Compound interest | Year 10

01 / LEARN

Understand the idea.

Rate and period must agree

For period rate $i$ and no extra transactions, compound balance is $A = P {(1 + i)}^{n}$ . Label both $i$ and $n$ with the same period: a monthly decimal rate needs a count of months, not years.

If a nominal annual rate $r$ is explicitly compounded $m$ times per year, use $i = \frac{r}{m}$ and $n = m t$ over $t$ years. Thus $A = P {(1 + \frac{r}{m})}^{m t}$ under that stated convention.

Report what was asked

The balance $A$ includes the original principal. When there are no other transactions, interest earned is $A - P$ . Retain full precision during exponentiation, then round the requested currency result to cents.

An effective annual rate already describes the factor over one whole year. Dividing it by $12$ does not generally give the equivalent monthly rate. Check whether a rate is nominal or effective before converting its period.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Annual compounding

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

Worked solution

1
Identify the annual rate and period count.
$i = 0.05, n = 3$
The given annual rate matches annual compounding over three years.
2
Calculate the final balance.
$A = 2000 {(1.05)}^{3} = 2315.25 dollars$
Each year multiplies the current balance by $1.05$ .
3
Separate the interest earned.
$I = A - P = 2315.25 - 2000 = 315.25 dollars$
Subtracting the original principal leaves only the growth.

Answer

2315.25 dollars; interest 315.25 dollars.

EXAMPLE 02

Monthly compounding

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

Worked solution

1
Convert the stated nominal rate.
$i = \frac{0.06}{12} = 0.005 per month$
Monthly compounding divides the nominal annual decimal rate into twelve period rates.
2
Count monthly periods.
$n = 12 \times 2 = 24$
The two-year duration contains twenty-four months.
3
Calculate without intermediate rounding.
$A = 1500 {(1.005)}^{24} = 1690.7396 \dots dollars$
The monthly factor is raised to the matching number of months.
4
Round the final balance to cents.
$A \approx 1690.74 dollars$
Currency to cents requires two decimal places.

Answer

1690.74 dollars.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

0.02 per quarter; 12 quarters.

Quarterly compounding gives $i = \frac{0.08}{4} = 0.02$ per quarter. Three years contain $4 \times 3 = 12$ quarters, so the matched pair is $0.02$ and $12$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet 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
$4000 {(1 + \frac{0.08}{4})}^{4} = 4329.72864$ .
4329.73 dollars
Give the rate and exponent for nominal 12% annually compounded monthly over 18 months.
2 marks
$\frac{0.12}{12} = 0.01$ 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
${(1 + \frac{0.05}{12})}^{12} > 1.05$ .
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 ↗

In this lesson

Understand the idea.

Rate and period must agree

Report what was asked

See the method in action.

Annual compounding

Identify the annual rate and period count.

Calculate the final balance.

Separate the interest earned.

Monthly compounding

Convert the stated nominal rate.

Count monthly periods.

Calculate without intermediate rounding.

Round the final balance to cents.

Try it yourself.

Check your understanding

Keep practising