Loans: differentiated and annuity payments
Take a loan and watch the debt grow and shrink: repaid in equal shares of the debt, or in equal payments. Compare what the borrower pays in the end.
- Russia: grade 11
Guide
Each year the debt grows by r % (it is multiplied by 1 + r/100), then the borrower makes a payment. All amounts here are in thousands of roubles.
With differentiated payments the debt falls by the same amount each year and the interest is charged on what is left, so the payments get smaller. With annuity payments all payments are equal; the payment x comes from the equation that the debt is zero after the last payment.
The overpayment is everything paid minus the loan. Common slips: charging the interest on the whole loan every year, calling the whole amount paid the overpayment, and forgetting the interest of the last year. This is advanced-level material.