Mathematics Advanced • Year 12 • Module 7 • Lesson 18
Savings Goals and Budgeting Models
Build fluency with required-saving and FV-with-starting-balance calculations and 50/30/20 budgeting.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete the formulas:
FV of regular savings (no starting balance): FV = ____________________
Required regular saving: a = ____________________
FV with starting balance A₀: FV = ____________________
Q1.2 A goal of $50,000 in 5 years is invested at 4% p.a. compounded monthly.
r = ____________ n = ____________
Q1.3 Under the 50/30/20 rule, state the three categories and the percentage allocated to each.
2. Worked example — house-deposit couple from the lesson
Follow every line. Each step has a short reason.
Problem. A couple wants $80,000 for a house deposit in 4 years. They have $10,000 already saved. Account pays 3.6% p.a. compounded monthly. Find the required monthly contribution a.
Step 1 — Convert annual rate and years.
r = 0.036 ÷ 12 = 0.003
n = 4 × 12 = 48 periods
Reason: r and n must use the same time unit (months).
Step 2 — Project the starting balance forward.
A₀(1+r)ⁿ = 10,000 × (1.003)⁴⁸ = 10,000 × 1.15426 = $11,542.59
Step 3 — Compute the annuity factor.
[(1.003)⁴⁸ − 1] / 0.003 = 0.15426 / 0.003 = 51.42
Step 4 — Solve for a.
80,000 = 11,542.59 + a × 51.42
a = (80,000 − 11,542.59) / 51.42 = 68,457.41 / 51.42 = $1,331.10
Conclusion. The couple must save $1,331/month to reach the $80,000 deposit in 4 years. (The lesson quotes ≈ $1,343 using slightly different rounding.)
3. Faded example — fill in the missing steps
Goal $15,000 in 2 years. Starting $0. Rate 4.8% p.a. compounded monthly. 4 marks
Step 1 — Identify r and n:
r = ______________ n = ______________
Step 2 — Compute (1+r)ⁿ − 1:
(1.____)²⁴ − 1 = ____________
Step 3 — Apply a = FV × r / [(1+r)ⁿ − 1]:
a = 15,000 × ______ / ______ = $______________
Conclusion. Required monthly saving = $______________ . Without interest, $15,000 / 24 = $______________ . Interest saves $______________/month.
4. Graduated practice — required savings and FV
Show the substitution and the final amount (to nearest cent or nearest dollar as indicated).
Foundation — single-step required saving (4 questions)
| Q | Scenario | Working & required a |
|---|---|---|
| 4.1 1 | Goal $12,000 in 1 year, rate 0% (no interest). | |
| 4.2 1 | Goal $12,000 in 1 year, rate 4.8% p.a. monthly. | |
| 4.3 1 | Goal $50,000 in 5 years, rate 4% p.a. monthly, start $0. | |
| 4.4 1 | Goal $20,000 in 3 years, rate 3% p.a. monthly, start $0. |
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 worked example. Goal = $80,000 deposit in 4 years; starting $10,000; rate 3.6% p.a. monthly. Find a. 3 marks
4.6 A student saves $400/month at 3.6% p.a. monthly for 4 years, starting with $5,000. Compute the FV and state whether the $25,000 emergency-fund goal is met. 3 marks
4.7 Goal $25,000 in 3 years at 4.8% p.a. monthly, starting $0. Find a. State the interest "discount" relative to the no-interest case. 2 marks
4.8 Goal $100,000 in 10 years at 5% p.a. monthly, starting $5,000. Find a. 2 marks
4.9 A first-home saver puts in $200/month at 5% p.a. monthly for 30 years from age 25. Find the FV at age 55. Comment briefly on the lesson's "consistency beats size" misconception. 2 marks
4.10 A worker on $4,500/month gross applies the 50/30/20 rule. State the dollar amounts for needs, wants and savings, and the goal value reached after 5 years at 4% p.a. monthly (start $0). 3 marks
Extension — combine concepts (2 questions)
4.11 A saver has $10,000 and wants $250,000 for a deposit in 15 years at 5% p.a. monthly. Find a. Then quote the FV that A₀ alone delivers over 15 years, so the saver can see what fraction of the goal is covered by interest on the starting balance. 3 marks
4.12 A 30-year-old wants $1,000,000 by age 65 in a fund returning 7% p.a. monthly, start $0. Find a. Then redo with the same goal but starting at age 25 instead. By what factor does delaying 5 years multiply the required monthly saving? 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 — Formulas
FV = a × [(1+r)ⁿ − 1] / r. a = FV × r / [(1+r)ⁿ − 1]. FV = A₀(1+r)ⁿ + a × [(1+r)ⁿ − 1] / r.
Q1.2 — Monthly rate and periods
r = 0.04 ÷ 12 = 0.003333. n = 5 × 12 = 60.
Q1.3 — 50/30/20
50% Needs (rent, food, utilities, minimum loan repayments). 30% Wants (discretionary). 20% Savings (emergency fund, goals, investments).
Q3 — Faded example: $15,000 in 2 years at 4.8% monthly
r = 0.004; n = 24. (1.004)²⁴ − 1 = 0.10063. a = 15,000 × 0.004 / 0.10063 = 60 / 0.10063 = $596.24/month. Without interest: 15,000 / 24 = $625.00/month. Interest saves $28.76/month, or about $690 over the 24 months.
Q4.1 — $12,000 in 1 year at 0%
a = 12,000 / 12 = $1,000/month.
Q4.2 — $12,000 in 1 year at 4.8% monthly
r = 0.004; n = 12. (1.004)¹² − 1 = 0.04907. a = 12,000 × 0.004 / 0.04907 = 48 / 0.04907 = $978.20/month. Interest saves about $22/month.
Q4.3 — $50,000 in 5 years at 4% monthly
r = 0.003333; n = 60. (1.003333)⁶⁰ − 1 = 0.22100. a = 50,000 × 0.003333 / 0.22100 = 166.67 / 0.22100 = $754.17/month (matches lesson).
Q4.4 — $20,000 in 3 years at 3% monthly
r = 0.0025; n = 36. (1.0025)³⁶ − 1 = 0.09405. a = 20,000 × 0.0025 / 0.09405 = 50 / 0.09405 = $531.63/month.
Q4.5 — Lesson's house-deposit example
r = 0.003; n = 48. A₀(1.003)⁴⁸ = 10,000 × 1.15426 = $11,542.59. Annuity factor = 0.15426 / 0.003 = 51.42. a = (80,000 − 11,542.59) / 51.42 = $1,331/month (lesson rounds to $1,343).
Q4.6 — $400/month + $5,000 start over 4 years at 3.6%
r = 0.003; n = 48. FV = 5,000 × 1.15426 + 400 × 51.42 = 5,771.32 + 20,568.00 = $26,339.32. Goal met: $25,000 emergency target is exceeded by $1,339.
Q4.7 — $25,000 in 3 years at 4.8% monthly
r = 0.004; n = 36. (1.004)³⁶ − 1 = 0.15440. a = 25,000 × 0.004 / 0.15440 = 100 / 0.15440 = $647.69/month (lesson quotes $649). Without interest: 25,000 / 36 = $694.44. Interest saves $46.75/month.
Q4.8 — $100,000 in 10 years at 5% monthly, start $5,000
r = 0.004167; n = 120. (1.004167)¹²⁰ = 1.64701. A₀(1+r)ⁿ = 5,000 × 1.64701 = $8,235.05. Annuity factor = 0.64701 / 0.004167 = 155.28. a = (100,000 − 8,235.05) / 155.28 = 91,764.95 / 155.28 = $590.96/month.
Q4.9 — $200/month for 30 years at 5% monthly
r = 0.004167; n = 360. (1.004167)³⁶⁰ = 4.46774. Annuity factor = 3.46774 / 0.004167 = 832.26. FV = 200 × 832.26 = $166,452. The lesson's "consistency beats size" misconception is supported: $200 a month for 30 years builds $167,000 — proving that the saving habit, not the saving amount, drives long-horizon growth.
Q4.10 — 50/30/20 on $4,500/month at 4% for 5 years
Needs = $2,250; Wants = $1,350; Savings = $900. With a = $900, r = 0.003333, n = 60: FV = 900 × [(1.003333)⁶⁰ − 1]/0.003333 = 900 × 66.30 = $59,670. The 20% bucket alone delivers nearly $60,000 in 5 years.
Q4.11 — $250,000 in 15 years starting with $10,000 at 5% monthly
r = 0.004167; n = 180. (1.004167)¹⁸⁰ = 2.11383. A₀(1+r)ⁿ = 10,000 × 2.11383 = $21,138 (covers only 8.5% of the goal). Annuity factor = 1.11383 / 0.004167 = 267.30. a = (250,000 − 21,138) / 267.30 = 228,862 / 267.30 = $856.20/month.
Q4.12 — $1M retirement: start at 30 vs start at 25
r = 0.07/12 = 0.005833. From 30 (n = 420): (1.005833)⁴²⁰ = 11.5469. Annuity factor = 10.5469 / 0.005833 = 1808.13. a = 1,000,000 / 1808.13 = $553.06/month. From 25 (n = 480): (1.005833)⁴⁸⁰ = 16.3878. Annuity factor = 15.3878 / 0.005833 = 2637.91. a = 1,000,000 / 2637.91 = $379.09/month. Delaying 5 years multiplies the required saving by 553.06 / 379.09 ≈ 1.46 — nearly 50% more out of pocket every month for the rest of the working life.