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14 January, 17:51

Chelsea took out three loans for a total of $76 comma 000 to start an organic orchard. Her business-equipment loan was at an interest rate of 11%, the small-business loan was at an interest rate of 5 %, and her home-equity loan was at an interest rate of 4.5 %. The total simple interest due on the loans in one year was $4380. The annual simple interest on the home-equity loan was $370 more than the interest on the business-equipment loan. How much did she borrow from each source?

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  1. 14 January, 18:05
    0
    Business equipment loan = $12,994

    small business loan = $23,027

    Home equity loan = $39,979

    Explanation:

    Let;

    x denote the business equipment loan amount

    y denote the small business loan amount

    z denote the home equity loan amount

    Therefore;

    x + y + z = $76,000 ... equation 1

    Simple interest on;

    loan x = 11% 0f x = 0.11x

    loan y = 5% of y = 0.05y

    loan z = 4.5% 0f z = 0.045z

    Total simple interest due on the loans = $4380

    therefore;

    0.11x + 0.05y + 0.045z = 4380 ... equation 2

    Annual interest on home equity loan (z) was $370 more than interest on business equipment loan (x),

    Therefore;

    0.045z = 0.11x + 370 ... equation 3

    z = 0.11x/0.045 + 370/0.045

    z = 2.444x + 8222.22 ... equation 4

    if we substitute equation 3 into equation 2, we have;

    0.11x + 0.05y + (0.11x + 370) = 4380

    0.22x + 0.05y = 4380 - 370

    0.22x + 0.05y = 4010

    0.05y = 4010 - 0.22x

    y = 4010/0.05 - 0.22x/0.05

    y = 80200 - 4.4x ... equation 5

    if we substitute equations 4 and 5 into equation 1, we have;

    x + y + z = 76000

    x + (80200 - 4.4x) + (2.444x + 8222.22) = 76,000

    80200 + 8222.22 - 76,000 = 4.4x - 3.444x

    12422.22 = 0.956x

    x = 12422.22/0.956

    x = 12993.96

    This means her business equipment loan was $12,994

    From equation 5

    y = 80200 - 4.4x

    y = 80200 - 4.4 (12993.96)

    y = 23026.59

    This means her small business loan was $23,027

    From equation 4

    z = 2.444x + 8222.22

    z = 2.444 (12993.96) + 8222.22

    z = 39,979.4

    This means her home equity loan was $39,979
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