How this math actually works

A loan isn't a purchase, it's a rental. You're renting someone else's money, and the rent is charged on whatever you still have out — not on what you originally borrowed.

That one detail explains almost everything else. Because the rent is charged on the remaining balance, every dollar you knock off the balance early cancels the rent on that dollar for the entire rest of the loan. A dollar paid in year one of a 30-year mortgage isn't a dollar of progress — it's a dollar of progress plus 29 years of avoided rent.

A fixed loan payment never changes, but what it's made of changes every single month. Early on the balance is huge, so the rent is huge, and only a sliver of your payment reaches the balance. Later the balance is small, the rent is small, and nearly the whole payment is progress.

Look at the first payment on that $350,000 mortgage:

The interest rate is the rent. The APR is the rent plus the fees, expressed as a yearly rate — it's the honest number for comparing two offers, and lenders are required to disclose it.

The monthly payment is neither. It's a function of three things — amount, rate, and term — and you can change it without changing the cost at all, simply by moving the term. That's why a payment quoted without a term tells you nothing, and why the Loan Offer Decoder exists.

Every calculator here ends the same way, on purpose:

• Interest saved — what the decision is worth in dollars. The number to compare against other things you could do with the money.
• Time saved — the same fact in the units people actually feel. “Six years sooner” lands where “$114,979” doesn't.
• Total cost — everything you hand over from first payment to last. The number no one selling you a loan will say out loud.

Learn to read those three on one page and you can read them on every page — and, more usefully, on any offer anyone ever puts in front of you.

These are clean models of messy things. Real loans have daily compounding, servicer rounding, escrow adjustments, fees, promotional rates that expire, and rate changes on anything adjustable. Each calculator lists exactly what it assumes at the bottom of its page — worth two minutes before you make a decision on it.

One honest caveat that applies everywhere: paying debt early isn't automatically the best use of a dollar. An employer match, an emergency fund, or a higher expected return elsewhere can all beat prepaying a low-rate loan. What's not in question is the guaranteed, tax-free return of not paying interest — which on a 20%+ credit card is very hard to beat anywhere.