Term Payments
Term Payments in NSW Year 10 Core maths looks at the cost of buying now and paying over time (NESA outcome MA5‑FIN‑C‑01). Buying on terms usually means a deposit and then regular repayments, and the flat-rate interest is charged on the balance owing for the whole loan.
You will find deposits, balances, total repayments and the extra cost compared with the cash price, and calculate monthly or weekly repayments with flat-rate interest. You will also work out the costs of buy now, pay later plans with late fees, interest-free offers and small loans with establishment and monthly fees, and find a flat interest rate, a deposit or a loan term working backwards. It is part of Year 10 Core Earning Money.
Every question with a fully worked solution.
- Term Payments - Video - Term Payments Watch
Theory
Buying on terms means paying a deposit now and the rest in regular repayments, which usually costs more than paying cash. This Year 10 maths page covers flat-rate loans, finding the interest and the rate, buy now, pay later, interest-free offers and the true cost of small loans.
Buying on terms (by instalments) means paying a deposit first, then the balance in regular repayments. The interest is the total paid minus the cash price.
A flat-rate loan charges simple interest on the balance owing after the deposit, for the whole time of the loan: \(I = Prn\).
Buy now, pay later (BNPL) splits a price into a few equal payments, often fortnightly, usually with no interest but with late fees. An interest-free offer charges interest only on any amount still owing when the offer ends.
A small loan (a short-term loan for a few hundred dollars) may charge an establishment fee and a monthly fee, each a percentage of the amount borrowed.
How to solve a term payment problem
- Find the deposit (often a percentage of the price) and the balance owing.
- Count the repayments: \(12\) a year if monthly, \(26\) if fortnightly, \(52\) if weekly. Leave out any months with no repayments.
- Add the deposit, all the repayments and any fees to get the total paid.
- Compare with the cash price to find the interest or extra cost.
Other questions you will meet
- Flat rate from repayments: interest \(\div\) (balance \(\times\) years) gives \(r\) (Example 3).
- How many years: each year the borrower repays (repayment \(\times 12\)) and is charged a year's interest. Solve balance \(+\) interest per year \(\times n =\) repaid per year \(\times n\).
- Working back to the price or deposit: if the deposit is \(25\%\), the repayments cover \(75\%\) of the price, so divide by \(0.75\). If flat interest adds \(30\%\) to the balance, the repayments are \(130\%\) of it, so divide by \(1.3\).
- Buy now, pay later: divide the price by the number of payments. Add any late fees and give the extra as a percentage of the price.
- Interest-free periods: to clear the debt in time, divide the price by the number of months. Anything left at the end starts to attract interest.
- Options: work out the full cost of each, such as cash with a discount, terms, a credit card at simple interest, or a small loan, then subtract.
(i) How much does he pay altogether?
(ii) How much more than the cash price is this?
Total on terms \(=\) deposit \(+\) all the repayments. \(2\) years is \(24\) months.
| \(\text{deposit}\) | \(=\) | \(0.15 \times 2450\) |
| \(\) | \(=\) | \(\$367.50\) |
| \(\text{repayments}\) | \(=\) | \(24 \times \$95\) |
| \(\) | \(=\) | \(\$2280.00\) |
| \(\text{total}\) | \(=\) | \(\$367.50 + \$2280.00\) |
| \(\) | \(=\) | \(\mathbf{\$2647.50}\) |
| \(\text{(ii) extra}\) | \(=\) | \(\$2647.50 - \$2450.00\) |
| \(\) | \(=\) | \(\mathbf{\$197.50}\) |
The extra \(\$197.50\) is the interest: the cost of not paying cash.
(i) each monthly instalment
(ii) the total cost of the car.
Flat-rate interest is simple interest on the balance owing after the deposit.
| \(\text{deposit}\) | \(=\) | \(0.2 \times 16\,500\) |
| \(\) | \(=\) | \(\$3300\) |
| \(\text{balance}\) | \(=\) | \(16\,500 - 3300\) |
| \(\) | \(=\) | \(\$13\,200\) |
| \(I\) | \(=\) | \(13\,200 \times 0.075 \times 3\) |
| \(\) | \(=\) | \(\$2970\) |
| \(\text{to repay}\) | \(=\) | \(13\,200 + 2970\) |
| \(\) | \(=\) | \(\$16\,170\) |
| \(\text{(i) monthly}\) | \(=\) | \(16\,170 \div 36\) |
| \(\) | \(=\) | \(449.166\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{\$449.17}\) |
| \(\text{(ii) total cost}\) | \(=\) | \(3300 + 16\,170\) |
| \(\) | \(=\) | \(\mathbf{\$19\,470.00}\) |
Charge the interest on the balance, not on the full price, and add the deposit back for the total cost.
(i) the interest she pays
(ii) the flat interest rate per annum charged on the balance.
(i) Interest \(=\) total paid \(-\) cash price.
| \(\text{total paid}\) | \(=\) | \(150 + 112 \times 12 = \$1494\) |
| \(I\) | \(=\) | \(1494 - 1350 = \mathbf{\$144}\) |
(ii) The interest is charged on the balance borrowed, \(1350 - 150 = \$1200\), for \(1\) year. Use \(I = Prn\).
| \(144\) | \(=\) | \(1200 \times r \times 1\) |
| \(r\) | \(=\) | \(144 \div 1200 = 0.12\) |
| \(\) | \(=\) | \(\mathbf{12\%} \text{ p.a.}\) |
For a loan over \(n\) years, divide by \(1200 \times n\) instead.
(i) Find the total fees.
(ii) He repays the loan and fees in \(6\) equal monthly payments. How much is each payment?
(iii) Write the fees as a flat interest rate per annum.
(i) Both fees are percentages of the amount borrowed.
| \(\text{establishment}\) | \(=\) | \(0.2 \times 300 = \$60\) |
| \(\text{monthly fees}\) | \(=\) | \(0.04 \times 300 \times 6 = \$72\) |
| \(\text{total fees}\) | \(=\) | \(\mathbf{\$132}\) |
| \(\text{(ii) payment}\) | \(=\) | \((300 + 132) \div 6 = \mathbf{\$72}\) |
(iii) The fees are \(\dfrac{132}{300} = 44\%\) of the loan for \(6\) months. There are two lots of \(6\) months in a year.
| \(44\% \times 2\) | \(=\) | \(\mathbf{88\%} \text{ p.a.}\) |
Short loans for small amounts look cheap in dollars but are very expensive as a yearly rate.
Common pitfalls
Frequently asked questions
What does buying on terms mean?
Buying on terms means paying a deposit and then the rest of the price in regular repayments over time. The total paid is usually more than the cash price, and the difference is the interest.
How do you calculate flat-rate interest?
Find the balance owing after the deposit and use \(I = Prn\) with the yearly rate and the number of years. The interest is the same each year because it is always worked out on the original balance.
How do you work out repayments on a loan?
For a flat-rate loan, add the interest to the balance, then divide by the number of repayments, for example \(36\) monthly repayments over \(3\) years.
Is buy now, pay later free?
Buy now, pay later often charges no interest if every payment is made on time. Late fees and account fees can add a lot, so work out the extra cost as a percentage of the price.
Why are small loans so expensive?
The fees are a large percentage of a small amount, charged over a short time. Written as a yearly interest rate, they are often far higher than a credit card or bank loan.