Mathematics Standard • Year 12 • Module 7 • Lesson 12
Module Review — Skill Drill
Build fluency across the whole module: simple/compound interest, effective rate, FV/PV of annuity, loan repayment and flat-vs-reducing — practise picking the right formula and applying it cleanly.
1. Quick recall — module formula sheet
Fill each blank with the correct formula or expression. 1 mark each
Q1.1 Simple interest: I = ____________. Amount: A = ____________.
Q1.2 Compound interest (single period type): A = ____________. With k compounding periods per year: A = ____________.
Q1.3 Effective annual rate = ____________ − 1.
Q1.4 FV of annuity: FV = ____________. PV of annuity: PV = ____________. Loan repayment: M = ____________.
Q1.5 Flat-rate true cost: r_reducing ≈ ____________.
2. Worked example — choose the formula, then calculate
Follow each line of working. Every step has a reason on the right.
Problem. $10,000 is invested at 5.4% p.a. compounded quarterly for 6 years. Find the FV and the effective annual rate.
Step 1 — Identify the formula.
Lump sum + compounding → A = P(1 + r/k)^(kn)
Reason: no regular contributions, no borrowing — just compounding on a lump sum, so the compound-interest formula applies.
Step 2 — Substitute and compute the FV.
A = 10,000 × (1 + 0.054/4)^(4×6) = 10,000 × (1.0135)²⁴ ≈ 10,000 × 1.3811 ≈ $13,811
Reason: with k = 4 (quarterly) and n = 6 years, kn = 24 quarters.
Step 3 — Effective annual rate.
r_eff = (1 + 0.054/4)⁴ − 1 = (1.0135)⁴ − 1 ≈ 0.0551 = 5.51% p.a.
Reason: gives a fair "per year" comparison number, useful when rates have different compounding frequencies.
Conclusion. FV ≈ $13,811; effective annual rate ≈ 5.51% p.a.
3. Faded example — pick the right formula from the lesson summary
Tag each problem with which formula it needs, then compute. Fill in each blank. 4 marks
(a) $500/month deposited at 6% p.a. compounded monthly for 10 years. Find FV.
Formula: ____________________
r = ____________ , n = ____________ , FV = $ ____________
(b) Borrow $25,000 at 7.2% p.a. compounded monthly over 5 years. Find M.
Formula: ____________________
M = $ ____________
(c) $20,000 at 4.8% flat over 4 years.
Formula: ____________________
Total interest = $ ____________
(d) Compare 5.8% compounded monthly vs 6% compounded semi-annually.
Formula: ____________________
5.8% monthly: r_eff = ____________ ; 6% semi-annual: r_eff = ____________ ; better = ____________
4. Graduated practice — mixed module questions
Show your working below each part. Round dollars to 2 dp.
Foundation — straight substitutions (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | $4,000 at 6% p.a. simple interest for 3 years. Find I. | |
| 4.2 1 | $4,000 at 6% p.a. compounded annually for 3 years. Find A. | |
| 4.3 1 | Effective annual rate for 6% p.a. compounded monthly. | |
| 4.4 1 | State the formula for the present value of an ordinary annuity. |
Standard — typical HSC difficulty (6 questions)
Name the formula in each answer before substituting.
4.5 $250/month contributed for 8 years at 5.4% p.a. compounded monthly. Find the FV. 2 marks
4.6 Find the present value of $800/month for 4 years at 6% p.a. compounded monthly. 2 marks
4.7 Find the monthly repayment on a $250,000 mortgage at 4.8% p.a. compounded monthly over 25 years. 2 marks
4.8 Compare effective annual rates for 6.2% p.a. compounded monthly vs 6.3% p.a. compounded quarterly. Which is the better rate? 2 marks
4.9 $15,000 car loan: 5.5% flat over 4 years versus 8.5% reducing balance compounded monthly over 4 years (M = $370.04). Find total interest under each. 2 marks
4.10 $300/month at 6% p.a. compounded monthly for 20 years. Find FV and total interest earned. 2 marks
Extension — multi-step rate-change problems (2 questions)
4.11 A $320,000 mortgage at 5.4% p.a. compounded monthly over 25 years has M = $1,949.37 (total interest $264,811). After 5 years, the rate drops to 4.8% and a new repayment is set for the remaining 20 years on the new balance of $296,889; the new M = $1,926.29. Find (a) total amount paid in the first 5 years, (b) total amount paid over the remaining 20 years, (c) total interest saved across the loan compared to the original schedule. 3 marks
4.12 A store offers 0% finance with a $400 fee on a $4,000 purchase over 12 months. (a) State the equivalent flat rate p.a. (b) Approximate equivalent reducing-balance annual rate using r_red ≈ 2·n·r_flat / (n + 1) with n = 12. 2 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1 — Module formula sheet
Q1.1 I = P r n; A = P(1 + rn). Q1.2 A = P(1 + r)ⁿ; A = P(1 + r/k)^(kn). Q1.3 Effective = (1 + r/k)^k − 1. Q1.4 FV = M × [(1+r)ⁿ − 1] / r; PV = M × [1 − (1+r)⁻ⁿ] / r; M = PV × r / [1 − (1+r)⁻ⁿ]. Q1.5 r_red ≈ 2·n·r_flat / (n + 1).
Q3 — Faded examples (pick the formula)
(a) FV of annuity. r = 0.005, n = 120. FV = 500 × [(1.005)¹²⁰ − 1] / 0.005 ≈ 500 × 163.88 ≈ $81,939.
(b) Loan repayment. r = 0.006, n = 60. M = 25,000 × 0.006 / [1 − (1.006)⁻⁶⁰] = 150 / 0.30166 ≈ $497.30/month.
(c) Simple/flat interest. Total interest = 20,000 × 0.048 × 4 = $3,840.
(d) Effective rate. 5.8% monthly: (1 + 0.058/12)¹² − 1 ≈ 5.96%; 6% semi-annual: (1 + 0.06/2)² − 1 ≈ 6.09%. Better = 6% semi-annual.
Q4.1 — Simple interest
I = 4,000 × 0.06 × 3 = $720.
Q4.2 — Compound amount
A = 4,000 × (1.06)³ ≈ 4,000 × 1.1910 ≈ $4,764.
Q4.3 — Effective annual rate
r_eff = (1 + 0.06/12)¹² − 1 ≈ 6.17% p.a.
Q4.4 — PV of ordinary annuity
PV = M × [1 − (1 + r)⁻ⁿ] / r.
Q4.5 — FV of $250/month
r = 0.0045, n = 96. FV = 250 × [(1.0045)⁹⁶ − 1] / 0.0045 ≈ 250 × 118.50 ≈ $29,625.
Q4.6 — PV of $800/month
r = 0.005, n = 48. PV = 800 × [1 − (1.005)⁻⁴⁸] / 0.005 ≈ 800 × 42.58 ≈ $34,064.
Q4.7 — Mortgage repayment
r = 0.004, n = 300. M = 250,000 × 0.004 / [1 − (1.004)⁻³⁰⁰] = 1,000 / 0.6975 ≈ $1,433.48/month.
Q4.8 — Effective rate comparison
Monthly: (1 + 0.062/12)¹² − 1 ≈ 6.38%. Quarterly: (1 + 0.063/4)⁴ − 1 ≈ 6.45%. Quarterly is the better rate.
Q4.9 — $15,000 car loan
Flat 5.5%: I = 15,000 × 0.055 × 4 = $3,300. Reducing 8.5%: Total = 370.04 × 48 = $17,762. Interest = $2,762. Reducing balance is cheaper, even though 8.5% > 5.5%.
Q4.10 — FV of $300/month
r = 0.005, n = 240. FV = 300 × [(1.005)²⁴⁰ − 1] / 0.005 ≈ 300 × 462.04 ≈ $138,611. Total contributed = 300 × 240 = $72,000. Interest earned = 138,611 − 72,000 = $66,611.
Q4.11 — Mortgage with mid-life rate cut
(a) First 5 yr = 1,949.37 × 60 = $116,962. (b) Remaining 20 yr = 1,926.29 × 240 = $462,310. (c) Actual total = 116,962 + 462,310 = $579,272. Original 25-yr total = 1,949.37 × 300 = $584,811. Interest saved ≈ $5,539.
Q4.12 — 0% finance with fee
(a) Extra = $400 on $4,000 over 1 year. r_flat = 400 / 4,000 = 10% p.a. flat. (b) r_red ≈ 2 × 12 × 0.10 / 13 = 2.4 / 13 ≈ 18.5% reducing balance — equivalent to a high-rate credit-card loan despite the "0%" headline.