Question

A natural history museum borrows $2,000,000 at simple annual interest to purchase new exhibits. some of the money is borrowed at 7%, some at 8.5%, and some at 9.5%. use a system of linear equations to determine how much is borrowed at each rate given that the total annual interest is $169,750 and the amount borrowed at 8.5% is four times the amount borrowed at 9.5%. solve the system of linear equations using matrices.

Answer 1

Let the amount borrowed at 9.5% be x. Now we have:

at 9.5% = x

at 8.5% = 4x

at 7% = 2,000,000 - 5x

0.095x + 0.085(4x) + 0.07(2,000,000-5x) = 169,750

0.095x + 0.34x + 140,000 - 0.35x = 169,750

0.085x = 29,750

x = 350,000

So the museum borrowed:

At 9.5% = 350K

At 8.5% = 350*4 = 1,400,000

At 7% = 2,000,000 - 5(350,000) = 250,000

Total borrowed

350,000 + 1,400,000 + 250,000 = 2,000,000 Correct

Interest paid:

0.095(350,000) + 0.085(1,400,000) + 0.07(250,000) = 169,750