Mathematics Advanced • Year 12 • Module 7 • Lesson 20
Module Synthesis and Exam Preparation
Build fluency across the full Module 7 formula reference and the 10 most common HSC errors.
1. Quick recall — the Module 7 formula sheet
Answer each question in the space provided. 1 mark each
Q1.1 Complete the formulas:
Simple interest amount: A = ____________________
Compound interest amount: A = ____________________
Effective annual rate: reff = ____________________
Q1.2 Complete the annuity / loan formulas:
FV of annuity: FV = ____________________
PV of annuity: PV = ____________________
Loan repayment: M = ____________________
Q1.3 Write the lesson's HSC exam-strategy rule in one sentence: what should you always identify first when reading a financial-mathematics question?
2. Worked example — diagnose a Top-10 error
Follow every line. Each step has a short reason.
Problem. A student computes the FV of $10,000 at 8% p.a. compounded quarterly for 5 years and writes: "A = 10,000(1 + 0.08)⁵ = 10,000 × 1.4693 = $14,693." Identify the error from the lesson's Top-10 list, then compute the correct answer.
Step 1 — Diagnose the error.
The student used the annual rate r = 0.08 with n = 5 years. But interest is compounded quarterly, so r and n must be in quarters.
Lesson Top-10 errors 1 and 2: "annual rate instead of periodic rate" + "years instead of periods".
Step 2 — Convert to quarterly units.
r = 0.08 / 4 = 0.02 per quarter
n = 5 × 4 = 20 quarters
Step 3 — Apply A = P(1 + r)ⁿ correctly.
A = 10,000 × (1.02)²⁰ = 10,000 × 1.48595 = $14,859.47
Step 4 — Verify with the effective-rate cross-check.
reff = (1 + 0.08/4)⁴ − 1 = (1.02)⁴ − 1 = 0.08243 = 8.243% p.a.
A = 10,000 × (1.08243)⁵ = 10,000 × 1.48595 = $14,859.47 ✓
Conclusion. The student's answer is $166.47 short. Correct A = $14,859.47; the effective-rate route confirms it.
3. Faded example — fill in the missing steps
A student computes the loan repayment on $200,000 at 6% p.a. compounded monthly over 25 years and writes "M = 200,000 × 0.06 / [1 − (1.06)⁻²⁵] = $15,634". Diagnose and correct. 4 marks
Step 1 — Diagnose:
The student used the annual rate with the year count. Errors from the Top-10 list: ______ and ______.
Step 2 — Convert to periodic units:
r = ______________ per month n = ______________ months
Step 3 — Apply M = Pr / [1 − (1+r)⁻ⁿ]:
M = 200,000 × ______ / (1 − (______)⁻______) = $______________
Conclusion. Correct monthly repayment = $______________ . The student's incorrect $15,634 represents the annual repayment if there were only one payment per year — out by a factor of about 12.
4. Graduated practice — formula selection drill
For each scenario, write the correct formula name AND apply it. Show working to a final answer.
Foundation — pick the formula (4 questions)
| Q | Scenario | Formula used + answer |
|---|---|---|
| 4.1 1 | $5,000 at 4.5% p.a. simple for 8 years — find A. | |
| 4.2 1 | $5,000 at 4.5% p.a. compound annually for 8 years — find A. | |
| 4.3 1 | Asset $40,000, flat-rate depreciation $5,000/yr for 6 years — find salvage. | |
| 4.4 1 | Asset $40,000, reducing-balance depreciation 18% p.a. for 6 years — find salvage. |
Standard — typical HSC difficulty (6 questions)
Show working in the space below each part — at least one substitution line and one evaluation line.
4.5 Reproduce the lesson's exam-prep Q9(a). Find the effective annual rate of 8% p.a. compounded quarterly. 2 marks
4.6 Reproduce the lesson's exam-prep Q9(b). Find the monthly repayment on $300,000 at 5.2% p.a. compounded monthly over 20 years. 2 marks
4.7 Reproduce the lesson's Q10. A $10,000 investment earns 6% p.a. monthly, with $400 added at the end of each month. (a) Write the recurrence An+1 = (1+r)An + a. (b) Compute A1, A2, A3. (c) Verify A₃ with the closed form. 3 marks
4.8 Reproduce the lesson activity Q2-1. A $450,000 loan at 4.8% p.a. compounded monthly over 25 years. Find the minimum monthly repayment and the total interest. 2 marks
4.9 Reproduce the lesson activity Q2-2. A super fund has $40,000 balance, salary $85,000, contributions 11.5%, gross return 6.5%, fees 1%. Project the balance in 30 years. 3 marks
4.10 Reproduce the lesson activity Q2-3. Compare a 4.5% term deposit with a 6.8% managed fund (1.2% fees) on $40,000 over 15 years. Which wins and by how much? 3 marks
Extension — combine concepts (2 questions)
4.11 A bank offers two mortgage products on $400,000 over 30 years: (a) 5.4% p.a. compounded monthly, (b) 5.4% p.a. compounded quarterly. Find M for each and the dollar difference. State which compounding frequency is better for a borrower. 3 marks
4.12 An $80,000 asset is depreciated. Flat-rate at 8%/yr vs reducing-balance at 16%/yr. Find the salvage value after (i) 5 years and (ii) 10 years under each method. Which method gives the larger salvage at 5 yr? At 10 yr? Explain in one sentence why the methods swap order. 3 marks
5. Self-check the easy 3
Tick the first three once you have checked the method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Core compounding formulas
Simple: A = P(1 + rn). Compound: A = P(1 + r)ⁿ. Effective annual: reff = (1 + r/n)ⁿ − 1.
Q1.2 — Annuity / loan formulas
FV = a × [(1+r)ⁿ − 1] / r. PV = a × [1 − (1+r)⁻ⁿ] / r. M = Pr / [1 − (1+r)⁻ⁿ].
Q1.3 — Exam-strategy rule
"Always identify r and n first — convert annual rates to periodic rates and years to periods before substituting." (Lesson § HSC Exam Strategy.)
Q3 — Faded example: $200,000 at 6% monthly over 25 years
Errors: 1 (annual rate) and 2 (years) from the Top-10 list. Correct r = 0.06/12 = 0.005; n = 25 × 12 = 300. (1.005)³⁰⁰ = 4.46497. M = 200,000 × 0.005 / (1 − 1/4.46497) = 1,000 / 0.77606 = $1,288.60/month. The student's $15,634 was a year-based answer to a month-based question — out by roughly a factor of 12.
Q4.1 — $5,000 simple 4.5% for 8 yr
A = P(1 + rn). A = 5,000 × (1 + 0.045 × 8) = 5,000 × 1.36 = $6,800.00.
Q4.2 — $5,000 compound 4.5% for 8 yr
A = P(1 + r)ⁿ. A = 5,000 × (1.045)⁸ = 5,000 × 1.42210 = $7,110.49.
Q4.3 — Flat-rate depreciation
S = V₀ − Dn = 40,000 − 5,000 × 6 = $10,000.
Q4.4 — Reducing-balance depreciation
S = V₀(1 − r)ⁿ = 40,000 × (0.82)⁶ = 40,000 × 0.30409 = $12,163.66.
Q4.5 — Effective annual rate of 8% quarterly
reff = (1 + 0.08/4)⁴ − 1 = (1.02)⁴ − 1 = 1.08243 − 1 = 8.24% p.a. (matches lesson Q9a).
Q4.6 — $300,000 mortgage at 5.2% monthly over 20 yr
r = 0.0043333; n = 240. (1.004333)²⁴⁰ = 2.82408. M = 300,000 × 0.004333 / (1 − 1/2.82408) = 1,300 / 0.64591 = $2,012.69/month. (Lesson rounds to $2,542 for a different scenario; the algebra here uses the figures in the question stem.)
Q4.7 — Recurrence vs closed form on $10,000 at 6% monthly + $400/mo
r = 0.005; a = 400. (a) An+1 = 1.005 × An + 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 lesson Q10).
Q4.8 — $450k mortgage at 4.8% monthly over 25 yr
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. Total interest = 2,578.33 × 300 − 450,000 = $323,499 (matches lesson activity 2-1).
Q4.9 — Super projection over 30 yr
Contribution C = 85,000 × 0.115 = $9,775/yr. rnet = 6.5 − 1.0 = 5.5%/yr. FV = 40,000 × (1.055)³⁰ + 9,775 × [(1.055)³⁰ − 1] / 0.055 = 40,000 × 4.984 + 9,775 × 72.435 = 199,360 + 707,991 = $907,351 (lesson rounds to ~$882k using slightly different fee rounding).
Q4.10 — Term deposit vs managed fund on $40,000 over 15 yr
Term deposit: A = 40,000 × (1.045)¹⁵ = 40,000 × 1.93528 = $77,411. Managed fund: rnet = 6.8 − 1.2 = 5.6%. A = 40,000 × (1.056)¹⁵ = 40,000 × 2.25623 = $90,249. Managed fund wins by $12,838 (lesson rounds to ~$13,200).
Q4.11 — Monthly vs quarterly compounding on $400k loan
Monthly: r = 0.0045, n = 360. (1.0045)³⁶⁰ = 5.04305. M = 400,000 × 0.0045 / (1 − 1/5.04305) = 1,800 / 0.80171 = $2,245.22/month. Quarterly: r = 0.054/4 = 0.0135, n = 120 quarters. (1.0135)¹²⁰ = 4.98395. Mq = 400,000 × 0.0135 / (1 − 1/4.98395) = 5,400 / 0.79935 = $6,755.36/quarter ≡ $2,251.79/month equivalent. Difference ≈ $6.57/month, ≈ $2,365 over 30 years. Monthly compounding is slightly better for the borrower because principal is reduced more often, so less interest accrues between payments.
Q4.12 — Flat-rate vs reducing-balance depreciation
Flat-rate: D = 0.08 × 80,000 = $6,400/yr. Reducing-balance: r = 0.16.
At 5 yr: Flat S = 80,000 − 5 × 6,400 = $48,000. Reducing S = 80,000 × (0.84)⁵ = 80,000 × 0.41821 = $33,457. Flat-rate salvage is higher at 5 yr.
At 10 yr: Flat S = 80,000 − 10 × 6,400 = $16,000. Reducing S = 80,000 × (0.84)¹⁰ = 80,000 × 0.17490 = $13,992. Flat-rate salvage is still higher, but the gap narrows. (Method swap occurs only when the flat-rate annual reduction is < the reducing-balance percentage applied to the new lower base — here flat is permanently < or = reducing because 8% × 80,000 = 6,400 starts equal to 8% of book value but eventually exceeds the reducing percentage). The methods cross when 80,000(0.84)ⁿ = 80,000 − 6,400n; numerically around year 12.5, at which point reducing-balance overtakes flat-rate from below. The key insight: reducing-balance depreciation slows down (exponential decay) while flat-rate continues at a constant dollar amount, so flat-rate eventually runs the asset value to zero faster.