Hey again,

Just want to see how you solve these two questions because they don't align with what I've been taught.

The answer for Q11 is C and the answer for Q12 is D.

Thanks again,

Luke

Your \(A + nR - P\) formula for Q11 rings a bell, but for some reason I feel like it's hitting something a bit off the mark. I can't quite recall what exactly it is, so I'd have to see some context for its usage (like say, a sample question).

I'm highly convinced Q12 is just a wrong question. I will note that I obtained your answer of \(-\$1068.49\) via

**both** an annuity formula, and by manually repeating the recurrence.

Mathematically, the question is wrong because clearly you now have a negative balance. Furthermore, after the

**fifth** payment (i.e. after only 2.5 years), the balance is already negative. But intuitively, the question doesn't make sense either. Try to think about it intuitively like this. Your debt is only $5000. Since you're only gonna pay 4.5% per annum, which becomes 2.25% per half-annum, your first interest charge will be \(\$5000 \times 0.0225 = \$112.50\). That's hardly any interest, when you're paying a whopping $1068.50 every six months. Your balance is getting reduced by something roughly $900.

With each subsequent payment, your owing debt just gets smaller and smaller, and you get charged progressively

**less** interest. So you're gonna get your balance reduced more rapidly, as time goes by. If your debt goes down by $900 (which is remarkably close to $1000) each 0.5 years, then intuitively you'd expect that around 2.5 years you would have a debt very close to $0.

So no way would it take a full 5 years to settle the debt. At most 3 years should've been enough intuitively. (Mathematically, it turns out 2.5 years is enough.)

Something in the question has to therefore be off. My first instinct was "what if one of those values was not correctly converted"?

- Attempt 1: Starting balance was $10000, not $5000. But this doubling of the opening balance definitely makes no sense, because you suddenly have to pay a lot more. Indeed, this did not remedy things.

- Attempt 2: Given payment was the (net) amount paid in the year; therefore the half-annum payment is actually $534.25. This yields a closer balance (after 3 years) of $2322.82, but is still pretty far off.

- Attempt 3: The 4.5% interest rate is actually a half-annum rate (meaning the p.a. rate was 9%). But this increased interest rate still isn't high enough, and we end up with the negative balance -$665.70 after 3 years.

- Attempt 4: Everything was actually just annual; nothing was half annum. Yearly payments of $1068.5, yearly compounding at 4.5% p.a.. This gives $2353.91; again quite close, but not close enough.

Therefore all my attempts at "fixing" the question failed either. So as it stands, I have no clue where they generated their values from.