Compound Interest

Chapter Overview

• Principal (P): The original sum of money borrowed or invested.
• Rate (R%): The percentage rate of interest per annum.
• Time (n or t): The duration for which the money is invested.
• Amount (A): The total money returned at the end of the time period (Amount = Principal + Compound Interest).
• Compound Interest (CI): CI = Amount − Principal.

Core Formulas & Conversion Rules

1. Annual Compounding Formula

When interest is compounded annually at R% per annum for n years:

Amount (A) = P(1 + R / 100)ⁿ

Compound Interest (CI) = P [(1 + R / 100)ⁿ − 1]

2. Compounding Frequency Variations

• Compounded Half-Yearly (Semi-Annually): Interest is calculated every 6 months.
  Rate = R / 2%,   Time = 2n
  A = P(1 + R / 200)²ⁿ
• Compounded Quarterly: Interest is calculated every 3 months.
  Rate = R / 4%,   Time = 4n
  A = P(1 + R / 400)⁴ⁿ

3. Different Rates for Successive Years

If the interest rates are different for different years (R₁% for year 1, R₂% for year 2, R₃% for year 3):

A = P(1 + R₁ / 100)(1 + R₂ / 100)(1 + R₃ / 100)

4. Special Shortcuts for 2 and 3 Years (CI vs SI Difference)

• For 2 Years:
  Difference between CI and SI = P · (R / 100)²
• For 3 Years:
  Difference between CI and SI = P · [R²(300 + R)] / 100³

5. Equal Annual Installments (CI)

If a debt of P is paid back in n equal annual installments of ₹x each at R% per annum compounded annually:

P = x / (1 + R / 100)¹ + x / (1 + R / 100)² + … + x / (1 + R / 100)ⁿ

Problem-Solving Methods

The Ratio / Fraction Method (Alternative to Formulas)

Instead of complex powers, convert the interest rate percentage into a fraction (R / 100 = numerator / denominator).

• Example: If Rate = 16⅔% = 1/6.
• This means every year, the Principal (6) grows by 1 to become the Amount (7).
• For 2 years:
  Principal : Amount = (6 × 6) : (7 × 7) = 36 : 49
  Interest = 49 − 36 = 13 units

Speed Shortcut

Convert R% into a simplified fraction a / b:

• Principal Ratio = bⁿ
• Amount Ratio = (b + a)ⁿ
• CI Ratio = (b + a)ⁿ − bⁿ

PYQ (Previous Year Questions)

1. Find the compound interest on ₹5000 at 10% per annum for 2 years, compounded annually. (SSC CGL 2025)

(A) ₹1000
(B) ₹1050
(C) ₹1100
(D) ₹1150

Answer: (B) ₹1050

Solution:

1. Using formula: A = P(1 + R / 100)ⁿ
2. A = 5000(1 + 10 / 100)² = 5000 × (11 / 10) × (11 / 10)
3. A = 5000 × (121 / 100) = 50 × 121 = ₹6050
4. CI = A − P = 6050 − 5000 = ₹1050

2. What will be the compound interest on ₹10,000 for 1 year at 20% per annum compounded half-yearly? (SSC CHSL 2025)

(A) ₹2000
(B) ₹2100
(C) ₹2200
(D) ₹2050

Answer: (B) ₹2100

Solution:

1. Compounded half-yearly: Rate = 20 / 2 = 10% per half-year, Time = 1 year × 2 = 2 half-years.
2. A = 10000 × (1 + 10 / 100)² = 10000 × (11 / 10) × (11 / 10) = ₹12100
3. CI = 12100 − 10000 = ₹2100

3. The difference between compound interest and simple interest on a certain sum of money at 5% per annum for 2 years is ₹25. Find the principal. (SSC CPO 2024)

(A) ₹5000
(B) ₹8000
(C) ₹10,000
(D) ₹12,000

Answer: (C) ₹10,000

Solution:

1. Use the 2-year difference formula: Difference = P · (R / 100)²
2. Substituting values: 25 = P · (5² / (100 × 100)) ⇒ 25 = P · (25 / 10000)
3. P = (25 × 10000) / 25 = ₹10,000

4. A sum of money placed at compound interest doubles itself in 4 years. In how many years will it amount to 8 times itself? (SSC GD 2024)

(A) 8 years
(B) 12 years
(C) 16 years
(D) 20 years

Answer: (B) 12 years

Solution:

1. In 4 years → becomes 2 times (2¹).
2. We need it to become 8 times (2³).
3. Multiplying the time period by the exponent of 2: Total Time = 4 years × 3 = 12 years.

5. Find the compound interest on ₹8000 at 15% per annum for 9 months, compounded quarterly. (SSC CGL 2025)

(A) ₹900
(B) ₹925
(C) ₹931
(D) ₹950

Answer: (C) ₹931

Solution:

1. Compounded quarterly: Rate = 15 / 4%, Time = 9 months = 9 / 12 years = 3 / 4 year.
2. Number of quarters (n) = (3 / 4) × 4 = 3 quarters.
3. Quarterly rate = 15 / 400 = 3 / 80.
4. Amount A = 8000 × (1 + 3 / 80)³ = 8000 × (83 / 80) × (83 / 80) × (83 / 80) ≈ ₹8931
5. CI = 8931 − 8000 = ₹931