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22 December, 13:20

You have $36,948.61 in a brokerage account, and you plan to deposit an additional $3,000 at the end of every future year until your account totals $280,000. you expect to earn 11% annually on the account. how many years will it take to reach your goal? round your answer to two decimal places at the end of the calculations.

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  1. 22 December, 14:27
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    Current amount in account

    P=36948.61

    Future value of this amount after n years at i=11% annual interest

    F1=P (1+i) ^n

    =36948.61 (1.11) ^n

    Future value of $3000 annual deposits after n years at i=11%

    F2=A ((1+i) ^n-1) / i

    =3000 (1.11^n-1) / 0.11

    We'd like to have F1+F2=280000, so forming following equation:

    F1+F2=280000

    =>

    36948.61 (1.11) ^n+3000 (1.11^n-1) / 0.11=280000

    We can solve this by trial and error.

    The rule of 72 tells us that money at 11% deposited will double in 72/11=6.5 years, approximately.

    The initial amount of 36948.61 will become 4 times as much in 13 years, equal to approximately 147800 by then.

    Meanwhile the 3000 a year for 13 years has a total of 39000. It will only grow about half as fast, namely doubling in about 13 years, or worth 78000.

    Future value at 13 years = 147800+78000=225800.

    That will take approximately 2 more years, or 225800*1.11^2=278000.

    So our first guess is 15 years, and calculate the target amount

    =36948.61 (1.11) ^15+3000 (1.11^15-1) / 0.11

    =280000.01, right on.

    So it takes 15.00 years to reach the goal of 280000 years.
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