Compound Interest: Notes और MCQ Practice

आसान भाषा में

Compound Interest (CI) का सीधा मतलब है "ब्याज पर ब्याज" (Interest on Interest)। Simple Interest (SI) में Principal हमेशा Same रहता है, लेकिन CI में हर Cycle का Interest मूलधन में जुड़कर नया Principal बना देता है। SSC CGL Exam में CI के सवाल Successive Percentage, Ratio Method और Tree Method से आसानी से Solve होते हैं।

Important Formulas & Rules

• Basic Amount Formula: Amount (A) = P × (1 + R ÷ 100)ⁿ
• Compound Interest: CI = Amount − Principal
• CI − SI Difference (2 Years): Difference = P × (R ÷ 100)²
• CI − SI Difference (3 Years): Difference = P × (R ÷ 100)² × (3 + R ÷ 100)
• Half-Yearly Compounding: Rate = R ÷ 2 %, Time = 2n
• Quarterly Compounding: Rate = R ÷ 4 %, Time = 4n

2 Quick Tricks

1. Effective Rate Method (2 Years): Effective CI % = A + B + (A × B) ÷ 100

Example: If Rate = 10%, Effective CI for 2 years = 10 + 10 + (10 × 10) ÷ 100 = 21%

1. Ratio Method: 2 Years के CI Calculation के लिए Ratio 2 : 1 use करें और 3 Years के लिए 3 : 3 : 1 use करें।

2 Common Mistakes

1. Half-Yearly / Quarterly Mistakes: Compounding Period बदलने पर Rate और Time को Adjust (Divide/Multiply) करना भूल जाना।
2. Amount vs CI Confusion: Question की Last Line ध्यान से न पढ़ना कि Option में 'CI' पूछा है या 'Total Amount'।

Compound Interest

1. Concept

Compound Interest (CI) is the interest earned on the original principal and on the interest that has already been added to that principal. Unlike simple interest, the amount grows faster because interest is calculated on a larger base each period.

2. Important formulas / rules / facts

• Simple Interest (SI) = P × R × T ÷ 100
• Amount (SI) = P + SI
• Compound Interest = P × [(1 + R/100)^n – 1]
• Amount (CI) = P × (1 + R/100)^n
• n = number of years (or periods) the money is invested/borrowed
• R = rate per period (in %)
• P = principal amount
• When interest is compounded half‑yearly: use rate = R/2 and n = 2×years.
• When compounded quarterly: use rate = R/4 and n = 4×years.
• Rule of 72 (approx.): Years to double ≈ 72 ÷ R (R in % per annum, compounded annually).

3. Solved examples (exam style)

Example 1

A sum of Rs. 8,000 is invested at 10% p.a. compounded annually. Find the amount after 3 years.

Solution

• P = 8000, R = 10, n = 3
• Amount = 8000 × (1 + 10/100)^3 = 8000 × (1.10)^3
• (1.10)^3 = 1.331
• Amount = 8000 × 1.331 = Rs. 10,648
• CI = Amount – Principal = 10,648 – 8,000 = Rs. 2,648

Example 2

Find the compound interest on Rs. 5,500 for 2 years at 8% p.a. compounded half‑yearly.

Solution

• Half‑yearly rate = 8 ÷ 2 = 4%
• Number of half‑yearly periods = 2 × 2 = 4
• CI = 5500 × [(1 + 4/100)^4 – 1]
• (1.04)^4 ≈ 1.16986
• CI = 5500 × (1.16986 – 1) = 5500 × 0.16986 ≈ Rs. 934.23
• Amount = 5500 + 934.23 ≈ Rs. 6,434.23

Example 3

A person wants to have Rs. 25,000 after 5 years. If the bank offers 12% p.a. compounded annually, what principal should he invest today?

Solution

• Desired amount = 25,000, R = 12, n = 5
• Amount formula: A = P × (1 + R/100)^n → P = A ÷ (1 + R/100)^n
• (1 + 12/100)^5 = (1.12)^5 ≈ 1.76234
• P = 25,000 ÷ 1.76234 ≈ Rs. 14,180 (rounded)

Example 4 (Multiple‑choice style)

If Rs. 1,200 is invested at 5% p.a. compounded quarterly, the amount after 2 years will be nearest to:

A) Rs. 1,320

B) Rs. 1,340

C) Rs. 1,360

D) Rs. 1,380

Solution

• Quarterly rate = 5 ÷ 4 = 1.25%
• Number of quarters = 4 × 2 = 8
• Amount = 1200 × (1 + 1.25/100)^8 = 1200 × (1.0125)^8
• (1.0125)^8 ≈ 1.1045
• Amount ≈ 1200 × 1.1045 = Rs. 1,325.4 → nearest to Rs. 1,320 (Option A).

4. Shortcuts and tricks

• Doubling time: Use Rule of 72 for quick estimate (e.g., at 9% p.a., years ≈ 72 ÷ 9 = 8 years).
• Using tables: For common rates (5%, 7.5%, 10%) memorize (1 + R/100)^n for n = 1‑5 to avoid repeated calculations.
• Approximation: For small rates, (1 + r)^n ≈ 1 + n·r (where r = R/100). Good for quick checks.
• Factor method: When the same rate is used for several years, multiply the yearly factor repeatedly instead of raising to power each time (e.g., for 3 years at 8%, factor = 1.08; amount = P × 1.08 × 1.08 × 1.08).

5. Common mistakes to avoid

1. Using simple interest formula for compound problems – always check if the interest is compounded.
2. Forgetting to adjust rate and periods when interest is compounded half‑yearly, quarterly, or monthly.
3. Incorrect power – raising to ‘n’ instead of ‘n‑1’ or vice‑versa; remember the exponent is the total number of compounding periods.
4. Mixing up percentage and decimal – always convert R% to r = R/100 before using in formulas.
5. Rounding too early – keep intermediate results to at least 4‑5 decimal places; round only at the final answer.