Compound-interest revision | Year 12

01 / LEARN

The idea, step by step

The balance grows by a multiplier

If the balance earns interest and that interest remains in the account, the next period starts with a larger amount. With a constant periodic rate i, each step multiplies the balance by 1+i. A rate of 4% means $i = 0.04,$ not 4. State when interest is credited before counting periods.

The model assumes no deposits, withdrawals, fees or rate changes. When any of those occur, use a recurrence or divide the calculation into stages. Keep the unrounded calculator value until the final money answer.

B_{n} = P {(1 + i)}^{n}

Read the result correctly

The future balance includes the original principal. Interest earned is $B_{n} - P$ ; it is not $B_{n}$ itself. A decreasing value can use the same multiplier idea, with 1−d for a depreciation rate d.

When reversing the model, divide by the complete growth factor. The rate and period count must match: monthly interest over three years requires 36 periods. A quoted annual nominal rate needs conversion to its stated compounding frequency.

A common trap

Using the annual rate with a monthly period count.

Instead: First convert the rate to one period, then count those same periods.

02 / WORKED EXAMPLES

See the thinking

Example 1

Build the model

AUD 1,200 earns 4% interest annually for 3 years. Find the final balance and total interest.

Use $i = 0.04$ and $n = 3$ .
Balance = $1200 {(1.04)}^{3}$ = AUD 1,349.84.
Subtract AUD 1,200 to find the interest.

AUD 1,349.84; interest AUD 149.84

Example 2

Use the model to make a decision

A tool costing AUD 2,500 loses 12% of its value each year. Estimate its value after 4 years.

Use the multiplier 0.88, because 88% of the value remains each year.
Value = $2500 {(0.88)}^{4}$ = AUD 1,499.24.

AUD 1,499.24

Check your understanding

AUD 900 grows to AUD 1,050. What interest has been earned?

Show answer

AUD 150.00

Interest = final balance − principal. $1050 - 900 = 150$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the balance of AUD 3,400 after 5 years at 3.5% compounded annually.
3 marks
AUD 900 grows to AUD 1,050. What interest has been earned?
3 marks
An account reaches AUD 8,000 after 6 years at 5% compounded annually. Find its original principal.
3 marks
AUD 2,000 earns a nominal annual rate of 6%, compounded monthly, for 2 years. Find the balance.
3 marks
A model uses $B = 6000 {(1.02)}^{8}$ . State the periodic rate and the number of periods.
3 marks
Compare AUD 5,000 invested for 4 years at 4% simple interest and at 4% annual compound interest.
3 marks

Read the worked solutions

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

Find the balance of AUD 3,400 after 5 years at 3.5% compounded annually.
3 marks
Use $i = 0.035$ and $n = 5$ .
$3400 {(1.035)}^{5}$ = AUD 4,038.13.
AUD 4,038.13
AUD 900 grows to AUD 1,050. What interest has been earned?
3 marks
Interest = final balance − principal.
$1050 - 900 = 150$ .
AUD 150.00
An account reaches AUD 8,000 after 6 years at 5% compounded annually. Find its original principal.
3 marks
Rearrange P = $B / {(1 + i)}^{n}$ .
P = $8000 / {(1.05)}^{6}$ = AUD 5,969.72.
AUD 5,969.72
AUD 2,000 earns a nominal annual rate of 6%, compounded monthly, for 2 years. Find the balance.
3 marks
The monthly rate is $0.06 / 12 = 0.005$ and there are 24 months.
$2000 {(1.005)}^{24}$ = AUD 2,254.32.
AUD 2,254.32
A model uses $B = 6000 {(1.02)}^{8}$ . State the periodic rate and the number of periods.
3 marks
The multiplier 1.02 equals 1+i, so $i = 0.02$ .
The exponent counts 8 periods. The formula alone does not identify whether the periods are months or years.
2% per period; 8 periods.
Compare AUD 5,000 invested for 4 years at 4% simple interest and at 4% annual compound interest.
3 marks
Simple balance = $5000 (1 + 0.04 \times 4) = 5800$ .
Compound balance = $5000 {(1.04)}^{4}$ = AUD 5,849.29.
Subtract the simple balance from the compound balance.
Simple: AUD 5,800.00; compound: AUD 5,849.29; compound is AUD 49.29 higher.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics financial models: investments and annuities, superannuation, income streams and borrowing. Original practice uses supplied hypothetical rates, taxes and fees; statutory or commercial conditions are not inferred.

Official curriculum source ↗

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