M
hscscience Maths Std · Y12
0/100daily goal
0
0
0 due
0
L1 · 0 XP
KJ
Your weak spots
Insights load after your first practice round.
Module 7 · L11 of 12 ~30 min MST-12-S2-02 ⚡ +90 XP available

Mixed Financial Problems

Real life does not present problems neatly labelled "compound interest" or "annuity." A typical financial decision involves multiple concepts simultaneously, comparing a car loan against saving up, deciding between extra mortgage repayments and investing in shares, or evaluating whether a balance transfer is worth the fee. This lesson brings together every concept from the module in integrated problems that mirror actual financial decisions.

Today's hook, You need a $\$25,000$ car. Option A: save $\$500$/month at 4%. Option B: 5-year loan at 7% reducing balance. Option C: dealer finance at 4% flat rate over 5 years. Which is cheapest overall? Predict first.
0/5QUESTS

Get oriented

Recall what you already know, meet the key ideas and settle the terms.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

You need a $\$25,000$ car. Option A: save $\$500$/month at 4% p.a. compounded monthly. Option B: take a 5-year loan at 7% reducing balance. Option C: dealer finance at 4% flat rate over 5 years. Which is cheapest overall? What information do you need?

Before reading on write your prediction. We will revisit this at the end of the lesson.

auto-saved
02
Key ideas, this lesson
reference

Mixed problems combine multiple financial formulas. The key is a systematic approach, not a new formula.

Integrated decisions: combine multiple financial concepts, simple interest, compound interest, annuities, and loan repayments, in one problem.

Total cost comparison: include all fees, interest charges, and opportunity costs when comparing options.

Break-even analysis: find the point where two options cost the same, before this point one option is better; after it the other wins.

Work through the same five steps for any calculation problem.
Key insight: the cheapest option is not always the best, consider flexibility, risk, liquidity, and personal circumstances alongside the pure mathematics.
Save vs borrow
Compare FV of an annuity (savings) against the total repayments on a reducing balance loan for the same amount.
Refinancing
Calculate total remaining cost at the old rate vs the new rate plus fees. Break-even = fee ÷ monthly saving.
Opportunity cost
Money used to pay off a debt cannot be invested. Compare the return you would earn against the interest you save.
03
What you will master
Know

Key facts

  • All module formulas
  • Fee structures and opportunity costs
  • Break-even concept
Understand

Concepts

  • How financial concepts interrelate
  • Why context matters in decisions
  • Trade-offs between options
Can do

Skills

  • Solve multi-step financial problems
  • Compare complex options quantitatively
  • Justify decisions with mathematics
04
Key terms
Opportunity costThe benefit foregone by choosing one option over another, e.g. investing instead of repaying debt.
Break-even pointThe time or amount at which two options have equal total cost.
Total costPrincipal + all interest + all fees over the full term.
RefinancingReplacing an existing loan with a new one, usually at a lower rate, often with a fee.
Debt avalancheStrategy of paying the highest-interest debt first to minimise total interest paid.
Salary sacrificeRedirecting pre-tax income to superannuation, reducing taxable income and being taxed at only 15%.

Work systematically

Use one repeatable method to take apart any mixed financial problem.

05
A systematic approach to mixed problems
core concept

For mixed financial problems, follow these five steps:

  1. Identify what you need to find: total cost, monthly payment, time to goal, or a comparison between options.
  2. List all given information: principal, rates, time periods, compounding frequency, fees.
  3. Choose the right formula(s): match the situation, is money growing (compound interest / FV annuity) or is money being borrowed and repaid (loan repayment / PV annuity)?
  4. Calculate each component separately: work step by step, keeping full precision until the final answer.
  5. Compare and conclude: state which option is cheaper or better, and give a brief justification.

Common combinations that appear in HSC questions:

  • Compare saving vs borrowing (FV of annuity vs total repayments on a loan)
  • Evaluate refinancing (compare total costs before and after, including fees)
  • Salary sacrifice vs after-tax investing (tax-adjusted returns)
  • Multiple debts, which to pay first (debt avalanche method)
Read the question carefully: "total cost," "total interest," "monthly repayment," and "amount saved" are different things. Circle what you are asked to find.

Mixed financial problems require a systematic approach: identify the scenario type (simple/compound interest, annuity, depreciation), list known variables, choose the correct formula, convert units, then calculate and verify.

Pause, copy the five-step approach: (1) identify scenario type; (2) list all known variables with units; (3) choose the correct formula; (4) convert rates and periods; (5) calculate and verify, applying it builds confidence for any mixed exam question into your book.

Quick check: When comparing a 5-year loan against saving up for the same item, which formula gives the total cost of the loan?

Decide: save or borrow

Work the car purchase both ways and say which one actually costs less.

Beyond the syllabus. The integrated core comparisons in this consolidation are worth doing. Refinancing analysis, debt-avalanche simulation, opportunity-cost optimisation and detailed super tax comparisons are extension — the exam draws on the core loan and investment skills.
06
The car purchase decision, save or borrow?
core concept

The five-step approach, identify scenario type, list known variables, choose formula, convert units, then calculate and verify, applies directly to the save-versus-borrow decision. To compare: calculate the FV of saving regularly (annuity FV formula) versus the total cost of a loan (repayment × n). Saving avoids all interest; borrowing incurs interest but provides immediate access to the asset.

Problem: You need a $\$25,000$ car. You can save $\$600$/month at 4.8% p.a. compounded monthly, or take a 5-year loan at 7.2% p.a. compounded monthly.

Saving option:

We need $FV = 25000$. Using $FV = M \times \frac{(1+r)^n - 1}{r}$ with $M = 600$ and $r = \frac{0.048}{12} = 0.004$:

$$25000 = 600 \times \frac{(1.004)^n - 1}{0.004}$$ $$\frac{25000 \times 0.004}{600} = (1.004)^n - 1 \implies (1.004)^n = 1.1\overline{6}$$ $$n = \frac{\ln 1.1\overline{6}}{\ln 1.004} \approx 38.7 \text{ months (round up to 39)}$$

Total contributed = $600 \times 39 = \$23{,}400$ (the rest is interest earned). You reach $25,000 in just over 3 years.

Borrowing option:

$r = \frac{0.072}{12} = 0.006$, $n = 60$.

$$M = 25000 \times \frac{0.006}{1 - (1.006)^{-60}} = \frac{150}{0.30265} \approx \$495.03 \text{ per month}$$

Total repayments = $495.03 \times 60 = \$29{,}702$. Interest paid = $\$4{,}702$.

Comparison: Saving costs $\$0$ interest but takes 3.25 years, you cannot drive the car now. Borrowing gives immediate access but costs $\$4,702$ in interest. The right choice depends on whether you need the car immediately and whether you can tolerate the wait.

Key insight: Saving is always cheaper in pure dollar terms. Borrowing has an "urgency premium", you pay interest for the convenience of having the item now.

Save vs borrow decision: calculate the future value of saving (FV annuity or compound growth) versus the total cost of borrowing (loan repayments × n). Saving avoids interest entirely; borrowing incurs interest but gives earlier access.

Pause, copy the save-vs-borrow framework: calculate FV of regular savings (annuity FV formula) vs total loan cost (repayment × n), and state the trade-off: saving costs nothing in interest but requires waiting; borrowing costs interest but gives immediate access into your book.

True or false: When comparing saving versus borrowing, saving always costs less in total dollars but requires waiting longer.

Work the examples

Follow two worked problems, then compare superannuation with after-tax investing.

Worked examples · reveal each step

PROBLEM 1 · REFINANCING DECISION

You have $\$350,000$ remaining on a mortgage at 5.4% p.a. compounded monthly with 20 years left. A new lender offers 4.8% p.a. but charges $\$1,500$ in fees. Should you refinance?

1
Current: $r = 0.054/12 = 0.0045$, $n = 240$. $M = \frac{350000 \times 0.0045}{1 - (1.0045)^{-240}} = \$2389.66$/month
Find current monthly repayment using the loan formula
1 of 4
PROBLEM 2 · MULTIPLE DEBTS, WHICH TO PAY FIRST?

A person has $\$5,000$ credit card debt at 20% p.a. and a $\$10,000$ car loan at 8% p.a. reducing balance. They can afford $400/month total. Should they split payments evenly (A), pay minimum on car and maximum on credit card (B), or minimum on credit card and maximum on car (C)?

1
Credit card charges $20\%/12 = 1.67\%$/month. Car loan charges $8\%/12 = 0.67\%$/month. Highest interest rate is the credit card.
Identify which debt is most expensive
1 of 4
08
Superannuation versus after-tax investing
core concept

The save-vs-borrow framework compares FV of savings against total repayments. The superannuation-vs-investment decision is a similar trade-off on a longer horizon: super contributions are taxed at only 15% going in (versus your marginal income tax rate), giving a significant compounding advantage over decades, but super funds are locked until the preservation age of around 60, so the cost is zero accessibility before then.

Problem: A 30-year-old can salary sacrifice $\$500$/month into super (taxed at 15%) earning 7% p.a. compounded monthly, OR invest $\$500$/month after-tax in shares earning 8% p.a. compounded monthly. Marginal tax rate = 30%.

Super option: Contribution is pre-tax. $M = 500$, $r = 7/12\% = 0.005833$, $n = 420$ months (35 years).

$$FV = 500 \times \frac{(1.005833)^{420} - 1}{0.005833} \approx 500 \times 1801 = \$900{,}400$$

Shares option: After $\$500$ is taxed at 30%, only $\$500 \times (1 - 0.30) = \$350.00$ is invested. Returns are also taxed, so effective return $\approx 8\% \times (1 - 0.30) = 5.6\%$. $r = 0.004667$, $n = 420$.

$$FV = 350 \times \frac{(1.004667)^{420} - 1}{0.004667} \approx 350 \times 1300 = \$455{,}000$$

Difference: Super wins by approximately $\$1.2$ million, entirely due to tax concessions on contributions and earnings.

Important limitation: Super is inaccessible until retirement (typically age 60). Shares can be sold at any time. The tax advantage of super must be weighed against this lack of flexibility.

Superannuation versus after-tax investing: super benefits from tax concessions on contributions (15% tax vs marginal rate) but funds are locked until preservation age (~60). After-tax investing is accessible but taxed at higher rates.

Pause, copy the super advantage: contributions taxed at 15% vs your marginal rate (e.g., 30% or 45%), and the trade-off: super funds cannot be accessed until preservation age (~60), so the tax advantage comes at the cost of long-term illiquidity into your book.

Fill the gap: A person on a 30% marginal tax rate takes $\$400$/month out of super salary sacrifice and invests it after tax instead. The actual amount invested each month is $\$$.

Drill the mixed problems

Run the quick-fire practice until choosing the method is automatic.

Quick-fire practice · 2 activities

1

Integrated problems. (a) Compare buying a $\$20,000$ car with cash (saved over 3 years at 4% p.a. monthly) vs a 5-year loan at 6.5% p.a. reducing balance. Find the total cost of each. (b) A credit card charges 19.99% p.a. on $\$3,000$. A personal loan offers 8% p.a. over 3 years. Compare total cost of each option for clearing the debt.

2

Break-even analysis. A bank offers a 0.3% lower mortgage rate with a $\$2,000$ fee. For a $\$300,000$ loan with $\$1,800$/month current repayment, the new repayment would be $\$1,740$/month. How many months until the fee is recovered?

Match each strategy to its description:

  • Debt avalanche
  • Refinancing
  • Opportunity cost
  • Break-even
  • Return foregone by choosing one option
  • Point where two options cost the same
  • Pay highest-interest debt first
  • Replace loan with lower-rate loan

Revisit your thinking

Return to your opening answer and name what has changed.

09
Revisit your thinking

Model answer for the Think First: Option A (save $\$500$/month at 4% monthly): $FV = 25000$. $n \approx 47$ months (3.9 years). Total contributed = $\$23{,}500$. Option B (7% reducing, 5-year loan): $M \approx \$495$/month. Total = $\$29{,}702$. Interest = $\$4{,}702$. Option C (4% flat rate): Interest = $\$25000 \times 0.04 \times 5 = \$5{,}000$. Total = $\$30{,}000$.

Ranking by total cost: Save ($\$25,000$) < Loan B ($\$29,702$) < Loan C ($30,000). But saving means waiting nearly 4 years. The best choice depends on whether you need the car now.

What has changed in your understanding? What did you get right? What surprised you?

auto-saved

Top 3 list: Name THREE non-mathematical factors that might influence a financial decision even when one option is clearly cheaper. For each, give a brief example.

Practise mixed problems

Answer the question bank, then write full short-answer responses.

01
Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next.

Q1. A 5-year loan at 7.2% p.a. compounded monthly on $\$25,000$ has monthly repayments of $\$495.03$. What is the total interest paid?

Q2. The "debt avalanche" method means:

Q3. A bank refinancing offer saves $\$90$/month but costs a $\$1,620$ fee. The break-even is:

Q4. Why does salary sacrificing into superannuation grow faster than investing the same after-tax dollars in shares (for someone on a 30% marginal rate)?

Q5. A family needs $15,000 for renovations. Option A: redraw from mortgage at 5.2% p.a. Option B: personal loan at 9.6% p.a. over 3 years. Which option has lower total interest, and why?

02
Short answer
ApplyBand 42 marks

SA 1. A family needs $\$15,000$ for home renovations. They can: (A) redraw $\$15,000$ from their mortgage at 5.2% p.a. over 5 years, (B) take a personal loan at 9.6% p.a. over 3 years, or (C) save $\$400$/month at 3.6% p.a. compounded monthly until they reach $\$15,000$. Compare all three options quantitatively and recommend. (2 marks)

auto-saved
ApplyBand 42 marks

SA 2. A person has $\$5,000$ credit card debt at 20% p.a. and a $\$10,000$ car loan at 8% p.a. reducing balance. They can afford $400/month total. Compare: (A) split payments evenly, and (B) pay minimum on car and maximum on credit card. Justify which strategy is better mathematically. (2 marks)

auto-saved
AnalyseBand 53 marks

SA 3. A 35-year-old has $\$100,000$ in super earning 7% p.a. compounded monthly. They salary sacrifice an extra $\$300$/month (pre-tax, 15% tax). (a) Calculate the super balance at age 60. (b) Alternatively, they take that same $\$300$/month as income instead and invest what is left after tax at a 30% marginal rate in shares at 8% p.a. compounded monthly, with returns also taxed at that rate. The existing $\$100,000$ stays in super either way, since it cannot be withdrawn at 35. Calculate this balance at age 60. (c) Compare and justify a recommendation, considering access restrictions. (3 marks)

auto-saved
Comprehensive answers (click to reveal)

MC 1, C: Total = 495.03 × 60 = $\$29,701.80$. Interest = $\$29,701.80$ − $25,000 = $4,701.80.

MC 2, B: Avalanche = highest interest first.

MC 3, A: Break-even = 1620 ÷ 90 = 18 months.

MC 4, D: More money is invested from the start because contributions are taxed at 15% not 30%.

MC 5, C: 5.2% < 9.6% so mortgage redraw has lower interest regardless of term.

SA 1 (2 marks): Comparison with all three totals [1 mark]; recommendation with justification [1 mark].

SA 2 (2 marks): Strategy B identified [1 mark]; dollar justification showing strategy B saves more [1 mark].

SA 3 (3 marks): The existing $\$100,000$ is common to both options, since it cannot leave super at 35: $\$100,000 \times (1.005833)^{300} ≈ \$572,500$. (a) the extra $\$300$/month is $\$255$ after the 15% contributions tax, so $\$255 \times [(1.005833)^{300}-1]/0.005833 ≈ \$206,600$, giving ≈ $\$779,000$ [1 mark]. (b) taken as income it is $\$300 \times 0.70 = \$210$/month, and the after-tax return is 8% × 0.70 = 5.6% p.a. (r = 0.004667), so $\$210 \times [(1.004667)^{300}-1]/0.004667 ≈ \$136,900$, giving ≈ $\$709,400$ [1 mark]. (c) salary sacrifice wins by ≈ $\$69,700$ on the extra contributions alone, but super is locked until preservation age while the shares are accessible [1 mark].