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Loans, credit and compound interest: QCE Essential Mathematics Unit 4

Syllabus dot point

“Calculate simple and compound interest, use the compound interest formula and tables or spreadsheets, compare loan and credit options including personal loans, car loans and credit cards, and calculate the total cost of a loan including interest and fees”

QCEEssential MathematicsUnit 4: Graphs, data and loans8 min read

Quick answer

Simple interest (I=PrtI = Prt) is charged on the original amount; compound interest (A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}) charges interest on interest. Loans charge interest on the balance owing, and credit cards charge high rates on unpaid balances. Compare loans by total repayments plus fees, the cost of borrowing, repayment size and term.

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  1. What this dot point is asking
  2. The answer
  3. Practice questions

What this dot point is asking

Unit 4 finishes with money: how interest works and how to compare ways of borrowing. You need to calculate simple and compound interest, understand how loans and credit cards charge interest, and compare options by total cost to make a sensible decision. Our heading follows the Unit 4 title; check your school's unit plan for the exact subject matter.

The answer

Simple interest

Interest is calculated only on the original amount (principal):

I=PrtI = Prt

where PP is the principal, rr the annual interest rate as a decimal, and tt the time in years.

Compound interest

Interest is added to the balance, and the next interest is calculated on the new balance, so interest earns interest:

A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}

where nn is the number of compounding periods per year (annually 1, quarterly 4, monthly 12, daily 365). The interest earned is A−PA - P. Check the version on your formula sheet.

Compounding more often

For the same annual rate, more frequent compounding gives a slightly larger final amount. Over long periods, compound growth is much larger than simple interest growth.

Loans and credit

  • Reducing-balance loans (personal loans, car loans, home loans): interest is charged on the amount still owing; regular repayments cover the interest and reduce the balance.
  • Credit cards: high interest rates on unpaid balances, often with an interest-free period if the full balance is paid on time. Paying only the minimum means the debt can take years to clear.
  • Buy now pay later: usually no interest, but late fees apply and it is easy to overspend.

Comparing loans

total cost=repayment×number of repayments+fees\text{total cost} = \text{repayment} \times \text{number of repayments} + \text{fees}

cost of borrowing=total cost−amount borrowed\text{cost of borrowing} = \text{total cost} - \text{amount borrowed}

Also consider: the size of repayments you can afford, the length of the loan, fees (establishment, monthly account fees, early repayment fees), balloon payments, and the comparison rate, which includes most fees.

Worked example

A credit card has a balance of 1500 dollars at 20% per year, compounding monthly. If no payments are made for 6 months:

  1. Monthly rate = 0.20 divided by 12 = 0.016667; periods = 6.
  2. A=1500(1.016667)6=1500×1.10426=1656.39A = 1500(1.016667)^6 = 1500 \times 1.10426 = 1656.39
  3. Interest = $156.39 in six months, which shows why paying off the full balance each month matters.
Common traps
Using the percentage instead of a decimal
6% is 0.06.
Forgetting to divide the rate by n and multiply the years by n
Comparing only repayment size
Lower repayments over a longer term usually cost more in total.

Practice questions

Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.

foundation3 marks
Mia invests 5000 dollars at 4% per year. Find the value after 3 years with (a) simple interest and (b) interest compounding annually.
Show worked solution →

(a) Simple interest = 5000 × 0.04 × 3 = $600, so value = $5600.

(b) A=5000(1.04)3=5000×1.124864=5624.32A = 5000(1.04)^3 = 5000 \times 1.124864 = 5624.32, so value = $5624.32.

Compound interest earns $24.32 more because interest earns interest.

Marking guide: 1 mark for (a), 2 marks for (b).

core4 marks
A 12 000 dollar car loan compounds monthly at 9% per year. If no repayments were made, how much would be owed after 2 years? Explain why real loans owe less than this.
Show worked solution →

n=12n = 12, r=0.09r = 0.09, t=2t = 2, so the monthly rate is 0.09 divided by 12 = 0.0075 and there are 24 periods.

A=12 000(1.0075)24=12 000×1.196414=14 356.96A = 12\,000(1.0075)^{24} = 12\,000 \times 1.196414 = 14\,356.96

With no repayments, $14 356.96 would be owed.

Real car loans require regular repayments. Each repayment reduces the balance, so interest is charged on a smaller amount each month (a reducing-balance loan), and the loan is fully repaid by the end of the term.

Marking guide: 1 mark for n and r, 1 mark for substitution, 1 mark for the amount, 1 mark for the explanation.

exam6 marks
Jordan needs 8000 dollars for a car. Option 1: a personal loan with monthly repayments of 260.50 dollars for 3 years plus a 250 dollar establishment fee. Option 2: a dealer loan with monthly repayments of 199 dollars for 4 years, no fees, and a final balloon payment of 1500 dollars. Compare the total cost of each option and recommend one, considering more than cost.
Show worked solution →
Option 1
repayments = 260.50 × 36 = $9378. Total paid = 9378 + 250 = $9628. Cost of borrowing = 9628 minus 8000 = $1628.
Option 2
repayments = 199 × 48 = $9552. Total paid = 9552 + 1500 = $11 052. Cost of borrowing = 11 052 minus 8000 = $3052.
Comparison
Option 1 costs $1424 less overall and is paid off a year earlier. Option 2 has lower monthly repayments ($199 against $260.50), which may suit a tight budget, but Jordan must find $1500 at the end and pays much more interest over the longer term.
Recommendation
If Jordan can afford $260.50 a month, Option 1 is better value. Option 2 only makes sense if the lower repayment is essential, and Jordan should plan to save for the balloon payment.

Marking guide: 2 marks for Option 1 totals, 2 marks for Option 2 totals, 1 mark for comparing cost and repayment size, 1 mark for a justified recommendation.

Practise this

Sources & how we know this

Based on the official QCAA syllabus. Not yet individually reviewed by a person; always check QCAA and your school's materials for current rules.

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