Mathematics Standard • Year 12 • Module 7 • Lesson 5

Future Value of Annuities — Skill Drill

Build fluency with FV = M[(1+r)^n − 1]/r and its rearrangement M = FV × r / [(1+r)^n − 1], plus the annuity-due adjustment.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Write the future-value formula for an ordinary annuity.

FV = M × ____________________________________________

Q1.2 Rearrange the FV formula to make M the subject.

M = FV × ____________________________________________

Q1.3 Write the annuity-due adjustment formula and the formula for total interest earned.

FV_due = FV_ord × ________ Total interest = ________ − ( ________ × ________ )

Stuck? Revisit lesson § Key Ideas — FV ordinary, annuity due, finding payment.

2. Worked example — Sarah's super target

Problem. Sarah wants $100,000 in her super in 15 years. Her fund earns 7.2% p.a. compounded monthly. Find her required monthly contribution and the total interest earned.

Step 1 — Identify FV, r per period, n.

FV = $100,000 r = 0.072/12 = 0.006 per month n = 15 × 12 = 180 months

Step 2 — Apply M = FV × r / [(1+r)^n − 1].

M = 100,000 × 0.006 / [(1.006)^180 − 1] = 600 / [2.93527 − 1] = 600 / 1.93527 = $310.04/month

Step 3 — Total contributed and interest.

Contributed = 310.04 × 180 = $55,806.88. Interest = $100,000 − $55,806.88 = $44,193.12.

Conclusion. Sarah needs $310.04/month; interest ≈ $44,193.12 (44% of the final amount).

3. Faded example — fortnightly super contributions

Tom deposits $150 fortnightly into super at 5.4% p.a. compounded fortnightly for 25 years. Find FV and total interest. Fill in the blanks. 4 marks

Step 1 — Identify M, r per period, n:

M = $ ________ r = 0.054 / ________ = ________ per fortnight n = 25 × ________ = ________ fortnights

Step 2 — Apply FV formula:

FV = ________ × [(1 + ________ )^( ________ ) − 1] / ________ = $ ____________

Step 3 — Total contributed and interest:

Contributed = $ ________ × ________ = $ ____________

Interest = $ ____________ − $ ____________ = $ ____________

Conclusion. FV = $ ____________ ; Interest = $ ____________.

Stuck? Revisit lesson § FV Formula — substitute then interpret.

4. Graduated practice — FV of annuities

Show one line of substitution and round dollar amounts to 2 decimal places.

Foundation — identify M, r, n (4 questions)

Q	Problem	Answer (M, r, n)
4.1 1	$100/month at 4.8% p.a. compounded monthly for 5 years.
4.2 1	$500/quarter at 6% p.a. compounded quarterly for 8 years.
4.3 1	$50/week at 5.2% p.a. compounded weekly for 10 years.
4.4 1	$180/month at 5.4% p.a. compounded monthly for 6 years.

Standard — FV calculations and find-the-payment (6 questions)

4.5 Calculate the FV of Q4.1 ($100/month at 4.8% monthly for 5 years). 2 marks

4.6 Calculate the FV of Q4.2 ($500/quarter at 6% quarterly for 8 years). 2 marks

4.7 Calculate the FV of Q4.4 ($180/month at 5.4% monthly for 6 years), and find the total interest earned. 2 marks

4.8 Find the monthly payment M required to reach $30,000 in 6 years at 5.4% p.a. compounded monthly. 2 marks

4.9 Find the fortnightly payment M required to reach $80,000 in 20 years at 6% p.a. compounded fortnightly (26 fortnights/year). 2 marks

4.10 $250/month is deposited at 4.8% p.a. compounded monthly for 4 years. Calculate (i) FV and (ii) total interest. 2 marks

Extension — annuity due and timing (2 questions)

4.11 For Q4.7 ($180/month at 5.4% monthly for 6 years), recompute the FV if the deposits are made at the start of each month (annuity due). 3 marks

4.12 Compare two savers: A deposits $200/month for 30 years at 6% p.a. compounded monthly; B deposits $400/month for 15 years at 6% p.a. compounded monthly. (i) Find each FV. (ii) State who ends up with more and by how much. 3 marks

Stuck on 4.12? Both contribute the same dollar total — but the one with more time will end with more.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I noted r = 0.004 per month and n = 60 (not 5).

For 4.8 I used the rearranged formula M = FV × r / [(1+r)^n − 1], not the FV formula directly.

For 4.10 I gave both FV and interest separately — and interest = FV − (M × n).

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — FV formula

FV = M × [(1 + r)^n − 1] / r.

Q1.2 — Payment formula

M = FV × r / [(1 + r)^n − 1].

Q1.3 — Annuity due and interest

FV_due = FV_ord × (1 + r). Total interest = FV − ( M × n ).

Q3 — Faded example (Tom: $150 fortnightly at 5.4% for 25 years)

Step 1: M = $150, r = 0.054/26 = 0.002077 per fortnight, n = 25 × 26 = 650.
Step 2: FV = 150 × [(1.002077)^650 − 1] / 0.002077 = 150 × [3.85097 − 1] / 0.002077 = 150 × 1372.6 = $205,894.99.
Step 3: Contributed = $150 × 650 = $97,500.00. Interest = $205,894.99 − $97,500 = $108,394.99.
Conclusion: FV ≈ $205,895; Interest ≈ $108,395.

Q4.1 — Identify (4.8% monthly, 5 years)

M = $100; r = 0.004; n = 60.

Q4.2 — Identify (6% quarterly, 8 years)

M = $500; r = 0.015; n = 32.

Q4.3 — Identify (5.2% weekly, 10 years)

M = $50; r = 0.001; n = 520.

Q4.4 — Identify (5.4% monthly, 6 years)

M = $180; r = 0.0045; n = 72.

Q4.5 — FV of $100/month at 4.8% monthly for 5 years

FV = 100 × [(1.004)^60 − 1] / 0.004 = 100 × [1.27049 − 1] / 0.004 = 100 × 67.62 = $6,762.36.

Q4.6 — FV of $500/quarter at 6% quarterly for 8 years

FV = 500 × [(1.015)^32 − 1] / 0.015 = 500 × [1.61032 − 1] / 0.015 = 500 × 40.69 = $20,344.27.

Q4.7 — FV of $180/month at 5.4% monthly for 6 years

FV = 180 × [(1.0045)^72 − 1] / 0.0045 = 180 × [1.38226 − 1] / 0.0045 = 180 × 84.95 = $15,290.70. Interest = $15,290.70 − (180 × 72) = $15,290.70 − $12,960 = $2,330.70.

Q4.8 — Find M for $30,000 in 6 years at 5.4% monthly

r = 0.0045, n = 72. M = 30,000 × 0.0045 / [(1.0045)^72 − 1] = 135 / 0.38226 = $353.16/month.

Q4.9 — Find M for $80,000 in 20 years at 6% fortnightly

r = 0.06/26 = 0.002308, n = 520. M = 80,000 × 0.002308 / [(1.002308)^520 − 1] = 184.6 / [3.3196 − 1] = 184.6 / 2.3196 = $79.60/fortnight.

Q4.10 — $250/month for 4 years at 4.8% monthly

(i) r = 0.004, n = 48. FV = 250 × [(1.004)^48 − 1] / 0.004 = 250 × [1.21039 − 1] / 0.004 = 250 × 52.60 = $13,149.66.
(ii) Deposited = $250 × 48 = $12,000. Interest = $13,149.66 − $12,000 = $1,149.66.

Q4.11 — Annuity due version of Q4.7

FV_due = FV_ord × (1 + r) = $15,290.70 × 1.0045 = $15,359.51. Difference = $68.81 — comes from each payment compounding for one extra month.

Q4.12 — Saver A vs Saver B at 6% monthly

(i) A: r = 0.005, n = 360. FV = 200 × [(1.005)^360 − 1] / 0.005 = 200 × [6.0226 − 1] / 0.005 = 200 × 1004.52 = $200,903.01.
B: r = 0.005, n = 180. FV = 400 × [(1.005)^180 − 1] / 0.005 = 400 × [2.4541 − 1] / 0.005 = 400 × 290.82 = $116,326.88.
(ii) A ends with more, $200,903.01 vs $116,326.88 — a gap of $84,576.13. Both contributed $72,000 total, but A's contributions had longer to compound.