Calculating Final Amounts and Interest with Simple and Compound Interest

Hello again. Last lesson dealt with gross earnings: separating ordinary hours, overtime, and allowances before adding the parts. Interest questions need the same careful reading. First identify the starting amount, rate, time, and whether the interest is simple or compound.

In this lesson, you will calculate both the interest earned and the final amount for savings or loans. This is a high-value exam skill because the maths is predictable once you choose the right formula and convert the percentage correctly.

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Start by decoding the financial language

These terms appear in almost every interest question:

Term	Meaning
Principal, $P$	The original amount invested or borrowed
Interest, $I$	The extra money earned on savings or charged on a loan
Rate, $r$	The interest rate per year, written as a decimal in formulas
Time, $t$	Time in years
Final amount, $A$	The total balance after interest has been added

For example, if AUD 2000 becomes AUD 2180, then:

$$\text{Interest}=2180-2000=180$$

The final amount is AUD 2180, while the interest is only AUD 180. A question may ask for either one, so circle the exact wording.

A quick decision guide:

If the question says...	Use...
“simple interest”	$I=Prt$
Interest is calculated only on the original deposit	Simple interest
“compounded annually,” “monthly,” or “daily”	Compound-interest formula
Interest is earned on previous interest	Compound interest

Watch this short explanation before working through the formulas. It gives a useful overview of both forms of interest.

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Simple interest: the original amount stays the base

With simple interest, the interest is always calculated from the original principal. It does not grow from year to year.

The formula for interest is:

$$I=Prt$$

where $r$ must be a decimal. Therefore:

$$4.5\%=0.045$$

If the question asks for the final amount, add the interest back onto the principal:

$$A=P+I$$

You can also combine these ideas:

$$A=P(1+rt)$$

However, in an exam, using $I=Prt$ first is often safer because it clearly distinguishes the interest from the final balance.

Worked example: savings account

A student deposits AUD 2400 in an account paying simple interest of 4.5% per annum for 3 years. Calculate the interest earned and final amount.

First, identify the values:

$$P=2400$$ $$r=0.045$$ $$t=3$$

Calculate the interest:

$$I=Prt$$ $$I=2400\times 0.045\times 3$$ $$I=324$$

The interest earned is AUD 324.00.

Now calculate the final amount:

$$A=P+I$$ $$A=2400+324$$ $$A=2724$$

Therefore, the final amount is AUD 2724.00.

Why simple interest grows evenly

The annual interest is the same every year because the principal remains AUD 2400.

$$\text{Interest each year}=2400\times 0.045=108$$

Over three years, the account earns three lots of AUD 108:

$$108\times 3=324$$

This regular growth is the defining feature of simple interest.

Time must be in years

If the rate is expressed per annum, convert months to years before using $I=Prt$.

For example:

$$18\text{ months}=\frac{18}{12}=1.5\text{ years}$$

If AUD 5000 is invested at simple interest of $6\%$ per annum for 18 months:

$$I=5000\times 0.06\times 1.5$$

Do not use $18$ as the value of $t$, because that would incorrectly treat 18 months as 18 years.

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Compound interest: interest earns interest

With compound interest, each new interest calculation uses the current balance, including interest earned earlier. This is why compound interest produces a higher final amount than simple interest when the principal, rate, and time are the same.

The key phrase is:

Interest is earned on interest already earned.

Compound interest compounded annually

When interest is compounded annually, use:

$$A=P(1+r{)}^t$$

Then find interest earned by subtracting the original principal:

$$I=A-P$$

Worked example: compounded annually

Use the same starting amount as the simple-interest example:

• Principal: AUD 2400
• Rate: $4.5\%$ per annum
• Time: 3 years
• Interest compounded annually

$$P=2400$$ $$r=0.045$$ $$t=3$$

Substitute into the formula:

$$A=2400(1+0.045{)}^3$$ $$A=2400(1.045{)}^3$$ $$A=2738.80$$

So the final amount is AUD 2738.80.

Now find the interest:

$$I=A-P$$ $$I=2738.80-2400$$ $$I=338.80$$

The compound interest earned is AUD 338.80.

Compare the two accounts:

Interest type	Final amount after 3 years	Interest earned
Simple interest	AUD 2724.00	AUD 324.00
Compound interest, annual	AUD 2738.80	AUD 338.80

Compound interest earns an extra AUD 14.80 in this example. The difference becomes much more noticeable with larger deposits, higher rates, or longer time periods.

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Monthly and daily compounding

If interest is compounded more than once per year, use the general compound-interest formula:

$$A=P{\left(1+\frac{r}{n}\right)}^{nt}$$

where:

• $A$ is the final amount;
• $P$ is the principal;
• $r$ is the annual interest rate as a decimal;
• $n$ is the number of times interest is compounded each year;
• $t$ is the time in years.

Common values of $n$:

Compounding frequency	$n$
Annually	$1$
Quarterly	$4$
Monthly	$12$
Daily	$365$

The formula makes two adjustments:

1. Divide the annual rate by $n$, because each interest period is shorter than a full year.
2. Multiply the number of years by $n$, because there are more interest calculations.

Worked example: compounded monthly

AUD 2000 is invested at $5\%$ per annum, compounded monthly, for 2 years. Find the final amount and interest earned.

Identify the values:

$$P=2000$$ $$r=0.05$$ $$n=12$$ $$t=2$$

Substitute carefully:

$$A=2000{\left(1+\frac{0.05}{12}\right)}^{12\times 2}$$ $$A=2000(1.0041667{)}^{24}$$ $$A=2209.88$$

The final amount is AUD 2209.88.

Find the interest:

$$I=2209.88-2000$$ $$I=209.88$$

Therefore, the interest earned is AUD 209.88.

Keep all calculator digits until the final answer, then round money to two decimal places. Rounding the monthly rate too early can slightly change the final amount.

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Exam method: choose, substitute, answer

For any interest question, use this layout.

1. Identify the type of interest

Look for key words such as:

• simple interest;
• compounded annually;
• compounded monthly;
• compounded daily.

Do not choose a formula just because the question involves a savings account. Savings accounts can use compound interest, while some other products may use simple interest.

2. Write the known values

For example:

$$P=6500,r=0.038,t=4$$

For monthly compounding, also write:

$$n=12$$

Writing these values first helps prevent a common error: entering $3.8$ instead of $0.038$.

3. Use the appropriate formula

For simple interest:

$$I=Prt$$ $$A=P+I$$

For compound interest:

$$A=P{\left(1+\frac{r}{n}\right)}^{nt}$$ $$I=A-P$$

For annual compounding, you may use:

$$A=P(1+r{)}^t$$

4. State exactly what was requested

Use a full financial conclusion, such as:

• “The interest earned is AUD 324.00.”
• “The final account balance is AUD 2738.80.”
• “The total amount owing is AUD 2209.88.”

For a loan with no repayments, the same mathematics applies, but interest is payable rather than earned. Real loans can also include repayments, fees, and changing rates; in an exam question, follow the conditions provided.

Common errors to avoid

• Leaving a percentage as $5$ instead of converting it to $0.05$.
• Giving the interest when the question asks for the final amount.
• Giving the final amount when the question asks for interest only.
• Using $t=18$ for 18 months instead of $t=1.5$ years.
• Treating monthly compounding as annual compounding.
• Rounding during intermediate calculator steps rather than at the end.

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Key takeaways

• Simple interest is calculated only from the original principal:

$$I=Prt$$

• For simple interest, the final amount is:

$$A=P+I$$

• Compound interest is calculated from the growing balance, so it includes interest on earlier interest.
• For annual compounding:

$$A=P(1+r{)}^t$$

• For monthly, quarterly, or daily compounding:

$$A=P{\left(1+\frac{r}{n}\right)}^{nt}$$

• To find compound interest earned, subtract the original principal:

$$I=A-P$$

This completes the consumer-arithmetic section of your revision. Next, you will move into measurement problems, using perimeter, area, volume, scale factors, similarity, and consistent units.