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QCE Stuff => QCE Mathematics Subjects => QCE Subjects + Help => QCE General Mathematics => Topic started by: RuiAce on February 20, 2019, 06:11:09 pm

Title: QCE General Maths Questions Thread
Post by: RuiAce on February 20, 2019, 06:11:09 pm
QCE GENERAL MATHS Q&A THREAD

What is this thread for?
If you have general questions about the QCE General Maths course (both Units 1&2 and 3&4) or how to improve in certain areas, this is the place to ask! 👌


Who can/will answer questions?
Everyone is welcome to contribute; even if you're unsure of yourself, providing different perspectives is incredibly valuable.

Please don't be dissuaded by the fact that you haven't finished Year 12, or didn't score as highly as others, or your advice contradicts something else you've seen on this thread, or whatever; none of this disqualifies you from helping others. And if you're worried you do have some sort of misconception, put it out there and someone else can clarify and modify your understanding! 

There'll be a whole bunch of other high-scoring students with their own wealths of wisdom to share with you, so you may even get multiple answers from different people offering their insights - very cool.


To ask a question or make a post, you will first need an ATAR Notes account. You probably already have one, but if you don't, it takes about four seconds to sign up - and completely free!
Title: Re: QCE General Maths Questions Thread
Post by: s110820 on May 14, 2020, 11:42:45 am
Hi ATAR Notes community,

I was just wondering if anyone has any advice on how to memorise formulas for maths? I'm just worried that I'm going to forget about the ones that are not on the formula sheet.

Thank you so much and have a great week!

Darcy Dillon.
Title: Re: QCE General Maths Questions Thread
Post by: Bri MT on May 19, 2020, 11:30:58 am
Hi ATAR Notes community,

I was just wondering if anyone has any advice on how to memorise formulas for maths? I'm just worried that I'm going to forget about the ones that are not on the formula sheet.

Thank you so much and have a great week!

Darcy Dillon.

Hey!

I find that making an effort to understand the formulas and what each part of it does really helps with this :)

hope this helps
Title: Re: QCE General Maths Questions Thread
Post by: s110820 on May 19, 2020, 11:49:17 am
Thanks so much, Bri! I really appreciated your help :)
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on July 25, 2020, 08:37:38 am
I was wondering if someone could help me with two questions I am currently stuck on. Please find them in the two attached images.

Note: I have attempted question 1, I think I got roughly 57 km and just want to see what is right and question 2 my teacher roughly went over, but not directly so the solution is kind of hazy.

Thanks for the help,
Luke

Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on January 06, 2021, 10:20:38 am
Hey,

I was just wondering if someone could help me with a couple of math problems that I have.

Number 1. A person of height 179 cm weighs 82 kg. If the regression equation is used to predict
their weight, weight = −96 + 0.95 × height, with r = 0.79, then the residual will be closest to...
For this I got 8 kg but my textbook is saying it is -3 kg.

Number 2. The rungs of a ladder diminish uniformly in length from 30 cm at the bottom of the
ladder to 22.5 cm at the top of the ladder. There are 16 rungs altogether. The length, in
centimetres, of the 10th rung up the ladder is...
I am pretty sure it is easier to calculate this with an arithmetic series than a sequence, but I am curious if/how it can be done through an arithmetic sequence formula.

Thanks in advanced ;D
Luke
Title: Re: QCE General Maths Questions Thread
Post by: keltingmeith on January 06, 2021, 02:00:03 pm
Hey,

I was just wondering if someone could help me with a couple of math problems that I have.

Number 1. A person of height 179 cm weighs 82 kg. If the regression equation is used to predict
their weight, weight = −96 + 0.95 × height, with r = 0.79, then the residual will be closest to...
For this I got 8 kg but my textbook is saying it is -3 kg.

Number 2. The rungs of a ladder diminish uniformly in length from 30 cm at the bottom of the
ladder to 22.5 cm at the top of the ladder. There are 16 rungs altogether. The length, in
centimetres, of the 10th rung up the ladder is...
I am pretty sure it is easier to calculate this with an arithmetic series than a sequence, but I am curious if/how it can be done through an arithmetic sequence formula.

Thanks in advanced ;D
Luke

Residual=difference between predicted y-value and actual y-value. At a quick glance, it looks like you calculated the residual for height - since the question is about predicting weight from height, you should calculate the residual for weight. (I also feel like they got the answer wrong, or they're using some weird formula I've never seen before)

I mean - the arithmetic series formula is just a special application of the arithmetic sequence formula - it's what you get when you add up the arithmetic sequence formula from your starting number up to n times. So they're not that different, really. Anyway, one approach could be to calculate the series up to 10 then subtract the same series up to 9, as you've suggested. I would instead suggest the following formula:

\[
a_n = a_1 + (n-1)d
\]

This formula predicts what the nth number in an arithmetic sequence will be. So, we're interested in the 10th rung - but we decrease instead of increase, so you need to remember to put a negative d, and we'll end up with:

\[
a_{10}=30+(10-1)(-0.5)=30-4.5=25.5
\]
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on January 30, 2021, 08:29:41 am
Hey ATARNotes Community,

Could someone help me with the question below.

In the following, each pair of locations lie on the same latitude. Find the approximate
shortest distance between them travelling along the small circle.
X (158°15′E, 40°45′N), Y(26°45′E, 40°45′N)

This is what I have done so far and I am not sure where I am going wrong. First, I changed the latitude and longitude to their respective angle measurements, which I got as 40.75oN, 158.25oE, 26.75oE.

Then I calculated the distance with the following formula:
\[
D = 111.2cos() * Angular Distance
\]

Which resulted in:

\[
D = 111.2cos(40.75) * (158.25 - 26.75)
\]
\[
D = 11077.72 km
\]
However, my textbook is saying the answer is 9898 km and I can't figure out how they came to it. Any help will be greatly appreciated.

Kind regards,
Luke
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 03, 2021, 02:12:07 pm
Hey all,

I was just wondering if someone could do this question for me because I don't understand what it is asking me to find. Plus, I have never come across a question like this before.

The answer is D btw

Thanks in advanced,
Luke
Title: Re: QCE General Maths Questions Thread
Post by: fun_jirachi on August 03, 2021, 03:32:55 pm
The question doesn't tell you how often the rate compounds, etc. The only thing you actually know is that every month you pay $1485, for twenty years, which means you make 240 payments of 1485 for a total of 356400. It's also a safe assumption that the starting value will compound to become larger than all the options listed (unless there's been a mistake somewhere, which there isn't).

Hope this helps :)
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 05, 2021, 09:18:26 am
How'd you get $356,400 because 360 X 1485 does not equal 356400?

Thanks,
Luke
Title: Re: QCE General Maths Questions Thread
Post by: fun_jirachi on August 05, 2021, 10:00:21 am
Oops, that is a typo - I'm correct in saying that you make monthly payments for twenty years which should have been 240 payments, rather than 360 as previously said. I'll fix that up. 240 * 1485 = 356400.
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 05, 2021, 10:12:58 am
Hey,

I am confused as to how to find the monthly repayments on reducing balance loans because every question I am doing is always a couple of dollars out. Could someone help me to figure out my issue.

The method I use is I sub in every value to the annuities formula and find M. However, doing it this way gave me $42.62 but the answer is $43.42.

Kind regards,
Luke
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 05, 2021, 10:13:35 am
Oops, that is a typo - I'm correct in saying that you make monthly payments for twenty years which should have been 240 payments, rather than 360 as previously said. I'll fix that up. 240 * 1485 = 356400.

Thanks! makes much more sense now.
Title: Re: QCE General Maths Questions Thread
Post by: fun_jirachi on August 05, 2021, 07:06:46 pm
-snip-

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 :)
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 06, 2021, 08:24:58 am
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.
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 06, 2021, 03:05:07 pm
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
Title: Re: QCE General Maths Questions Thread
Post by: fun_jirachi on August 06, 2021, 03:27:23 pm
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 :)
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 07, 2021, 09:08:31 am
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.
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 07, 2021, 01:59:10 pm
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
Title: Re: QCE General Maths Questions Thread
Post by: fun_jirachi on August 07, 2021, 05:11:33 pm
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).

Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 07, 2021, 06:10:53 pm
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   
Title: Re: QCE General Maths Questions Thread
Post by: fun_jirachi on August 07, 2021, 06:51:52 pm
-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 :)

Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 07, 2021, 07:56:59 pm
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
Title: Re: QCE General Maths Questions Thread
Post by: fun_jirachi on August 07, 2021, 10:36:28 pm
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.
Title: Re: QCE General Maths Questions Thread
Post by: RuiAce on August 08, 2021, 05:29:15 pm
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.
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 10, 2021, 02:46:41 pm
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
Title: Re: QCE General Maths Questions Thread
Post by: RuiAce on August 10, 2021, 05:44:42 pm
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
Title: Re: QCE General Maths Questions Thread
Post by: Luke_8064 on August 11, 2021, 02:47:29 pm
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