Compound interest and compounding periods | Year 11

01 / LEARN

Understand the idea.

Compound growth

Compound interest adds each period's interest to the balance before calculating the next period's interest. A periodic rate $i$ therefore multiplies the balance by $1 + i$ . Repeating the same multiplication for $n$ periods gives $A = P {(1 + i)}^{n}$ .

For a nominal annual rate $r$ compounded $m$ times a year, use periodic rate $i = \frac{r}{m}$ and period count $n = m t$ . Both adjustments are needed: the rate and exponent must describe the same-sized period.

State the assumptions

The single-deposit formula assumes the rate stays constant and no money is added or removed during the term. Deposits, withdrawals or fees require balance updates at their stated dates because those cash flows change the amounts that can earn later interest.

Set up the periodic rate and number of periods before entering the power. Keep full internal precision until the final cents unless the problem explicitly rounds each interest credit. For a positive rate, later periods earn more interest because the balance has grown.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

Find the balance on AUD 2500 at 6% compounded annually for 3 years.

Worked solution

1
Identify the annual rate and period count.
$P = 2500, i = 0.06, n = 3$
Annual compounding for three years gives three periods, each with a 6% rate.
2
Build the repeated-growth calculation.
$A = 2500 {(1.06)}^{3}$
Each year's closing balance becomes the following year's interest base.
3
Evaluate and round once.
$A = AUD 2977.54$
The exact calculation already gives a cent amount, so no earlier rounding is needed.

Answer

AUD 2,977.54

EXAMPLE 02

Worked example

Find the balance on AUD 4000 at 4.8% nominal p.a., compounded monthly for 2 years.

Worked solution

1
Convert the nominal annual rate to a monthly rate.
$i = \frac{0.048}{12} = 0.004$
A 4.8% nominal annual rate supplies 0.4% per month.
2
Count the monthly periods.
$n = 12 (2) = 24$
Two years contain 24 monthly interest credits.
3
Apply monthly compound growth.
$A = 4000 {(1.004)}^{24} \approx 4402.193201$
The monthly growth factor is used once for each of the 24 months.
4
Round to cents.
$A \approx AUD 4402.19$
The third decimal place is 3, so the cent value rounds down.

Answer

AUD 4,402.19

03 / YOUR TURN

Try it yourself.

Check your understanding

At 6% nominal compounded quarterly for 3 years, what are i and n?

Show answer & reasoning

i=0.015;n=12

Quarterly compounding has four periods per year. Thus $i = \frac{0.06}{4} = 0.015$ and $n = 4 (3) = 12$ , so each period earns 1.5% and there are twelve periods.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Balance on AUD 1000 at 5% compounded annually for 2 years?
2 marks
Compound interest on AUD 6000 at 4% annually for 3 years?
2 marks
Balance on AUD 1500 at 8% nominal compounded half-yearly for 2 years?
2 marks
Balance on AUD 8000 at 6% nominal compounded quarterly for 18 months?
2 marks
Compare AUD 2000 at 10% simple and 10% annual compound interest after 2 years.
2 marks
A monthly model uses $B_{next} = 1.005 B$ . State monthly and nominal annual rates.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Balance on AUD 1000 at 5% compounded annually for 2 years?
2 marks
$A = 1000 {(1.05)}^{2} = 1102.50$ .
AUD 1,102.50
Compound interest on AUD 6000 at 4% annually for 3 years?
2 marks
$A = 6000 {(1.04)}^{3} = 6749.184$ .
$Interest = A - 6000$ .
AUD 749.18
Balance on AUD 1500 at 8% nominal compounded half-yearly for 2 years?
2 marks
$i = 0.04; n = 4$ .
$A = 1500 {(1.04)}^{4}$ .
AUD 1,754.79
Balance on AUD 8000 at 6% nominal compounded quarterly for 18 months?
2 marks
$i = 0.015; n = 6$ .
$A = 8000 {(1.015)}^{6}$ .
AUD 8,747.55
Compare AUD 2000 at 10% simple and 10% annual compound interest after 2 years.
2 marks
$Simple = 2000 (1 + 0.1 \times 2)$ .
$Compound = 2000 {(1.1)}^{2}$ .
AUD 2400 simple; AUD 2420 compound
A monthly model uses $B_{next} = 1.005 B$ . State monthly and nominal annual rates.
2 marks
$i = 0.005$ .
$12 i = 0.06$ .
0.5% monthly;6% nominal p.a.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: investing and borrowing. Supplied educational models cover interest, investment comparisons, shares and credit; they are not recommendations of real products or statements of personal tax liability.

Official curriculum source ↗

In this lesson

Understand the idea.

Compound growth

State the assumptions

See the method in action.

Worked example

Identify the annual rate and period count.

Build the repeated-growth calculation.

Evaluate and round once.

Worked example

Convert the nominal annual rate to a monthly rate.

Count the monthly periods.

Apply monthly compound growth.

Round to cents.

Try it yourself.

Check your understanding

Keep practising