# 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. [3] (a) 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.

[2]

(b)

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?

[1]

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

[2]

(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.

[2]

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