Interest Rate Solver
Solve for the missing variable in a loan: find the interest rate, monthly payment, or loan term given the other two, plus total interest paid.
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How to use this calculator
M = monthly payment, P = principal, r = monthly interest rate, n = total months. Rearrange algebraically or iteratively to solve for the unknown variable.
- 1
Choose what you want to solve for: interest rate, payment amount, or loan term.
- 2
Enter the loan principal and the two known values (rate + term, or payment + term, or rate + payment).
- 3
The calculator solves for the missing value using exact financial mathematics.
- 4
Use the rate solver to find what APR a lender is actually charging if you know the payment and term.
Frequently asked questions
What equation connects the three modes?
This tool uses M = P × r ÷ [1 − (1 + r)^−n], where r is the annual rate ÷ 100 ÷ 12 and n is the term in years × 12. Payment mode evaluates it directly, term mode rearranges it to n = −ln(1 − P × r ÷ M) ÷ ln(1 + r), and rate mode searches for the r that makes the equation match your payment.
Is the rate it reports an APR or an effective rate?
A nominal annual rate: the monthly rate multiplied by 12. A result of 0.62% a month appears as 7.42% a year. Converting to an effective annual rate would give a slightly higher figure, and fees, points or insurance are not included, so it will not necessarily match a lender’s disclosed APR.
How does it solve for the rate?
It brackets the monthly rate between 0% and 500% and halves the interval 300 times until the formula’s payment equals yours, so the answer repeats exactly and has no starting-guess dependence. For 20,000 repaid at 400 a month over five years the result is 7.42%; at 500 a month it is 17.27%.
What if total payments exactly equal the principal?
The rate is 0.00%. Repaying 12,000 at 1,000 a month for one year means no interest was charged. If the payments total less than the principal, the tool shows an error, because that would need a negative rate; and a 0% rate in payment mode simply divides the principal by the number of months.
Why does term mode show a final payment?
The term is rounded up to whole months, and the last instalment is whatever clears the balance. With 20,000 at 7% and 400 a month the term is 60 months, but the last payment is 115.34, so total repaid is 23,715.34 and not 24,000. If the payment is no larger than the monthly interest, the loan never ends and an error appears.
What is the Payment if Rate +1% line?
In payment mode, the tool reruns the formula with the annual rate raised by one percentage point and shows the new payment and the monthly increase. For 250,000 over 30 years, 6.5% gives 1,580.17 a month and 7.5% gives 1,748.04, an extra 167.87 a month.
What does the tool not cover?
Variable rates, interest-only periods, extra repayments, fees, taxes and insurance. All repayments are equal and monthly, and results come from the numbers you enter, not from any lender’s offer. They are illustrations, not advice.
Finding the rate, payment or term of an instalment loan
Three unknowns, one relationship
A level-payment loan links four numbers: the amount borrowed, the monthly payment, the rate and the term. Fix any three and the fourth is determined. This solver covers the three most useful cases: working out the rate from a quoted payment, the payment from a quoted rate, and the time to repay from a payment you can afford. Each result also shows total repaid and interest.
Worked example: the rate hidden in a payment
Borrow 20,000 and repay 400 a month for five years: that is 24,000 in total, so 4,000 of interest, and the implied rate is 7.42% a year (0.62% a month). Raise the payment to 500 and the rate jumps to 17.27%, with 10,000 of interest. A 30-year loan of 250,000 repaid at 1,800 a month implies 7.80% and $398,000.00 of interest.
Rate, payment and term compared
The table assembles runs from all three modes. The payment rows show how a rate change alters the monthly cost and total interest, the 0% row confirms the formula collapses to division, and the term rows show the final-payment effect of rounding to whole months.
| Mode and inputs | Result | Total interest |
|---|---|---|
| Rate: 20,000, 400 a month, 5 years | 7.42% a year | $4,000.00 |
| Rate: 250,000, 1,800 a month, 30 years | 7.80% a year | $398,000.00 |
| Payment: 20,000, 7%, 5 years | $396.02 a month | $3,761.44 |
| Payment: 250,000, 6.5%, 30 years | $1,580.17 a month | $318,861.22 |
| Payment: 10,000, 0%, 2 years | $416.67 a month | $0.00 |
| Term: 20,000, 400 a month, 7% | 60 months (last payment $115.34) | $3,715.34 |
| Term: 20,000, 450 a month, 7% | 52 months (last payment $268.70) | $3,218.70 |
Why a small payment can break the loan
Each month the lender charges interest on the outstanding balance, and only the remainder of the payment reduces it. On 20,000 at 7% the interest is about 116.67 in month one, so a 100 payment would leave the balance growing. The term mode checks for this and returns an error, because no number of months repays such a loan.
Treating the answers as estimates
The solver assumes a fixed rate, equal monthly payments and no charges beyond interest. Real contracts add fees and may change the rate, so use the figures to compare options and understand the arithmetic, not to predict a specific offer. Nothing here is a forecast or financial advice. Each mode leaves the other fields untouched, so after solving for a rate you can switch to payment mode with that rate and check that the instalment comes back to your original figure, which confirms the inputs were consistent.
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Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →