2018 ACJC BT1 Q8

Timothy Gan

2018 ACJC BT1 Question 8

A man took a housing loan of $\text{ }\!\!\$\!\!\text { } P$ 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  $\text{ }\!\!\$\!\!\text { }R$ to his loan, leaving a balance of $\text{ }\!\!\$\!\!\text { }\left(\left(\frac{6013}{6000}\right)P-R\right)$ 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.



If he has taken a housing loan of $\text{ }\!\!\$\!\!\text  {  } 300 000$ and he intends to repay his loan in 30 years, find to the nearest cent, the minimum amount of monthly repayment he should pay.



Beginning from 1 January 2001, he begins saving $\text{ }\!\!\$\!\!\text  {  } 50$ every month in the year 2001. For each subsequent year, he saves $\text{ }\!\!\$\!\!\text  {  }50$ more every month than the previous year, so that he saves $\text{ }\!\!\$\!\!\text  {  } 100$ every month in the year 2002, $\text{ }\!\!\$\!\!\text  {  } 150$ every month in the year 2003, and so on. The man has also taken a housing loan of $\text{ }\!\!\$\!\!\text  {   } 300 000$ on 1 January 2000 and his monthly repayment is $\text{ }\!\!\$\!\!\text  {  } 1300$.

(i) What is the outstanding balance of his loan, to the nearest cent, on 31 December 2010 after repayment for that month has been made?


(ii) What is the total amount of money that he has saved by 31 December 2010?


(iii) 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 $\text{ }\!\!\$\!\!\text  {  } 1300 $ on 31 December 2010 and continues to pay  $\text{ }\!\!\$\!\!\text  {  } 1300 $ at the end of every subsequent month.

Determine how many more months would he take to complete repaying his loan.


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Published: 15th August 2022

Written by

Timothy Gan

This is Tim. Tim loves to teach math. Tim seeks to improve his teaching incessantly! Help Tim by telling him how he can do better.

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