Mary has a monthly income of $\$4000$. She is considering applying for a car loan of $\$40,000$ for

Question

Mary has a monthly income of $\$4000$. She is considering applying for a car loan of $\$40,000$ for $6$ years which charges an interest rate of $3.00\%$ per annum, compounded monthly. Interest is chargeable immediately when the loan sum is drawn out. The monthly payment, $\$m$, is fixed throughout the loan tenure. Show that the calculated loan balance at the end of the ${{n}^{\mathrm{th}}}$ loan month, after the monthly repayment is made, is given by $40000{{\left( \frac{401}{400} \right)}^{n}}-400m\left[ {{\left( \frac{401}{400} \right)}^{n}}-1 \right]$. [3] By legislation, banks can approve a car loan only if the monthly repayment does not exceed $15\%$ of an applicant’s monthly income. Prove that Mary will not be able to apply for the car loan. [3] If the interest rate for all car loans by the banks is compounded monthly, find the range of interest rates chargeable which will enable Mary to apply for the car loan successfully. Give your answer in the form of $r\%$ per annum, correct to $1$ decimal place. [3]

Accepted Answer

(i) Balance = 40000(401/400)^n − 400m[(401/400)^n − 1]; (ii) Least m = $607.75, Mary cannot apply; (iii) 0% < r% ≤ 2.5%

2013 AJC Promo Q12

2013 AJC Promo Q12

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