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Math Help - geometrical progression

  1. #1
    Senior Member furor celtica's Avatar
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    geometrical progression

    im not sure where to post this
    so
    a bank loan of 500 pounds is arranged to be repaid in two years by equal monthly instalments. Interest, CALCULATED MONTHLY, is charged at 11% per annum on the remaining debt. Calculate the monthly repayment if the first repayment is to be made one month after the loan is granted.

    I've been working for almost two hours on this problem, ending up with the equation (500-m)(1211/1200)^24 - (1200m/11) ((1211/1200)^24 -1)
    m being the equal monthly repayment
    from there i ended up with m= 21.something, apparently not far from the truth but nevertheless incorrect; can somebody show me how they would do this?
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  2. #2
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    P = Ai / [1 - 1/(1 + i)^n]

    A = Amount borrowed (500)
    i = interest per month (.11/12)
    n = number of payments (24)
    Last edited by Wilmer; September 8th 2009 at 11:26 PM. Reason: none
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  3. #3
    Senior Member furor celtica's Avatar
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    thanks but i can't use this thing, i need to use geometric progression methods
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  4. #4
    Senior Member furor celtica's Avatar
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    please i really need help here, ive posted this cos ive tried everything already
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  5. #5
    Senior Member furor celtica's Avatar
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    ok so i got 23.29 when my textbook says 23.31; is this a negligible difference or a symptom of a BIG problem? this is my work
    ahem
    S1= (500-m monthly payment) x (1211/1200)
    S2= (S1 -m) x (1211/1200)
    = (500-m)(1211/1200)^2 - (1211/1200)m
    and so on
    so i take as general equation for finding m
    0= (500-m)(1211/1200)^24 - m[(1211/1200)^23 -1]/[(1211/1200)-1]

    from there i get 23.29 pounds
    is there any fault in my working?
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  6. #6
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    Hello furor celtica
    Quote Originally Posted by furor celtica View Post
    ok so i got 23.29 when my textbook says 23.31; is this a negligible difference or a symptom of a BIG problem? this is my work
    ahem
    S1= (500-m monthly payment) x (1211/1200)
    S2= (S1 -m) x (1211/1200)
    = (500-m)(1211/1200)^2 - (1211/1200)m
    and so on
    so i take as general equation for finding m
    0= (500-m)(1211/1200)^24 - m[(1211/1200)^23 -1]/[(1211/1200)-1]

    from there i get 23.29 pounds
    is there any fault in my working?
    I make it 23.30 to the nearest penny, so I reckon you're about OK. I don't know that I really understand your working, though. Here's how I did it:

    At the end of month 1, the amount owing = 500\times\frac{1211}{1200}-m

    At the end of month 2, when we simplify the expression, the amount owing = 500\times\Big(\frac{1211}{1200}\Big)^2-\Big(m+m\times\frac{1211}{1200}\Big)

    At the end of month 3, the amount owing = 500\times\Big(\frac{1211}{1200}\Big)^3-\Big(m+m\times\frac{1211}{1200}+m\times\Big(\frac{  1211}{1200}\Big)^2\Big)

    ... and so on.

    The expression in the bracket is a GP, first term m, common ratio \frac{1211}{1200}, sum to 24 terms = m\frac{\Big(\dfrac{1211}{1200}\Big)^{24}-1}{\dfrac{1211}{1200}-1}

    If the sum owing at the end of month 24 is zero, then:

    500\times\Big(\frac{1211}{1200}\Big)^{24} - m\frac{\Big(\dfrac{1211}{1200}\Big)^{24}-1}{\dfrac{1211}{1200}-1}=0

    ...which you then solve, giving 23.30 to the nearest penny.

    Grandad
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  7. #7
    Senior Member furor celtica's Avatar
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    are you sure its not to 23 terms?
    what do you not understand in my working?
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  8. #8
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    Grandad is correct: 23.30391.....

    Plus he IS using the equivalent of 23 terms:
    the divisor (1211/1200 - 1) can be moved to the numerator as (1211/1200 - 1)^(-1)
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  9. #9
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    Hello furor celtica
    Quote Originally Posted by furor celtica View Post
    are you sure its not to 23 terms?
    what do you not understand in my working?
    Since you ask...

    First, since you're a regular contributor to Math Help Forum, learn to use LaTeX - it makes things so much easier to follow!

    Second, explain your working more clearly; say what you mean by S1, S2, ... for instance.

    As to your actual method, if S1 means the amount still owing after the first monthly payment has been made, this is not correct. You have taken off the payment first, and then worked out the interest payable on the balance. This would be correct if the first payment were made at the beginning of the month (i.e. as soon as the loan is taken out), but it's not. It's made at the end of the month. Therefore interest is charged on the whole 500 for the first month. And so on in each subsequent month.

    This is why your repayments are a penny or two less than they should be.

    And, yes, I am sure that the GP has 24 terms, not 23.

    Grandad
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