Country days scholarship rounds receive a gift of $135000. The money is invested in stock, bonds, and CDs. CDs pay 2.75% interest, bonds pay

Question

Country days scholarship rounds receive a gift of $135000. The money is invested in stock, bonds, and CDs. CDs pay 2.75% interest, bonds pay 4.5% interest, and stocks pay 10.4% interest. Country days invests $70000 more in bonds than CDs. If the annual income from the investments is $8555, how much was invested in stocks, bonds, and CDs?

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3 days 2021-09-13T15:43:38+00:00 1 Answer 0

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  1. Emma
    0
    2021-09-13T15:44:38+00:00

    Answer:

    • CDs — $10,000
    • bonds — $80,000
    • stocks — $45,000

    Step-by-step explanation:

    Let the variables c, b, s represent the dollar amounts invested in CDs, stocks, and bonds, respectively. Then the problem statement gives us 3 relations between these 3 variables:

      c + b + s = 135000 . . . . . . . . . . . . . . . . . total invested

      0.0275c +.045b +0.104s = 8555 . . . . . total income earned

      -c + b = 70000 . . . . . . . . . . . . . . . . . . . . . 70,000 more was in bonds than CDs

    Using the third equation to write an expression for b, we can substitute into the other two equations.

      b = 70000 +c . . . . . . . . . . . . . . . . expression we can substitute for b

      c + (70000 +c) +s = 135000 . . . . substitute for b in the first equation

      2c +s = 65000 . . . . . . . . . . . . . . . . [eq4] simplify

      .0275c +.045(70000 +c) +.104s = 8555 . . . . . substitute for b in 2nd eqn

      .0725c +.104s = 5405 . . . . . . . . . . [eq5] simplify

    Using [eq4], we can write an expression for s that can be substituted into [eq5].

      s = 65000 -2c . . . . . . . expression we can substitute for s

      0.0725c +0.104(65000 -2c) = 5405

      -0.1355c = -1355 . . . . . . . . . . . . . . . . . . . . subtract 6760, simplify

      c = 1355/.1355 = 10,000

      s = 65000 -2×10000 = 45,000

      b = 70000 +10000 = 80,000

    The amounts invested in stocks, bonds, and CDs were $45,000, $80,000, and $10,000, respectively.

    _____

    Alternatively, you can reduce the augmented matrix for this problem to row-echelon form using any of several calculators or on-line sites. That matrix is …

    \left[\begin{array}{ccc|c}1&1&1&135000\\0.0275&0.045&0.104&8555\\-1&1&0&70000\end{array}\right]

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