Mathematics Standard • Year 12 • Module 7 • Lesson 11
Mixed Financial Problems — Problem Set
Apply integrated financial reasoning to realistic Australian decisions — save vs borrow, refinance, debt strategies, salary sacrifice vs after-tax investing.
Problem 1 — A $20,000 car: save or borrow?
Eli needs a $20,000 car. Option A: save $600/month at 4% p.a. compounded monthly. Option B: take a 5-year loan at 7% p.a. compounded monthly (M = $396.02). Option C: use $400/month dealer flat-rate finance at 5% over 5 years.
Set up: What are we solving for?
(i) Approximately how many months will Option A take? Use FV = M × [(1+r)ⁿ − 1] / r and trial-and-error (r = 0.04/12). 2 marks
(ii) Calculate the total amount paid under Option B and under Option C (flat: interest = P × r_flat × n; total = P + I; M = total / 60). 2 marks
(iii) Recommend the cheapest option and explain in one sentence why time-to-have-the-car may make Eli choose differently. 2 marks
Stuck? Revisit lesson § Example: Save or Borrow? — the $25,000 car comparison.Problem 2 — Refinancing a $250,000 mortgage
Selena has $250,000 remaining on a 5.4% p.a. mortgage with 15 years left (M = $2,036.07). A competitor offers 4.5% for the remaining 15 years (M = $1,912.48) but charges $1,500 in switching fees.
Set up: What are we solving for?
(i) Calculate the total amount Selena would pay under each option over the remaining 15 years. 2 marks
(ii) Calculate the monthly saving from refinancing and the break-even month (fees ÷ monthly saving). 2 marks
(iii) Recommend whether Selena should refinance, naming the dollar interest saved over the 15-year term. 2 marks
Stuck? Revisit lesson § Example: Refinancing — the $350,000 break-even at 13 months.Problem 3 — Two debts, one budget
Marco has two debts: a $4,000 credit-card balance at 20% p.a. (monthly r = 0.20/12) and a $12,000 car loan at 8% p.a. compounded monthly (monthly r = 0.08/12). He can afford a total of $500/month. Compare two strategies:
Strategy A (even split): $250/month to each debt.
Strategy B (avalanche): Minimum payment $100 on the car loan; the rest ($400) to the credit card until it is cleared, then $500/month to the car loan.
Set up: What are we solving for?
(i) Calculate the Month-1 interest on each debt. 2 marks
(ii) Under Strategy B, approximately how many months until the $4,000 card is cleared at $400/month? (Use trial — month 1 principal ≈ $400 − $67; allow for the falling balance.) 2 marks
(iii) In one or two sentences, explain mathematically why Strategy B (avalanche) saves more total interest than Strategy A. 2 marks
Stuck? Revisit lesson § Activity 1 Q3 — credit card 20% vs personal loan 8% comparison.Problem 4 — Salary sacrifice vs after-tax investing
Priya is 35 and considers contributing an extra $400/month towards super (taxed at 15% on contributions, returns 7% p.a. compounded monthly) for 30 years, versus taking the $400 as after-tax pay (marginal tax 32.5%, so $270 reaches her bank) and investing it in shares at 8% p.a. compounded monthly for 30 years.
Set up: What are we solving for?
(i) Calculate the FV at age 65 if Priya salary-sacrifices $400/month at 7% p.a. (use FV-of-annuity, r = 0.07/12, n = 360). 2 marks
(ii) Calculate the FV at age 65 if she invests $270/month after-tax at an effective 8% × (1 − 0.325) ≈ 5.4% (r = 0.054/12). 2 marks
(iii) Recommend the better option for retirement savings, naming the dollar difference and one realistic trade-off. 2 marks
Stuck? Revisit lesson § Worked Example — the 30-year-old salary-sacrifice vs after-tax shares comparison.Problem 5 — Fortnightly vs monthly repayments
A $400,000 mortgage at 5.0% p.a. compounded monthly over 30 years has monthly repayment $2,147.29. The same borrower could instead pay half ($1,073.65) every fortnight — that's 26 fortnights = 13 full monthly equivalents per year (one extra payment per year compared to 12 monthly payments).
Set up: What are we solving for?
(i) Calculate the total amount paid per year under (a) monthly payments and (b) fortnightly payments. 2 marks
(ii) Calculate the dollar difference per year and state how this difference reduces the balance compared to the monthly schedule. 2 marks
(iii) Under fortnightly payments, the term drops to about 25 years (with total interest ~$340,000 vs the monthly schedule's ~$373,000). Calculate the interest saving and explain in one sentence why "just paying fortnightly" is so powerful. 2 marks
Stuck? Revisit lesson § Q8 short-answer — the fortnightly trick on a $500,000 mortgage.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Save vs borrow vs dealer flat
Set up. Each option uses a different formula. Compute and rank dollar costs.
(i) r = 0.04/12 ≈ 0.003333. Solve 600 × [(1.003333)ⁿ − 1] / 0.003333 = 20,000 → (1.003333)ⁿ ≈ 1.1111 → n ≈ ln(1.1111)/ln(1.003333) ≈ 31.6 → about 32 months.
(ii) Option B: Total = 396.02 × 60 = $23,761. Interest = $3,761. Option C: I_flat = 20,000 × 0.05 × 5 = $5,000. Total = $25,000. M = 25,000 / 60 = $416.67.
(iii) Saving (Option A) is cheapest at ~$19,200 in contributions + interest; Option B costs $3,761 more; Option C costs $5,000 more. But Option A means waiting 32 months for the car, so a buyer who needs transport now may rationally pick Option B.
Problem 2 — Refinance break-even
Set up. Compute totals; find monthly saving and break-even.
(i) Old: 2,036.07 × 180 = $366,493. New: 1,912.48 × 180 = $344,246; with $1,500 fees = $345,746.
(ii) Monthly saving = 2,036.07 − 1,912.48 = $123.59. Break-even = 1,500 / 123.59 ≈ 12.1 months.
(iii) Yes, refinance. She is ahead from month 13, and over 15 years saves $366,493 − $345,746 = $20,747 in interest after fees.
Problem 3 — Debt avalanche
Set up. Monthly interest on each, then compare strategies.
(i) Card: 4,000 × 0.20/12 = $66.67. Car: 12,000 × 0.08/12 = $80.00.
(ii) Month 1 principal on card = 400 − 66.67 ≈ $333. Balance drops quickly; full payoff in roughly 11-12 months with total card interest ~$200.
(iii) Every extra dollar paid against the 20% card saves 20¢/year of interest; the same dollar against the 8% car loan saves only 8¢/year. Strategy B accelerates payoff of the highest-rate debt first, so total interest charged across both debts is minimised.
Problem 4 — Salary sacrifice vs after-tax shares
Set up. Two FV-of-annuity calculations; compare on the same "value at age 65" basis.
(i) r = 0.07/12 ≈ 0.005833, n = 360. FV = 400 × [(1.005833)³⁶⁰ − 1] / 0.005833 ≈ 400 × 1219.97 ≈ $487,988.
(ii) r = 0.054/12 = 0.0045, n = 360. FV = 270 × [(1.0045)³⁶⁰ − 1] / 0.0045 ≈ 270 × 904.65 ≈ $244,255.
(iii) Salary sacrifice wins by about $243,733. Trade-off: super is locked until preservation age (currently around 60), so the money is not accessible for emergencies or a home deposit before retirement.
Problem 5 — Fortnightly trick
Set up. Compute annual totals; compare interest.
(i) Monthly per year = 2,147.29 × 12 = $25,767.48. Fortnightly per year = 1,073.65 × 26 = $27,914.90.
(ii) Extra paid per year = 27,914.90 − 25,767.48 = $2,147.42 — essentially one extra full monthly payment per year, all of which goes against principal.
(iii) Interest saved ≈ $373,000 − $340,000 = $33,000. One extra full monthly payment per year, repeated every year, snowballs through compound interest into roughly five years of term shaved and tens of thousands of dollars saved.