Mathematics Standard • Year 12 • Module 7 • Lesson 12
Module Review — Past-Paper Style
Practise HSC Mathematics Standard 2-style writing across the whole Investment & Loans module — three short-answer questions and one longer scenario with marking criteria.
1. Short-answer questions
1.1 $250 is contributed monthly to a savings plan at 5.4% p.a. compounded monthly for 8 years.
(a) Calculate the future value using FV = M × [(1+r)ⁿ − 1] / r.
(b) State the total amount the saver actually contributed. 2 marks Band 3
1.2 A $320,000 mortgage at 5.4% p.a. compounded monthly has monthly repayment $1,949.37 over 25 years.
(a) Calculate the total amount repaid over the 25 years.
(b) Calculate the total interest paid.
(c) Briefly explain in one sentence why the total interest is more than 80% of the original loan amount. 3 marks Band 3-4
1.3 Two savings accounts are advertised: Account X at 6.2% p.a. compounded monthly, Account Y at 6.3% p.a. compounded quarterly.
(a) Calculate the effective annual rate of each account.
(b) State which account is better for a customer carrying a balance, and by how many percentage points.
(c) Briefly explain why the higher-nominal-rate account is not automatically the better account. 4 marks Band 4
2. Extended response
2.1 Mia is 30 years old and earns $70,000. She wants to use $400/month spare to grow long-term wealth over 35 years (to age 65). The expected return on each option is given.
Option A: Salary sacrifice $400/month into super (taxed at 15% on contributions; super earns 7% p.a. compounded monthly).
Option B: Take $400/month as after-tax income (marginal rate 32.5%, so $270 reaches her bank), invest in shares at 8% p.a. compounded monthly.
Option C: Use $400/month as extra mortgage repayments at 5.4% p.a. compounded monthly on a $400,000 mortgage over 25 years.
(a) Calculate the FV of Option A at age 65 using the FV-of-annuity formula with r = 0.07/12, n = 420.
(b) Calculate the FV of Option B at age 65 using r = 0.08/12, n = 420 and the after-tax contribution of $270/month.
(c) For Option C, the original mortgage M ≈ $2,432.69; the extra $400/month is projected to cut interest by approximately $130,000 over the loan's life. State this as the dollar value of Option C.
(d) Recommend the best long-term strategy for Mia, naming the option, quoting the dollar difference vs Option B, and noting one realistic trade-off (e.g. super preservation age, market risk, mortgage illiquidity). 7 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — correct r and n.
• 1 mark — correct FV using 400 × [(1+r)ⁿ − 1] / r.
Part (b) — 2 marks
• 1 mark — uses the after-tax contribution $270 (not the full $400).
• 1 mark — correct FV using 270 × [(1+r)ⁿ − 1] / r with r = 0.08/12.
Part (c) — 1 mark
• 1 mark — states the $130,000 saving as the relevant comparison value for Option C.
Part (d) — 2 marks
• 1 mark — recommendation names the chosen option AND quotes the dollar difference vs Option B.
• 1 mark — names one realistic trade-off relevant to the chosen option.
Your response:
Stuck on (d)? After computing the three values, identify the largest, name the option, quote the dollar gap vs Option B, AND add a realistic trade-off sentence to score the full 2 marks.How did this worksheet feel?
What I'll revisit before next class:
1.1 — FV of $250/month at 5.4% for 8 years (2 marks)
(a) Sample response. r = 0.0045, n = 96. FV = 250 × [(1.0045)⁹⁶ − 1] / 0.0045 ≈ 250 × 118.50 ≈ $29,625.
(b) Sample response. Contributed = 250 × 96 = $24,000.
Marking notes. (a) 1 mark — correct FV using the FV-of-annuity formula. (b) 1 mark — correct total contributed. Common error: writing 250 × 8 = $2,000 (forgets months) scores 0 for (b).
1.2 — Mortgage total and interest (3 marks)
(a) Sample response. Total = 1,949.37 × 300 = $584,811.
(b) Sample response. Interest = 584,811 − 320,000 = $264,811.
(c) Sample response. Over 25 years, interest compounds on a balance that takes decades to amortise — early payments are mostly interest, so the cumulative interest charge ends up around 83% of the original loan.
Marking notes. (a) 1 mark — correct total = M × n. (b) 1 mark — correct interest. (c) 1 mark — explanation links the long term to high cumulative interest.
1.3 — Effective rate comparison (4 marks)
(a) Sample response. X: (1 + 0.062/12)¹² − 1 ≈ 6.38%. Y: (1 + 0.063/4)⁴ − 1 ≈ 6.45%.
(b) Sample response. Account Y is better, by about 0.07 percentage points.
(c) Sample response. Compounding frequency matters: even though X has a higher nominal rate, Y's quarterly compounding combined with the higher nominal rate produces a slightly higher realised annual return — the effective rate is the apples-to-apples comparison.
Marking notes. (a) 1 mark — X correct. 1 mark — Y correct. (b) 1 mark — names winner with difference. (c) 1 mark — explanation references the effective-rate comparison.
2.1 — Mia's long-term strategy (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Option A — salary sacrifice.
r = 0.07/12 ≈ 0.005833, n = 420. [1 mark — correct r and n.]
FV = 400 × [(1.005833)⁴²⁰ − 1] / 0.005833 ≈ 400 × 1,766.8 ≈ $706,720. [1 mark — correct FV using the FV-of-annuity formula.]
(b) Option B — after-tax shares.
After-tax monthly contribution = $400 × (1 − 0.325) = $270. [1 mark — uses $270, not $400.]
r = 0.08/12 ≈ 0.006667, n = 420. FV = 270 × [(1.006667)⁴²⁰ − 1] / 0.006667 ≈ 270 × 2,295 ≈ $619,650. [1 mark — correct FV using 270/month at the right rate.]
(c) Option C — mortgage extra repayments.
Total interest saved ≈ $130,000 (as given). [1 mark — states the comparison figure for Option C.]
(d) Recommendation.
Comparison: Option A ≈ $706,720; Option B ≈ $619,650; Option C ≈ $130,000 saved in interest. Option A is by far the largest dollar outcome at age 65.
Conclusion: Mia should choose Option A (salary sacrifice into super) — it grows to about $706,720, around $87,070 more than Option B's $619,650 by age 65. [1 mark — names Option A and quotes the dollar gap vs Option B.] The trade-off is that super is locked until preservation age (currently around 60), so the money is not accessible in case of a short-term emergency or a home deposit need — unlike the share portfolio in Option B. [1 mark — one realistic trade-off named.]
Total: 7/7.
Band descriptors for marker.
Band 3: Option A FV correct; Option B uses the full $400 (forgets tax); Option C ignored. ≈ 2-3 marks.
Band 4: All three numerical values correct, but recommendation absent or no dollar gap stated. ≈ 5 marks.
Band 5: Full numerical solution and recommendation that names Option A with the dollar saving vs Option B, but no trade-off discussion. ≈ 6 marks.
Band 6: Complete: three numerical values, recommendation with dollar gap, AND a realistic trade-off. 7/7.