Mathematics Advanced • Year 12 • Module 7 • Lesson 20
Module Synthesis and Exam Preparation
Practise HSC-style writing on the full Module 7 toolkit — compound interest, depreciation, loans, super and product comparison.
1. Short-answer questions
1.1 $15,000 is invested at 4.8% p.a. compounded annually for 5 years. (a) Find the final amount. (b) Find the total interest earned. (c) State the simple-interest amount on the same principal/rate/time and the dollar difference. 3 marks Band 3
1.2 A loan of $300,000 is taken at 8% p.a. compounded quarterly over 20 years. (a) Find the effective annual rate to two decimal places. (b) Find the monthly-equivalent rate? (No — the loan is monthly-repaid but quarterly-compounded; use the quarterly rate r = 0.08/4 and convert n to quarters.) Find the quarterly repayment M using M = Pr / [1 − (1+r)⁻ⁿ]. 3 marks Band 3-4
1.3 $10,000 is invested at 6% p.a. compounded monthly, with $400 added at the end of each month. (a) Write the recurrence An+1 = (1+r)An + a. (b) Compute A₁, A₂ and A₃. (c) Verify A₃ with the closed-form An = P(1+r)ⁿ + a × [(1+r)ⁿ − 1]/r. 4 marks Band 4
Stuck on 1.3? Match the lesson's exam-prep Q10 — A₁ = $10,450, A₂ = $10,902.25, A₃ = $11,356.76.2. Extended response
2.1 An investor at age 35 has the following position:
Mortgage: $500,000 outstanding at 5.4% p.a. compounded monthly, 25 years remaining.
Super balance: $80,000.
Annual super contributions: $12,000 (employer 11.5% on $104,000 salary).
Super net return: 6.0% p.a. (after fees and 15% internal tax).
Extra surplus: $1,000/month, currently uncommitted.
The investor is considering three uses for the $1,000/month surplus over the next 30 years:
Strategy A: Pay extra $1,000/month off the mortgage.
Strategy B: Salary-sacrifice $12,000/yr into super (additional contribution).
Strategy C: Invest $1,000/month in a balanced fund earning 6.0% p.a. net.
(a) Compute the current monthly mortgage repayment M and the total interest under the do-nothing baseline.
(b) For Strategy C, find the FV at age 65 of $1,000/month for 30 years at 6% p.a. compounded monthly. For Strategy B, find the additional super FV at age 65 of $1,000/month at 6% p.a. compounded monthly. For Strategy A, estimate the interest saved on the mortgage by paying it down 1,000/month faster (use the lesson's approximation: extra repayments avoid the loan-rate interest on each dollar from the date of payment to the original loan-end date).
(c) Rank the three strategies by dollar outcome. Write 2–3 sentences on which strategy a Band-6 student should recommend, referencing both the dollar ranking and the rate-comparison principle (when does "invest" beat "repay"?). 8 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — correct M ≈ $3,038 with r = 0.0045, n = 300.
• 1 mark — correct total interest ≈ $411,400.
Part (b) — 4 marks
• 1 mark — Strategy C FV ≈ $1,004,500 with annuity formula shown.
• 1 mark — Strategy B same FV figure (same rate, contribution, term) ≈ $1,004,500, plus correct note that super tax-wrapper is already baked into the 6.0% net rate.
• 1 mark — Strategy A interest saved approximated by 1,000 × [(1.054/12)ᶜⁿ − 1]/r-style annuity at the 5.4% rate over 25 years (≈ $750,000).
• 1 mark — all three figures comparable in same units (dollar at age 65 / value of avoided interest).
Part (c) — 2 marks
• 1 mark — correct ranking (B/C tied above A, given investment rate > loan rate).
• 1 mark — explicit "invest beats repay when rinvestment > rloan" rule.
Your response:
Stuck on (c)? Quote the specific numbers — e.g. "Strategy C delivers $1.00M vs Strategy A's $750k of avoided interest, a $250k gap that hinges on the 0.6 pp rate advantage".How did this worksheet feel?
What I'll revisit before next class:
1.1 — $15,000 at 4.8% for 5 years (3 marks)
Sample response. (a) A = 15,000 × (1.048)⁵ = 15,000 × 1.26397 = $18,959.60. (b) Interest = 18,959.60 − 15,000 = $3,959.60. (c) Simple: A = 15,000 × (1 + 0.048 × 5) = 15,000 × 1.24 = $18,600.00. Compound earns $359.60 more.
Marking notes. 1 mark — correct compound A. 1 mark — correct interest. 1 mark — correct simple A and difference. (Lesson states $18,939.60 / $3,939.60 with marginal rounding; either is acceptable if working is shown.)
1.2 — $300,000 loan at 8% quarterly over 20 yr (3 marks)
Sample response. (a) reff = (1 + 0.08/4)⁴ − 1 = (1.02)⁴ − 1 = 1.08243 − 1 = 8.24% p.a. (lesson Q9a). (b) Quarterly rate r = 0.02; n = 80 quarters. (1.02)⁸⁰ = 4.87544. M = 300,000 × 0.02 / (1 − 1/4.87544) = 6,000 / 0.79490 = $7,548.10/quarter (≈ $2,516/month equivalent). The lesson rounds the monthly figure to $2,542 using slightly different rounding.
Marking notes. 1 mark — correct reff. 2 marks — correct quarterly r, n and M with substitution shown.
1.3 — Recurrence vs closed form (4 marks)
Sample response. r = 0.005; a = 400. (a) An+1 = 1.005An + 400. (b) A₁ = 1.005 × 10,000 + 400 = $10,450.00. A₂ = 1.005 × 10,450 + 400 = $10,902.25. A₃ = 1.005 × 10,902.25 + 400 = $11,356.76. (c) Closed form: A₃ = 10,000 × (1.005)³ + 400 × [(1.005)³ − 1]/0.005 = 10,150.75 + 400 × 3.01503 = 10,150.75 + 1,206.01 = $11,356.76 ✓ — matches the recurrence exactly (matches lesson Q10).
Marking notes. 1 mark — correct recurrence. 1 mark — correct A₁ and A₂. 1 mark — correct A₃. 1 mark — closed-form verification with arithmetic shown.
2.1 — Mortgage / super / surplus integrated decision (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Baseline mortgage repayment.
r = 0.054 / 12 = 0.0045; n = 25 × 12 = 300. (1.0045)³⁰⁰ = 3.83894. M = 500,000 × 0.0045 / (1 − 1/3.83894) = 2,250 / 0.73954 = $3,042.46/month. [1 mark]
Total paid = 3,042.46 × 300 = $912,738. Total interest = 912,738 − 500,000 = $412,738. [1 mark]
(b) Strategy outcomes at age 65 (30-year horizon).
Strategy C. r = 0.06/12 = 0.005; n = 360. (1.005)³⁶⁰ = 6.02257. Annuity factor = 5.02257/0.005 = 1,004.515. FV = 1,000 × 1,004.515 = $1,004,515. [1 mark]
Strategy B. Same r = 0.06/12 = 0.005, same a = $1,000/month, same n = 360 (super tax wrapper is already baked into the 6.0% net rate). FV = $1,004,515. [1 mark]
Strategy A. The $1,000/month extra repayment avoids interest at 5.4% over the original mortgage horizon. Approximating using the same annuity formula with r = 0.054/12 = 0.0045 and n = 300 (mortgage runs out at age 60), then re-investing the residual surplus for 5 more years at 6.0%: FV ≈ 1,000 × [(1.0045)³⁰⁰ − 1]/0.0045 × (1.06)⁵ = 1,000 × 631.32 × 1.3382 = $844,632. [1 mark]
Units check. All three figures are dollar values at age 65 (or, for Strategy A, the equivalent value of avoided interest + reinvested surplus). They are comparable. [1 mark]
(c) Ranking and recommendation.
Ranking: C ≈ B ($1.00M) > A ($845k). [1 mark]
A Band-6 student should recommend Strategy B (salary sacrifice into super) on a tie-breaker over C: the dollar FV is identical to C, but the super contribution is taxed at the entry-side concessional rate of 15% rather than the marginal 32.5–37% the investor would pay on the same dollar going into Strategy C — so each $1,000 of pre-tax surplus buys more units in super than in the open-market fund. Strategy A loses on dollar outcome because the loan rate (5.4%) is below the investment rate (6.0%) — the synthesis principle is invest the surplus whenever rinvestment > rloan on an after-fee, after-tax basis; pay down the loan only when the comparison flips. [1 mark]
Total: 8/8.
Band descriptors for marker.
Band 3: Correct M, total interest stated; one of three FVs computed. ≈ 3 marks.
Band 4: M and interest correct, Strategies B and C FVs both correct (and identified as equal), Strategy A estimated but not converted to comparable units. (c) names a ranking but no rule. ≈ 5 marks.
Band 5: All three FVs correct and comparable, ranking justified, (c) names the rinvestment > rloan rule but does not pick a tie-breaker between B and C. ≈ 6–7 marks.
Band 6: Full calculations, units check, ranking with explicit tie-breaker (super entry-side tax beats marginal-rate investing), and the synthesis rule articulated as an if-then on the rate comparison. 8/8.