Mathematics Advanced • Year 12 • Module 7 • Lesson 18
Savings Goals and Budgeting Models
Apply required-saving and 50/30/20 calculations to realistic goal-planning scenarios.
Problem 1 — The car-fund "Think First" scenario
A saver needs $20,000 in 3 years for a car. Two strategies:
Option A: $555/month, no interest.
Option B: $520/month into an account at 3% p.a. compounded monthly.
Set up: What are we solving for?
(i) Compute the FV of each option after 36 months. 3 marks
(ii) State which option, if either, reaches $20,000 and by how many dollars each falls short or overshoots. 2 marks
(iii) Find the exact monthly saving at 3% p.a. monthly that hits $20,000 on the nose. 2 marks
Stuck? Revisit lesson § Think First and § Worked Example.Problem 2 — House deposit on a four-year plan
A couple has $10,000 saved and wants $80,000 for a deposit in 4 years. Their savings account pays 3.6% p.a. compounded monthly.
Set up: What are we solving for?
(i) Find the required monthly contribution a using FV = A₀(1+r)ⁿ + a × [(1+r)ⁿ − 1] / r. 3 marks
(ii) Suppose the couple can only afford $1,000/month. Recompute the FV and the shortfall. 2 marks
(iii) To close the gap from (ii) without changing the contribution, the couple extends the timeframe. Find the number of months n needed (use trial-and-error against 80,000 = 10,000(1.003)ⁿ + 1,000 × [(1.003)ⁿ − 1]/0.003). 3 marks
Problem 3 — Budget reality check (50/30/20)
A recent graduate earns $5,000/month gross. They want to save $30,000 for an emergency fund + house-deposit starter in 3 years at 4.5% p.a. monthly.
Set up: What are we solving for?
(i) Find the required monthly saving a. 2 marks
(ii) State what percentage of the $5,000 gross this represents, and whether it fits inside the 50/30/20 "savings" bucket. 2 marks
(iii) If the savings bucket is hard-capped at 20% of gross, recompute the new "achievable" goal value after 3 years. Suggest one realistic adjustment (extra income, lower wants, longer timeframe) and quantify it. 3 marks
Stuck? Revisit lesson § The 50/30/20 Budget Rule.Problem 4 — Design your goal
Choose a real savings goal. Worked example: a $15,000 European trip in 30 months. Starting balance $1,500. Account at 4.2% p.a. compounded monthly.
Set up: What are we solving for?
(i) Find the monthly saving needed. 2 marks
(ii) Halfway through (at 15 months), the saver checks progress. Find the projected balance at that point and state whether they are on track relative to a linear $15,000 / 30 = $500/month checkpoint. 3 marks
(iii) Now design your own goal (holiday, car, deposit, emergency fund). State the goal value, timeframe, starting balance, assumed rate, and the required monthly saving. 3 marks
Problem 5 — Two strategies for a long-horizon milestone
A 30-year-old wants $200,000 by age 50 (20 years). Two strategies are on the table:
Strategy I: Saver A puts $400/month into a 5% p.a. monthly account.
Strategy II: Saver B puts $0/month but a one-off lump sum L0 today into the same account.
Set up: What are we solving for?
(i) Find the FV of Strategy I after 20 years. 2 marks
(ii) Find the lump sum L0 Strategy II needs today to reach exactly $200,000 in 20 years at 5% p.a. monthly. 2 marks
(iii) Compare the total contributions of the two strategies (Strategy I = 400 × 240; Strategy II = L0). Which involves more money out of pocket, and by how much? Write one sentence on what this says about starting capital vs the savings habit. 3 marks
Stuck on (ii)? Use L₀ = FV / (1 + r)ⁿ — the present-value lens.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Car fund
Set up. We are computing the FV of two saving schedules and the exact contribution needed at 3% to hit the goal.
(i) Option A: 555 × 36 = $19,980. Option B: r = 0.0025, n = 36; (1.0025)³⁶ − 1 = 0.09405; FV = 520 × 0.09405/0.0025 = 520 × 37.62 = $19,562.40.
(ii) A falls short of $20,000 by $20; B falls short by $437.60. Neither option hits the target.
(iii) a = 20,000 × 0.0025 / 0.09405 = 50 / 0.09405 = $531.63/month. Only $11.63/month more than Option B's $520 — interest helps but does not eliminate the need for disciplined saving (lesson key insight).
Problem 2 — House deposit, 4-year plan
Set up. We are sizing the monthly contribution against a fixed target, then re-solving for n when the contribution is constrained.
(i) r = 0.003; n = 48. A₀(1.003)⁴⁸ = 10,000 × 1.15426 = $11,542.59. Annuity factor = 51.42. a = (80,000 − 11,542.59) / 51.42 = $1,331/month (lesson rounds to $1,343).
(ii) With a = $1,000: FV = 11,542.59 + 1,000 × 51.42 = 11,542.59 + 51,420 = $62,962.59. Shortfall = 80,000 − 62,962.59 = $17,037.
(iii) Trial n = 60: FV = 10,000 × (1.003)⁶⁰ + 1,000 × [(1.003)⁶⁰ − 1]/0.003 = 10,000 × 1.19668 + 1,000 × 65.56 = 11,966.80 + 65,560 = $77,527 — short. Trial n = 63: (1.003)⁶³ = 1.20754; FV = 12,075.40 + 1,000 × 69.18 = 81,254 ✓. The couple needs ≈ 63 months (5 years 3 months) at $1,000/month.
Problem 3 — Budget reality check on $5,000 gross
Set up. We are sizing required saving, comparing to the 20% bucket, and adjusting the plan when the bucket caps out.
(i) r = 0.045/12 = 0.00375; n = 36. (1.00375)³⁶ − 1 = 0.14515. a = 30,000 × 0.00375 / 0.14515 = 112.50 / 0.14515 = $775.06/month.
(ii) 775.06 / 5,000 = 15.5% of gross — comfortably inside the 20% savings bucket. The plan is feasible without breaking the budget.
(iii) If saving is hard-capped at 20% × 5,000 = $1,000/month: new FV = 1,000 × 0.14515/0.00375 = 1,000 × 38.70 = $38,704 after 3 years. The graduate can either accept the larger achievable goal, or extend the timeframe to 4 years to lower the required a (with a = 30,000 × 0.00375 / [(1.00375)⁴⁸ − 1] = 112.50 / 0.19668 = $571.95/month, only 11.4% of gross).
Problem 4 — Design your goal
Set up. We are sizing a personal goal and checking the half-way milestone against a naive linear schedule.
(i) r = 0.042/12 = 0.0035; n = 30. A₀(1.0035)³⁰ = 1,500 × 1.11038 = $1,665.57. (1.0035)³⁰ − 1 = 0.11038. Annuity factor = 31.54. a = (15,000 − 1,665.57) / 31.54 = 13,334.43 / 31.54 = $422.78/month.
(ii) At 15 months: (1.0035)¹⁵ = 1.05384. FV = 1,500 × 1.05384 + 422.78 × (0.05384/0.0035) = 1,580.76 + 422.78 × 15.383 = 1,580.76 + 6,503.27 = $8,084. Linear checkpoint = $500 × 15 = $7,500. The saver is $584 ahead of the linear schedule at the halfway point — because the early contributions have already earned a little interest.
(iii) Self-designed answer. Sample: goal $25,000 car, 5 years, start $2,500, rate 4.5% p.a. monthly. r = 0.00375; n = 60. (1.00375)⁶⁰ − 1 = 0.25319. Annuity factor = 67.52. a = (25,000 − 2,500 × 1.25319) / 67.52 = (25,000 − 3,132.97) / 67.52 = $323.85/month.
Problem 5 — Monthly saving vs lump sum at 30
Set up. We are comparing the FV of regular contributions with the present-value lump sum that would deliver the same target.
(i) r = 0.05/12 = 0.004167; n = 240. (1.004167)²⁴⁰ = 2.71264. Annuity factor = 1.71264 / 0.004167 = 411.03. FV = 400 × 411.03 = $164,412.
(ii) L₀ = 200,000 / (1.004167)²⁴⁰ = 200,000 / 2.71264 = $73,727.
(iii) Strategy I total contributions = 400 × 240 = $96,000. Strategy II contribution = $73,727. Strategy I pays $22,273 more out of pocket, yet finishes below the $200,000 target — showing that starting capital is worth more than the savings habit at any given total spend, because compounding has 20 unbroken years to work on the lump sum but only an average of 10 years on each regular contribution.