The news about the $2-$pot retirement system which kicked$-$off in the country on $1$ September $2024$ has brought mixed feelings to many people. The system will allocate one$-$third of past and future retirement contributions to a savings component and the remaining two$-$thirds will go into a retirement component. In a panicked state, your $35-$year$-$old co$-$worker wants to know the size of the annual contributions that she must make for the next $25$ years so that she can retire at age $60.$ She expects to live to an age of $80$ and wants to be able to start making an annual withdrawal of R$600000$ from the retirement account on her $61$st birthday through to the $80$th$.$ The account is expected to earn $10\%$ per year for the entire period and the interest will be compounded monthly.

Required:

$3\mathrm{.}1.$ Calculate the monthly contribution that she must make to the retirement fund over the next $25$ years. $(6)$

$3\mathrm{.}2.$Calculate the difference in the monthly contribution if she starts saving only $15$ years before retirement and her life expectancy after retirement increases to $30$ years. The return remains unchanged. $(4)$

$3\mathrm{.}3.$ A retirement planning exercise is useful in that it provides an indication of the level of investment required, of course, based on certain assumptions. It is however a difficult exercise in practice. Name two factors that make it such a difficult exercise. $(2)$

$3\mathrm{.}4.$ When compounding occurs, the effective annual interest rate is greater than the annual percentage rate. Evaluate this statement and substantiate your answer. $(3)$