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.
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(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.
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