Financial applications (QCE Math Methods Unit 1)

Syllabus dot point

“Apply simple interest, compound interest and depreciation models to financial calculations, including future value, present value and effective annual rate”

8 min readUpdated 7 Oct 2026

Quick answer

Simple interest grows on the principal only ( $I = Prt$ ); compound interest grows on the running balance ( $A = P(1 + r/n)^{nt}$ ), with effective annual rate $(1 + r/n)^n - 1$ . Present value rearranges the compound formula, and depreciation is straight-line (arithmetic) or declining-balance (geometric, $V_t = P(1-r)^t$ ).

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What this dot point is asking
Simple interest
Compound interest
Effective annual rate
Present value
Depreciation
In one sentence
Exam-style questions

Key fact

Check against your course. This topic is not confirmed as part of the 2025 Mathematical Methods syllabus (simple interest, present value and effective annual rate are not in the published Unit 1 topic list). Check your teacher's unit plan and QCAA's syllabus before spending revision time here. This page stays in Unit 1 so its link does not change.

What this dot point is asking

QCAA wants you to apply the standard financial formulas (simple interest, compound interest and depreciation), compute future and present values, find an effective annual rate, and interpret the results in everyday contexts such as savings, loans and asset values. Financial mathematics is a recurring modelling context, and the key skills are choosing the right model and keeping rates and time periods consistent.

Simple interest

Simple interest is calculated only on the original principal, never on accumulated interest:

I = P r t, \qquad A = P + I = P(1 + rt),

where $P$ is the principal, $r$ is the annual rate as a decimal, and $t$ is the time in years. Because the interest each period is constant, the balance grows arithmetically.

Compound interest

Compound interest adds the interest to the balance at the end of each compounding period, so later interest is earned on earlier interest:

A = P\left(1 + \frac{r}{n}\right)^{nt},

where $n$ is the number of compounding periods per year ( $n = 1$ annually, $2$ half-yearly, $4$ quarterly, $12$ monthly, $365$ daily). The balance grows geometrically with common ratio $1 + \tfrac{r}{n}$ per period. For continuous compounding the limit is $A = Pe^{rt}$ .

Effective annual rate

The same nominal annual rate produces different real returns depending on how often it compounds. The effective annual rate (EAR) is the single annual rate that gives the same growth:

\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1.

For a nominal $5\%$ compounded monthly, $\text{EAR} = (1 + 0.05/12)^{12} - 1 \approx 0.05116$ , that is about $5.12\%$ . The EAR lets you compare products quoted at different compounding frequencies on a fair basis.

Present value

Present value answers "how much must I invest now to reach a target amount $A$ ?" by rearranging the compound-interest formula:

P = \frac{A}{(1 + r/n)^{nt}} = A\left(1 + \frac{r}{n}\right)^{-nt}.

This discounts a future amount back to today and underlies lump-sum settlements and savings goals. Because the discount factor $(1 + r/n)^{-nt}$ shrinks as the rate or the time grows, money promised further in the future is worth less now, the core idea of the time value of money.

Depreciation

Assets lose value over time, and two models are standard. Straight-line depreciation loses a constant dollar amount each year, an arithmetic model:

V_t = P - dt,

where $d$ is the fixed annual loss. Declining-balance depreciation loses a constant percentage of the current value each year, a geometric model:

V_t = P(1 - r)^t.

Declining balance falls quickly at first and then more slowly, which often matches real assets such as vehicles and machinery.

Comparing compounding frequencies

$\$8000$ is invested at a nominal $6\%$ per annum for $5$ years. Compare the final amounts under annual and quarterly compounding.

Step 1: Set up the compound-interest formula for each scenario

The compound-interest formula is $A = P\!\left(1 + \dfrac{r}{n}\right)^{nt}$ , where $r = 0.06$ and $t = 5$ . The only difference between the two scenarios is $n$ , the number of compounding periods per year. For annual compounding $n = 1$ , so the periodic rate is $r/n = 0.06$ and there are $nt = 5$ periods. For quarterly compounding $n = 4$ , so the periodic rate is $r/n = 0.015$ and there are $nt = 20$ periods.

Step 2: Evaluate the annual case

A_{\text{annual}} = 8000(1.06)^5 = 8000 \times 1.33823 \approx \$10\,705.80.

Step 3: Evaluate the quarterly case

Because interest is added four times a year, each period earns interest on a slightly larger balance than the last:

A_{\text{quarterly}} = 8000(1.015)^{20} = 8000 \times 1.34686 \approx \$10\,774.84.

Step 4: Compare and interpret

Quarterly compounding earns approximately $\$69$ more over five years. The extra growth comes from interest being credited more often, so each batch of interest itself begins earning interest sooner. This difference grows with higher rates and longer terms, which is why the effective annual rate, not the nominal rate, is the fair basis for comparison.

Final answer: Annual compounding gives $\approx \$10\,705.80$ ; quarterly compounding gives $\approx \$10\,774.84$ .

Common traps

Mixing annual and periodic rates: If $r$ is the annual rate and $n = 12$ , the time $t$ stays in years; do not also divide $t$ by $12$ .
Forgetting to subtract the principal: Total interest is $A - P$ , not $A$ .
Choosing simple interest where compound is intended: Australian savings accounts, loans and credit cards almost always compound.
Percent versus decimal: Use $0.05$ for $5\%$ in the formulas; substituting $5$ gives a nonsensical answer.

In one sentence

Simple interest is $I = Prt$ on the principal only, compound interest is $A = P(1 + r/n)^{nt}$ with effective annual rate $(1 + r/n)^n - 1$ , present value is $A(1 + r/n)^{-nt}$ , and depreciation is either straight-line (arithmetic) or declining-balance (geometric, $V_t = P(1 - r)^t$ ).

Exam-style questions

Questions in the style of QCAA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

Practice4 marks

Paper 2 (complex familiar).

\$5000

is invested at

5\%

per annum compounded monthly for

3

years. (a) Determine the future value. (b) Determine the total interest earned.

Show worked answer →

Compound interest: $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$ with $P = 5000$ , $r = 0.05$ , $n = 12$ , $t = 3$ .

(a) $A = 5000\left(1 + \dfrac{0.05}{12}\right)^{36} = 5000(1.004167)^{36} \approx 5000 \times 1.16147 \approx \$5807.36.$

(b) Total interest $= A - P = 5807.36 - 5000 = \$807.36.$

Markers reward identifying the variables, the monthly compounding ( $n = 12$ ), and subtracting the principal for the interest.

Practice5 marks

Paper 2 (complex familiar). A machine costing

\$40\,000

depreciates by

15\%

of its value each year (declining balance). (a) Determine its value after

4

years. (b) Determine the first year in which its value first falls below

\$15\,000

Show worked answer →

Declining balance: $V_t = P(1 - r)^t$ with $P = 40\,000$ , $r = 0.15$ , so $V_t = 40\,000(0.85)^t$ .

(a) $V_4 = 40\,000(0.85)^4 = 40\,000 \times 0.52200625 \approx \$20\,880.$

(b) Solve $40\,000(0.85)^t < 15\,000$ , so $(0.85)^t < 0.375$ . Taking logs, $t > \dfrac{\ln 0.375}{\ln 0.85} \approx 6.03$ , so the value first falls below $\$15\,000$ during year $7$ .

Markers reward the declining-balance model, the evaluated value, and solving the inequality with logarithms (rounding up to a whole year).

Practise this

Sources & how we know this

Mathematical Methods General senior syllabus 2025 — QCAA