Question 22
A reducing balance loan for $15 000 has an interest rate of 8.4% p.a. calculated monthly with a $250 repayment at the end of every month.
Use the monthly interest rate to write a recurrence relation for the loan balance after n months.
[2 marks]Calculate the loan balance after two months.
[1 mark]Use the reduction in the loan balance and the total repayments to determine the amount of interest paid in the first two months.
[3 marks]Every repayment pays the interest before it touches the loan
Each bar is one $250 repayment, drawn to scale. Watch how much of it goes on interest each month, then watch the two repayments regroup for part c).
Each repayment covers that month’s interest first, and only the rest comes off the loan. So the interest is the total repaid minus the reduction in the balance.
Build the relation, run it twice, then split the repayments
Choose the multiplier, what happens to the $250 and the order each month. The panel writes your relation and iterates it twice, rounding to the cent only at the end. Then choose what to give as the interest for part c).
Choose a multiplier, what happens to the repayment and an order. The stem gives you each one: “8.4% p.a. calculated monthly”, the word “loan” and “at the end of every month”.
0.007 is the monthly interest rate, $i$. Multiplying by it keeps only one month’s interest and throws the balance away, so the loan goes negative straight away. The multiplier is $r=1+i=1.007$. Keeping $i$ and $r$ apart is exactly what the subject report praised in this question.
1.084 charges the whole year’s 8.4% every month, so the balance grows even with repayments going in. One step of the relation is one month, so the rate must be monthly: $i=\frac{8.4}{100\times12}=0.007$, and $r=1.007$.
Adding 250 makes the debt grow. A repayment reduces what is owed, so it is subtracted. The formula book writes a reducing balance loan as $A_{n+1}=rA_n-d$.
The repayment is made at the end of every month, so the month’s interest is charged on the whole balance first. Paying first undercharges the interest, and the balance after two months comes out a few dollars too low.
Your relation is right. The calculator shows $A_2=14\,708.985$, exactly half a cent, so it rounds up to $14 708.99. Now choose what to give as the interest for part c).
291.01 is the reduction, the part of your two repayments that actually paid off the loan. Part c) asks for the other part, the interest. Take the reduction away from the total repayments.
500 is everything you repaid in two months. Some of it covered interest and the rest reduced the loan by 291.01. The interest is what is left after you take the reduction out.
Adding each month’s interest gives the same $208.99, and the marking guide gives full marks for this method too. The question names the reduction and the total repayments, though, so the safest working is the one that shows both.
That’s it. The two repayments total $500, the balance fell by $291.01, and the rest, $208.99, was interest. That is one mark for each of the three lines.
Two lines for part a), three short ones for part c)
Try each step yourself before you reveal it. Part a) has a mark for the monthly rate and one for the relation. Part b) is one mark. Part c) has a mark each for the reduction, the total repayments and the interest.
The question tells you to use the monthly interest rate, and that rate earns a mark of its own. Divide the yearly percentage by 100 and by 12.
$$i=\frac{8.4}{100\times12}=0.007$$
Write it as its own line with a label, as the sample response does, so the marker can find it.
Marker: correctly calculates the monthly interest rate
Turn the rate into the multiplier, then substitute into the formula book’s reducing balance loan, $A_{n+1}=rA_n-d$, with the repayment $d=250$.
$$r=1+i=1.007$$
$$A_{n+1}=1.007A_n-250,\qquad A_0=15\,000$$
The 2024 formula sheet and this marking guide call the repayment $R$ and write $A_{n+1}=rA_n-R$. It is the same rule with a different letter. Writing $(1+i)$ in place of 1.007 is also accepted, as long as you defined $i$ earlier. Follow-through applies, so a slip in the monthly rate used correctly here still earns this mark. The guide doesn’t need $A_0=15\,000$ for part a), but it starts part b) with it, so write it now.
Marker: determines recurrence relation
Run the relation twice, one line per month.
$$A_1=1.007\times15\,000-250=14\,855$$
$$A_2=1.007\times14\,855-250=14\,708.985$$
The exact value lands on half a cent, so round up. The loan balance after two months is $14 708.99.
The guide accepts appropriate rounding, such as to the nearest cent, nearest 5 cents or nearest dollar. On your calculator, enter 15 000, then type 1.007 × Ans − 250 and press equals once for each month.
Marker: determines loan balance after two months
The reduction in the loan balance is the starting balance minus your part b) answer.
$$\text{reduction}=15\,000-14\,708.99=291.01$$
This mark may be implied by your later working, but label the line anyway so the marker can see it.
Marker: determines reduction in loan balance in first two months
Two repayments of $250 were made.
$$\text{total repayments}=2\times250=500$$
Marker: determines total repayments in first two months
Whatever you repaid that didn’t reduce the loan went on interest.
The guide also gives full marks for a second method, one mark for each month’s interest and one for the total: $0.007\times15\,000=105$, $0.007\times14\,855=103.99$, and $105+103.99=208.99$. It makes a good check, but the question names the reduction and the total repayments, so show those.
Marker: determines total amount of interest paid in the first two months
Each phrase of the question gave you something
A reducing balance loan written as a recurrence relation is standard Unit 4 work, and every number you need is in the first sentence. Part c) even names the two quantities to use. Two months of a recurrence relation is quick on a scientific calculator with the Ans key.
If I had to guess where you might lose marks, it would be the first two lines of part a). The 2024 subject report showed a response that kept the monthly rate $i=0.007$ separate from the multiplier $r=1.007$, and mixing those two up wrecks every later number. Then watch the rounding. $A_2$ lands exactly on half a cent, so it rounds up to $14 708.99, and the same report asks you to keep money to the cent. In part c), 291.01 and 500 are both on your page, and neither of them is the interest.
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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