Reducing Balance Loans
Learn reducing balance loans for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-08). With home loans and car loans, interest is charged on the amount still owing each period, and every regular repayment covers that interest and reduces the debt.
You will learn to track the balance owing period by period, write a recurrence for the amount owing, find the repayment needed to clear a loan, how much can be borrowed, and the total paid and total interest — real decisions about borrowing.
Theory
A reducing balance loan is a compound-interest loan repaid by regular instalments. This Year 12 Mathematics Advanced topic (MAV-12-08) tracks the balance owing.
With rate \(r\) per period and repayment \(M\): \(A_{n}=A_{n-1}(1+r)-M\) — add interest, then subtract the repayment. Interest is charged on the reducing balance. Total paid \(=M\times\) (number of repayments); total interest \(=\) total paid \(-\) amount borrowed.
Method
- Add interest: multiply the balance by \((1+r)\).
- Subtract the repayment \(M\).
- Repeat each period.
| \(A_1\) | \(=\) | \(1000(1.10)-400=700\) |
| \(A_2\) | \(=\) | \(700(1.10)-400=370\) |
| \(\text{total}\) | \(=\) | \(500\times24=\$12\,000\) |
| \(\text{interest}\) | \(=\) | \(7000-5000=\$2000\) |
Common pitfalls
Frequently asked questions
What is a reducing balance loan?
A compound-interest loan repaid by regular instalments, where interest is charged each period on the balance still owing.
How do you find the amount owing after a repayment?
Multiply the previous balance by 1 plus the rate, then subtract the repayment.
How do you find the total interest paid?
Subtract the amount borrowed from the total of all repayments.
Why does the balance reduce faster over time?
As the balance falls, the interest each period is smaller, so more of each repayment reduces the principal.