Compound Interest Explained: The Math Behind Both Your Savings and Your Debt

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The same formula that quietly grows an investment for decades is the one that makes a loan feel endless. Here is compound interest in plain arithmetic, the Rule of 72, and why early loan payments barely touch the balance.

Interest on Interest

Simple interest is a flat charge: you pay (or earn) a fixed percentage of the original amount each period. Compound interest is different and far more powerful — each period, the interest is added to the balance, and the next period's interest is calculated on that bigger number. You earn interest on your interest. That one recursion, repeated over years, is what turns modest sums into large ones and small debts into crushing ones.

The formula is short:

``
A = P × (1 + r/n)^(n×t)
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P is the principal you start with, r the annual rate as a decimal, n how many times a year interest compounds, and t` the number of years. Put 1,000 at 5% compounded yearly for 10 years and you do not get 1,000 + 500 = 1,500 (that would be simple interest); you get about 1,629. The extra 129 is interest-on-interest — the whole game.

Why It Only Looks Magic in Reverse

Compound growth is invisible early and dramatic late, which trips people up. In year one, 5% of 1,000 is a forgettable 50. The balance creeps; you convince yourself nothing is happening. Then, after a decade or two, the same 5% is applied to a base that has swollen, and each year adds more than every earlier year combined. The curve is exponential, but it feels flat right up to the moment it isn't. This is why starting to save early beats saving more later: the years, not the amount, do the compounding.

The Rule of 72

You don't need the formula to estimate doubling time. Divide 72 by the annual percentage rate and you get roughly how many years for the money to double. At 6%, about 12 years. At 24% on a credit card, about three. At 1% in a lazy savings account, over 70 years. The rule is an approximation, but a remarkably good one across the everyday rates, and it instantly tells you whether a rate is working for you or against you — the same arithmetic that makes a mutual fund grow is what makes high-rate debt explode.

Loans: Why the Balance Barely Moves at First

Now flip it to the borrower's side, and the same compounding explains a maddening experience. On an amortising loan (a mortgage, a car loan), each fixed monthly payment is split between interest and principal — and early on, the interest slice is huge because it is charged on the full balance you owe. If you owe 300,000 at a rate that works out to, say, 1,250 a month in interest, and your payment is 1,600, then only 350 of that first payment actually reduces the debt. The balance shrinks slowly, so next month's interest is barely lower, and the pattern repeats.

Only as the principal grinds down does the interest portion fall and the principal portion rise — the crossover happens surprisingly late. This is why making the minimum payment on a card can leave the balance almost untouched for years, and why an extra payment early in a loan is worth far more than the same extra payment later: you attack the balance while it is still generating the most interest.

Two Numbers They Don't Show You

When comparing loans, ignore the headline "interest rate" and look at the APR (Annual Percentage Rate), which folds in fees and charges so different lenders become comparable. And remember that a rate quoted monthly (like "1% per month") compounds to a far larger effective annual figure than the naive ×12 — that gap is exactly how "low monthly rate" offers hide their true cost.

The honest way to see all this is to run the numbers yourself rather than estimate. Our free loan calculator shows the monthly payment, the total interest over the life of a loan, and how the interest-versus-principal split shifts each year, while the percentage calculator handles the everyday "what is X% of Y" and increase/decrease questions that compound interest is built from.

See also

• What a Central Bank Actually Does — Interest Rates, Money and the Last Resort
• Understanding Exchange Rates — What a Quote Actually Means
• Percentages, Ratios and Discounts: The Mental Maths of Everyday Life
• How Inflation Quietly Changes What Your Money Buys