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May 20, 2022, 08:22:00 am

Author Topic: QCE General Maths Questions Thread  (Read 9499 times)

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Luke_8064

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Re: QCE General Maths Questions Thread
« Reply #15 on: August 06, 2021, 08:24:58 am »
+1
Using the formula, we have that \(500 = M \times \frac{1-(1+0.042)^{-1}}{0.042}\). This implies that the amount of money we have to pay is \(M = $521\). Since we have to make monthly payments, each payment is just \(\frac{521}{12} \approx $43.42\).

Hope this makes sense :)

Thanks! The thing I was doing wrong was making n 12 because every time we've done these types of questions the interest rate compounds the same amount of times as the repayment amount. If I got more then 6 hours of sleep yesterday, I would've easily seen this trick XD.
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Luke_8064

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Re: QCE General Maths Questions Thread
« Reply #16 on: August 06, 2021, 03:05:07 pm »
0
Back at it again with another question!

Yeah, the wording on this question is throwing me off and I don't really know what it is asking me to do. Would appreciate it if you could explain what this question is asking me to find (I am assuming A in the annuities formula but I keep getting $20,693) or just show me how to do it.

ANSWER IS D btw

Kind regards,
Luke
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fun_jirachi

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Re: QCE General Maths Questions Thread
« Reply #17 on: August 06, 2021, 03:27:23 pm »
+3
If the rate is 5%pa, and we compound it half-yearly, the relevant rate that we should be considering is 2.5%/6 months. There are 12 periods, each of 6 months for a total of six years.

Then, we have that \(A = 1500 \times \frac{1-1.025^{-12}}{0.025}\).

Hope this helps :)
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Luke_8064

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Re: QCE General Maths Questions Thread
« Reply #18 on: August 07, 2021, 09:08:31 am »
+1
If the rate is 5%pa, and we compound it half-yearly, the relevant rate that we should be considering is 2.5%/6 months. There are 12 periods, each of 6 months for a total of six years.

Then, we have that \(A = 1500 \times \frac{1-1.025^{-12}}{0.025}\).

Hope this helps :)

Huh... that's weird because I swear I tried this yesterday and got a different answer. What's also annoying about this question is it says invested, which made me keep using the other formula for Annuities as it sounded like this problem was an investment rather than a loan. 

Thanks again for your help! I really appreciate it.
« Last Edit: August 07, 2021, 09:12:40 am by Luke_8064 »
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Luke_8064

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Re: QCE General Maths Questions Thread
« Reply #19 on: August 07, 2021, 01:59:10 pm »
0
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
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fun_jirachi

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Re: QCE General Maths Questions Thread
« Reply #20 on: August 07, 2021, 05:11:33 pm »
+1
What have you been taught? I'm a bit more curious as to what your thought processes are with these questions.

For the first one: there's a lot of irrelevant information you can ignore (consider why this is the case)
For the second one: a bit of a tougher question, so I want to see what you've tried first. Check also for typos in the question, something seems a bit off to me (unless it's just me not being able to read).

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Luke_8064

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Re: QCE General Maths Questions Thread
« Reply #21 on: August 07, 2021, 06:10:53 pm »
+1
What have you been taught? I'm a bit more curious as to what your thought processes are with these questions.

For the first one: there's a lot of irrelevant information you can ignore (consider why this is the case)
For the second one: a bit of a tougher question, so I want to see what you've tried first. Check also for typos in the question, something seems a bit off to me (unless it's just me not being able to read).

So when it comes to reducing balance loans, I have been taught there are 4 ways to solve them. First, is by using a payment schedule where you go through each repayment period and calculate the interest rate, the interest paid, the withdrawal amount, and the principle reduction (or increase if it is an investment). The second way I have been taught is to use recurrence relations, which is:

Quote
A0 = the principle, An+1 = r x An - R
With r being 1 + the interest rate (in decimal form), R = repayment amount

The third method I have been taught is to use the annuities formula. In this case,  I used the one with -n in it.

The final method I have been told is to use a formula that combines the compound interest formula with the annuities formula, which looks like this:

Quote
A = P(1+i)n-M([(1+i)n-1] divided by i)

So, for Question 11, I initially was trying to find A (in both the annuities and the formula above) but this wasn't going anywhere. So, I randomly tried 2215.42 \times 100 (because of the compounding periods) - 120000 and this gave me the answer. But, I don't know why because I have been taught to find the total interest you need to do this formula:
Quote
A + n x R - P

I did try this formula but I kept getting weird A values (like a negative or a value in the millions).

But, for question 12, I tried both the mixed compound interest and annuities formula and just the annuities formula by itself and I got like $-536.604 (with the mixed formula) and $89042.82 (with the annuities formula). I even tried just doing the recurrence relations method but the principle was just decreasing too fast for it to be even close to $2136.07 (I got -1068.49 somehow).

The main thing that trips me up, other than the formula issue, is I don't know what to do to the interest rate when it says adjusted half-yearly. If it said compounding half-yearly, I'll immediately divide the interest rate by 200 (as this transforms the nominal rate to the compounding rate). I did this with both the questions, but it didn't work, which led me to believe that you do something different to the interest when it says adjusted.

Up until this Jacaranda worksheet, I was completely confident with reducing balance loans and annuities. But them labeling questions like these as simple familiar is making me worried as I don't feel as though I have covered these types of questions. The questions I normally get are something like the one attached.

Hopefully I am making sense,
Luke   
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fun_jirachi

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Re: QCE General Maths Questions Thread
« Reply #22 on: August 07, 2021, 06:51:52 pm »
+1
-snip-

Good to see you elaborating on your thinking :) - this really helps. Shows that you're taking away a lot and just getting tripped up on things you haven't come across

Q11 - the reason the thing you tried works is that you are told that you can pay the money off with that particular amount every so often. This implies that the difference between the money you were loaned originally and the money you actually paid back is interest (it can't possibly be anything else).

Q12 - Usually the questions are the same but with wackier wording. I think at some point, ask your teacher (who I guarantee will be more familiar with your syllabus/study design/whatever you call it in QLD) if something like this is likely to appear and how they would approach it. Intuitively, the question seems wrong (and it appears you agree) (textbooks aren't unsusceptible, I've seen many riddled with typos). I haven't personally come across adjusted rates, but judging by context it just means that you're given an interest rate every year, but you need to adjust it to the time period stipulated. Again, if these are supposed to be familiar, I would check with your teacher :)

Apologies if this isn't the most helpful or if it generally brushes away your question (I'm not familiar with this course and subject, I'm working mostly off context here). In any case, definitely take away the response for Q11 (a lot of questions will misdirect you with extra information, but no question will give you incorrect information apart from admitted typos). Hope this helps in some shape or form :)

« Last Edit: August 07, 2021, 07:39:00 pm by fun_jirachi »
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Luke_8064

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Re: QCE General Maths Questions Thread
« Reply #23 on: August 07, 2021, 07:56:59 pm »
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Heya,

I can tell from the footer of your account that you didn't do the equivalent of general in HSC (I am assuming) so I completely understand that you are not familiar with it. I'm still super grateful for all the support you are providing me. Also, I just read the whole answering a good question thread and I apologise for not showing my working and thus potientially coming off as needy. For future posts (and I'll probs go back and add working to the Q12 problem), I'll try to make sure to show working.

Thanks for everything again,
Luke
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fun_jirachi

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Re: QCE General Maths Questions Thread
« Reply #24 on: August 07, 2021, 10:36:28 pm »
+2
Heya,

I can tell from the footer of your account that you didn't do the equivalent of general in HSC (I am assuming) so I completely understand that you are not familiar with it. I'm still super grateful for all the support you are providing me. Also, I just read the whole answering a good question thread and I apologise for not showing my working and thus potientially coming off as needy. For future posts (and I'll probs go back and add working to the Q12 problem), I'll try to make sure to show working.

Thanks for everything again,
Luke

There's nothing to worry or be sorry about :D (in general most people, myself included will be more lenient if you haven't asked that many questions). It's more when you ask more questions we want to a) check you're actually taking something tangible out of the answers provided and b) know exactly where you're stuck. Context is always important and it's frustrating for everyone involved if it should be there but isn't; I'm okay with dealing with a lack of it every once in a while but not too often (which is why I apply this 'rule' if a user has asked more than a reasonable number of questions). You've got nothing to worry about, you've provided all the context required and shown work when prompted, don't stress :D.
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RuiAce

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Re: QCE General Maths Questions Thread
« Reply #25 on: August 08, 2021, 05:29:15 pm »
+5
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.

Luke_8064

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Re: QCE General Maths Questions Thread
« Reply #26 on: August 10, 2021, 02:46:41 pm »
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Hey RuiAce,

Thanks for your explanation, it helped a bunch. Attached to this reply is an example question that uses the interest formula I was talking about for Q11, just in case you wanted to see if Q11 could be done through the formula. I might've not got the right answer because I rearranged the formula a bit, but I don't see this as being a problem.

Kind regards,
Luke
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RuiAce

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Re: QCE General Maths Questions Thread
« Reply #27 on: August 10, 2021, 05:44:42 pm »
+2
Actually, now that I see the formula used in a question it does make sense to me. I also got it to work; not too sure how you obtained your values.

With \(n=4\times 25 = 100\), \(R = 2215.42\), \(A_0 = 125000\), and \(A_n = 0\), I compute:
\[ nR - (A_0 - A_n) = 100\times 2215.42 - 120000 = 101542. \]
-Rui

Luke_8064

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Re: QCE General Maths Questions Thread
« Reply #28 on: August 11, 2021, 02:47:29 pm »
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Actually, now that I see the formula used in a question it does make sense to me. I also got it to work; not too sure how you obtained your values.

With \(n=4\times 25 = 100\), \(R = 2215.42\), \(A_0 = 125000\), and \(A_n = 0\), I compute:
\[ nR - (A_0 - A_n) = 100\times 2215.42 - 120000 = 101542. \]
-Rui

I see where I went wrong. I thought, for some reason, the loan would still continue after 25 years, despite it clearly saying it would be fully repaid.

Thanks for your help,
Luke
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