2018 ACJC BT1 Q8

A man took a housing loan of \(\mathrm{ }\!\!\$\!\!\text { } P\) on 1 January 2000. At the end of each month, an interest of \(\frac{13}{60}\)% is charged to the outstanding balance of his loan before deducting his monthly repayment of  \(\mathrm{ }\!\!\$\!\!\text { }R\) to his loan, leaving a balance of \(\mathrm{ }\!\!\$\!\!\text { }\left(\left(\frac{6013}{6000}\right)P-R\right)\) at the end of the first month. Prove that the outstanding balance of his loan at the end of the first \(n\) months is \({{\left( \frac{6013}{6000} \right)}^{n}}P-\left( \frac{6000}{13} \right)\left( {{\left( \frac{6013}{6000} \right)}^{n}}-1 \right)R\), showing detailed working on how you arrive at this results.[3]

1. If he has taken a housing loan of \(\mathrm{ }\!\!\$\!\!\text  {  } 300 000\) and he intends to repay his loan in 30 years, find to the nearest cent, the minimum amount of monthly repayment he should pay.[2]
2. Beginning from 1 January 2001, he begins saving \(\mathrm{ }\!\!\$\!\!\text  {  } 50\) every month in the year 2001. For each subsequent year, he saves \(\mathrm{ }\!\!\$\!\!\text  {  }50\) more every month than the previous year, so that he saves \(\mathrm{ }\!\!\$\!\!\text  {  } 100\) every month in the year 2002, \(\mathrm{ }\!\!\$\!\!\text  {  } 150\) every month in the year 2003, and so on. The man has also taken a housing loan of \(\mathrm{ }\!\!\$\!\!\text  {   } 300 000\) on 1 January 2000 and his monthly repayment is \(\mathrm{ }\!\!\$\!\!\text  {  } 1300\).
   1. What is the outstanding balance of his loan, to the nearest cent, on 31 December 2010 after repayment for that month has been made?[1]
   2. What is the total amount of money that he has saved by 31 December 2010?[2]
   3. He decides to use the amount of money he has saved from 1 January 2001 to 31 December 2010 to make a lump sum payment in addition to his monthly repayment of \(\mathrm{ }\!\!\$\!\!\text  {  } 1300 \) on 31 December 2010 and continues to pay  \(\mathrm{ }\!\!\$\!\!\text  {  } 1300 \) at the end of every subsequent month.
      Determine how many more months would he take to complete repaying his loan.[2]