What is the present value of this compound interest loan? ?

I did most of this sheet, but am stuck at compound and continuous interest. Can someone help me fill out these 3? I’m sure it doesn’t take long but it’s simply not clicking for me as this is a relatively new subject to me. 5 points if anyone helps our, thanks in advance. 

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1 Answer

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  • 3 weeks ago

    Compound interest equation is:

    a(t) = P(1 + r/n)^(nt)

    Where:

    a(t) is the value after "t" years

    P is the initial amount

    r is the annual interest rate

    n is the number of times per year interest is compounded

    t is the number of years

    We are given the future value is $29,494.12 after 4 years, so a(4) = 29494.12.

    P is the unknown.

    r = 0.0414

    n = 12

    Substitute and solve for the unknown:

    29494.12 = P(1 + 0.414/12)^(12 * 4)

    29494.12 = P(1 + 0.00345)^48

    29494.12 = P(1.00345)^48

    29494.12 = P(1.1797647)

    I rounded the value above but didn't in my calculator to reduce errors due to rounding.

    P = 29494.12 / 1.1797647

    P = $25,000.00 (rounded to the nearest penny)

    ------

    Continuous interest is:

    a(t) = P e^(rt)

    Where:

    a(t) is the amount after t years

    P is the initial amount

    r is the interest rate

    t is the number of years

    We are given P = 23000, r = 0.0488, t = 6. Solve for a(6):

    a(6) = 23000 e^(0.0488 * 6)

    a(6) = 23000 e^(0.2928)

    a(6) = 23000(1.340175)

    a(6) = $30,824.02 (rounded to nearest penny)

    And the last line uses the same equation as the last one but we are given a(4) = 20214.77, r = 0.0433, and need to solve for P:

    a(t) = P e^(rt)

    20214.77 = P e^(0.0433 * 4)

    20214.77 = P e^(0.1732)

    20214.77 = P(1.1891039)

    P = 20214.77 / 1.1891039

    P = $17,000.00 (rounded to nearest penny)

    ----------------

    Let's go back to your other compound interest that you tried. I don't think that's right. Starting from:

    a(t) = P(1 + r/n)^(nt)

    using:

    a(8) = unknown

    P = 22000

    r = 0.0487

    n = 12

    t = 8

    a(8) = 22000(1 + 0.0487/12)^(12 * 8)

    a(8) = 22000(1 + 0.00405833)^96

    a(8) = 22000(1.00405833)^96

    a(8) = 22000(1.47522655)

    a(8) = $32,454.98 (rounded to the nearest penny)

    Hope this helps. Please give best answer if it does.

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