Simple versus compound growth | Year 10

01 / LEARN

Understand the idea.

Which amount earns interest?

With simple interest, each period earns interest on the original principal $P$ . At annual decimal rate $r$ for $n$ years, interest is $P r n$ and balance is $A = P (1 + r n)$ ; the yearly dollar increase stays constant.

With annual compound interest, each new period uses the current balance: $A = P {(1 + r)}^{n}$ . Earlier interest remains in the account and can earn more interest, so a fixed percentage generally produces increasing annual dollar gains.

Use matching assumptions

A fair comparison matches the starting principal, duration and rate convention. State whether the model includes deposits, withdrawals, fees or tax, and when interest and transactions occur. A headline percentage alone does not define all these choices.

Distinguish the final balance from interest earned. In a no-transaction model, interest is $A - P$ . These supplied classroom calculations describe stated assumptions; their outputs are not real account quotations.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Three-year comparison

Compare 1000 dollars at 10% annually for 3 years using simple and annual compound interest.

Worked solution

1
Calculate the simple-interest total.
$A_{s} = 1000 (1 + 0.10 \times 3) = 1300 dollars$
Simple interest adds $100$ on the original principal in each of the three years.
2
Calculate the compound-interest total.
$A_{c} = 1000 {(1.10)}^{3} = 1331 dollars$
Annual compounding applies the factor $1.10$ to each current balance.
3
Compare the balances.
$1331 - 1300 = 31 dollars$
The additional $31$ comes from interest earned on earlier interest, with the same principal, rate and duration.

Answer

1300 dollars simple; 1331 dollars compound.

EXAMPLE 02

Recognise a pattern

A balance is 500, 525 and 551.25 dollars at years 0, 1 and 2. Is it consistent with adding 25 annually or multiplying by 1.05?

Worked solution

1
Compare the dollar increases.
$525 - 500 = 25, 551.25 - 525 = 26.25$
Adding a fixed $25$ would require the increases to be equal.
2
Compare the consecutive multipliers.
$\frac{525}{500} = 1.05, \frac{551.25}{525} = 1.05$
Both year-to-year ratios match a $5 %$ compound increase.
3
Confirm the second-year value.
$500 {(1.05)}^{2} = 551.25 dollars$
Multiplying by $1.05$ fits both supplied transitions.

Answer

Multiplication by 1.05.

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the yearly simple interest on 200 dollars at 5%.

Show answer & reasoning

10 dollars

Yearly simple interest is $P r = 200 (0.05) = 10$ dollars. It remains $10$ each year because the principal used in this model stays $200$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Find the yearly simple interest on 800 dollars at 6%.
2 marks
Find the total after 4 years on 800 dollars at 6% simple interest.
2 marks
Find the total after 2 years on 800 dollars at 6% annual compound interest.
2 marks
Balances are 1000, 1040, 1080 at consecutive years. Identify the simple model rate.
2 marks
Compare simple and annual compound totals for 500 dollars at 10% annually for 2 years.
3 marks
Why do the two models agree after exactly one year when principal and annual rate match?
3 marks

Read the worked solutions

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

Find the yearly simple interest on 800 dollars at 6%.
2 marks
$800 \times 0.06 = 48$ .
48 dollars
Find the total after 4 years on 800 dollars at 6% simple interest.
2 marks
$800 (1 + 0.06 \times 4) = 992$ .
992 dollars
Find the total after 2 years on 800 dollars at 6% annual compound interest.
2 marks
$800 {(1.06)}^{2} = 898.88$ .
898.88 dollars
Balances are 1000, 1040, 1080 at consecutive years. Identify the simple model rate.
2 marks
The yearly increase is 40, and $\frac{40}{1000} = 0.04$ .
4% annually
Compare simple and annual compound totals for 500 dollars at 10% annually for 2 years.
3 marks
Simple: $500 (1 + 0.1 \times 2) = 600$ . Compound: $500 {(1.1)}^{2} = 605$ .
600 dollars simple; 605 dollars compound.
Why do the two models agree after exactly one year when principal and annual rate match?
3 marks
At $n = 1$ , both formulas equal $P (1 + r)$ .
Both apply the rate once to the original principal.

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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.

Which amount earns interest?

Use matching assumptions

See the method in action.

Three-year comparison

Calculate the simple-interest total.

Calculate the compound-interest total.

Compare the balances.

Recognise a pattern

Compare the dollar increases.

Compare the consecutive multipliers.

Confirm the second-year value.

Try it yourself.

Check your understanding

Keep practising