Chelsea took out three loans for a total of ​$ 78 comma 000 78,000 to start an organic orchard. Her​ business-equipment loan was at an inter

Question

Chelsea took out three loans for a total of ​$ 78 comma 000 78,000 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 5​%, and her​ home-equity loan was at an interest rate of 4.5 4.5​%. The total simple interest due on the loans in one year was ​$ 4605 4605. The annual simple interest on the​ home-equity loan was ​$ 105 105 more than the interest on the​ business-equipment loan. How much did she borrow from each​ source?

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Melanie 1 week 2021-09-14T06:46:48+00:00 1 Answer 0

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    2021-09-14T06:48:21+00:00

    Answer:

    Business-equipment loan, A=$15000

    Small-business loan, B=$24000

    home-equity loan, C= $39000

    Step-by-step explanation:

    Let the Business-equipment loan at an interest rate of​ 11%=A

    Let the​ small-business loan was at an interest rate of 5​% =B

    Let her​ home-equity loan was at an interest rate of 4.5​%.=C

    Since her total Loan=$78000

    A+B+C=78000….(I)

    Simple Interest = (P X R X T)/100

    Since the Time, T=1 year

    Interest on A = 0.11A

    Interest on B = 0.05B

    Interest on C = 0.045C

    The total simple interest due on the loans in one year was ​$4605.

    0.11A+0.05B+0.045C=4605….(II)

    The annual simple interest on the​ home-equity loan was ​$105 more than the interest on the​ business-equipment loan.

    0.045C = 0.11A +105…(III)

    We proceed to solve the simultaneous equations.

    A+B+C=78000….(I)

    0.11A+0.05B+0.045C=4605….(II)

    0.045C = 0.11A +105…(III)

    From (III), 0.045C = 0.11A +105

    Substitute 0.045C = 0.11A +105 into (II).

    0.11A+0.05B+0.11A +105=4605

    0.22A+0.05B=4500

    From (III),

    C=\frac{22A}{9}+\frac{7000}{3}

    Substitute into (I)

    A+B+\frac{22A}{9}+\frac{7000}{3}=78000

    \frac{31A}{9}+B=78000-\frac{7000}{3}

    \frac{31A}{9}+B=\frac{227000}{3}

    \frac{31A+9B}{9}=\frac{227000}{3}

    3(31A+9B)=9 X 227000

    31A+9B=681000

    0.22A+0.05B=4500 (Multiply by 9)

    31A+9B=681000 (Multiply by 0.05)

    1.98A+0.45B=40500

    1.55A+0.45B=34050

    Subtracting

    0.43A=6450

    A=$15000

    From (III)

    C=\frac{22A}{9}+\frac{7000}{3}

    C=\frac{22X15000}{9}+\frac{7000}{3} =39000

    C=$39000

    from (I)

    A+B+C=78000

    15000+B+39000=78000

    B=$24000

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