The Verification Venue · a mechanism hiding in your statement
The Minimum That Never Ends
Paying the minimum on a credit card feels like slow progress. If the minimum were a pure percentage of the balance, it would be — mathematically — no progress at all: the balance shrinks by a fixed factor each month and approaches zero without ever reaching it.
Here is the whole mechanism in one line. Each month interest is added at the monthly rate r = APR / 12, and you pay a minimum of p (say 2%) of the balance. So next month's balance is:
That single factor (1 + r − p) decides everything. If your monthly rate is below the minimum percentage, the balance decays — but geometrically, so it never quite hits zero. If the rate equals the percentage, the balance is frozen forever: your whole payment is interest. If the rate is higher, the balance grows — the "minimum" doesn't even cover the interest. Set the dials and watch the curve.
Months to pay off
70
≈ 5.8 years
Total interest paid
$735
$1,735 paid on $1,000
Decay factor (1+r−p)
0.9975
decays toward zero
A bigger balance keeps the 2% minimum above the floor for longer.
At a 2% minimum, 24% is the knife-edge: at and above it the pure-percentage balance never shrinks at all.
Real cards use 1–3%. The minimum is also where the floor compares.
Minimum = the greater of the % and the floor. Set it to none and watch the loan refuse to end.
Switch the floor to none at 24% or higher and the readout stops giving a number: it says never, because it is true. The line crawls toward the bottom of the chart and flattens into an asymptote that never touches it. Then put the $25 floor back. The moment the percentage minimum would fall below $25, the payment becomes a flat $25 — and a flat payment that beats the interest does reach zero, in finite time. The floor is the only reason your card ever gets paid off.
The check — every number recomputed in front of you
The chart and readouts above come from a month-by-month ledger run live in your browser — the same ledger the offline verifier runs. For your current dials:
This is exactly the box the Credit CARD Act of 2009 forces onto every US statement — the minimum-only payoff time and cost, beside a 3-year payoff for comparison:
The verifier recomputes all of these — the geometric factor, the 70-month / $735 headline, the 36-month comparison, and the "never terminates with no floor" result — and exits non-zero if any disagrees. Run it yourself: node research/the-minimum-that-never-ends/verify-the-minimum-that-never-ends.mjs
What's idealised here, and what's exactly true
Exactly true. The recurrence B_next = B·(1 + r − p) is exact for a flat percentage minimum with monthly compounding. A geometric sequence with factor in (0, 1) is strictly positive for every finite term — so a pure-percentage minimum never clears the balance in finite time; it only approaches zero as a limit. At r = p (with a 2% minimum, an APR of exactly 24%) the factor is 1 and the balance is frozen; above it the balance grows. The fixed-floor phase is a constant payment, and a constant payment that exceeds the interest clears the balance in N = −ln(1 − B·r/F)/ln(1+r) months — finite, and shorter the larger the floor.
Free choices, named. (1) The payment convention. We treat the minimum as p of the balance and add interest separately, so the floor compares against p·B. Real issuers vary: many use "1% of balance plus that month's interest and fees," which always covers interest and so always terminates — a different, gentler dynamic than the pure-percentage rule shown here. (2) Monthly compounding at r = APR/12; real cards compute interest on the average daily balance, which shifts the totals by cents to a few dollars but not the mechanism. (3) The last payment is partial, so we round the month count up — the closed form gives 69.4 months for the headline case, the ledger reports 70. (4) No new charges, no rate changes, no fees — the same assumptions the CARD Act statement box is required to make.
Where it breaks from a real statement. Issuers' minimum formulas, floors ($25–$35 is typical but not universal), and APRs all vary, and federal guidance actually forbids a minimum so low it causes negative amortization — so a real "minimum" usually does cover interest, even if barely. The "never ends" result is therefore a statement about the pure-percentage rule in isolation, which is the honest framing: the percentage rule alone would never finish; the floor is what saves you. The CARD Act numbers this page mirrors are themselves estimates the regulation allows to be off by up to two months.