General Mathematics · Past QCAA questions All simple familiar questions
2023 · Paper 1 · Simple familiar

Question 19

[4 marks] Scientific calculator
Topic guidesPremiumTopic guide 8 · Comparing options
Game plansPremiumGame plan 5 · Finance
The question

Ngarra compares two investment options and decides option A will provide the better return.

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Option A: 5.60% p.a. compounding monthly

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Option B: 5.62% p.a. compounding quarterly

Use the effective annual rate of interest formula to evaluate the reasonableness of Ngarra’s decision.

Watch the situation first

The lower nominal rate wins by a whisker

Watch one year of interest build up under each option. Monthly steps are small but come often. Quarterly steps are big but rare. Then zoom in on the finish line.

Turn each nominal rate into a rate per period Let each option compound for one year Zoom in on the end of the year Calculate, compare, conclude A: 5.60% ÷ 12, so i = 0.0046667 per month and k = 12 B: 5.62% ÷ 4, so i = 0.01405 per quarter and k = 4 0% 2% 4% 6% 0 3 6 9 12 months interest earned so far, as a % of the amount invested B: one big step each quarter A: a small step every month After 12 months the two lines are almost level 5.75% 5.74% 5.73% A 0.057459… B 0.057395… A is higher by 0.0064 percentage points 0.057459… > 0.057395…, so A gives the better return. The decision is reasonable.

Option B starts with the higher nominal rate. Option A compounds three times as often, and by the end of the year that is enough to put it just ahead.

Now for the mathematics

Match each rate to its compounding, then keep enough decimals

For each option, choose the rate per period $i$ and the number of periods in a year $k$. The panel works out $(1+i)^k-1$ live. Then choose how far to round before you compare, and make the call on Ngarra’s decision.

Option A: 5.60% p.a. compounding monthly
Rate per period, $i$
Periods per year, $k$
Option B: 5.62% p.a. compounding quarterly
Rate per period, $i$
Periods per year, $k$
Round both rates before comparing
Is Ngarra’s decision reasonable?
Option A
Option B
Compare

QCAA marking guide · 4 marks

Two effective rates, one comparison, one sentence

Try each step yourself before you reveal it. There is a mark for setting up the formula, one for each effective rate, and one for the decision.

Step 1 · substitute1 mark

Write the formula book’s rule, then substitute for option A. Compounding monthly means 12 periods a year, so the rate per period is the annual rate divided by 12.

$$i_{\text{effective}}=(1+i)^k-1,\qquad i=\frac{0.056}{12},\quad k=12$$

$$i_{\text{effective}}=\left(1+\frac{0.056}{12}\right)^{12}-1$$

One correct substitution, for either option, earns this mark. The 2023 formula sheet and the marking guide write the rule as $\left(1+\frac{i}{n}\right)^n-1$ with $i=0.056$ as the annual rate and $n=12$. That is the same calculation, because its $\frac{i}{n}$ is the 2025 book’s $i$ and its $n$ is the 2025 book’s $k$.

Marker: correctly substitutes into appropriate rule for either option

Step 2 · option A1 mark

Type it in one line, brackets and all, and write down more decimal places than feels necessary.

$$\left(1+\frac{0.056}{12}\right)^{12}-1\approx0.057459\ldots$$

That is an effective rate of about 5.746% p.a. Follow-through applies, so if your substitution had a slip, evaluating your own expression correctly still earns this mark.

Marker: calculates effective interest rate for option A

Step 3 · option B1 mark

Quarterly means 4 periods a year, so $i=\frac{0.0562}{4}=0.01405$ and $k=4$.

$$\left(1+\frac{0.0562}{4}\right)^{4}-1\approx0.057395\ldots$$

That is about 5.740% p.a. The same follow-through applies. Notice that both effective rates sit above their nominal rates, which is a quick check that you compounded rather than divided.

Marker: calculates effective interest rate for option B

Step 4 · decision1 mark

Compare the two calculated rates in writing, then answer the question that was asked.

$$0.057459\ldots>0.057395\ldots$$

Ngarra’s decision is reasonable, because option A has the higher effective interest rate.

The sentence has to be linked to the effective rates. “A is better” on its own, or a reason based on the nominal rates, does not earn the mark. If you round, 5.75% against 5.74% still shows the difference. Rounded to one decimal place both are 5.7%, and then you have nothing to compare.

Marker: provides a statement of reasonableness linked to effective interest rate

Putting it all together

Each phrase of the question gave you something

“compares two investment options”
Two effective rates to calculate, one for each option, and they must be compared.
“decides option A will provide the better return”
The claim you are testing. For an investment, the better return is the higher effective rate.
“5.60% p.a. compounding monthly”
$i=0.056\div12$ per month and $k=12$.
“5.62% p.a. compounding quarterly”
$i=0.0562\div4=0.01405$ per quarter and $k=4$.
“Use the effective annual rate of interest formula”
The formula book’s $i_{\text{effective}}=(1+i)^k-1$, once per option. Comparing the nominal rates is ruled out.
“evaluate the reasonableness of Ngarra’s decision”
Calculate, compare, conclude. Say which effective rate is higher, then say whether the decision is reasonable.
What makes this simple familiar

The question names the formula for you and gives every rate and compounding period. All that is left is two substitutions, a comparison and a sentence. The 2023 subject report listed comparing investment options with the effective annual rate among the things students did well. The rule looks a little different in the 2025 formula book, $(1+i)^k-1$ with $i$ per period, but you type the same thing into your calculator.

If I had to guess where you might lose marks, it would be the last one. The report’s sample earned it by writing the two effective rates side by side to identify the higher one before concluding, so put $0.057459\ldots>0.057395\ldots$ on the page. The rates differ by only 0.0064 percentage points, so rounding to one decimal place too early makes them look equal. And the question is built around the nominal rates pointing the wrong way: 5.62% looks better than 5.60% until the compounding is counted.

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 2023 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.

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