2024 HCI Promo Q13

At the start of January 2014, Leon took a home loan of \(\$500\,\,000\) from a bank that offers a fixed monthly interest rate of \(0.3\%\), to buy an apartment unit in a condominium. The interest was added to the amount owed on the \({{15}^{\mathrm{th}}}\) of every month, with the first interest amount added on 15 January 2014. Leon made a monthly repayment of \(\$x\) on the 20\(^{\mathrm{th}}\) of every month, with the first repayment on 20 January 2014.

1. Show that the amount (in dollars) that Leon owed at the end of \(n\) months was
   \(500000{{\left( 1.003 \right)}^{n}}-\frac{1000x}{3}\left( {{1.003}^{n}}-1 \right)\).[3]

Leon planned to repay the loan fully in \(360\) monthly repayments.

2. Find the value of \(x\), giving your answer to the nearest cents.[2]

The estate management collects a monthly maintenance fee from each apartment unit within the condominium at the start of every month. The monthly maintenance fee on 1 January 2014 was \(\$400\). To cope with the rising cost of maintenance over time, the estate management increased the monthly maintenance fee by \(\$k\) every \(2\) years such that the monthly maintenance fee increased to \(\$\left(400+k\right)\) on 1 January 2016 and to \(\$\left(400+2k\right)\) on 1 January 2018 and so on.

3. Show that the total maintenance fee (in dollars) that Leon paid at the end of \(10\) years was of the form \(s+tk\), where \(s\) and \(t\) are positive constants to be determined.[3]

The value of Leon’s apartment unit increased to \(\$750\,\,000\) at the end of December 2023. It was also known that \(k=50\).

4. By considering the amount Leon owed, along with the repayment and maintenance fee that he paid over the \(10\) years, explain, with justification, whether Leon should sell his apartment unit at the end of December 2023 if he received an offer of \(\$750\,\,000\) from an interested buyer.[4]