Reducing-Balance Loans (up to 3 periods)
Learn how a reducing balance loan works for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-03). Each period, interest is charged on the amount still owing, the repayment is taken off, and the closing balance becomes the next period's opening balance.
You will treat this kind of loan as compound interest with regular repayments taken off, use the rate for each period to fill in a table for the first three periods, and see why the balance and the interest both fall each month — the way car loans and personal loans really work.
Theory
Reducing-balance loans are part of the Depreciation and loans topic in Year 12 Standard 1 (NSW). Interest is charged each month on the amount still owing, the repayment is subtracted, and the closing balance becomes the next month's opening balance. This guide shows how to work through a loan table for up to three months.
A reducing-balance loan charges interest each month on the opening balance — the amount still owing at the start of the month. The repayment is then subtracted to give the closing balance.
The closing balance carries forward as the next month's opening balance. Because the repayment is bigger than the interest, the balance falls each month, and because interest is charged on a smaller balance, the interest also falls each month.
Like compound interest, the interest is worked out on the current balance — but here a repayment is taken off each period. In this Year 12 Standard 1 (NSW) topic you complete loan tables for up to three periods.
A larger or extra repayment (subtract the total paid that month, e.g. \(\$1000+\$500\)) lowers the balance faster, so later interest is smaller: the loan is repaid sooner with less total interest. The rate does not change.
Each month, with the monthly rate \(r\) written as a decimal:
| Month | Opening ($) | Interest ($) | Repayment ($) | Closing ($) |
|---|---|---|---|---|
| 1 | 5000.00 | 100.00 | 900.00 | 4200.00 |
| 2 | 4200.00 | 84.00 | 900.00 | 3384.00 |
| 3 | 3384.00 | 67.68 | 900.00 | 2551.68 |
How to work through a reducing-balance loan
- Write the rate as a decimal: \(1\%=0.01\), \(1.5\%=0.015\), \(2\%=0.02\).
- Interest: multiply the rate by the month's opening balance and round to the nearest cent.
- Closing balance: opening \(+\) interest \(-\) repayment.
- Carry forward: use the closing balance as the next month's opening balance, and repeat for each month.
Interest on the opening balance, then add it and subtract the repayment.
| \(\text{Interest}\) | \(=\) | \(0.01 \times 7500 = 75\) |
| \(\text{Closing}\) | \(=\) | \(7500 + 75 - 600\) |
| \(\) | \(=\) | \(7575 - 600\) |
| \(\) | \(=\) | \(6975\) |
The interest is \(\$75\) and the balance owing is \(\$6975\).
Complete month \(1\) first; its closing balance is the month-\(2\) opening balance.
| \(\text{Month 1 interest}\) | \(=\) | \(0.015 \times 6400 = 96\) |
| \(\text{Month 1 closing}\) | \(=\) | \(6400 + 96 - 700 = 5796\) |
| \(\text{Month 2 interest}\) | \(=\) | \(0.015 \times 5796\) |
| \(\) | \(=\) | \(86.94\) |
The interest charged in month \(2\) is \(\$86.94\).
Interest \(=0.02\times\) opening; closing \(=\) opening \(+\) interest \(-\) repayment, carried down.
| Month | Opening ($) | Interest ($) | Repayment ($) | Closing ($) |
|---|---|---|---|---|
| 1 | 4500.00 | 90.00 | 800.00 | 3790.00 |
| 2 | 3790.00 | 75.80 | 800.00 | 3065.80 |
| 3 | 3065.80 | 61.32 | 800.00 | 2327.12 |
| \(\text{Month 3 interest}\) | \(=\) | \(0.02 \times 3065.80 = 61.32\) |
| \(\text{Closing}\) | \(=\) | \(3065.80 + 61.32 - 800\) |
| \(\) | \(=\) | \(2327.12\) |
The balance owing after \(3\) months is \(\$2327.12\).
Work out both months' interest and compare.
| \(\text{Month 1 interest}\) | \(=\) | \(0.01 \times 8500 = 85\) |
| \(\text{Month 1 closing}\) | \(=\) | \(8500 + 85 - 1000 = 7585\) |
| \(\text{Month 2 interest}\) | \(=\) | \(0.01 \times 7585 = 75.85\) |
| \(75.85\) | \(\lt\) | \(85\) |
The month-\(2\) interest is less (\(\$75.85\)), because it is charged on the smaller opening balance — the rate itself has not changed.
Common pitfalls
Frequently asked questions
What is a reducing-balance loan?
A reducing-balance loan charges interest each period on the amount still owing. The repayment is then taken off, so the balance owing, and the interest charged on it, get smaller over time.
How do you calculate interest on a reducing-balance loan?
Multiply the monthly interest rate, written as a decimal, by that month's opening balance. For example, 2 percent of $5000 is 0.02 times 5000, which is $100.
How do you find the balance owing after a repayment?
Add the month's interest to the opening balance, then subtract the repayment. For $7500 at 1 percent per month with a $600 repayment, the balance is 7500 plus 75 minus 600, which is $6975.
Why does the interest go down each month?
The rate stays the same, but each repayment lowers the balance owing. The next month's interest is worked out on this smaller balance, so it is less.
Is a reducing-balance loan the same as compound interest?
It is similar, because interest is worked out on the current balance each period. The difference is that a repayment is subtracted every period, so the balance goes down instead of up.
What happens if I pay more than the usual repayment?
Subtract the total paid that month, for example $1000 plus $500. A larger or extra repayment lowers the balance faster, so later interest is smaller: the loan is repaid sooner with less total interest. The rate does not change.