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Math Help - Money Problem

  1. #1
    Senior Member sakonpure6's Avatar
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    Money Problem

    Jack's parents have been paying $450 per month into a retirement fund for the last 30 years. The fund is now worth $450 000. What annual interest rate, compounded monthly are Jack's parents earning? Round to two decimal places.

    Please any one, how do you solve this?
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  2. #2
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    Re: Money Problem

    Hello, sakonpure6!

    Where did this question come from?
    It cannot be solved by elementary means.


    Jack's parents have been paying $450 per month into a retirement fund for the last 30 years.
    The fund is now worth $450,000. .What annual interest rate, compounded monthly,
    are Jack's parents earning? .Round to two decimal places.

    This is an annuity.
    The formula is: . A \;=\;D\,\frac{(1+i)^n-1}{i}

    . . where: . \begin{Bmatrix} A &=& \text{final balance} \\ D &=& \text{periodic deposit} \\ i &=& \text{periodic interest rate} \\ n &=& \text{number of periods} \end{Bmatrix}


    We have: . A \,=\,450,\!000;\;D \,=\,450;\;n \,=\,360

    We wish to solve for i, the monthly interest rate.

    Substitute: . 45,\!000 \;=\;450\,\frac{(1+i)^{360} - 1}{i}

    . . \frac{(1+i)^{360}-1}{i} \:=\:100 \quad\Rightarrow\quad (1+i)^{360}-1 \:=\:100i

    . . (1+i)^{360} - 100i - 1 \:=\:0


    Now you must solve a 360-degree polynomial equation.

    Good luck!
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  3. #3
    Junior Member Barioth's Avatar
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    Re: Money Problem

    I'm not so sure about this one.

    When I was younger I had a sweet formula for these problem, however I cant remeber it...
    I'll guess that 5% interest calculated monthly mean that after 1 month you have 105% of what you first had. If not.... well... =)

    Now to find back the formula let's try this:

    A_n = \frac{100+x}{100}*A_{n-1}+450

    we know that A_{0}=450, we'll use n as the nomber of month that have passed. in 30 years there is 360 month. So we know that A_{360}=450000

    we need to find the recurence equation solution that goes with A_n

    so.. there I cheated I asked wolfram (I'm not that good with recurence equation solution... You can give it a try on your side maybe you're better than me at it!)

    anyway here it is..

    A_n=\frac{100(((c_{1}+450)x+45000)(\frac{x}{100}+1  )^{n}-450(x+100))}{x(x+100)}

    You know the value for n=0 and n=360. so you could put them in there and find the value of c and x.

    I think this MAY work

    If you've not seen recurrence equation yet, I don't know how to solve it without them. =(

    edit: Soroban solution seem much more sweet!
    Last edited by Barioth; January 17th 2013 at 07:27 PM.
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  4. #4
    Senior Member sakonpure6's Avatar
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    Re: Money Problem

    alright I will try it both ways , thank you very much!
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  5. #5
    Junior Member Barioth's Avatar
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    Re: Money Problem

    I have no idea how you could solve a 360 degre polynomial equation. I know that wolfram doesn't have enough computer time for free users and I dont have maple/mathlab/etc.. on this computer..


    Altough I guess you could try brute forcing it. since they want the answer rounded to 2 decimal. it wouldn't be too long to find an answer that fit there.

    so I did take 10 min to write a code in C# and took a minute to find a solution that worked.

    the answer should be
    Spoiler:
    The interest should be 0.49804


    I added the code below its very basic in C#.
    Code:
    namespace MoneyProblem{
        class Program
        {
            const int NUMBER_MONTH = 360
            const float MONTLY_PAYMENT = 450f;
            const float THE_NUMBER_WE_LOOKING_FOR= 0.49804f;
    
    
            static void Main(string[] args)
            {
                float[] Tableau = new float[NUMBER_MONTH];
                
                Tableau[0] = 450f;
                
                for (int i = 0; i < NUMBER_MONTH-1; i++)
                {
                    Tableau[i + 1] = FonctionRecursive(Tableau[i]);
                }
                Console.WriteLine(Tableau[NUMBER_MONTH]);
            }
    
    
            static float FonctionRecursive(float val)
            {
                float Reponse;
                Reponse =((100f+THE_NUMBER_WE_LOOKING_FOR)/100f)*val+MONTLY_PAYMENT;
                return Reponse;
            }
        }
    }
    edit: I substitued my value in soroban equation and it does work!
    Last edited by Barioth; January 17th 2013 at 08:21 PM.
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  6. #6
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    Re: Money Problem

    An easy way to find i in this problem is to change monthly payments to yearly.Then you look at regular 100 year interest tables for a sinking fund factor R/S=5400/45000=0.012 and find that 6% table has a SFF of 0.01265 at 30 years.Use of the standard formula given by Soroban can then be used to find monthly interest to three significant figures (0.498) Annual (5.98)using successive approximations
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