Credit & Debt
How To Pay Off Debt
Debt payoff looks like a division problem and is not. Interest is charged on what is left each month, so the balance falls more slowly than a naive division suggests.
Quick Answer
months = -ln(1 - r x balance / payment) / ln(1 + r)
- B
- The balance outstanding at the start
- r
- Monthly interest rate, the annual rate divided by twelve
- P
- The fixed payment made every month
- n
- Number of payments needed to clear the balance
£5,000 at 18% with £250 a month takes about 24 months and costs £989 in interest. Adding £75 a month, paying £325 in total, clears it in about 17.6 months and cuts the interest to £727.
What Is How To Pay Off Debt?
Amortising a balance with fixed payments is not a division problem. Dividing the balance by the payment gives the right answer only when the interest rate is zero, and the true answer is always a longer schedule.
The reason is that interest is charged on the outstanding balance every month. Each payment first covers that month's interest and only the remainder reduces the balance, so the principal falls more slowly than the payment.
The closed form uses logarithms. Setting the balance after n payments to zero and solving for n gives the number of payments directly, with no need to simulate month by month.
The formula has a boundary condition that matters. If the payment does not exceed the interest that accrues in a month, the balance never falls and no finite number of payments clears it.
That boundary is easy to reach on a high-rate balance. £3,000 at 29.99% accrues about £75 of interest a month, so a £60 payment leaves the balance growing rather than shrinking.
The first payment is almost all interest. On £5,000 at 18%, the first payment of £250 covers £75 of interest and reduces the balance by only £175, which is why the early months feel like they achieve nothing.
The split flips as the balance falls. The last payment before clearance covers almost no interest and reduces the balance by almost the whole payment, so the final months make rapid visible progress.
Total interest is total paid minus the original balance. It is not a separate charge but an unavoidable consequence of paying off over time, and it is the number that an extra payment actually reduces.
An extra payment does two things at once. It removes principal permanently, which reduces every future month's interest, and it shortens the schedule, which removes whole months of interest from the end.
That double effect is why an extra £75 a month on the loan above saves £262 rather than the £225 it removes from the principal. The savings include every month of interest that no longer accrues.
The effect grows with the rate and the balance. The same £75 on a 6% loan saves far less than on an 18% one, because the interest being avoided was smaller in the first place.
Order matters when there are several balances. Paying the smallest balance first clears a debt quickly and frees a payment, while paying the highest rate first saves the most money. Both are defensible and neither is simply right.
Minimum payments are designed to keep the debt alive. A card quoting a minimum of 2% of the balance or a fixed floor, whichever is larger, can take decades and cost more in interest than the original purchase.
A balance transfer to a lower rate reduces the interest directly, but a transfer fee and a promotional period that ends before the balance is cleared can make the position worse than it started.
The calculator solves for the number of payments, the total paid, the total interest and the share of the total that interest represents, and reports the interest saved by any extra payment you add.
Formula
n = -ln(1 - r x B / P) / ln(1 + r)
Setting the balance after n payments to zero and solving for n gives the number of payments directly. The argument of the logarithm must be positive, which is the condition that the payment covers the interest.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| r | Monthly rate | percent | Annual rate divided by twelve. |
| B | Balance | currency | The amount outstanding at the start. |
| P | Payment | currency | The fixed payment made each month. |
| n | Payments | payments | How many payments clear the balance. |
total interest = P x n - B
Everything paid goes to either the balance or the interest. Subtracting the original balance from the total paid isolates the interest exactly.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Payment | currency | The fixed payment made each month. |
| n | Payments | payments | How many payments are made. |
| B | Balance | currency | The amount outstanding at the start. |
| I | Total interest | currency | The cost of the borrowing over the whole term. |
first-month interest = r x B
Interest for the month is charged on the opening balance, so multiplying the balance by the monthly rate gives the interest before any of the payment is applied.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| r | Monthly rate | percent | Annual rate divided by twelve. |
| B | Balance | currency | The amount outstanding at the start. |
| I_1 | First-month interest | currency | Interest charged in the first month. |
How To Calculate How To Pay Off Debt
- 1
Convert the annual rate to a monthly one
Divide the annual percentage rate by twelve. On 18% the monthly rate is 1.5%, and that figure is what each month's interest is charged on.
- 2
Check the payment covers the interest
Multiply the balance by the monthly rate. If the payment does not exceed that figure the balance grows rather than shrinks, and no schedule will clear it.
- 3
Solve for the number of payments
Apply the logarithmic formula. A £5,000 balance at 18% with £250 a month gives roughly 24 payments, well beyond the 20 a zero-interest division suggests.
- 4
Multiply out to find the total paid
Payment times number of payments gives everything paid. Subtracting the original balance leaves the total interest, which is the cost of the schedule.
- 5
Repeat with the extra payment
Add an extra payment and solve again. The new schedule is shorter, and the interest saved is the difference between the two totals rather than the extra payments themselves.
Examples
Example 1: £5,000 at 18% with £250 a month
- Balance
- £5,000
- APR
- 18%
- Monthly payment
- £250
- Extra payment
- £0
| Step | Calculation | Result |
|---|---|---|
| Monthly rate | 18% / 12 | 1.50% |
| Interest in the first month | £5,000 x 0.015 | 75 |
| Principal in the first payment | £250 - £75 | 175 |
| Payments needed | -ln(1 - 0.015 x 5000/250) / ln(1.015) | 23.96 |
| Total paid | £250 x 23.96 | 5989.06 |
| Total interest | £5,989.06 - £5,000 | 989.06 |
Result: At £250 a month the £5,000 balance at 18% takes about 23.96 months to clear. The first payment covers £75 of interest and only £175 of the balance, and the total interest comes to £989.06.
Example 2: The same balance with £75 more each month
- Balance
- £5,000
- APR
- 18%
- Monthly payment
- £250
- Extra payment
- £75
| Step | Calculation | Result |
|---|---|---|
| Monthly rate | 18% / 12 | 1.50% |
| Effective payment | £250 + £75 | 325 |
| Payments needed | -ln(1 - 0.015 x 5000/325) / ln(1.015) | 17.62 |
| Total paid | £325 x 17.62 | 5727.09 |
| Total interest | £5,727.09 - £5,000 | 727.09 |
| Interest saved | £989.06 - £727.09 | 261.97 |
Result: Paying £325 instead of £250 clears the £5,000 balance in about 17.62 months rather than 23.96. Total interest falls from £989.06 to £727.09, saving £261.97, which is more than the extra payments themselves.
Calculator
Payments needed
17.6218
- Years to clear
- 1.4685
- Total paid
- $5,727.09
- Total interest
- $727.09
- Interest as a share of total
- 12.70%
- Interest saved by the extra payment
- $261.97
- Payment too low flag
- 0
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the How To Pay Off Debt calculator page.
Common Mistakes
Dividing the balance by the payment
That only works at a zero interest rate. On any real balance it understates the schedule, because each payment first covers a month of interest.
Assuming any payment reduces the balance
If the payment does not exceed the monthly interest the balance grows. On a high-rate card with a small minimum this is exactly what happens.
Paying the minimum because it is affordable
Minimums are set to keep the balance alive. A balance that takes decades and costs more in interest than the original purchase is the normal outcome, not a worst case.
Adding the extra payments to get the saving
The saving is larger than the extra payments, because each one also removes every future month of interest that would have accrued on it.
Ignoring the rate when choosing which balance to attack first
Attacking the highest rate saves the most money per pound, while attacking the smallest balance frees a payment soonest. Both are valid but they are different strategies.
Rolling a balance to a card with a shorter promotional period
A lower rate helps only if the balance is cleared before the promotion ends. Otherwise the standard rate applies to a balance that has barely moved.
Forgetting that a new purchase restarts the clock
Paying a card down and then spending on it returns the balance to where it was, with the same interest rate and the same schedule ahead of it.
FAQ
How do I calculate how long it will take to pay off debt?
Use the logarithmic formula with the monthly rate, the balance and the payment. The answer is always longer than dividing the balance by the payment, because interest accrues on what is left each month.
How much should I pay to clear debt quickly?
As much as you can without creating a second problem. Even a modest extra payment shortens the schedule and removes whole months of interest from the end, so the saving is larger than the extra amount.
Which debt should I pay off first?
Highest rate first saves the most money, smallest balance first frees a payment soonest. The second approach is more motivating for many people, and a method that gets followed beats one that does not.
What happens if my payment is less than the monthly interest?
The balance grows rather than shrinking, and no number of those payments will ever clear it. The payment has to be raised above the monthly interest before any progress is possible.
Does paying more each month change the interest rate?
No, but it changes the balance faster, and interest is charged on the balance. A lower balance means less interest each month, which is why the saving compounds rather than merely accumulating.
Is a balance transfer worth it?
If the balance is cleared before the promotional period ends, yes. If it is not, the standard rate applies afterwards and the transfer fee makes the overall cost higher than staying put.
References
- [1]Consumer Financial Protection Bureau, Understanding how to pay down debt — https://www.consumerfinance.gov/ask-cfpb/how-do-i-pay-down-my-debt-en-1435/
- [2]Federal Reserve Education, Amortization and loan payoff — https://www.federalreserveeducation.org/
- [3]Consumer Financial Protection Bureau, Minimum payments and long-term debt cost — https://files.consumerfinance.gov/f/documents/cfpb_credit-card-survey-report_final.pdf