Mathematics Advanced • Year 12 • Module 7 • Lesson 20
Module Synthesis and Exam Preparation
Apply the full Module 7 toolkit to integrated saving, borrowing, super and comparison scenarios.
Problem 1 — Integrated home-loan scenario (lesson Q2-1)
A $450,000 loan at 4.8% p.a. compounded monthly over 25 years.
Set up: What are we solving for?
(i) Find the minimum monthly repayment M using M = Pr / [1 − (1+r)⁻ⁿ]. 2 marks
(ii) Find the total interest paid over the life of the loan. 2 marks
(iii) Use the recurrence An+1 = (1+r)An − M to find the loan balance after 5 years (60 months). Compare with the closed-form An = P(1+r)ⁿ − M × [(1+r)ⁿ − 1]/r. 3 marks
Stuck? Revisit lesson § Mixed Problems Q1.Problem 2 — Superannuation projection (lesson Q2-2)
A 35-year-old has $40,000 in super. Salary $85,000. Employer contributions 11.5% of salary, paid at the end of each year. Fund returns 6.5% p.a. gross with 1.0% fees, all returns reinvested annually.
Set up: What are we solving for?
(i) State the annual contribution C and the net return rnet. 1 mark
(ii) Project the balance at age 65 using FV = A₀(1+r)³⁰ + C × [(1+r)³⁰ − 1] / r. 3 marks
(iii) Re-run the projection assuming fees rise to 2.0% (rnet = 4.5%). State the new FV and the dollar cost of the fee increase. 2 marks
Problem 3 — Investment-product comparison (lesson Q2-3)
$40,000 to invest over 15 years. Two options:
Option I: Term deposit at 4.5% p.a. (no fees).
Option II: Managed fund at 6.8% p.a. gross, 1.2% fees.
Set up: What are we solving for?
(i) Find the FV of each option. 2 marks
(ii) State which option wins and by how many dollars. 1 mark
(iii) If the investor is taxed at the marginal 32.5% rate on the managed-fund earnings (term deposit is held in a tax-free retirement wrapper), recompute the after-tax winner. Does the choice flip? 3 marks
Stuck? Apply rafter-tax = rnet × (1 − t) from Lesson 16.Problem 4 — Exam-simulation timing (lesson Q3)
Time yourself: 20 minutes for the three problems above. Record your timing and self-mark.
Set up: What are we solving for?
(i) Record your timing per problem. Did you finish all three within 20 minutes? Which was hardest? 2 marks
Problem 1 time: ____________ Problem 2 time: ____________ Problem 3 time: ____________
(ii) Mark your own work harshly against the answer key. Tally marks earned out of (10 + 6 + 6) = 22. Where did you lose marks — formula selection, periodic-rate conversion, arithmetic, or interpretation? 2 marks
(iii) Identify 3 topics to revisit before the exam, with the lesson references that cover them. 3 marks
Problem 5 — Save vs repay (the synthesis question)
A 30-year-old has $10,000 spare per year and faces a choice. Option A: pay $10,000 extra off a $400,000 mortgage at 5.4% p.a. monthly (current M = $2,245.22). Option B: invest $10,000/yr in a balanced fund at 6.0% p.a. (net) for 30 years.
Set up: What are we solving for?
(i) For Option B, find the FV of the investment after 30 years using FV = a × [(1+r)ⁿ − 1] / r with a = 10,000 and r = 0.06 (annual). 2 marks
(ii) For Option A, a 30-yearly $10,000 lump-sum extra repayment is mathematically equivalent to reducing the loan principal by a present-value chain at 5.4% p.a. Approximate the "interest avoided" as a × [(1+r)ⁿ − 1] / r with r = 0.054. State the figure. 2 marks
(iii) Compare the Option B FV with the Option A "interest avoided" figure. Which gives the bigger dollar outcome? State the lesson principle this illustrates — when does "invest" beat "repay" and vice versa? 3 marks
Stuck? The decision pivots on rinvestment > rloan (after fees and tax).How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — $450,000 mortgage at 4.8% over 25 years
Set up. We are computing M, total interest, and the balance after 5 years via both recurrence and closed form.
(i) r = 0.004; n = 300. (1.004)³⁰⁰ = 3.31350. M = 450,000 × 0.004 / (1 − 1/3.31350) = 1,800 / 0.69821 = $2,578.33/month.
(ii) Total paid = 2,578.33 × 300 = $773,499. Total interest = 773,499 − 450,000 = $323,499 (matches lesson).
(iii) Closed form at n = 60: (1.004)⁶⁰ = 1.27050. A₆₀ = 450,000 × 1.27050 − 2,578.33 × (0.27050/0.004) = 571,725 − 2,578.33 × 67.625 = 571,725 − 174,358 = $397,367. The lesson rounds to ≈ $390,000 using a slightly different fixed-rate iteration; either value is acceptable. Note that after 5 years of $154,700 in payments, principal has only shrunk by $52,633 — the remaining $102,067 was interest.
Problem 2 — Super projection from age 35 to 65
Set up. We are projecting a starting balance plus annual contributions at a fee-adjusted return, then re-running for the fee-shock scenario.
(i) C = 85,000 × 0.115 = $9,775/yr. rnet = 6.5 − 1.0 = 5.5%/yr.
(ii) (1.055)³⁰ = 4.98395. FV = 40,000 × 4.98395 + 9,775 × (3.98395/0.055) = 199,358 + 9,775 × 72.435 = 199,358 + 708,051 = $907,409. (Lesson rounds to $882,000 using slightly different rounding.)
(iii) rnet = 4.5%. (1.045)³⁰ = 3.74532. FV = 40,000 × 3.74532 + 9,775 × (2.74532/0.045) = 149,813 + 9,775 × 61.007 = 149,813 + 596,343 = $746,156. The extra 1% fee costs $161,253 over 30 years — roughly 18% of the smaller balance.
Problem 3 — TD vs managed fund on $40,000 over 15 yr
Set up. We are comparing two FVs and then re-running with after-tax adjustment.
(i) TD: A = 40,000 × (1.045)¹⁵ = 40,000 × 1.93528 = $77,411. MF: rnet = 5.6%. A = 40,000 × (1.056)¹⁵ = 40,000 × 2.25623 = $90,249.
(ii) Managed fund wins by $12,838 (lesson rounds to ~$13,200).
(iii) After-tax MF rate = 5.6 × (1 − 0.325) = 3.78%. AMF,AT = 40,000 × (1.0378)¹⁵ = 40,000 × 1.74336 = $69,734. Term deposit (no tax) = $77,411. After tax, term deposit wins by $7,677 — the choice flips because the marginal tax bite (32.5%) eats almost all of the managed fund's gross-return advantage.
Problem 4 — Exam-simulation timing (self-assessed)
Set up. Self-assessed. Sample marker notes:
(i) Target: Problem 1 ≈ 8 min, Problem 2 ≈ 7 min, Problem 3 ≈ 5 min. If you exceed 25 min, identify the slowest step and practise that formula in isolation.
(ii) Common weak spots from the lesson: transposition for n, loan recurrence tables, fee-adjusted super.
(iii) Sample revision list: (a) periodic-rate conversion (Lesson 1 § Critical Rule), (b) annuity closed-form vs recurrence (Lesson 10), (c) after-tax / after-fee FV (Lesson 16). Tailor to your own weakest areas.
Problem 5 — Save vs repay on $10,000/yr
Set up. We are comparing the FV of investing the surplus against the "interest avoided" by paying down a 5.4% mortgage.
(i) r = 0.06, n = 30. (1.06)³⁰ = 5.7435. FV = 10,000 × (5.7435 − 1)/0.06 = 10,000 × 79.058 = $790,582.
(ii) r = 0.054, n = 30. (1.054)³⁰ = 4.8338. "Interest avoided" ≈ 10,000 × (4.8338 − 1)/0.054 = 10,000 × 70.996 = $709,963.
(iii) Option B (invest) wins by $790,582 − $709,963 = $80,619. Principle: when the after-tax, after-fee investment return exceeds the loan rate (here 6% vs 5.4%), the rational mathematics says invest the surplus and let compounding outpace the mortgage interest. If the rates flip (e.g. loan at 7% vs investment at 5%), extra repayments win — and the closer the rates, the more behavioural factors (risk tolerance, certainty preference) tip the decision.