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Before we dive into formulas, I'm curious—when you think about money growing in a bank account, what's your mental model of how that works?

That's a solid starting point! You're describing "simple interest"—the bank pays you a fixed percentage of your original deposit each year.

But here's where it gets interesting. Let me ask you this:

If you deposit $1,000 at 10% interest, after one year you'd have $1,100. Now, in year two—should the bank pay you 10% of $1,000... or 10% of $1,100?

Exactly! And that's the key insight behind compound interest—you earn interest on your interest.

Let's trace through what happens:
- Year 0: $1,000
- Year 1: $1,000 × 1.10 = $1,100
- Year 2: $1,100 × 1.10 = $1,210
- Year 3: $1,210 × 1.10 = $1,331

Notice how the growth accelerates? Each year you're multiplying by 1.10, not adding $100.

The general formula is:

$$A = P(1 + r)^t$$

Where:
- $P$ = principal (starting amount)
- $r$ = interest rate as decimal (10% = 0.10)
- $t$ = time in years
- $A$ = final amount

Quick check: Using this formula, what would $1,000 become after 3 years at 10%?

Perfect! You've got it.

Now here's a practical question that comes up all the time: "How long until my money doubles?"

You could solve $2P = P(1+r)^t$ for $t$, but there's a beautiful shortcut called the Rule of 72.

Just divide 72 by your interest rate:

$$\text{Doubling time} \approx \frac{72}{r\%}$$

At 6% interest: 72 ÷ 6 = 12 years to double
At 8% interest: 72 ÷ 8 = 9 years to double
At 12% interest: 72 ÷ 12 = 6 years to double

Why 72? It's a convenient approximation that comes from $\ln(2) \approx 0.693$, but 72 is close enough and much easier to calculate mentally!

What would be the doubling time at 4% interest?

Exactly!

Now you understand why compound interest is called "the eighth wonder of the world."

The key insights to remember:
1. Compound = interest on interest (exponential growth)
2. Rule of 72 = quick mental math for doubling time
3. Time is your friend - the earlier you start, the more compounding cycles you get

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