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12 December, 00:30

A retired woman has $120,000 to invest. She has chosen one relatively safe investment fund that has an annual yield of 9% and another riskier fund that has a 13% annual yield. How much should she invest in each fund if she would like to earn $12,000 per year from her investments?

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  1. 12 December, 00:39
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    She should invest $90,000 in the safe fund and $30,000 in the riskier fund

    Step-by-step explanation:

    * Lets explain how to solve the problem

    - A retired woman has $120,000 to investment

    - She has two options:

    # Save investment fund that has an annual yield of 9%

    # Riskier fund that has a 13% annual yield

    - She would like to earn $12,000 per year from her investments

    * Assume that she invest $x in the safe fund and $y in the riskier fund

    ∵ The total investment is $120,000

    ∵ x represents the amount of money invested in the safe fund

    ∵ y represents the amount of money invested in the riskier fund

    ∴ x + y = 120,000 ⇒ (1)

    ∵ The interest I = Prt, where P is the invested amount, r is the

    percentage of the interest per year in decimal, t is the time

    ∵ The interest of x amount is 9%

    ∴ I = x * 9/100 * 1 = 0.09 x

    ∴ The interest of y amount is 13%

    ∴ I = y * 13/100 * 1 = 0.13 u

    ∵ She would like to earn $12,000 per year from her investment

    ∴ 0.09 x + 0.13 y = 12,000 ⇒ (2)

    * Lets solve the two equations to find x and y

    - Multiply equation (1) by - 0.13 to eliminate y

    ∵ - 0.13 x + - 0.13 y = - 15600 ⇒ (3)

    - Add equations (2) and (3)

    ∴ (0.09 x + - 0.13 x) + (0.013 y + - 0.13 y) = 12,000 + - 15,600

    ∴ - 0.04 x = - 3,600

    - Divide both sides by - 0.04

    ∴ x = 90,000

    - Substitute the value of x in equation (1)

    ∴ 90,000 + y = 120,000

    - Subtract 90,000 from both sides

    ∴ y = 30,000

    ∵ x represents the amount invested in the safe fund

    ∵ y represents the amount invested in the riskier fund

    ∴ She should invest $90,000 in the safe fund and $30,000 in

    the riskier fund
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