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11 February, 13:10

You want to be able to withdraw $20,000 each year for 15 years. Your account earns 7% interest. a) How much do you need in your account at the beginning? $ b) How much total money will you pull out of the account? $ c) How much of that money is interest?

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  1. 11 February, 13:20
    0
    a) How much do you need in your account at the beginning?

    $182,158.28

    b) How much total money will you pull out of the account?

    $300,000

    c) How much of that money is interest?

    $117,841.72

    Explanation:

    We need to calculate the present value of the annuity:

    PV of annuity = P x {[1 - (1 + r) ⁻ⁿ] / r}

    P = periodic payment (annuity) = $20,000 r = interest rate = 7% n = number of periods = 15

    PV of annuity = $20,000 x {[1 - (1 + 7%) ⁻¹⁵] / 7%}

    = $20,000 x {[1 - 1.07⁻¹⁵] / 7%} = $20,000 x (0.63755398 / 7%) = $20,000 x 9.107914 = $182,158.28

    total money pulled out = $20,000 x 15 payments = $300,000

    total interest received = $300,000 - $182,158.28 = $117,841.72
  2. 11 February, 13:25
    0
    Answer: A) The amount needed at the beginning would be $146,342

    B) The total money you would pull out would be $300,001

    C) The interest earned would total $153,659

    Explanation: If you planned on withdrawing 20000 per year for 15 years, that would be a total of 300000 at the end of 15 years. The interest you would have earned is calculated as;

    I = PRT

    Note that the addition of your interest and the sum invested (Principal) shall be a total of 300000. Hence,

    I + P = 300000

    Making P the subject of the equation,

    P = 300000 - I.

    So we have, R as 0.07, T as 15 and P as 300000 - I, we can now substitute for the values as follows;

    I = PRT

    I = (300000 - I) x 0.07 x 15

    I = (300000 - I) x 1.05

    I = 315000 - 1.05I

    Collecting like terms we now have

    I + 1.05I = 315000

    2.05I = 315000

    Divide both sides of the equation by 2.05

    I = 153659 (approximately)

    If the interest earned is $153,659, and the amount you wish to withdraw at the end of 15 years is a total of $300,000, then the principal (P) invested would be

    I = PRT

    I/RT = P

    153659 / (0.07 x 15) = P

    146341.9 = P

    P≈ 146342

    Therefore, interest earned = $153,659

    Principal invested = $146,342

    Total withdrawal = $300,001
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