Mathematics Standard • Year 12 • Module 7 • Lesson 5
Future Value of Annuities — Past-Paper Style
Practise HSC-style short answers and one extended response on future value of annuities, with full marking criteria.
1. Short-answer questions
1.1 Calculate the future value of monthly deposits of $300 made for 8 years into a fund paying 4.8% per annum compounded monthly. 3 marks Band 3
1.2 A saver wants to have $25,000 in 5 years. The account pays 5.4% p.a. compounded monthly.
(a) Calculate the required monthly contribution.
(b) Calculate the total interest the fund will earn over the 5 years. 3 marks Band 3-4
1.3 Hayden invests $250 at the end of each month into a fund paying 6% p.a. compounded monthly for 20 years.
(a) Calculate the FV of his account at the end of 20 years.
(b) Hayden then leaves the balance to compound (with no further deposits) for another 10 years at the same rate. Calculate the value of his account at the end of the 30 years. 4 marks Band 4
2. Extended response
2.1 Two friends, Layla and Jordan, both target a retirement balance at age 65. Both funds pay 7% p.a. compounded monthly.
Layla: Contributes $400 at the end of every month from age 25 to age 35 (10 years), then stops contributing. The balance keeps compounding to age 65.
Jordan: Contributes $400 at the end of every month from age 35 to age 65 (30 years) continuously.
(a) Calculate the FV of Layla's account at age 35 (the moment she stops contributing).
(b) Calculate the value of Layla's account at age 65 by treating the balance from (a) as a lump sum that compounds for a further 30 years.
(c) Calculate the FV of Jordan's account at age 65.
(d) Calculate each person's total contribution (M × n).
(e) Recommend who is mathematically better off at age 65 and state the dollar gap, then comment in one sentence on why the result might surprise someone who only looks at total contributions. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — correct FV of Layla's annuity at age 35 (10 years).
Part (b) — 1 mark
• 1 mark — correct compounding of the lump sum over 30 years using A = P(1 + r/k)^(kn).
Part (c) — 1 mark
• 1 mark — correct FV of Jordan's annuity at age 65 (30 years).
Part (d) — 1 mark
• 1 mark — correct totals: Layla $48,000 and Jordan $144,000.
Part (e) — 3 marks
• 1 mark — explicit comparison of the two age-65 figures with the dollar gap.
• 1 mark — recommendation sentence naming the higher-balance person.
• 1 mark — comment that links the surprising result to the much longer compounding time for Layla's early deposits.
Your response:
Stuck on (e)? Compare the two age-65 figures, name the winner, then say in one sentence: "Layla's $48,000 had ___ years to compound, which beat Jordan's $144,000 with only ___ years."How did this worksheet feel?
What I'll revisit before next class:
1.1 — FV of $300/month at 4.8% monthly for 8 years (3 marks)
Sample response. r = 0.004, n = 96. FV = 300 × [(1.004)^96 − 1] / 0.004 = 300 × [1.46637 − 1] / 0.004 = 300 × 116.594 = $34,978.21.
Marking notes. 1 mark — correct r and n. 1 mark — correct substitution. 1 mark — correct numerical answer (must include $ sign and 2 d.p.).
1.2 — Required monthly contribution + interest (3 marks)
(a) Sample response. r = 0.0045, n = 60. M = 25,000 × 0.0045 / [(1.0045)^60 − 1] = 112.5 / [1.30945 − 1] = 112.5 / 0.30945 = $363.55/month.
(b) Sample response. Total contributed = $363.55 × 60 = $21,813.05. Interest = $25,000 − $21,813.05 = $3,186.95.
Marking notes. (a) 1 mark — correct M with substitution shown. 1 mark — correct answer. (b) 1 mark — correct interest from total − contributed.
1.3 — Hayden's 20-year then lump-sum compounding (4 marks)
(a) Sample response. r = 0.005, n = 240. FV = 250 × [(1.005)^240 − 1] / 0.005 = 250 × [3.31020 − 1] / 0.005 = 250 × 462.04 = $115,510.32.
(b) Sample response. Treat $115,510.32 as a lump sum for 10 more years: A = 115,510.32 × (1.005)^120 = 115,510.32 × 1.81940 = $210,159.74.
Marking notes. (a) 1 mark — correct r, n. 1 mark — correct FV. (b) 1 mark — correct application of compound-interest formula to the lump sum. 1 mark — correct final value at year 30.
2.1 — Layla vs Jordan retirement comparison (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Layla's FV at age 35.
r = 0.07/12 = 0.005833, n = 120. FV = 400 × [(1.005833)^120 − 1] / 0.005833 = 400 × [2.00966 − 1] / 0.005833 = 400 × 173.084 = $69,233.57. [1 mark.]
(b) Layla's value at age 65.
A = 69,233.57 × (1.005833)^360 = 69,233.57 × 8.11649 = $562,008 (to nearest dollar). [1 mark — lump-sum compounding for 30 years.]
(c) Jordan's FV at age 65.
r = 0.005833, n = 360. FV = 400 × [(1.005833)^360 − 1] / 0.005833 = 400 × [8.11649 − 1] / 0.005833 = 400 × 1220.0 = $488,001 (to nearest dollar). [1 mark.]
(d) Total contributions.
Layla: $400 × 120 = $48,000. Jordan: $400 × 360 = $144,000. [1 mark — both contribution totals correct.]
(e) Comparison, recommendation, comment.
Difference at 65: $562,008 − $488,001 = $74,007. [1 mark — explicit comparison and gap.]
Layla is mathematically better off at age 65, with $74,007 more than Jordan despite contributing one-third as much in total. [1 mark — recommendation sentence names Layla and the gap.]
Comment: Layla's contributions had 30 extra years of compounding compared to Jordan's last-deposited dollar, and that long compounding window beats the larger total contribution Jordan made later in life. [1 mark — links the surprise to the long compounding time.]
Total: 7/7.
Band descriptors for marker.
Band 3: Computes Layla's age-35 FV and Jordan's age-65 FV but does not compound Layla's lump sum to age 65, so the comparison is meaningless. ≈ 2-3 marks.
Band 4: All four numerical values correct but recommendation is missing or unjustified. ≈ 5 marks.
Band 5: Full numerical solution with comparison and recommendation, but no explanatory comment about compounding time. ≈ 6 marks.
Band 6: Complete: all numbers, comparison, recommendation naming Layla and the dollar gap, AND a one-sentence comment linking the surprise to compounding duration. 7/7.