How this instrument works
This quiz drills the two-step calculation at the center of every fixed-rate loan: first derive the standard monthly payment from the loan amount, rate, and term, then use that payment to find how much principal remains after a chosen number of months. Finance students, mortgage-loan-originator exam candidates, and tutors checking a homework key use it the same way — enter the four inputs a textbook problem gives, and see whether a hand-worked payment and balance match the instrument's.
The two formulas chain together because a payment only makes sense once its size is fixed to zero the loan out over its stated term. The payment formula, PMT = L·r(1+r)^N ⁄ ((1+r)^N − 1), spreads that goal evenly across N months. The balance formula then asks a narrower question: if that exact payment had gone out for k of those months, what is left? It compounds the original principal forward by (1+r)^k, the growth it would show untouched, then subtracts the future value of the k payments already sent, PMT·((1+r)^k − 1) ⁄ r.
Both formulas assume the payment computed here — not a lender's rounded or fee-adjusted figure — was sent on time every single month, at a rate that never moved. A loan with a skipped payment, extra principal thrown at it, or an adjustable rate needs the site's separate loan-balance instrument, which takes the real payment as an input instead of deriving one; this quiz exists to check the textbook math itself, not to reconstruct a real account's history.
- Enter the loan's original amount under Loan amount, $ — the sum borrowed at the start, before any payments.
- Set Annual interest rate, % to the note's stated rate, and Loan term, years to the term used to size the standard payment.
- Enter Months already paid — how many monthly payments into that term you want to check.
- Read Monthly payment, $ — the standard payment the amortization formula derives from the first three entries.
- Read Remaining balance, $ and compare it against a hand-worked answer or a homework key.
Worked example — $300,000 at 6% after 60 payments
Take a $300,000 loan at a 6% annual rate on a standard 30-year term: Loan amount, $ = 300000, Annual interest rate, % = 6, Loan term, years = 30. The payment formula first returns Monthly payment, $ = 1,798.65, the figure that retires this loan in exactly 360 equal installments. Set Months already paid = 60, five years in, and the balance formula returns Remaining balance, $ = 279,163.07.
Sixty payments of $1,798.65 add up to $107,919.00 sent to the lender, yet the balance has only dropped by $20,836.93, from $300,000 to $279,163.07 — meaning roughly $87,082 of those five years' payments went to interest and less than a quarter went to principal. Checking that split, not just the final balance, is usually where a hand-worked homework answer goes wrong.
Questions
Why does this quiz chain two separate formulas instead of one?
Because a loan's balance at any month depends on a payment that has to be solved for first. The instrument runs the payment formula on Loan amount, $, Annual interest rate, %, and Loan term, years to get Monthly payment, $, then feeds that payment into the balance formula along with Months already paid — the same order a textbook problem set expects you to follow by hand.
How is this different from the site's loan-balance calculator?
That instrument takes an actual monthly payment as an input, because real accounts often send more or less than a textbook schedule calls for. This quiz derives the payment itself from the loan amount, rate, and term, then checks the balance against that derived figure — built to verify the standard amortization formula, not to reconstruct an account whose real payment history is unknown.
What should Remaining balance, $ show once Months already paid equals the full term in months?
Zero, or a figure close enough to zero to read as a rounding artifact — a 30-year term run out to Months already paid = 360 leaves nothing owed, since the payment was sized precisely to retire the loan by then. If a hand-worked answer sits hundreds of dollars from zero at the full term, the payment was likely computed or rounded incorrectly earlier in the problem.
Why is the balance still above 90% of the original loan after five years?
Because the earliest payments are charged against the largest remaining balance, so interest claims most of each one. On the $300,000, 6%, 30-year example above, five years of $1,798.65 payments — $107,919 total — knock only $20,836.93 off the principal, leaving $279,163.07 owed. The share going to principal grows every month after that, but slowly at first.
Can Months already paid exceed the loan term?
The field will not stop you from entering it, but the result stops meaning anything once k passes N — the payment was only sized to reach zero at N months, so pushing k further produces a negative balance rather than a real payoff. Keep Months already paid at or below Loan term, years times 12 for an answer that matches a genuine amortization schedule.
Who actually uses a quiz like this one?
Candidates studying for a mortgage-loan-originator or accounting licensing exam use it to check timed practice problems, and instructors use it to generate a fresh answer key by changing the four inputs. Loan officers new to amortization math run it the way a lab student checks a sensor reading — confirming the formula by hand before trusting software that hides the arithmetic.
References
- Consumer Financial Protection Bureau — Owning a Home: loan options and amortization
- Federal Reserve — Consumer Credit statistical release (G.19)
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.