×ⁿ · Compound Interest
Simple interest adds interest; compound interest multiplies interest onto interest. The gap explodes as time stretches.
At 7% a year for 30 years, how many times bigger is your money?
What are we trying to understand?
Save at 7% a year for 30 years — how much do you end with? Thinking in simple interest, 7% × 30 years = 210%, about 3x. But compound interest gives 7.6x.
The difference is the structure of 'interest earning interest'. Each year's interest becomes next year's principal and grows along with it. Not repeated addition — repeated multiplication.
For the first few years, simple and compound look nearly identical. That's why people underestimate compounding. In this lab you'll see both the 'looks the same' stretch and the 'pulls away' stretch with your own eyes.
Change one assumption at a time
The result follows the inputs and assumptions. Check the units and model before treating it as an actual outcome.
Predict first — grow $10,000 at 7% a year for 30 years. How many times bigger does it get?
Deposit in both banks at once — advance +5 years
Simple Bank (interest set aside)
$10,000
Compound Bank (interest reinvested)
$10,000
Read the calculation another way
7% a year for 30 years means 7×30 = 210%, roughly 3x.
That's the simple-interest calculation. Compound interest multiplies 1.07 thirty times, giving about 7.6x — because interest is reinvested every year and snowballs.
Double the rate, double the final amount.
The rate goes into the base, not just the multiplier. 4% for 30 years gives 3.2x; 8% for 30 years gives 10.1x — more than 3x the result, not 2x.
Why the calculation works, and when
Write the graph's straight line (simple) and curve (compound) as formulas, and the difference between addition and multiplication shows itself.
Simple interest — the world of addition
Each year, interest accrues only on the principal P at rate r. Over n years the interest is added n times — hence the straight line.
Compound interest — the world of multiplication
Each year multiplies by (1+r). Because n sits in the exponent, the graph bends upward into an exponential curve.
The Rule of 72
An approximation for how long the principal takes to double at r% a year. At 6% it's 12 years; at 9%, 8 years. The reason this works lies in logarithms.
Try the idea in another setting
Pensions and long-term investing
$1,000 in your 20s and $1,000 in your 40s differ severalfold in value at retirement. Compounding is the mathematical basis of 'start early'.
Loan interest compounds too
Compounding doesn't only grow assets. Revolving credit and missed payments — interest charged on unpaid interest — follow the exact same exponential curve.
Inflation
At a steady 3% inflation, prices double in 24 years (72÷3). Your cash's purchasing power halves at the same speed.
Subscriptions, accumulated
What would that monthly subscription have become if invested at 7% for 30 years? Compounding gives you a new lens on spending decisions.
Now explain it in your own words
Follow the mathematics behind the calculation
Related lab
Logarithms
'How many years to double?' means solving (1+r)^n = 2 for n — exactly what logarithms do. The identity of the Rule of 72 lives in the Log Lab.
Go to the Logarithms lab →Related lab
Sequences
A balance growing by (1+r)x every year is a geometric sequence, plain and simple. If you understand compounding, you already know geometric sequences.
Go to the Sequences lab →Previous lab
Recommended next
Loans
If compounding is multiplication working for your assets, a loan is the bank's multiplication working on you. See how each monthly payment splits, in the next lab.
Go to the Loans lab →