Mathematics Advanced • Year 12 • Module 7 • Lesson 18
Savings Goals and Budgeting Models
Practise HSC-style writing on required-saving, FV-with-starting-balance and 50/30/20 budgeting.
1. Short-answer questions
1.1 A saver targets $25,000 in 3 years. Account pays 4.8% p.a. compounded monthly. Starting balance is $0. (a) State r and n. (b) Find the required monthly contribution a, to the nearest cent. (c) State how much less than the no-interest contribution this represents. 3 marks Band 3
1.2 A couple has $5,000 already saved. They put $400/month into a 3.6% p.a. monthly account for 4 years. (a) Show that the FV ≈ $26,179. (b) State whether their $25,000 deposit goal is reached, and by how much. 3 marks Band 3-4
1.3 A graduate wants $100,000 in 5 years at 4% p.a. compounded monthly, starting from $0. (a) Find the required monthly saving a. (b) State the gross monthly income at which this saving represents exactly 20% of income under the 50/30/20 rule. (c) Find the new monthly saving if the timeframe is extended to 6 years. 4 marks Band 4
Stuck on 1.3(c)? Reuse a = FV × r / [(1+r)ⁿ − 1] with n = 72 and watch the contribution drop.2. Extended response
2.1 A young professional has $5,000 already saved and wants $60,000 in 5 years for a house-deposit starter and emergency-fund combo. The savings account pays 4.5% p.a. compounded monthly. Their gross income is $5,500/month.
Goal: FV target = $60,000.
Starting balance: A₀ = $5,000.
Rate: 4.5% p.a. compounded monthly.
Income: $5,500/month gross.
(a) Find r and n, then compute the required monthly saving a using FV = A₀(1+r)ⁿ + a × [(1+r)ⁿ − 1] / r. Show working.
(b) Express a as a percentage of gross income and decide whether it fits inside the 50/30/20 savings bucket.
(c) Suppose the professional can manage only $800/month. (i) Compute the achievable FV at 5 years. (ii) Find the number of months required at $800/month to still reach $60,000. (iii) Explain in 2–3 sentences how the lesson's "the mathematics is unforgiving" principle applies when a > 0.2 × income, referencing both a longer timeframe and increased income as legitimate corrections. 8 marks Band 5-6
Explicit marking criteria
Part (a) — 3 marks
• 1 mark — explicit r = 0.00375 and n = 60.
• 1 mark — correct A₀(1+r)ⁿ ≈ $6,259 and annuity factor ≈ 67.15.
• 1 mark — correct a ≈ $799–$801/month with substitution shown.
Part (b) — 2 marks
• 1 mark — correct percentage of gross (≈ 14.5%).
• 1 mark — explicitly states the plan fits inside the 20% savings bucket.
Part (c) — 3 marks
• 1 mark — correct FV at $800/month after 60 months (≈ $60,000).
• 1 mark — correct extended n (or equivalent reasoning if FV already meets goal).
• 1 mark — discussion connects "a > 0.2 × income ⇒ budget breaks" to either timeframe or income corrections.
Your response:
Stuck on (c)? Quote the inequality a > 0.20 × income and link it to either income up or timeframe out.How did this worksheet feel?
What I'll revisit before next class:
1.1 — $25,000 in 3 years at 4.8% monthly (3 marks)
Sample response. (a) r = 0.048/12 = 0.004, n = 36. (b) (1.004)³⁶ − 1 = 0.15440. a = 25,000 × 0.004 / 0.15440 = 100 / 0.15440 = $647.69/month. (c) Without interest: 25,000 / 36 = $694.44/month. Interest saves $46.75/month, or about $1,683 over the 36 months.
Marking notes. 1 mark — explicit r and n. 1 mark — correct a (accept $647–$650 with valid rounding; lesson rounds to $649). 1 mark — correct comparison to the no-interest baseline.
1.2 — Couple's deposit progress (3 marks)
Sample response. (a) r = 0.003, n = 48. (1.003)⁴⁸ = 1.15426. A₀(1+r)ⁿ = 5,000 × 1.15426 = $5,771.32. Annuity factor = 0.15426 / 0.003 = 51.42. FV = 5,771.32 + 400 × 51.42 = 5,771.32 + 20,568.00 = $26,339.32 (lesson rounds to $26,177). (b) Goal of $25,000 is exceeded by $1,339.32 — yes, the couple meets the target.
Marking notes. 1 mark — correct A₀(1+r)ⁿ. 1 mark — correct annuity component. 1 mark — explicit comparison to $25,000 goal. Accept the lesson's $26,177 with internally consistent rounding.
1.3 — $100,000 in 5 years (4 marks)
Sample response. (a) r = 0.04/12 = 0.003333; n = 60. (1.003333)⁶⁰ − 1 = 0.22100. a = 100,000 × 0.003333 / 0.22100 = 333.33 / 0.22100 = $1,508.30/month. (b) Required income = 1,508.30 / 0.20 = $7,541.50/month gross (≈ $90,500 p.a.) — a senior-graduate or trade-supervisor salary range. (c) With n = 72: (1.003333)⁷² − 1 = 0.27130. a = 100,000 × 0.003333 / 0.27130 = 333.33 / 0.27130 = $1,228.62/month, a 19% drop for one extra year of saving.
Marking notes. 1 mark — correct a at 5 years. 1 mark — correct income calculation using the 20% rule. 1 mark — correct a at 6 years. 1 mark — explicit comparison showing the contribution drops.
2.1 — Young professional, $5,000 → $60,000 in 5 years (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Required monthly saving.
r = 0.045 / 12 = 0.00375 per month; n = 5 × 12 = 60 periods. [1 mark]
(1.00375)⁶⁰ = 1.25179. A₀(1+r)ⁿ = 5,000 × 1.25179 = $6,258.95. Annuity factor = (1.25179 − 1) / 0.00375 = 0.25179 / 0.00375 = 67.143. [1 mark]
a = (60,000 − 6,258.95) / 67.143 = 53,741.05 / 67.143 = $800.40/month. [1 mark]
(b) Percentage of gross and 50/30/20 check.
800.40 / 5,500 = 0.1456 = 14.56% of gross. [1 mark]
This is well inside the 20% savings bucket, leaving 5.44% of gross (≈ $300/month) as buffer for over-saving or other goals. The plan is feasible without compromising needs or wants. [1 mark]
(c) The "only $800/month" scenario.
(i) FV at $800/month for 60 months: FV = 5,000 × 1.25179 + 800 × 67.143 = 6,258.95 + 53,714.40 = $59,973.35, only $26.65 short of the $60,000 target — essentially on track at 5 years. [1 mark]
(ii) To hit $60,000 exactly at $800/month: trial n = 61: (1.00375)⁶¹ = 1.25649; FV = 5,000 × 1.25649 + 800 × (0.25649/0.00375) = 6,282.45 + 800 × 68.397 = 6,282.45 + 54,717.60 = $60,999.95. So 61 months (5 years 1 month) suffices at exactly $800/month. [1 mark]
(iii) The lesson's principle "if a > 0.2 × income, the budget does not balance" means there are only three valid corrections when the required saving exceeds the 20% bucket: (1) increase income (here, going from $5,500 to ≈ $5,800 gross would absorb the original $800 requirement at exactly 20%), (2) decrease wants and divert that into savings (reducing the 30% wants bucket by ≈ 5 pp), or (3) extend the timeframe — here, just 1 extra month at $800 already meets the goal. Cutting needs is mathematically possible but socially destructive (rent, utilities, food), so the realistic answer in most plans is a combination of timeframe and income. [1 mark]
Total: 8/8.
Band descriptors for marker.
Band 3: r and n correct, A₀ component computed but annuity factor wrong; a calculated but not connected to budget. ≈ 3–4 marks.
Band 4: Correct a, percentage of gross stated but no explicit 20% check; (c) shows correct FV but no n re-solve. ≈ 5–6 marks.
Band 5: All calculations correct including extended n, but (c)(iii) restates rather than connects to the three legitimate corrections. ≈ 6–7 marks.
Band 6: Full calculations to the cent, explicit 20% check, and (c)(iii) names at least two of {timeframe, income, wants cut} with numeric illustration of how each closes the gap. 8/8.