How can changing the value of m affect the percentage p?

[Astro]

Easy to understand problem (please read; I promise you'll understand):

I've created my own financial spreadsheet. One of its functions is to allow me to automatically determine how much money I need to save every month m in order to accumulate x dollars in y amount of time.

Normally, the monthly amount m is calculated by:
1. Dividing the wanted total by the number of months n (n is anywhere from 1-12). (In the case where accumulating y would take greater than 12 months then the spreadsheet simply multiplies the annual accumulated amount by 1, 2, 3, etc. years respectively to show what the accumulated amount would be over a greater length of time.) Note that: The amount saved per month is a constant.
2. Calculating what percentage pof my gross monthly income m is. My wage rate and weekly hours worked are constants; thus p=m/(gross monthly income)*100%.
3. The final value of m is calculated thusly:
m=p*(gross monthly income) .
(Note: You might ask, 'why go through all that trouble to calculate m if that was already determined in step#1? The answer is simple: In the reality of life, my monthly income fluctuates and is not a perfect constant. One of my main objectives in setting up a budget was to determine a way to divide and distribute income to different items based on their priority. Using a relative system (ie. percentage based) to divide income rater than an absolute system (ie. constant dollar values) made it possible to adapt the budget to the realities of life.)

For each item that I save money for, I have the categories "savings goal" and "current savings".


Now, here is where I start to get confused:
I thought to myself, what if I setup the calculation of m so that instead of it being a flat rate it were to change depending on the value of the "current savings" for the item in question?

So now, m is calculated like this:

1. m=\frac{(savings goal)-(current savings)}{n}
2. p=m/(gross monthly income)*100% (Note: this step is unchanged.)
3. m=p*(gross monthly income) (Note: this step is also unchanged.)

You will note that the values of m and p are now in flux and change with each iteration. You will also note that the initial values of m and p are the same, regardless of whether m and p are constant or variable. Logically, unless the initial rate of p is higher in the case of variable m's and p's then I don't see how the "savings goal" could be meet in the time allotted, and yet, on the otherhand it seems that it would work in practice. I'm missing/not-understanding something here. Can someone please help me out and explain to me what I'm not quite grasping?

(Note: In my effort to answer this for myself, I tried to derive a formula so that I could graph the variable value of m and be able to compare it to when it was a constant. Well, that failed tragically as I ended up with a paradox for where the initial value of m equals itself divided by 100 [ie. m=p*(gross monthly income) ]. Obviously, things like that do not make sense. I'm pretty sure my calculations are accurate so, once again, I believe my underlying problem in one of logic rather than mathematical adeptness. Anyway...help please? Someone? :3 )

 

[dodo]

Honesty, I'm not sure I understand the question.

In the following example, given the table

G = Goal = 12000
N = Total months to goal = 12

E = Months elapsed    0     1     2     3     4     5     6     7     8     9    10    11    12
R = Months remaining  12    11    10    9     8     7     6     5     4     3    2     1     0

(first case) G/N      1000  1000  1000  1000  1000  1000  1000  1000  1000  1000 1000  1000  1000
A = Accum. savings    0     1000  2000  3000  4000  5000  6000  7000  8000  9000 10000 11000 12000

(second case) (G-A)/R 1000  1000  1000  1000  1000  1000  1000  1000  1000  1000 1000  1000  1000
A = Accum. savings    0     1000  2000  3000  4000  5000  6000  7000  8000  9000 10000 11000 12000

Is your question, "why the numbers look the same in both cases, if the step formula is different"?

 

[CRGreathouse]

I made a quick Excel sheet for you... does this help?

 

[Astro]

Hello,

I want to thank you both for trying to help. :) However, you didn't quite grasp what I'm trying to say. I will therefore try harder to explain.

So, to help you understand what I'm saying, I've included pictures and matching explanations. My question will hopefully make sense to you after you take a look.

Please see attached ZIP file.

 

[dodo]

(Sounds like Xeno's paradox - that's why I was interested in the first place.)

I still have a question. In the variable-amount case, do you still divide by a fixed 11 (months), in your example? I think you shouldn't; as time passes and the remaining amount decreases, you need to divide by the remaining months, in my opinion. (In which case the situation is as depicted in post#2.)

 

[CRGreathouse]

I'm still having trouble understanding.

Does your paradox still manifest itself with my quick-n-dirty spreadsheet?

 

[Astro]

(Sounds like Xeno's paradox - that's why I was interested in the first place.)

I still have a question. In the variable-amount case, do you still divide by a fixed 11 (months), in your example? I think you shouldn't; as time passes and the remaining amount decreases, you need to divide by the remaining months, in my opinion. (In which case the situation is as depicted in post#2.)

You're absolutely right! It makes sense now; I was keeping the number of months constant when they should have been variable. Thank you!

I looked up Xeno's paradox and it sounds like fascinating reading! While on the one hand I'm relieved that my question was resolved, I regret that it was not on the same league as Xeno's paradox since it sounds that you would have relished the opportunity to tackle a truly challenging problem. Perhaps one day I'll be able to live up to the challenge. ^_^

Dodo:
Thank you for making your quick-n-dirty spreadsheet. Having looked at it I do not believe the problem manifests in your spreadsheet. If I understand your calculations correctly it seems you thought I was performing interest-rate calculations which I wasn't. It's a useful spreadsheet (and elegantly programed) but it's purpose is not the same as I was describing and so not really comparable. The percentages I'm talking about are not interest rates; they are proportional rates. If you take a careful look at the ZIP file I attached in my previous post I think you will understand the gist of what I'm saying.

Thank you all for your help!

 

[CRGreathouse]

Dodo:
Thank you for making your quick-n-dirty spreadsheet. Having looked at it I do not believe the problem manifests in your spreadsheet. If I understand your calculations correctly it seems you thought I was performing interest-rate calculations which I wasn't. It's a useful spreadsheet (and elegantly programed) but it's purpose is not the same as I was describing and so not really comparable. The percentages I'm talking about are not interest rates; they are proportional rates. If you take a careful look at the ZIP file I attached in my previous post I think you will understand the gist of what I'm saying.

(I'm not Dodo, but I *am* the one who wrote and posted the spreadsheet.)

If you look at cell B8, it has the percentage you are talking about. The interest rate (cell B6) is actually 0%; I put that in there just in case you later wanted to figure that.