SOLUTION: Chelsea took out three loans for a total of $120,000 to start an organic orchard. Her bank loan was at an interest rate of 8%, the small business loan was at an interest rate of 5%

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Question 697292: Chelsea took out three loans for a total of $120,000 to start an organic orchard. Her bank loan was at an interest rate of 8%, the small business loan was at an interest rate of 5% and the mortgage on her house was at an interest rate of 4%. The total simple interest due on the loans in one year was $5750.The annual simple interest on the mortgage was 1600 more than the interest on the bank loan.How much did she borrow from each source?
Thank you!!!
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Chelsea took out three loans for a total of $120,000 to start an organic orchard. Her bank loan was at an interest rate of 8%, the small business loan was at an interest rate of 5% and the mortgage on her house was at an interest rate of 4%. The total simple interest due on the loans in one year was $5750.The annual simple interest on the mortgage was 1600 more than the interest on the bank loan.How much did she borrow from each source?
Thank you!!!

Let the bank, business, and mortgage loans be D, B, and M, respectively
Then: D + B + M = 120,000 ------- eq (i)
Also, .08D + .05B + .04M = 5,750 ----- eq (ii)
Additionally, .04M = .08D + 1,600 ----- .08D - .04M = - 1,600 ----- eq (iii)
- .05D - .05B - .05M = - 6,000 ----- Multiplying eq (i) by - .05 ---- eq (iv)

.03D - .01M = - 250 ---- Adding eqs (ii) & (iv) ------ eq (v)
- .12D + .04M = 1,000 ----- Multiplying eq (v) by - 4 ---- eq (vi)
.08D - .04M = - 1,600 ----- eq (iii)
- .04D = - 600 ------ Adding eqs (iii) & (vi)

D, or bank loan received = %28-+600%29%2F-+.04, or $highlight_green%2815000%29

.03(15,000) - .01M = - 250 -------- Substituting 15,000 for D in eq (v)
450 - .01M = - 250
- .01M = - 250 – 450
- .01M = - 700

M, or mortgage loan received = %28-+700%29%2F-+.01, or $highlight_green%2870000%29

15,000 + B + 70,000 = 120,000 ------- Substituting 15,000 for D, and $70,000 for M in eq (i)
B + 85,000 = 120,000

B, or business loan received = 120,000 – 85,000, or $highlight_green%2835000%29

OR

If you're familiar with matrices, you can also determine the various loan amounts utilizing that method.

You can do the check!!

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