📈 Compound Interest - Complete Theory

Master Compound Interest from basics to advanced. Understand the power of compounding!

🎯 What is Compound Interest?

Compound Interest (CI) is the interest calculated on the principal plus accumulated interest from previous periods.

Key Difference from Simple Interest:

• SI: Interest only on principal
• CI: Interest on principal + previous interest (Interest on Interest!)

Albert Einstein: “Compound interest is the eighth wonder of the world”

📐 Basic Formula

Amount (A) = P(1 + R/100)^T

Compound Interest (CI) = A - P
                        = P[(1 + R/100)^T - 1]

Where:
P = Principal
R = Rate of interest per annum
T = Time in years

🔍 Understanding Compounding

Year-by-Year Breakdown

Example: ₹1,000 at 10% p.a. for 3 years

Year 1:

• Principal: ₹1,000
• Interest: ₹100 (10% of 1000)
• Amount: ₹1,100

Year 2:

• Principal: ₹1,100 (previous amount becomes new principal!)
• Interest: ₹110 (10% of 1100)
• Amount: ₹1,210

Year 3:

• Principal: ₹1,210
• Interest: ₹121 (10% of 1210)
• Final Amount: ₹1,331

Total CI = ₹1,331 - ₹1,000 = ₹331

Compare with SI = (1000×10×3)/100 = ₹300 CI is ₹31 more!

📊 Compounding Frequency Formulas

1. Annual Compounding (Standard)

A = P(1 + R/100)^T

2. Half-Yearly (Semi-Annual) Compounding

Interest is compounded every 6 months:

A = P(1 + R/200)^(2T)

Rate becomes R/2 per half-year
Time becomes 2T half-years

Example: 10% p.a. compounded half-yearly = 5% per 6 months

3. Quarterly Compounding

Interest is compounded every 3 months:

A = P(1 + R/400)^(4T)

Rate becomes R/4 per quarter
Time becomes 4T quarters

4. Monthly Compounding

A = P(1 + R/1200)^(12T)

Rate becomes R/12 per month
Time becomes 12T months

💡 Solved Examples

Example 1: Basic CI Calculation

Q: Find CI on ₹5,000 at 8% p.a. for 2 years compounded annually.

Solution:

P = ₹5,000, R = 8%, T = 2 years

A = P(1 + R/100)^T
A = 5000(1 + 8/100)²
A = 5000(1.08)²
A = 5000 × 1.1664
A = ₹5,832

CI = A - P = 5832 - 5000 = ₹832

Answer: CI = ₹832

Example 2: Half-Yearly Compounding

Q: Find CI on ₹8,000 at 10% p.a. for 1 year compounded half-yearly.

Solution:

P = ₹8,000, R = 10%, T = 1 year

For half-yearly:
A = P(1 + R/200)^(2T)
A = 8000(1 + 10/200)²
A = 8000(1.05)²
A = 8000 × 1.1025
A = ₹8,820

CI = 8820 - 8000 = ₹820

Note: If it was annual compounding, CI would be only ₹800!

Example 3: Time Period in Fraction

Q: Find CI on ₹10,000 at 20% p.a. for 1.5 years compounded annually.

Solution:

P = ₹10,000, R = 20%, T = 1.5 years = 1 + 1/2 years

For fractional years:
First calculate for 1 year:
A₁ = 10000(1 + 20/100)
A₁ = 10000 × 1.2 = ₹12,000

For remaining 0.5 years, use SI:
SI = (12000 × 20 × 0.5)/100 = ₹1,200

Final Amount = 12000 + 1200 = ₹13,200
CI = 13200 - 10000 = ₹3,200

Answer: CI = ₹3,200

🔄 Important Variations & Formulas

1. Difference between CI and SI

For 2 Years:

CI - SI = P(R/100)²

For 3 Years:

CI - SI = P(R/100)²(3 + R/100)

Example:

P = ₹10,000, R = 10%, T = 2 years

CI - SI = 10000(10/100)²
        = 10000 × 0.01
        = ₹100

2. When CI for 2 Years is Given

If CI for 2 years = X
And CI for 3 years = Y

Then: Y - X = Interest on X for 1 year
      R = [(Y - X) / X] × 100

3. Population Growth/Depreciation

Same Formula!

Growth:

Final Population = Initial × (1 + R/100)^T

Depreciation:

Final Value = Initial × (1 - R/100)^T

Note the minus sign for depreciation!

📈 Real-Life Applications

1. Bank Fixed Deposits

Most FDs use quarterly compounding → higher returns than annual!

2. Loan EMI Calculations

Home loans, car loans use monthly compounding

3. Investment Growth

Mutual funds, stocks show compound growth over time

4. Inflation

Prices increase with compound effect year-over-year

⚡ Quick Calculation Methods

Method 1: For Small Rates and Short Time

When R is small (≤10%) and T≤2:

CI ≈ SI + (SI × R × T)/(200)

Method 2: Using (1 + R/100)^T Table

Memorize common values:

• (1.05)² = 1.1025
• (1.10)² = 1.21
• (1.10)³ = 1.331
• (1.20)² = 1.44

⚠️ Common Mistakes

❌ Mistake 1: Using SI Formula

Wrong: CI = (P × R × T)/100
Right: CI = P[(1 + R/100)^T - 1]

❌ Mistake 2: Wrong Compounding Frequency

For half-yearly: Use R/200 and 2T (not R/100 and T)
For quarterly: Use R/400 and 4T

❌ Mistake 3: Fractional Years

For 2.5 years:
Calculate CI for 2 years, then SI for 0.5 years
Don't use (1 + R/100)^2.5 directly in exams!

🎯 Shortcuts for IBPS Exams

Shortcut 1: Doubling Time (Rule of 72)

Approximate time to double ≈ 72/R years

Example:
At 8% p.a.: Time ≈ 72/8 = 9 years
At 12% p.a.: Time ≈ 72/12 = 6 years

Shortcut 2: 2-Year CI Trick

For 2 years:
CI = SI + (SI)²/(100P)
   = SI + (P × R²)/10000

Shortcut 3: 3-Year Quick Formula

For 3 years at R%:
CI/P = 3R/100 + 3R²/10000 + R³/1000000

🔗 Comparison: SI vs CI

📝 Practice Problems

Level 1:

1. Find CI on ₹4,000 at 10% p.a. for 2 years
2. Find CI on ₹5,000 at 8% p.a. for 2 years compounded half-yearly
3. Difference between CI and SI on ₹8,000 at 5% for 2 years

Level 2:

4. ₹10,000 becomes ₹13,310 in 3 years at CI. Find rate.
5. Find CI on ₹12,000 at 10% for 2.5 years compounded annually
6. A sum doubles in 5 years at CI. When will it become 4 times?

Level 3:

7. CI for 2 years is ₹410 and for 3 years is ₹623.05. Find principal and rate.
8. Population increases by 10% annually. If current population is 50,000, what was it 2 years ago?

🔗 Related Topics

Prerequisites:

• Simple Interest - Must know before CI
• Percentage - For rate calculations

Related:

• Profit & Loss - Uses similar growth concepts
• Population/Depreciation problems (same formula!)

Practice:

• Compound Interest Questions
• Formula Sheet

Remember: Compound Interest = Interest on Interest! Master this for banking exams! 💪