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📖 Lesson 6 ⏱ ~30 min Year 10 · Unit 1 ⚡ +50 XP

Compound Interest

Discover why Einstein reportedly called compound interest the eighth wonder of the world. Learn how interest earns interest, and how this transforms small savings into large sums over time.

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Worksheet

Worksheet

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Warm-up
Think First
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Q1 · How is compound interest different from simple interest?

Q2 · If you invest $1,000 at 5% compounded annually for 3 years, do you think the interest will be more or less than simple interest? Why?

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From the lesson
Intentions

Learning Intentions

Know

  • The compound interest formula and the meaning of each variable.
  • Common compounding periods: annually, quarterly, monthly, daily.

Understand

  • How compound interest differs from simple interest: interest is earned on interest.
  • Why more frequent compounding yields higher returns.

Can Do

  • Calculate compound interest using the formula.
  • Compare simple and compound interest for the same principal, rate and time.
  • Calculate the total amount using $A = P(1 + R)^n$.
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From the lesson
Success Criteria

Success Criteria

  • I can use the compound interest formula $A = P(1 + R)^n$ correctly.
  • I can adjust the rate and number of periods for different compounding frequencies.
  • I can calculate and compare simple vs compound interest for the same scenario.
  • I can interpret compound interest in savings accounts, superannuation and loans.
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From the lesson
Key Terms

Key Terms

Compound interest — Interest calculated on the principal plus any interest already earned.
Compounding period — How often interest is calculated and added: annually, quarterly, monthly, etc.
Annual rate ($R$) — The yearly interest rate, divided by the number of compounding periods per year.
Number of periods ($n$) — The total number of times interest is compounded: years $\times$ periods per year.
Principal ($P$) — The initial amount invested or borrowed.
Total amount ($A$) — The principal plus all compound interest earned.
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From the lesson
Misconceptions

Common Mistakes to Avoid

Wrong: “Compound interest and simple interest give the same result for the first year.” This is true, but after the first year compound interest pulls ahead because it earns interest on interest.

Right: After year 1, both give the same amount. From year 2 onward, compound interest grows faster. The gap widens over time.

Wrong: Forgetting to divide the annual rate by the number of compounding periods. A 6% p.a. rate compounded quarterly uses $1.5\%$ per quarter, not $6\%$.

Right: For quarterly: $R = \dfrac{\text{annual rate}}{4}$ and $n = \text{years} \times 4$.

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Concept
The Compound Interest Formula
+5 XP

Compound interest is the secret behind long-term wealth building. Unlike simple interest, each period's interest is added to the principal, so the next period's interest is calculated on a larger amount.

Compound Interest Formula
$A = P(1 + R)^n$
where $A$ = total amount, $P$ = principal, $R$ = interest rate per compounding period, $n$ = total number of compounding periods
$\text{Compound Interest} = A - P$

Compounding frequency matters:

  • Annually: $R = \text{annual rate}$, $n = \text{years}$
  • Quarterly: $R = \dfrac{\text{annual rate}}{4}$, $n = \text{years} \times 4$
  • Monthly: $R = \dfrac{\text{annual rate}}{12}$, $n = \text{years} \times 12$
  • Daily: $R = \dfrac{\text{annual rate}}{365}$, $n = \text{years} \times 365$
Heads up

Real-World Anchor: Australian high-interest savings accounts typically compound interest monthly. The ING Savings Maximiser and ubank Save Account both compound monthly. Australian superannuation funds compound earnings daily or monthly, which is why starting early makes such a dramatic difference to retirement balances.

What to write in your book
  • Compound interest formula: $A = P(1 + R)^n$, where $R$ is the rate per period and $n$ is the total number of periods.
  • Compound interest earned = $A - P$.
  • More frequent compounding gives a higher return because interest earns interest sooner.
  • Adjust $R$ and $n$ for the compounding frequency: quarterly = $\div 4$, monthly = $\div 12$, daily = $\div 365$.
$8,000 is invested at 3% p.a. compounded annually for 2 years. What is the total amount?
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From the lesson
Worked Example 1
Worked Example 1 - Compound Interest (Annual Compounding)
1
Given: Ava invests $\$8{,}000$ at $4.5\%$ p.a. compounded annually for 3 years.
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Find: The total amount and the compound interest earned.
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Method: $A = 8{,}000 \times (1 + 0.045)^3 = 8{,}000 \times 1.141166 = 9{,}129.33$. Interest = $9{,}129.33 - 8{,}000 = 1{,}129.33$.
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Answer: Total amount is $\mathbf{\$9{,}129.33}$ and compound interest is $\mathbf{\$1{,}129.33}$.
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From the lesson
Worked Example 2
Worked Example 2 - Quarterly Compounding
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Given: A business invests $\$25{,}000$ at $6\%$ p.a. compounded quarterly for 2 years.
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Find: The total amount.
3
Method: Quarterly rate = $6\% / 4 = 1.5\% = 0.015$. Number of quarters = $2 \times 4 = 8$. $A = 25{,}000 \times (1.015)^8 = 25{,}000 \times 1.126493 = 28{,}162.32$.
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Answer: The total amount is $\mathbf{\$28{,}162.32}$.
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From the lesson
Worked Example 3
Worked Example 3 - Comparing Simple and Compound Interest
1
Given: $\$10{,}000$ invested at $5\%$ p.a. for 5 years.
2
Find: The difference between simple and annually compounded interest.
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Method: Simple: $I = 10{,}000 \times 0.05 \times 5 = 2{,}500$. Total = $12{,}500$. Compound: $A = 10{,}000 \times (1.05)^5 = 12{,}762.82$. Difference = $12{,}762.82 - 12{,}500 = 262.82$.
4
Answer: Compound interest earns $\mathbf{\$262.82}$ more than simple interest over 5 years.
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From the lesson
Interactive

Interactive: Compound Interest Visualiser

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From the lesson
Practice

Your Turn

Question 1: Calculate the compound interest on $6,000 at 4% p.a. compounded annually for 4 years.

Question 2: $15,000 is invested at 5.2% p.a. compounded quarterly for 3 years. Calculate the total amount.

Question 3: Compare simple and compound interest on $20,000 at 6% p.a. for 10 years (compounded annually). What is the difference?

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From the lesson
Revisit

Revisit Your Thinking

Look back at your Think First answer about $1,000 at 5% compounded annually for 3 years. Calculate the compound interest amount and compare it to simple interest. Was your initial prediction correct? What is the difference, and why does it exist?

Reflect
Revisit your thinking
reflect

Earlier you were asked: What was your first thought on this topic?

Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?

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From the lesson
MCQ 1
MCQ2 marks

$5,000 is invested at 4% p.a. compounded annually for 3 years. What is the total amount?

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From the lesson
MCQ 2
MCQ2 marks

An investment of $10,000 earns 6% p.a. compounded quarterly for 2 years. What is the value of $n$ in the formula $A = P(1 + R)^n$?

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From the lesson
MCQ 3
MCQ2 marks

Which compounding frequency will give the highest return for the same principal, rate and time?

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From the lesson
MCQ 4
MCQ2 marks

$8,000 invested at 5% p.a. simple interest for 4 years earns the same total interest as how much invested at 5% p.a. compounded annually for 4 years?

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From the lesson
MCQ 5
MCQ2 marks

$12,000 is invested at 3.6% p.a. compounded monthly for 2 years. What is the total amount correct to the nearest cent?

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From the lesson
SAQ 1
Short Answer3 marks

Zoe invests $9,000 in a high-interest savings account paying 3.2% p.a. compounded quarterly. Calculate the total amount after 2.5 years, correct to the nearest cent.

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From the lesson
SAQ 2
Short Answer4 marks

Harrison has $20,000 to invest for 5 years. Bank A offers 4.5% p.a. simple interest. Bank B offers 4.2% p.a. compounded annually.

(a) Calculate the total amount from Bank A. (2 marks)

(b) Calculate the total amount from Bank B. (2 marks)

(c) Which bank should Harrison choose? Justify with calculations. (1 mark - bonus)

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From the lesson
SAQ 3
Short Answer5 marks

A superannuation fund compounds earnings monthly. Priya's balance is $45,000 and the fund returns 7.2% p.a.

(a) Calculate the monthly interest rate as a decimal. (1 mark)

(b) Calculate the balance after 1 year. (2 marks)

(c) Calculate how much interest is earned in the first year. (1 mark)

(d) Explain why the actual interest earned is slightly more than $45,000 × 0.072. (1 mark)

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Recap
Quick Review

Compound amount

$A = P(1 + R)^n$

Compound interest

Interest = $A - P$

Quarterly compounding

$R = \frac{r}{4}$, $n = 4t$

Monthly compounding

$R = \frac{r}{12}$, $n = 12t$

More frequent compounding

Higher total return because interest earns interest sooner

Simple vs compound

Same result after year 1; compound interest wins after that

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From the lesson
Real-Life Link

Real-Life Link

Compound interest is the engine behind long-term wealth creation. If you contribute $5,000 per year to superannuation from age 25 to 65 at 7% p.a. compounded, you will retire with over $1 million. Start at 35 instead and you will have roughly half as much. This is why Australian financial advisors stress starting early - the extra years of compounding matter more than the amount you contribute. Understanding compound interest helps you make informed decisions about savings, loans and investments throughout your life.

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From the lesson
Game

Game Time!

Test your compound interest skills in an interactive challenge.

Play Compound Interest Challenge
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From the lesson
Continue
Continue to Lesson 7: Depreciation and Financial Decision Making →
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