💰 Complete Compound Interest Guide 💰
All Formulas & Concepts for K-12 Students
🌟 The Compound Interest Formula 🌟
A = P(1 + r/n)nt
Where money grows exponentially over time!
Understanding the Variables
A = Final Amount (Future Value) - Total money after time period
P = Principal - Initial amount invested or borrowed
r = Annual Interest Rate (as decimal) - Divide percentage by 100
n = Number of times interest is compounded per year
t = Time in years
I = Interest Earned - The amount of growth (I = A - P)
Important Note: Always convert percentage rates to decimals!
Example: 5% = 0.05, 12% = 0.12, 0.5% = 0.005
Example: 5% = 0.05, 12% = 0.12, 0.5% = 0.005
Complete Formula Collection
1. Standard Compound Interest
A = P(1 + r/n)nt
Used when interest compounds multiple times per year
2. Continuous Compounding
A = Pert
Used when interest compounds continuously (infinite compounding)
e ≈ 2.71828 (Euler's number)
e ≈ 2.71828 (Euler's number)
3. Simple Interest (for comparison)
A = P(1 + rt)
I = Prt
Interest calculated only on principal, not on accumulated interest
4. Interest Earned
I = A - P
The difference between final amount and principal
5. Finding Principal (Present Value)
P = A / (1 + r/n)nt
How much to invest now to reach a future goal
6. Finding Time
t = ln(A/P) / (n × ln(1 + r/n))
How long it takes to reach a financial goal
7. Finding Rate
r = n[(A/P)1/(nt) - 1]
What interest rate is needed to reach a goal
8. Annual Percentage Yield (APY)
APY = (1 + r/n)n - 1
The effective annual rate, accounting for compounding
9. Effective Interest Rate
reff = (1 + r/n)n - 1
True rate after compounding; same as APY
Compounding Frequencies
The value of n depends on how often interest is compounded:
| Compounding Period | Value of n | Formula |
|---|---|---|
| Annually | n = 1 | A = P(1 + r)t |
| Semi-Annually | n = 2 | A = P(1 + r/2)2t |
| Quarterly | n = 4 | A = P(1 + r/4)4t |
| Monthly | n = 12 | A = P(1 + r/12)12t |
| Weekly | n = 52 | A = P(1 + r/52)52t |
| Daily | n = 365 | A = P(1 + r/365)365t |
| Continuously | n = ∞ | A = Pert |
💡 Key Insight:
The more frequently interest compounds, the more money you earn! However, the difference between daily and continuous compounding is very small.
Simple Interest vs. Compound Interest
📉 Simple Interest
A = P(1 + rt)
I = Prt
✓ Interest calculated only on principal
✓ Linear growth
✓ Easier to calculate
✗ Earns less money
📈 Compound Interest
A = P(1 + r/n)nt
✓ Interest calculated on principal + accumulated interest
✓ Exponential growth
✓ "Interest on interest"
✓ Earns more money over time
Worked Examples
Example 1: Basic Compound Interest (Monthly Compounding)
You invest $5,000 at 6% annual interest, compounded monthly for 3 years. How much will you have?P = $5,000
r = 6% = 0.06
n = 12 (monthly)
t = 3 years
A = 5000(1 + 0.06/12)12×3
A = 5000(1 + 0.005)36
A = 5000(1.005)36
A = 5000(1.19668)
A = $5,983.40
Final Amount: $5,983.40
Interest Earned: $5,983.40 - $5,000 = $983.40
Interest Earned: $5,983.40 - $5,000 = $983.40
Example 2: Comparing Compounding Frequencies
$10,000 invested for 5 years at 8% with different compounding:Annually (n=1): A = 10000(1.08)5 = $14,693.28
Quarterly (n=4): A = 10000(1.02)20 = $14,859.47
Monthly (n=12): A = 10000(1.00667)60 = $14,898.46
Daily (n=365): A = 10000(1.000219)1825 = $14,918.25
Continuously: A = 10000e0.4 = $14,918.25
Difference between annual and daily compounding: $224.97
More frequent compounding = More money!
More frequent compounding = More money!
Example 3: Finding Present Value
How much do you need to invest now to have $20,000 in 6 years at 5% compounded quarterly?A = $20,000
r = 0.05, n = 4, t = 6
P = A / (1 + r/n)nt
P = 20000 / (1 + 0.05/4)24
P = 20000 / (1.0125)24
P = 20000 / 1.34735
P = $14,845.43
You need to invest $14,845.43 now
Example 4: Simple vs. Compound Interest Comparison
$8,000 at 7% for 4 years - comparing both methods:Simple Interest: A = 8000(1 + 0.07×4) = 8000(1.28) = $10,240
Compound (Annual): A = 8000(1.07)4 = 8000(1.31080) = $10,486.40
Difference: $246.40 more with compound interest!
This advantage grows larger over time.
This advantage grows larger over time.
Example 5: Continuous Compounding
$15,000 invested at 4.5% compounded continuously for 8 years:P = $15,000, r = 0.045, t = 8
A = Pert
A = 15000 × e0.045×8
A = 15000 × e0.36
A = 15000 × 1.43333
A = $21,499.95
Final Amount: $21,499.95
Interest Earned: $6,499.95
Interest Earned: $6,499.95
Advanced Concepts & Applications
1. Rule of 72 (Doubling Time Shortcut)
Doubling Time ≈ 72 / interest rate
Quick way to estimate how long it takes to double your money!
Example: At 8% interest, money doubles in approximately 72/8 = 9 years
Note: Works best for rates between 6% and 10%
Example: At 8% interest, money doubles in approximately 72/8 = 9 years
Note: Works best for rates between 6% and 10%
2. Rule of 69.3 (More Accurate)
Doubling Time = 69.3 / interest rate
More accurate for continuous compounding
3. Exact Doubling Time
t = ln(2) / (n × ln(1 + r/n))
Precise calculation using logarithms
4. Tripling Time
t = ln(3) / (n × ln(1 + r/n))
How long to triple your investment
5. Monthly Payment with Compound Interest
M = P[r(1+r)n] / [(1+r)n - 1]
Used for loans and mortgages
M = monthly payment, P = loan amount, r = monthly rate, n = number of payments
M = monthly payment, P = loan amount, r = monthly rate, n = number of payments
6. Growth Factor
Growth Factor = (1 + r/n)n
How much $1 grows to in one year
🧮 Interactive Compound Interest Calculator
Calculate Your Investment Growth
🎯 Helpful Shortcuts & Tips
💡 Converting Percentages
Always divide by 100:
• 5% → 0.05
• 12.5% → 0.125
• 0.75% → 0.0075
💡 Common Values
e ≈ 2.71828
ln(2) ≈ 0.693
ln(3) ≈ 1.099
💡 Which Method to Use?
• Savings accounts → Usually daily or monthly
• Investments → Often quarterly or monthly
• Bonds → Typically semi-annually
• CDs → Varies, check terms
💡 The Power of Time
Starting early makes a HUGE difference!
$1000 at 7% for 40 years = $14,974
$1000 at 7% for 20 years = $3,870
Starting 20 years earlier = 3.9× more money!
📋 Quick Reference Table
| What You Want to Find | Formula to Use |
|---|---|
| Future Value (A) | A = P(1 + r/n)nt |
| Present Value (P) | P = A / (1 + r/n)nt |
| Interest Earned (I) | I = A - P |
| Time to Goal (t) | t = ln(A/P) / (n × ln(1 + r/n)) |
| Required Rate (r) | r = n[(A/P)1/(nt) - 1] |
| Doubling Time | ≈ 72 / interest rate (%) |
| APY | (1 + r/n)n - 1 |
| Continuous Compounding | A = Pert |