General Mathematics · Past QCAA questions All simple familiar questions
2025 · Paper 1 · Simple familiar

Question 22

[4 marks] Scientific calculator
Topic guidesPremiumTopic guide 10 · Perpetuities
Game plansPremiumGame plan 5 · Finance
The question

A $50 000 perpetuity earning fortnightly interest at 4.94% p.a. provides a regular fortnightly payment.

a)

Calculate the fortnightly payment.

[2 marks]
b)

Calculate the perpetuity’s effective annual rate of interest as a percentage.

[2 marks]
Watch the situation first

Pay out the interest and the balance never moves

First watch three fortnights of a perpetuity paying out its interest. Then watch the same rate compound for a full year to find the effective rate.

a) Each fortnight earns one fortnight of interest Pay that interest out and the $50 000 never shrinks b) Now leave the interest in for a whole year Interest on interest lifts 4.94% to 5.06% $50 000 Balance A Paid out Fortnight 1 Fortnight 2 Fortnight 3 + $95 interest + $95 interest + $95 interest $95 $95 $95 4.94% p.a., compounding fortnightly i = 0.0494 ÷ 26 = 0.0019 A = di , so d = A × i d = 50 000 × 0.0019 = $95 per fortnight Check: 50 000 × 1.0019 − 95 = 50 000 so the payments can go on forever $50 000 $52 500 0 13 26 Fortnights Compounded: $52 529.56 Simple: $52 470 ieffective = (1 + i)k − 1 i = 0.0019, k = 26 = (1 + 0.0019)26 − 1 ≈ 0.050591 ≈ 5.06% p.a. Simple: 26 × $95 = $2470, only 4.94% Interest on interest adds the other $59.56

A perpetuity only ever pays out its interest. The effective rate says what that interest would grow to over a year if it stayed in and compounded.

Now for the mathematics

Match the rate to the period, then let it compound

Part a) needs one fortnight’s interest. Part b) needs what 26 fortnights of compounding are worth. Choose a part and build the calculation. The panel works it out and checks it.

What you know
Choose a calculation for the payment
Rate per period, $i$
Periods per year, $k$

Pick one of the calculations. A perpetuity pays out exactly one period’s interest, so the right payment leaves the balance where it started.

That is a whole year of interest, 4.94% of $50 000. The payment is fortnightly, so you need one fortnight’s interest. Divide the rate by 26 first.

Dividing by 12 gives a monthly rate. There are 26 fortnights in a year, so the fortnightly rate is $\frac{0.0494}{26}=0.0019$.

That’s it. One fortnight’s interest is $95, so paying out $95 leaves the $50 000 untouched, fortnight after fortnight.

That rearranges $A=\frac{d}{i}$ the wrong way. Multiply both sides by $i$ to get $d=A\times i$. A payment of over 26 million dollars from a $50 000 fund should set off alarm bells.

Pick a rate per period and a number of periods. The effective rate only works when $i$ and $k$ describe the same period.

That’s it. The fortnightly rate, compounded 26 times, gives $0.050591\ldots$. The question asks for a percentage, so the answer is 5.06%. Leaving it as 0.0506 doesn’t answer “as a percentage”.

With the annual rate and $k=1$ you just get 4.94% back. That is the same as working out $26\times95\div50\,000$, which ignores interest earned on interest. The effective rate has to include the fortnightly compounding.

That charges the whole annual rate every period, so the answer is far too big. Divide 0.0494 by 26 first. The effective rate of a 4.94% investment should be a little above 4.94%.

That is monthly compounding. The answer comes out close to 5.06%, which is why this slip is easy to miss. This perpetuity earns interest fortnightly, so use $\frac{0.0494}{26}$ with $k=26$.

That is weekly compounding. It happens to round to 5.06% too, but the working describes a different investment and the working mark is for this one. Use $\frac{0.0494}{26}$ with $k=26$.

With $k=1$ you compound only once, so the answer is just the rate for one period. The effective annual rate compounds every period in the year, which here is 26 times.

Your $i$ and $k$ describe different periods. If $i$ is a rate per fortnight, $k$ must count the fortnights in a year. Match them as $\frac{0.0494}{26}$ with $k=26$.

QCAA marking guide · 4 marks

The working earns half of each part

Try each step yourself before you reveal it. Each part has one mark for the working and one for the answer, so the working has to be on the page.

Step 1 · part a)1 mark

Interest is paid fortnightly, so turn the annual rate into a rate per fortnight. There are 26 fortnights in a year.

$$i=\frac{0.0494}{26}=0.0019$$

The formula book gives the perpetuity as $A=\frac{d}{i}$. Multiply both sides by $i$ to make the payment the subject.

$$d=A\times i$$
$$=50\,000\times0.0019$$

The guide writes this in one line as $\frac{0.0494}{26}\times50\,000$, which is just as good.

Marker: correctly provides mathematical reasoning or working to support the answer

Step 2 · part a)1 mark

Multiply it out and give the answer as a fortnightly payment in dollars.

$$d=\text{\textdollar}95\text{ per fortnight}$$

As a check, $1.0019\times50\,000-95=50\,000$, so the balance never drops. Follow-through applies, so if your rate in step 1 had a slip you can still earn this mark by using it correctly.

Marker: calculates fortnightly payment

Step 3 · part b)1 mark

Use the effective annual rate from the formula book. Here $i$ is the rate per fortnight from part a) and $k=26$ is the number of fortnights in a year.

$$i_{\text{effective}}=(1+i)^k-1$$
$$=(1+0.0019)^{26}-1$$
$$\approx0.050591$$

Type $1.0019^{26}-1$ in one line and keep every digit until the end. The guide writes the formula as $\left(1+\frac{i}{n}\right)^n-1$ with $i=0.0494$ and $n=26$. That is the same calculation, and either form is fine.

The guide also accepts a longer route. Grow $50 000 for a year with $A=P(1+i)^n$ to get $52 529.56, then divide the interest, $2529.56, by $50 000.

Marker: correctly provides mathematical reasoning or working to support the answer

Step 4 · part b)1 mark

The question asks for a percentage, so multiply by 100.

$$0.050591\times100\%\approx5.06\%$$

Effective annual rate $\approx5.06\%$ p.a.

Any number of decimal places is accepted, rounded or truncated, so 5.059% or 5.1% also earns the mark. Follow-through applies. As a check, 5.06% sits a little above 4.94%, which is exactly what compounding should do.

Marker: calculates effective interest rate as a percentage

Putting it all together

Each phrase of the question gave you something

“A $50 000 perpetuity”
This is $A$ in $A=\frac{d}{i}$. A perpetuity pays out only its interest, so $A$ never changes.
“earning fortnightly interest at 4.94% p.a.”
An annual rate that compounds 26 times a year, so $i=\frac{0.0494}{26}=0.0019$.
“a regular fortnightly payment”
Payments match the compounding period, so each payment is one fortnight’s interest, $d=A\times i$.
“effective annual rate of interest”
The formula book’s $(1+i)^k-1$ with the same $i$ and $k=26$. A perpetuity is treated like any other compounding investment.
“as a percentage”
Multiply the decimal by 100 and write the % sign, 5.06%.
“[2 marks]” on each part
One mark for working, one for the answer. A bare answer can only earn half.
What makes this simple familiar

Both parts come straight from the formula book, the perpetuity $A=\frac{d}{i}$ and the effective annual rate $(1+i)^k-1$. The only setting up is turning 4.94% p.a. into a rate per fortnight, and part b) reuses it.

If I had to guess where you might lose marks, it would be part b). The 2025 subject report asked for students to use the effective rate formula for a perpetuity the same way as for any other compounding investment, so $i$ is the fortnightly rate and $k=26$. Writing 4.94% back down, or working out $26\times95\div50\,000$, ignores the compounding. Stopping at 0.0506 misses “as a percentage”.

Only mark this done when you could do it without help. Reading the solution does not count.

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Question wording and marking-guide steps are from the 2025 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.

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