Question 20
Determine the monthly repayment on a $350 000 home loan over 25 years with 6.5% p.a. fixed interest compounded monthly.
Make everything monthly, then solve for the one payment that clears the loan
Watch the rate and the term turn monthly, the annuity formula take the loan, and the balance fall to zero after the last payment.
The repayment is the one monthly amount that takes the balance to exactly zero at payment 300. A smaller one leaves money owing, a bigger one finishes early.
Set up the annuity, then run the loan to test your repayment
Choose $i$, $n$ and the method. The panel solves for the repayment, then runs the real loan at 6.5% p.a. with your repayment for 300 months. Only the right repayment lands on zero. Then choose how to write the answer.
Choose $i$, $n$ and a method first. The stem gives you everything: the loan, the term in years, the yearly rate and the word “monthly”, twice.
6.5% is the yearly rate. Charging it every month gives a repayment of about $22 750.00, and the check shows the loan gone in well under two years. The interest compounds monthly, so divide by 12: $i=\frac{0.065}{12}$.
0.0054 is $i$ rounded too early. The repayment drops to $2358.85, a few dollars short each month, and over 300 months the check shows thousands still owing. Store $0.065\div12$ in your calculator, or type it into the formula in one line.
25 counts years, but with a monthly rate $n$ must count months. Paying the loan off in 25 monthly repayments needs about $15 007.13 a month, and the check finishes about two years in. Use $n=25\times12=300$.
The future value annuity answers “what do regular payments grow to?”. A loan is money handed over now, so it is a present value. The future value version gives only $467.39, which doesn’t even cover the first month’s interest, so the debt grows.
Sharing $350 000 over 300 months, $1166.67 each, ignores the interest completely. The check shows the balance climbing, because each month’s interest alone is bigger than that.
Simple interest charges 6.5% a year on the full $350 000 for all 25 years, giving $3062.50 a month. But the balance falls as you pay, so you are charged interest only on what you still owe. The check finishes years early. Use the annuity.
The set-up is right, and the check lands on zero at month 300. Now choose how to write the final answer.
That chops off the last digits. The repayment is 2363.225 06…, and the digit after the cents is 5, so the cents round up to 23. The 2020 subject report used this exact amount as its rounding example: $2363.225 ≈ $2363.23.
Money is paid in whole cents, so a repayment has two decimal places. Three decimal places is an amount nobody can pay. Round to the nearest cent.
Right number, no units. The last mark is for correct units and appropriate rounding together, and the subject report asked for units when stating a quantity. Write it with the dollar sign and “per month”.
That’s all four marks. Monthly $i$, $n=300$, the present value annuity, and the answer rounded to the cent with units. The check lands on zero after payment 300.
Three marks for the method, and the fourth is in the last line
Try each step yourself before you reveal it. There is one mark each for $i$ and $n$, the substitution, the repayment, and the way you state it.
Repayments are monthly and interest compounds monthly, so one period is one month. Turn the yearly rate and the term into monthly values.
$$i=\frac{0.065}{12}=0.005\,416\,6\ldots,\qquad n=25\times12=300$$
Keep $i$ unrounded. The guide writes it as 0.005416…, and the dots matter. This mark may be implied, so if your substitution shows the right $i$ and $n$ you get it even without this line.
Marker: correctly determines the $i$ and $n$ values
The $350 000 is borrowed now, so it is the present value of all the repayments. Use the formula book’s present value annuity and substitute.
$$A_{PV}=d\left(\frac{1-(1+i)^{-n}}{i}\right)$$
$$350\,000=d\left(\frac{1-(1+0.005\,416\,6\ldots)^{-300}}{0.005\,416\,6\ldots}\right)=d\times148.102\ldots$$
The 2020 formula sheet wrote this as $A=M\left(\frac{1-(1+i)^{-n}}{i}\right)$, the same formula with $M$ for the payment, so the guide’s $M$ is your $d$. The guide accepts $350\,000=M\times148.102$ or $M=\frac{350\,000}{148.102}$ for this mark, and follow-through applies if your $i$ or $n$ had a slip.
Marker: substitutes into appropriate annuity rule
Divide to get the repayment on its own.
$$d=\frac{350\,000}{148.102\ldots}=2363.225\ldots$$
Divide by the stored, unrounded factor. On a scientific calculator you can type the whole thing in one line: 350 000 × (0.065 ÷ 12) ÷ (1 − (1 + 0.065 ÷ 12)^(−300)). This mark may be implied by your final answer.
Marker: determines monthly repayment
Round to the nearest cent only now, and answer in dollars per month. The digit after the cents is 5, so round up.
The guide accepts equivalent answers with appropriate rounding, and allows follow-through for the units and rounding, so a correctly rounded answer from an earlier slip can still earn this mark.
Marker: states solution with correct units and appropriate rounding
Each phrase of the question gave you something
A home loan repayment is the standard present value annuity question. One formula from the formula book, one unknown, and the compounding period already matches the repayments. The only thinking is turning the yearly figures into monthly ones.
If I had to guess where you might lose marks, it would be the last one. The 2020 subject report used this very answer as its rounding example, $2363.225 ≈ $2363.23 to the nearest cent, and also asked for units on every quantity. Leaving 2363.225, or chopping it to 2363.22, are easy slips to make in the last line. The report also flagged “calculating interest rates for different compounding periods”, so check that your $i$ is a monthly rate before you substitute.
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 2020 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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