Overview
This lecture provides a comprehensive guide to compound interest, covering key concepts, methods for calculation, differences from simple interest, shortcuts, formulae, and a variety of exam-style problems.
Simple Interest vs. Compound Interest
- Simple Interest (SI): Interest is calculated only on the original principal each period.
- In SI, the amount of interest earned is the same each year.
- Compound Interest (CI): Interest is calculated on both the principal and accumulated interest.
- In CI, interest increases every year as it is calculated on a growing base (principal + previous interest).
- CI always yields more interest than SI for the same rate and time.
Calculating Compound Interest
- CI for n years at rate r% per year: Amount = Principal × (1 + r/100)^n.
- Multiplying factor per year: (1 + r/100).
- For non-integer percent rates, convert to fraction for easier calculation (e.g., 16 2/3% = 1/6; multiplying factor: 7/6).
Successive Percentage Change
- CI problems often use "successive percentage change": x + y + (xy/100) for two years, extend for more years.
- For 3 years: x + y + z + (xy + yz + zx)/100 + (xyz/10000).
Special Cases & Shortcuts
- When CI is compounded more frequently (half-yearly, quarterly), adjust rate and period accordingly:
- Half-yearly: r/2 per period, n×2 periods.
- Quarterly: r/4 per period, n×4 periods.
- Effective annual rate for half-yearly compounding: [1 + (r/2)/100]^2 - 1.
Ratios in CI Problems
- When comparing amounts after different years, the ratio can be expressed as (multiplying factor)^(years difference).
- For example, if A grows 3/2 times in 3 years, the per-year factor is the cube root of 3/2.
Standard Rates & Memorization Tips
- Memorize standard CI values for common rates (% over 2–3 years), e.g.:
- 10% for 2 years: 21%.
- 10% for 3 years: 33.1%.
- 20% for 2 years: 44%.
- 20% for 3 years: 72.8%.
Tree Method & Golden Ratio
- Tree method helps break down CI over multiple years, showing interest on principal and interest on interest.
- Golden ratios for years: 2 years (2:1), 3 years (3:3:1), 4 years (4:6:4:1).
Problem Solving Techniques
- Use ratio method, percentage shortcuts, or formula method based on the question type.
- For "find the rate," use nth root: If amount grows k times in n years, annual factor = k^(1/n).
- When given CI and SI difference: Difference = Principal × (r^2)/100 for 2 years.
Key Terms & Definitions
- Principal — The original sum of money invested or borrowed.
- Simple Interest (SI) — Interest calculated only on the principal.
- Compound Interest (CI) — Interest calculated on principal plus accumulated interest.
- Amount — Total of principal plus interest accrued.
- Multiplying Factor — The factor by which principal increases each period (1 + r/100).
- Effective Annual Rate — Actual interest earned/paid in a year after compounding.
- Tree Method — Visual representation of CI calculation year-by-year.
Action Items / Next Steps
- Review and memorize standard CI values for quick calculations.
- Practice ratio and successive percentage shortcuts.
- Download the provided class PDF and attempt the given homework questions.
- Take live tests on the application as announced.