2021 ACJC Promo Q12

Mrs Tan plans to start a business which requires a start-up capital of \(\$700,000\). She decided to first save \(\$200,000\) by depositing money every month into a savings plan. For the remaining \(\$500,000\), she intends to take a loan from a finance company.

She deposited \(\$3000\) into the savings plan in the first month and on the first day of each subsequent month, she deposited \(\$100\) more than the previous month. Mrs Tan will continue depositing money into the savings plan until the total amount in her savings plan reaches \(\$200,000\). It is given that this savings plan pays no interest.

1. Find the month in which Mrs Tan’s monthly deposit will exceed \(\$6550\).[2]
2. Find the number of months that it will take for Mrs Tan to save \(\$200,000\) and hence find the amount that she would have deposited in the last month.[4]

After Mrs Tan has saved \(\$200,000\), she took a loan of \(\$500,000\) from a finance company. To repay the loan from the finance company, Mrs Tan would have to pay a monthly payment of \(\$x\) at the beginning of each month, starting from the first month. An interest of \(0.3\%\) per month will be charged on the outstanding loan amount at the end of the month.

3. Show that the outstanding amount at the end of \({{n}^{\mathrm{th}}}\) month, after the interest has been charged, is \(A\left( {{1.003}^{n}} \right)-Bx\left( {{1.003}^{n}}-1 \right)\), where \(A\) and \(B\) are exact constants to be determined.[3]
4. Find the amount of \(\$x\), to \(2\) decimal places, if Mrs Tan wants to fully repay her loan in \(8\) years.[2]
5. Using the value of \(x\) found in part (iv), calculate the total interest that the finance company will earn from Mrs Tan at the end of \(8\) years.[2]