7.0% actually 3.69%?

Bridlington1

OceanSound said:

AmityNeon said:

RG2015 said:

I was actually intrigued how @AmityNeon derived it (or any other pertinent formula) from the GCSE maths syllabus.

mrn(n+1)/24 + m(r+x)(12-n)(12-n+1)/24 + m(r-y)n(12-n)/12 > mr6.5

We're familiar with m * r * 6.5, which is based on consecutively adding incremental monthly interest contributions (up to month 12). It represents the last part (right side) of the formula, of which the left side needs to be greater than the right, in order for closing/renewing our RS to be considered worthwhile.

For the left side, we need a similar formula but for a variable number of months, n.

• 1 month + 2 months + 3 months + 4 months + 5 months...
• = 1 + 2 + 3 + 4 + 5...
• = 1, 3, 6, 10, 15...

The formula for finding the value for the nth month is n(n + 1) / 2. You can visualise this intuitively like a triangle being doubled to a rectangle:

n o o o o o
n n o o o o
n n n o o o
n n n n o o
n n n n n o

In the fifth month, there are five rows of <b>n</b>, and inversely there are also five rows of <b>o</b>, but it's a rectangle as there are <b>n + 1</b> columns. So however many months we've held our RS for, the formula <b>n(n + 1) / 2</b> calculates how many monthly contributions of interest we've accrued. At 12 months: <b>12 * 13 / 2 = 78</b>.

We know interest is calculated daily, but we approximate using monthly interest, so the annual interest rate is divided by 12. 78 monthly contributions earning 1/12 of the annual interest rate = m * (r / 12) * 78 = m * r * 6.5.

The two formulas are then merged. Instead of calculating 78, we leave the formula for n as it is, because the number of months we've held our RS for is variable (and less than 12 for this purpose).

m * (r / 12) * n(n + 1) / 2

mrn(n + 1) / 24

Dividing by 2 and then dividing by 12 is the same as dividing by 24.

So that's the first part of the left side figured out; it calculates the amount of interest accrued in our current RS. Then we close/renew...

When we restart our RS, we calculate its length up to the end of month 12 of the old RS. So if we closed our old RS at 5 months, we need to calculate the amount of interest accrued in the new RS for 7 months. This ensures that the total amount of monthly contributions earning interest on both sides of the formula is equal to 78 over 12 months (for a fair mathematical comparison).

n
n n
n n n
n n n n
n n n n n

e e e e e   S
e e e e e   S S
e e e e e   S S S
e e e e e   S S S S
e e e e e   S S S S S
e e e e e   S S S S S S
e e e e e   S S S S S S S

We can see that the 78 monthly contributions consist of three parts:

• The original 5 months, represented by the upper n triangle.
• The next 7 months, represented by the lower-right S triangle.
• The 'excess' 5 months of contributions moved into an easy-access saver, represented by the e rectangle.

We already know the formula to calculate interest over a variable number of months: mrn(n+1)/24

What's different about the S triangle? It has an increased rate of interest, so we use r + x, where x is the increase over r. The number of months is also 12 - n. We plug those values in to achieve the middle part of the left side:

m(r + x)(12 - n)(12 - n + 1) / 24

Calculating the interest for the e rectangle means we do not divide by 24, but by 12. We can visually see the dimensions of the rectangle: n * (12 - n). The reduced rate of interest is expressed as r - y. This achieves the last part of the left side and completes the formula:

m(r - y)n(12 - n) / 12

Part 1            Part 2                        Part 3
mrn(n+1)/24   +   m(r+x)(12-n)(12-n+1)/24   +   m(r-y)n(12-n)/12   >   mr6.5

The full formula can be entered into a spreadsheet for easy calculation.

It can be simplified if desired to the variants below, but that requires a stronger grasp of expanding, factoring and rearranging.

m * ((n − 12) * ((n − 13) * x + (2 * n * y)) + (156 * r)) / 24 > mr6.5

We can eliminate m and r as they're both constant on both sides of the comparison (although we lose actual interest figures as a result):

(n − 12) * ((n − 13) * x + (2 * n * y)) > 0

Please explain this bit:

"..the formula n(n + 1) / 2 calculates how many monthly contributions of interest we've accrued. At 12 months: <b>12 * 13 / 2 = 78</b>..."

At 12 months we've accrued 78 monthly contributions of interest? I can't get my head round this.

I think this is based on the idea that each £300 earning 1 month's worth of interest.

In month 1 you have 1 lot of £300 earning interest
In month 2 you have 2 lots of £300 earning interest
...
In month 12 you have 12 lots of £300 earning interest

Summing these you get 1+2+3+...+11+12 lots of £300 earning interest over the entire year, which comes to 78 lots of interest in total.

n(n+1)/2 is just a quick way of summing the first n natural numbers. In this case we have 12 months so n=12. Plugging this in to the formula yields 12*13/2=78 as required.

AmityNeon

Eco_Miser said:

You keep talking about 'logic'. Which logic? Aristotelian? Boolean? Something else?

You are aware that there's a flaw in the logical basis of mathematics?

Is the set of all sets that are not members of themselves a member of itself?
If it is, it isn't. If it isn't it is.

Is that not an example of logic itself revealing those contradictions and paradoxes, so such inconsistencies can be avoided? It's like common reasoning fallacies and misconceptions, which are a result of faulty logic, whereas a mathematical proof like Fermat's Last Theorem will never be 'disproven'.

Sea_Shell

I haven't read the last 15 pages...wow, still going...shall I come back in another 15? 
😉

Nebulous2

AmityNeon said:

Eco_Miser said:

You keep talking about 'logic'. Which logic? Aristotelian? Boolean? Something else?

You are aware that there's a flaw in the logical basis of mathematics?

Is the set of all sets that are not members of themselves a member of itself?

If it is, it isn't. If it isn't it is.

Is that not an example of logic itself revealing those contractions and paradoxes, so such inconsistencies can be avoided? It's like common reasoning fallacies and misconceptions, which are a result of faulty logic, whereas a mathematical proof like Fermat's Last Theorem will never be 'disproven'.

I've skipped much of this, and probably wouldn't understand it anyway, but I'm reminded of Harold Macmillan's comment about Enoch Powell. 

"Poor Enoch, driven mad by the remorselessness of his own logic."

Bridlington1

Sea_Shell said:

I haven't read the last 15 pages...wow, still going...shall I come back in another 15? 
😉

It's been quite a thread hasn't it?

Eco_Miser

OceanSound said:

Eco_Miser said:

AmityNeon said:

zagfles said:

AmityNeon said:

zagfles said:

AmityNeon said:

zagfles said:
You seem to be saying people don't need teaching maths or be shown mathematical techniques because they should be able to work it all out themselves.

Really? Where did I say that?

"It's not a case of they 'should' be able to solve everyday maths problems with innate logic; they can do it. "

Can do it (not 'should'); the context being adults are already educated in basic arithmetic and algebra due to compulsory maths education.

So not "innate logic", but what you've been taught! You seem to agree with me now!

(PS my last final word on this issue otherwise we'll end up going round in circles and some people may find it booooring )

Surely you understand how logic is integral to the foundation and application of maths? The ‘rules of math’ were not derived arbitrarily; it’s a formal system where logic is used to develop these ‘techniques’. One has to apply their learned knowledge with logic to solve new problems. It’s the difference between blindly following instructions using a calculator/spreadsheet and not understanding the result, and being able to decide for yourself which functions to use and why they’re appropriate for the task at hand. These decisions are all made with logic.

We also did use spreadsheets to visualise interest being accrued at discrete intervals with specific examples and daily interest, but it was useful having a generic formula that everyone could use (in the absence of a calculator).

You keep talking about 'logic'. Which logic? Aristotelian? Boolean? Something else?

You are aware that there's a flaw in the logical basis of mathematics?

Is the set of all sets that are not members of themselves a member of itself?

If it is, it isn't. If it isn't it is.

Definitely not boolean. That's to do with 1's and 0's. digital equipment like computers work according to boolean logic. 

Anyone who studies/studied electronics can confirm.

Wasn't Russell's paradox solved by Zermelo, Franekel, and Skolem (ZFC)?

Boolean logic is about the truth or falsity of assertions.

Very conveniently, that can be applied to the 0s and 1s of binary, represented in computers by assorted binary states, such as high (>3V) or low voltage, current or no current, north pole or south pole, hole or no hole, reflecting or not.

Zermelo et al sidestepped Russell's paradox by redefining the axioms of set theory.  That's rather like changing the rules of a game when you're losing.

OceanSound

Eco_Miser said:

OceanSound said:

Eco_Miser said:

AmityNeon said:

zagfles said:

AmityNeon said:

zagfles said:

AmityNeon said:

zagfles said:
You seem to be saying people don't need teaching maths or be shown mathematical techniques because they should be able to work it all out themselves.

Really? Where did I say that?

"It's not a case of they 'should' be able to solve everyday maths problems with innate logic; they can do it. "

Can do it (not 'should'); the context being adults are already educated in basic arithmetic and algebra due to compulsory maths education.

So not "innate logic", but what you've been taught! You seem to agree with me now!

(PS my last final word on this issue otherwise we'll end up going round in circles and some people may find it booooring )

Surely you understand how logic is integral to the foundation and application of maths? The ‘rules of math’ were not derived arbitrarily; it’s a formal system where logic is used to develop these ‘techniques’. One has to apply their learned knowledge with logic to solve new problems. It’s the difference between blindly following instructions using a calculator/spreadsheet and not understanding the result, and being able to decide for yourself which functions to use and why they’re appropriate for the task at hand. These decisions are all made with logic.

We also did use spreadsheets to visualise interest being accrued at discrete intervals with specific examples and daily interest, but it was useful having a generic formula that everyone could use (in the absence of a calculator).

You keep talking about 'logic'. Which logic? Aristotelian? Boolean? Something else?

You are aware that there's a flaw in the logical basis of mathematics?

Is the set of all sets that are not members of themselves a member of itself?

If it is, it isn't. If it isn't it is.

Definitely not boolean. That's to do with 1's and 0's. digital equipment like computers work according to boolean logic. 

Anyone who studies/studied electronics can confirm.

Wasn't Russell's paradox solved by Zermelo, Franekel, and Skolem (ZFC)?

Boolean logic is about the truth or falsity of assertions.

Very conveniently, that can be applied to the 0s and 1s of binary, represented in computers by assorted binary states, such as high (>3V) or low voltage, current or no current, north pole or south pole, hole or no hole, reflecting or not.

Zermelo et al sidestepped Russell's paradox by redefining the axioms of set theory.  That's rather like changing the rules of a game when you're losing.

So,  do you think boolean logic applies here? 

Zermelo et al sidestepped Russell's paradox by redefining the axioms of set theory.

Didn't Russell himself try to alter the logical language in order to solve?

 That's rather like changing the rules of a game when you're losing.

Is it?  Oh darn! These water pumps caused a cholera epidemic. We've tried everything to solve, we can try removing the handles from the pumps, but hey that'll be changing  the rules of a game when we're losing!