SI = P \times R \times T / 100) | Short‑term loans, savings certificates, or where interest isn’t reinvested. |
| Compound Interest (CI) | Interest earned on the principal and on any interest that has already accumulated. | (\displaystyle A = P \left(1 + \dfrac{R}{n}\right)^{nT}) (CI = A - P) |
Bank deposits, long‑term loans, investment plans. |
| Effective Annual Rate (EAR) | The real return after accounting for compounding frequency. | (\displaystyle EAR = \left(1 + \dfrac{R}{n}\right)^n - 1) | Comparing different compounding options. |
- (P) – Principal (initial amount)
- (R) – Annual nominal rate (%).
- (T) – Time in years.
- (n) – Number of compounding periods per year.
- (A) – Amount after time (T).
Rule of 72 – Approximate number of years to double an investment at rate (R):
(\displaystyle \text{Years} \approx \frac{72}{R}) (useful for quick mental checks).
2. Key Points to Remember
| Point | Details |
|---|---|
| Compounding Frequency | (n = 1) (annually), 2 (semi‑annually), 4 (quarterly), 12 (monthly), 365 (daily). |
| Partial Years | Treat (T) as a fraction (e.g., 6 months = 0.5 years). |
| Unequal Periods | If the compounding period isn’t a full year, adjust (T) accordingly. |
| Convert Rate | For (R) in % convert to decimal by dividing by 100. |
| Compound Interest vs. Simple | CI ≥ SI for the same (P, R, T). |
| Present Value (PV) | (PV = \dfrac{A}{(1+R)^T}) – useful for loan repayments. |
| Annuities | Future value of an ordinary annuity: (FV = P \left[\dfrac{(1 + r)^n - 1}{r}\right]). |
| Quick Checks | Use Rule of 72 for doubling time; use Rule of 69.3 for continuous compounding. |
| Special Cases | 0% rate → no interest; 100% rate → amount doubles each period. |
3. Worked Example
Problem
A person deposits ₹10,000 in a bank that offers 10% per annum interest, compounded quarterly.
Find:
- The amount after 3 years.
- The compound interest earned.
- The simple interest that would have been earned for the same period.
Solution
| Step | Calculation | Result |
|---|---|---|
| 1. Convert parameters | (P = ₹10,000) (R = 10% = 0.10) (n = 4) (quarterly) (T = 3) years |
— |
| 2. Calculate amount using CI formula | (A = 10,000 \times \left(1 + \dfrac{0.10}{4}\right)^{4 \times 3}) (= 10,000 \times \left(1 + 0.025\right)^{12}) (= 10,000 \times (1.025)^{12}) |
(A \approx 10,000 \times 1.34489 = ₹13,448.90) |
| 3. Compound Interest | (CI = A - P = 13,448.90 - 10,000 = ₹3,448.90) | ₹3,448.90 |
| 4. Simple Interest | (SI = P \times R \times T = 10,000 \times 0.10 \times 3 = 10,000 \times 0.30 = ₹3,000) | ₹3,000 |
| 5. Comparison | (CI - SI = 3,448.90 - 3,000 = ₹448.90) | ₹448.90 extra with compounding |
Answer
- Amount after 3 years: ₹13,448.90
- Compound interest earned: ₹3,448.90
- Simple interest for the same period: ₹3,000
4. Quick Revision Summary
- Simple Interest: (SI = PRT/100).
- Compound Interest: (A = P(1 + R/n)^{nT}); (CI = A - P).
- Compounding Frequency: yearly (n=1), semi‑annual (n=2), quarterly (n=4), monthly (n=12), daily (n=365).
- Rule of 72: (\text{Years to double} \approx 72/R).
- Effective Annual Rate: (\left(1 + \frac{R}{n}\right)^n - 1).
- Present Value: (PV =