Answer
100 000 dollars at 9%, 250 000 dollars at 10%, and 150 000 dollars at 12%
Work Step by Step
Firstly, we are given that the total amount borrowed is 500 000 dollars, and a part of it was borrowed at 9%, another at 10%, and the rest at 12%.
Let the 9%=x, 10%=y and 12%=z.
The first equation is:
$x+y+z=500 000$
Secondly, we have that the total annual interest was 52 000 dollars, and we know that it is made up of the interest gained from the 9%, 10%, and 12% interest loans. The second equation is therefore:
$0.09x+0.1y+0.12z=52 000$
Thirdly, we are told that the amount borrowed at 10% is 2.5 times the amount borrowed at 9%, therefore the third equation can be written as:
$2.5x-y=0$
The system of equations:
$x +y +z =500 000$
$0.09x+0.1y+0.12z=52 000$
$2.5x -y =0$
Augmented matrix:
(See attached image for the augmented matrix)
Elementary row operations:
(See the attached image for the row operations in the matrices)
1) $Row2×100$
$Row3×2$
2) $Row2-(Row1×9)$
$Row3-(Row1×5)$
4) $Row3+(Row2×7)$
5) $Row3×\frac{1}{16}$
Rewrite the system of equations now in Row-Echlon form:
$x+y+z=500 000$
$y+3z=700 000$
$z=150 000$
By solving for the variables using back substitution we get: $z=150 000$, $y=250 000$, and $x=100 000$
Solution: 100 000 dollars at 9%, 250 000 dollars at 10%, and 150 000 dollars at 12%
