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11 April, 08:45

Barbara made a bad investment. Rather than earning interest, her money is decreasing in value by 11% each week! After just one week, she is down to just $142.40. How much money did she start with? If she does not withdraw her money, how long will it be before she has less than half of what she originally invested?

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  1. 11 April, 08:47
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    Answer: She started with $160.

    It will take 6 weeks before she has less than half of what she originally invested.

    Step-by-step explanation:

    If her money is decreasing in value by 11% each week, it means that the rate at which it is decreasing is exponential.

    We would apply the formula for exponential decay which is expressed as

    A = P (1 - r) ^t

    Where

    A represents the value of the investment after t weeks.

    t represents the number of weeks.

    P represents the initial value of the investment.

    r represents rate of depreciation.

    From the information given,

    A = $142.40

    r = 11% = 11/100 = 0.11

    t = 1

    Therefore

    142.40 = P (1 - 0.11) ^1

    142.40 = P (0.89)

    P = 142.4/0.89

    P = 160

    For her to have half of what she invested originally, then

    80 = 160 (0.89) ^t

    80/160 = (0.89) ^t

    0.5 = (0.89) ^t

    Taking log of both sides to base 10

    Log 0.5 = log0.89^t = tlog0.89

    - 0.3010 = - 0.051t

    t = - 0.3010 / - 0.051

    t = 5.9

    Approximately 6 weeks
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