Beer induced opinion and reckoning follows.
The question:
A debt is repaid with an annuity of semi-annual payments of 1,680.39. If the 5th and 15th principal repayments are 1,047.17 and 1,407.30, respectively, how much is the interest portion of the 10th installment?
A simulation with known quantities might be a good idea.
Take the following amortization schedule where you know that [imath]A = 52,000[/imath] is the loan amount, [imath]R = 4,571.63[/imath] is the periodic payment (monthly in this case), and [imath]i[/imath] is the effective monthly rate.
You know from the table what the principal repaid on the 7th & 10th payment are.
Suppose you don't know the loan amount and the monthly rate.
You can set up a system of equations to solve for [imath]A[/imath] and [imath]i[/imath]; once you have a value for [imath]i[/imath], you can then determine the interest portion of any nth payment.
Software is highly recommended for a somewhat complicated system of equations considering that your course (unless I'm mistaken) is just basic finance mathematics. I think I remember seeing TKHunny (or maybe Denis) tackling something similar a few years back but I can't seem to recall how those two did it back then given my recent fever bout that lasted for several days.
The complicated scenario I have in mind for now goes something like
[imath]R-\left[A(1+i)^{10-1}-R\frac{(1+i)^{10-1}-1}{i}\right]*i=4,459.22[/imath]
and
[imath]R-\left[A(1+i)^{7-1}-R\frac{(1+i)^{7-1}-1}{i}\right]*i=4,349.57[/imath]
where [imath]R = 4,571.63[/imath] and you let a software solve for [imath]A[/imath] and [imath]i[/imath].
Adapting this simulation to your problem should be easy.
