Asked by walter white

what formula can be used to explain the total cost of a large purchase with regular payments

All Answers 1

Answered by GPT-5 mini AI
Use the loan-amortization (annuity) formula. Define:

- P = purchase price
- D = down payment (if any)
- L = loan principal = P − D + any financed fees/taxes
- r = periodic interest rate (annual APR divided by payments per year)
- n = total number of payments
- PMT = payment per period

Fixed-period payment (amortizing loan):
PMT = L * [ r / (1 − (1 + r)^−n) ]

Total amount paid on the loan = PMT × n
Total cost of the purchase = D + (PMT × n) + any non‑financed fees
Total interest paid = (PMT × n) − L

You can also rearrange to find the loan amount from a known payment:
L = PMT * [ (1 − (1 + r)^−n) / r ]

Example:
$20,000 purchase, $2,000 down, finance L = $18,000, 5% APR, monthly payments for 5 years → r = 0.05/12 ≈ 0.0041667, n = 60.
PMT ≈ 18,000 * [0.0041667 / (1 − (1.0041667)^−60)] ≈ $339.30
Total paid on loan ≈ $339.30 × 60 = $20,358 → total cost ≈ $2,000 + $20,358 = $22,358
Interest paid ≈ $20,358 − $18,000 = $2,358

Notes:
- Use the periodic rate matching payment frequency (monthly, weekly, etc.).
- APR may be quoted differently (compounding) — convert to the correct periodic rate.
- If payments vary or interest is simple/interest‑only, the formulas differ and you’d use an amortization schedule or appropriate rate formulas.