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Math Help - [SOLVED] please help!

  1. #1
    abcd19
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    Question [SOLVED] please help!

    I can't find the answer to this question, I don't even know where to start, can anyone help?!

    A family has a mortgage their house on a 25 year amortization period. The nominal interest rate is 5.95%, the interest is compounded semi-annually. The family pays $3107.77 monthly towards their mortgage, and their current balance is $237562.52 Assuming the interest rate stays the same over the duration of the mortgage, answer the following questions:

    a) how many more months will it take until the mortgage is paid off?
    b) what was the original amount of the mortgage loan?

    Thank you!!!
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  2. #2
    GAMMA Mathematics
    colby2152's Avatar
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    Quote Originally Posted by abcd19 View Post
    I can't find the answer to this question, I don't even know where to start, can anyone help?!

    A family has a mortgage their house on a 25 year amortization period. The nominal interest rate is 5.95%, the interest is compounded semi-annually. The family pays $3107.77 monthly towards their mortgage, and their current balance is $237562.52 Assuming the interest rate stays the same over the duration of the mortgage, answer the following questions:

    a) how many more months will it take until the mortgage is paid off?
    b) what was the original amount of the mortgage loan?

    Thank you!!!
    We need to value everything at the same period. Our payment period is 25 years worth of months, or 300 months. We need to find the monthly interest rate by using this equality:

    (1 + \frac{i^{(12)}}{12})^{12} = (1 + \frac{i^{(2)}}{2})^2
    (1 + \frac{i^{(12)}}{12})^{12} = (1 + \frac{.0595}{2})^2
    (1 + \frac{i^{(12)}}{12})^{12} = 1.060385
    (1 + \frac{i^{(12)}}{12}) = 1.004897764
    \frac{i^{(12)}}{12} = .004897764

    The equation for a constant payment annuity-immediate is:

    a_{n} = \frac{1 - v^n}{i}

    For an interest rate of .4898% each month with payments of $3107.77 monthly, and the current balance, we can figure out the number of months...

    237562.52 = 3107.77\frac{1 - (.995126)^n}{.004897}
    .3743 = 1 - .995126^n
    .62566 = .995126^n

    Take natural log of both sides...

    -.46894 = -.004885917n
    n = 95.98 months which is about 8 years.

    The original amount of the loan becomes pretty easy... we have 300 months in the entire loan, so:

    A = 3107.77\frac{1 - (.995126)^{300}}{.004897} = 488,092.52
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