Compound Interest - All Formulas & Shortcuts

Quick reference guide for all Compound Interest formulas with calculation shortcuts

📘 Basic Formulas

1. Standard Compound Interest Formula

Amount (A):

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

Compound Interest (CI):

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

Where:

• P = Principal amount
• R = Rate of interest per annum (%)
• T = Time period (years)
• A = Final amount
• CI = Compound Interest

2. Compound Interest for Different Compounding Frequencies

a) Half-Yearly Compounding:

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

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

b) Quarterly Compounding:

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

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

c) Monthly Compounding:

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

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

d) Daily Compounding:

A = P(1 + R/36500)^(365T)

3. When Time is in Fraction

Example: 2 years 3 months

Method 1: Convert to decimal

T = 2 + 3/12 = 2.25 years A = P(1 + R/100)^2.25

Method 2: Separate calculation

A₁ = P(1 + R/100)² [for 2 years] A₂ = A₁(1 + R/400) [for 3 months = 1 quarter]

🎯 Special Formulas & Shortcuts

4. 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)

Quick Shortcut for 2 years:

CI - SI = SI² / (100 × P) OR CI - SI = (SI for 1 year)² / P

5. Population/Depreciation Formula

Growth (Population Increase):

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

Depreciation (Value Decrease):

Final Value = P(1 - R/100)^T

Different rates for different years:

Final Value = P(1 + R₁/100)(1 + R₂/100)(1 + R₃/100)

Note: Use minus (-) for depreciation

6. When CI for certain years is given

If CI for 1st year = I₁:

P = (I₁ × 100) / R

If CI for 2nd year = I₂ and CI for 1st year = I₁:

R = [(I₂ - I₁) / I₁] × 100

If CI for 2 years = C₂ and CI for 1 year = C₁:

R = [(C₂ - C₁) / C₁] × 100

7. Finding Time Period

When Principal, Amount, and Rate are known:

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

Take log both sides:

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

Shortcut for small rates: If rate is small (≤10%), use approximation:

T ≈ [(A/P) - 1] / (R/100)

⚡ Quick Calculation Shortcuts

8. Compound Interest for 2 Years

Instead of: A = P(1 + R/100)²

Use:

A = P × [1 + 2R/100 + R²/10000]

Even Faster:

CI for 2 years = SI for 2 years + (SI for 1 year)²/P

9. Compound Interest for 3 Years

Instead of: A = P(1 + R/100)³

Use:

A = P × [1 + 3R/100 + 3R²/10000 + R³/1000000]

Faster method:

CI₃ = SI₃ + (R/100) × [SI₂ + SI₁]

10. Rule of 72 (Doubling Time)

To find time when money doubles:

T ≈ 72 / R

Example: At 8% per annum, money doubles in approximately:

T = 72/8 = 9 years

More Accurate: Rule of 69

T = 0.35 + 69/R

11. Rule of 114 (Tripling Time)

To find time when money triples:

T ≈ 114 / R

Example: At 6% per annum, money triples in approximately:

T = 114/6 = 19 years

12. Successive Percentage Change

When rate changes every year:

Method 1:

Overall change = R₁ + R₂ + (R₁ × R₂)/100

For 3 years with same rate R:

Overall % change = 3R + 3R²/100 + R³/10000

📊 Important Power Values (Memorize!)

(1 + R/100)^T Values

• (1.05)² = 1.1025 → Remember: “11025” (easy to recall)
• (1.10)² = 1.21 → Doubles every ~7 years
• (1.20)² = 1.44 → Almost 50% increase in 2 years
• (1.25)² = 1.5625 → Increases by more than 50%

🎓 Advanced Shortcuts

13. When Principal is Same, Find Rate

If amount doubles in T years:

R = [(2^(1/T) - 1)] × 100

Approximation for small T:

R ≈ 100/T

14. Present Worth (Reverse Calculation)

Present worth of amount A payable after T years:

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

15. Installment Formula

When equal installments are paid:

P = I/(1+R/100) + I/(1+R/100)² + … + I/(1+R/100)^T

Shortcut:

P = I × [(1+R/100)^T - 1] / [R/100 × (1+R/100)^T]

💡 Exam-Specific Quick Tricks

Trick 1: Mental Calculation for 2 Years

Question: Find CI on ₹5000 at 10% for 2 years

Instead of: 5000(1.1)² = 5000 × 1.21 = 6050

Do this:

• Year 1 interest: 500
• Year 2 principal: 5500
• Year 2 interest: 550
• Total CI: 500 + 550 = 1050

Trick 2: Quick CI - SI Difference

Question: CI - SI on ₹10,000 at 5% for 2 years

Instead of calculating both:

CI - SI = P(R/100)² = 10000 × (5/100)² = 10000 × 0.0025 = 25

Mental: “5% of 5% of 10000 = 25”

Trick 3: Depreciation Quick Check

Question: Value after 20% depreciation for 2 years

Instead of: P(0.8)² = 0.64P

Think:

• Year 1: Loses 20%, left with 80% = 0.8P
• Year 2: Loses 20% of 0.8P = 0.16P, left with 0.64P
• Total loss: 36%

📝 Common Exam Patterns

Pattern 1: Find Principal

Given: CI, Rate, Time Formula: P = CI / [(1+R/100)^T - 1]

Pattern 2: Find Rate

Given: P, A, Time Formula: R = [(A/P)^(1/T) - 1] × 100

Pattern 3: Find Time

Given: P, A, Rate Formula: T = log(A/P) / log(1+R/100)

Pattern 4: CI vs SI Difference

Given: P, R, T=2 Formula: Difference = P(R/100)²

Pattern 5: Population Growth

Given: Initial, Rate, Time Formula: Final = Initial(1+R/100)^T

🎯 Speed Calculation Tips

1. Memorize squares: (1.05)², (1.10)², (1.20)², (1.25)²
2. Use Rule of 72 for doubling questions
3. For 2 years: Use CI = SI + SI²/P
4. For depreciation: Remember (0.9)² = 0.81, (0.8)² = 0.64
5. Mental math: Break into year-by-year for 2-3 years

🔗 Related Resources

Practice Questions:

• Compound Interest Practice Questions

Theory:

• Compound Interest Theory & Concepts

Related Topics:

• Simple Interest
• Percentage

Study Resources:

• All-in-One Formula Sheet
• Shortcuts & Tricks Compilation

Download this formula sheet as PDF (feature coming soon)

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Master these formulas and you’ll solve CI questions in under 60 seconds! ⚡