The formula, in one paragraph
Start with the balance, the APR and the monthly payment. The month’s interest is the balance times the APR divided by 12, the payment covers the interest first and then the balance, and the payoff is the month-by-month simulation of the balance falling until it reaches zero. The bigger the payment, the more of it reaches the balance, and the faster the balance falls.
Step 1: Work out the month’s interest
The interest is the balance times the APR divided by 12, because the APR is the yearly rate and the loan charges a twelfth of it each month. At the example numbers that is $25,000 times 7.5% divided by 12, which is $156.25 of interest in month 1. At 5% the same balance costs only $104.17 of interest in month 1, which is $52.08 less, and at 10% it costs $208.33, which is $52.08 more.
Step 2: Work out what reaches the balance
The payment covers the interest first, so only the rest touches the balance. At $500 a month and $156.25 of interest, $343.75 reaches the balance in month 1. At $600 a month, $443.75 reaches the balance, and at $400 a month, only $243.75 does. That is why the payment matters so much: the part that reaches the balance is the part that shortens the loan.
Step 3: Simulate month by month
Each month, add the interest to the balance and subtract the payment, then work out the next month from the new balance. After month 1 the balance is $25,000 plus $156.25 minus $500.00, which is $24,656.25 — only $343.75 of the payment touched the balance. The interest shrinks with the balance, so more of the $500 payment reaches the balance every month, and the loan takes 61 months to reach zero.
Step 4: Read the totals
Add up the interest over all the months for the total interest, and add the balance for the total paid. At the example numbers the interest over 61 months is $5,069.24, and the total paid is $25,000 plus $5,069.24, which is $30,069.24 — and the last payment is $69.24, smaller than the usual $500. The standard 60-month plan is $500.95 a month with $5,056.91 of interest, so paying $500 a month costs $12.33 more interest and takes 1 extra month against it.
That box is the whole formula.
Worked example: the default loan
Step 2: balance after month 1 = $25,000 + $156.25 − $500.00 = $24,656.25.
Step 3: repeat for 61 months — interest $5,069.24, total $30,069.24.
Step 4: last payment $69.24, standard plan $500.95/mo, interest $5,056.91.
Answer: 61 months (5 yr 1 mo), $5,069.24 of interest, $30,069.24 paid in all.
What changes the payoff most
The payment moves the payoff fastest. Paying $600 a month takes 49 months and costs $4,051.02 of interest, and paying $400 a month takes 80 months and costs $6,799.84 — a $200 a month difference that costs 31 more months and $2,748.82 more interest. The balance is next: a $20,000 balance takes 47 months and costs $3,086.63, and a $30,000 balance takes 76 months and costs $7,718.05. The APR is third: at 5% the $25,000 balance takes 57 months and costs $3,092.30, which is $1,976.94 less than at 7.5%, and at 10% it takes 65 months and costs $7,474.45.
Paying extra: the same loan, different numbers
A bigger payment and the standard plan use the same first month’s interest: $156.25 either way. The numbers part company at the payment. The standard plan is $500.95 a month and takes 60 months, with $5,056.91 of interest. Paying $500 a month takes 61 months and costs $5,069.24 — a $12.33 loss and 1 extra month. Paying $600 a month, a hundred dollars extra, takes 49 months and costs $4,051.02, saving 12 months and $1,018.22. Paying $750 a month takes 38 months and costs $3,121.88, saving 23 months and $1,947.36. Every extra dollar goes straight to the balance, because the interest is already covered.
Work backwards from a payoff date you want
When the payoff date is fixed, run the formula in reverse. A $777.66 a month payment clears the $25,000 loan in 36 months, $604.48 a month clears it in 48 months, $500.95 a month clears it in 60 months, and $432.26 a month clears it in 72 months. At $500 a month, the biggest balance gone in 60 months is $24,952.65, and in 48 months it is $20,679.19. Each reverse payment is rounded up to the cent: one cent less takes one month longer. Check the payment against the interest first: at $500 a month and 7.5% APR, any balance at or above $80,000 never pays off, because the first month’s interest alone is $500.00 or more. Pay more than the interest every month, or the loan never shrinks.