2022 ACJC Promo Q11

Antonio used a credit card to buy a diamond ring to propose to his girlfriend. On 1st October 2022, he charged \(\$7500\) to the credit card. The credit card charges a \(2\%\) monthly interest rate that is applied on the last day of every month. From 1st November 2022, Antonio started paying off his credit card debt by paying \(\$x\) at the start of each month, where \(x\) is a constant.

1. Find the amount of Antonio’s credit card debt left after his first repayment, giving your answer in terms of \(x\).[1]
2. Show that the amount of Antonio’s credit card debt after his \(n\)th repayment is
   \(\left( 7500-50x \right){{\left( 1.02 \right)}^{n}}+50x\).[3]
3. Find the minimum amount that Antonio must consistently pay each monthly repayment for him to fully pay off his credit card debt with his \(60\)th repayment. Find also the amount of credit card interest he would have paid in this case.[4]
4. Suppose that up to and including 1st October 2023, Antonio had faithfully repaid \(\$500\) a month to service his credit card debt. As a promotional reward, the credit card company decided to lower Antonio’s monthly interest rate to \(1\%\) with immediate effect from October 2023, provided he maintains his \(\$500\) monthly repayments.
   Find the date of Antonio’s last repayment to the credit card company with this new interest rate, and the amount of his last repayment.[4]

Antonio and his fiancée are deciding between two venues for their wedding banquet by comparing the costs of the catering at each venue. At the Gomez Hotel, the catering is priced at \(\$1500\) for each table of \(10\) guests. At the Grande Hotel, the catering is priced as follows:

5. How many tables will the couple need at least, for it to be cheaper to hold their wedding banquet at the Grande Hotel?[3]