Compound interest | Year 10

01 / LEARN

The idea, step by step

Rate and period must agree

For decimal rate $i$ per period, after $n$ periods the no-payment model is $A = P {(1 + i)}^{n}$ . Both $i$ and $n$ refer to the same period.

If a nominal annual rate $r$ is explicitly compounded $m$ times a year, use period rate $r / m$ and $m t$ periods over $t$ years: $A = P {(1 + r / m)}^{m t}$ .

Report what was asked

The balance includes the original principal. With no other transactions, interest is $A - P$ . Keep full calculator precision until the requested final currency rounding.

An effective annual rate is already a one-year factor. Do not divide it by 12 as though it were a nominal annual rate. Read the wording before substituting.

A common trap

Use an annual rate with a monthly exponent.

Instead: That applies the annual percentage every month. Convert the rate only under the stated nominal-rate convention.

02 / WORKED EXAMPLES

See the thinking

Example 1

Annual compounding

2000 dollars compounds annually at 5% for 3 years. Find the balance and interest.

$A = 2000 {(1.05)}^{3} = 2315.25$ .
Interest $= 2315.25 - 2000 = 315.25$ .

2315.25 dollars; interest 315.25 dollars.

Example 2

Monthly compounding

A classroom model uses 1500 dollars at nominal 6% annually, compounded monthly for 2 years. Find its balance to cents.

Monthly rate $= 0.06 / 12 = 0.005$ .
Number of periods $= 24$ .
$A = 1500 {(1.005)}^{24} = 1690.7396 \dots$ .

1690.74 dollars.

Check your understanding

A nominal rate is 8% annually, compounded quarterly for 3 years. Give the period rate and number of periods.

Show answer

0.02 per quarter; 12 quarters.

$0.08 / 4 = 0.02$ and $4 \times 3 = 12$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the total of 1000 dollars after 2 years at 4% annual compound interest.
2 marks
Find the interest in that model.
2 marks
Find the one-year balance of 4000 dollars at nominal 8% annually compounded quarterly. Round to cents.
3 marks
Give the rate and exponent for nominal 12% annually compounded monthly over 18 months.
2 marks
An annual-compounding model grows 2500 dollars to 3025 dollars in 2 years. Find its rate.
3 marks
Why is 5% nominal annual interest compounded monthly different from 5% effective annually?
3 marks

Read the worked solutions

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

Find the total of 1000 dollars after 2 years at 4% annual compound interest.
2 marks
$1000 {(1.04)}^{2} = 1081.6$ .
1081.60 dollars
Find the interest in that model.
2 marks
Subtract the original 1000 dollars.
81.60 dollars
Find the one-year balance of 4000 dollars at nominal 8% annually compounded quarterly. Round to cents.
3 marks
$4000 {(1 + 0.08 / 4)}^{4} = 4329.72864$ .
4329.73 dollars
Give the rate and exponent for nominal 12% annually compounded monthly over 18 months.
2 marks
$0.12 / 12 = 0.01$ and the duration is already in months.
1% monthly; exponent 18.
An annual-compounding model grows 2500 dollars to 3025 dollars in 2 years. Find its rate.
3 marks
${(1 + r)}^{2} = 1.21$ ; the positive factor is $1.1$ .
10% annually
Why is 5% nominal annual interest compounded monthly different from 5% effective annually?
3 marks
${(1 + 0.05 / 12)}^{12} > 1.05$ .
Monthly interest itself earns interest during the year.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Simple versus compound growth; compound interest; depreciation; repeated percentage change; savings/loan scenarios with supplied models; changing assumptions and checking results; spreadsheets/digital models.

Official curriculum source ↗

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