Question

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?

Answer 1

Answer:

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