Simple & Compound Interest – Study Notes
Program: Delhi Police Sub‑Inspector (SI)
Subject: Quantitative Aptitude
Topic: Simple & Compound Interest
1. Simple Explanation
| Concept | What It Means | When It Is Used |
|---|---|---|
| Simple Interest (SI) | Interest calculated only on the initial principal. | Short‑term loans, fixed‑rate savings accounts, etc. |
| Compound Interest (CI) | Interest calculated on the principal and on the interest that has been added in previous periods. | Bank deposits, loans with periodic compounding, investment growth. |
Key Idea:
Simple = “add interest only once.”
Compound = “interest earns interest.”
2. Key Points & Formulae
2.1 Simple Interest
-
Formula
[ \text{SI} = P \times r \times t ]- (P) = Principal amount (₹)
- (r) = Rate of interest per annum (in decimal form)
- (t) = Time period (in years)
-
Total Amount (A) [ A = P + \text{SI} = P(1 + rt) ]
-
Rate (r) [ r = \frac{\text{SI}}{P \times t} ]
-
Time (t) [ t = \frac{\text{SI}}{P \times r} ]
2.2 Compound Interest
-
Formula
[ A = P\left(1 + \frac{r}{n}\right)^{nt} ]- (n) = Number of compounding periods per year (e.g., (n=1) for annual, (n=4) for quarterly)
- If compounding is continuous, use (A = P e^{rt}).
-
Compound Interest (CI) [ \text{CI} = A - P ]
-
Effective Annual Rate (EAR)
[ \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 ] -
Finding the Number of Periods
[ nt = \frac{\ln\left(\frac{A}{P}\right)}{\ln\left(1 + \frac{r}{n}\right)} ]
2.3 Common Pitfalls
| Mistake | Fix |
|---|---|
| Using rate per month but keeping (t) in years | Convert rate to yearly or convert (t) to months |
| Mixing up (n) (compounding frequency) and (t) (years) | Keep (n) fixed, compute (nt) as compounding periods |
| Forgetting to convert percentage to decimal | Divide by 100 |
| Using the wrong formula for continuous compounding | Use (A = P e^{rt}) when indicated |
3. Worked Example
Problem (SI):
A person deposits ₹50,000 in a bank at an annual simple interest rate of 8 % for 4 years. Find the total amount at the end of the period.
- Convert rate to decimal: (r = \frac{8}{100} = 0.08)
- Apply SI formula:
[ \text{SI} = 50,000 \times 0.08 \times 4 = 16,000 ] - Total amount:
[ A = 50,000 + 16,000 = 66,000 ]
Answer: ₹66,000
Problem (CI):
₹10,000 is invested at an annual rate of 6 % compounded quarterly. Find the amount after 3 years.
- Rate per period: (r/n = 0.06/4 = 0.015)
- Number of periods: (nt = 4 \times 3 = 12)
- Apply CI formula:
[ A = 10,000 \times (1 + 0.015)^{12} ] [ A \approx 10,000 \times 1.195618 \approx 11,956.18 ]
Answer: ₹11,956.18
4. Quick Revision Summary
| Concept | Formula | Typical Usage | Quick Tip |
|---|---|---|---|
| Simple Interest | ( \text{SI} = P r t ) | Short‑term loans, savings | Remember: only once |
| Total Amount (SI) | ( A = P(1 + rt) ) | Compute final balance | Add to principal directly |
| Compound Interest | ( A = P\left(1 + \frac{r}{n}\right)^{nt} ) | Savings, investments | Compute (nt) first |
| Effective Rate | ( \text{EAR} = (1+r/n)^n-1 ) | Compare schemes | Useful for annual comparison |
| Continuous CI | ( A = P e^{rt} ) | Scientific/finance contexts | Use when “continuous” is specified |
- Conversion Rule: % → Decimal: divide by 100.
- Time Units: Match rate units with time units.
- Compounding Frequency: Check problem statement (annual, half‑annual, quarterly, monthly).
- Common Calculation: Start with (nt) (total periods) → compute factor ((1 + r/n)) → raise to power → multiply by (P).
Final Tip for SI Exam
- Read the question carefully: Look for words like “simple”, “compound”, “quarterly”, “continuous”.
- Write down all given values in a small table to avoid confusion.
- Check units: If time is given in months, convert to years or adjust rate accordingly.
- Do a quick sanity check: Result should be greater than principal for positive rates.
Good luck!