2013 PJC P1 Q11

Ms. Tan took up a car loan of \(\$100000\) from a bank in the beginning of a year and she was offered the following bank loan payment packages:

Package \(A\):

A monthly installment of \(\$500\) is to be paid in the beginning of the first month and increases by \(\$50\) for every subsequent month. It is given that the bank charges a one -time administrative fee of \(\$12000\) for Package \(A\).

Package \(B\):

A monthly instalment of \(\$x\) is to be paid at the beginning of each month and a bank interest of \(5\%\) per annum will be charged on the outstanding loan amount at the end of the year.

1. On which month would Ms. Tan pay up her loan completely under Package \(A\)?[4]
2. Show that the outstanding amount at the end of \({n}^{\mathrm{th}}\) year under Package B is
   \( \left(100000 \right){{1.05}^{n}}+252x\left( 1-{{\left( 1.05 \right)}^{n}} \right)\).[3]
3. Deduce the least amount \(\$x\) corrected to \(2\) decimal places if Ms. Tan wishes to pay up her loan completely within \(7\) years under Package \(B\).[2]
4. Based on your calculations in (iii), determine which package will be cheaper for Ms. Tan.[2]