Question 21
A perpetuity earns interest quarterly at 5.2% p.a. and pays $975 each quarter.
Determine the quarterly interest rate.
[1 mark]Calculate the value of the perpetuity.
[2 marks]Pay out exactly the interest and the money lasts forever
Watch four quarters of payments leave a balance that never shrinks. Then work backwards from the payment to the balance that makes that possible.
A perpetuity only ever pays out its interest. So the value is whatever balance earns exactly one payment of interest each period.
Match the payment and the rate to the same period
Part a) needs a rate per quarter. Part b) feeds it into the perpetuity formula. Choose a part and build the calculation. The panel works it out and tests it against one quarter of the real perpetuity.
Pick one of the calculations. A rate per quarter, used four times, should rebuild the 5.2% a year.
That’s it. The quarterly rate is $i=0.013$, which is 1.3% per quarter. Four quarters of it give back the 5.2% a year.
Right rate, risky form. 1.3 is a percentage, 1.3%, and the guide accepts “1.3%” for part a). Write the % sign. Before you substitute into any formula, change it to the decimal 0.013.
Dividing by 12 gives a monthly rate. This perpetuity earns interest quarterly, and there are 4 quarters in a year, so divide by 4.
Multiplying makes the rate bigger than the whole year’s rate. A quarter is a quarter of a year, so its rate is a quarter of 5.2%. Divide by 4.
Pick a payment, a rate and a way to combine them. The formula book gives the perpetuity as $A=\frac{d}{i}$, with $d$ and $i$ for the same period.
That’s it. The quarterly payment over the quarterly rate gives $75 000. One quarter’s interest on that is exactly $975, so the balance never changes and the payments can go on forever.
This works too. A year of payments, $3900, over the annual rate gives the same $75 000, because it is the quarterly calculation with top and bottom both multiplied by 4. Payment and rate still describe the same period. The quarterly version follows on from part a), so it is the tidier choice here.
That puts a quarterly payment over a yearly rate. Look at the check: the interest on $18 750 for one quarter is far less than $975, so the balance falls every quarter and the fund runs out. Use the quarterly rate from part a), 0.013.
That puts a yearly payment over a quarterly rate, so the value comes out four times too big. The check shows a fund this size grows every quarter, which means it is worth more than the payments need. Match the periods: 975 with 0.013, or 3900 with 0.052.
1.3 is the percentage, not the decimal. Dividing by 1.3 instead of 0.013 makes the answer 100 times too small, and a fund that small could never pay $975 a quarter. Use $i=0.013$.
Multiplying gives the interest that the payment itself would earn, which is not what the question asks. The perpetuity formula is $A=\frac{d}{i}$, so divide the payment by the rate.
A rate, a substitution and a division
Try each step yourself before you reveal it. Part b) has a mark for substituting into the rule and one for the answer, so write the rule down first.
Interest is earned quarterly, and there are 4 quarters in a year. Divide the annual percentage by 100 to make it a decimal and by 4 to make it quarterly.
The guide also accepts 1.3%. Either way, label it as the quarterly rate so the marker can see what it is.
Marker: correctly determines the quarterly interest rate
Write the perpetuity rule from the formula book, then substitute the quarterly payment and the quarterly rate.
$$A=\frac{d}{i}=\frac{975}{0.013}$$
The guide writes the rule as $A=\frac{M}{i}$, using the payment letter from the old formula sheet, which had no perpetuity formula at all in 2024. The 2025 formula book calls the payment $d$. Same rule, different letter.
The guide accepts another route. A perpetuity’s payment is one period of simple interest, so $I=Pin$ with $n=1$ quarter gives $975=P\times0.013\times1$, which leads to the same division. Follow-through applies, so a slip in part a) still lets you earn this mark if you use your rate correctly.
Marker: substitutes into appropriate rule
Divide, and give the answer in dollars.
$$A=\text{\textdollar}75\,000$$
This one divides exactly, but the guide accepts rounding to the nearest cent, 5 cents or dollar. As a check, 1.3% of $75 000 is $975, exactly one payment, so the balance never moves.
Marker: calculates value of perpetuity
Each phrase of the question gave you something
It is a one-rule question. The payment and the interest are both quarterly, so once you have the quarterly rate the formula book does the rest, and part a) hands you that rate as the first step.
If I had to guess where you might lose marks, it would be the rate you put into part b). Dividing 975 by the annual rate, 0.052, gives $18 750, and dividing by 1.3 instead of 0.013 gives $750. Both are easy slips. The 2024 subject report asks teachers to help students write the formula and identify the values before substituting, and that habit catches both.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2024 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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