Question 13
Determine the effective annual rate of interest for a two-year investment, for which the interest compounds quarterly at 3.64% p.a.
Not quite. 3.67% is $\left(1+\tfrac{0.0364}{2}\right)^2-1$, which is six-monthly compounding. Quarterly means four periods a year, so $k=4$ and $i=\tfrac{0.0364}{4}$. In 2025, 17.7% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2025, 44.9% of Queensland students chose (B).
Not quite. 7.28% is $2\times3.64\%$. Doubling the rate for two years is simple interest over the whole term, and the question wants a rate for one year. The two years is there to tempt you. In 2025, 20.8% of Queensland students chose (C).
Not quite. 7.52% is $(1.0091)^8-1$, the growth over the full two years. An effective annual rate always covers one year, so the power is the number of periods in a year, $k=4$. In 2025, 15.6% of Queensland students chose (D).
This is the question’s compounding. After one year, $100 has grown to $103.69, so the effective annual rate is 3.69%. Carry on to two years and you reach 7.52%, the growth over the whole term.
Compounding , the effective annual rate is %. More periods a year nudge the rate up, but only a little. Yearly compounding gives back the nominal 3.64%.
Quarterly means four compounding periods a year. Split the annual rate across them.
$$i=\frac{0.0364}{4}=0.0091,\qquad k=4$$
Here $i$ is the rate per quarter and $k$ is the number of quarters in one year.
Use the formula book’s effective rate. Type it in one line.
$$\begin{aligned}i_{\text{effective}}&=(1+i)^k-1\\&=(1.0091)^4-1\\&=0.036\,899\ldots\approx3.69\%\end{aligned}$$
You may also see it written as $\left(1+\frac{r}{n}\right)^n-1$. It’s the same calculation.
Ignore the two years. An effective annual rate describes one year of growth, whatever the length of the investment.
Check the size: compounding more than once a year lifts the rate slightly above the nominal 3.64%. Anything near 7% can’t be a yearly rate here.
$$\begin{aligned}i&=\frac{0.0364}{4}=0.0091,\quad k=4\\i_{\text{effective}}&=(1+i)^k-1\\&=(1.0091)^4-1\\&=0.036\,899\ldots\approx3.69\%\end{aligned}$$
The rate per quarter is compounded four times to give one year of growth. The two-year term doesn’t enter the calculation.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2025 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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