2018 SAJC P1 Q10

Albert and Betty each took a study loan of \(\$100\,000\) from a bank on 1 January 2014 and both graduated on 31 December 2017. The bank only starts charging an annual interest rate of \(5\%\) on the outstanding loan at the end of each year from 2018 onwards. Albert pays the bank \(\$x\) on the \(11^{\mathrm{th}}\) day of every month starting in January 2018. Let \(n\) be the number of years after 2017 that repayment of the study loan has begun.

1. Find an expression for the amount of outstanding loan at the end of \(n\) years.[4]
2. Find the minimum value of \(x\) if Albert wishes to complete his repayment of the loan at the end of 2027, giving your answer to the nearest dollar.[2]

An investment fund pays out a constant \(r\%\) dividend per annum on 31 December every year based on the amount of funds held to maturity from 1 January till 30 December of the same year.

On 1 January 2017, Betty decided to invest \(\$50\,000\) in the fund to finance her repayment of her study loan using the annual dividend payout. Her repayment is once per year which she uses the full amount of the annual dividend payout. The schedule of repayment is fixed on 11 January of each year, starting with effect from 2018.

Find the minimum value of \(r\) such that Betty will be able to complete her repayment of the loan at the end of 2027.[5]

[Note that the principal sum of Betty’s investment remains unchanged at \(\$50\,000\) throughout the repayment period.]