Chapter 7, Problem 77RE

Solution

To calculate: The amount borrowed at each rate if the annual interest was $46,250 and the amount borrowed at 9% was twice the amount borrowed at 4%.

The amount and rate of each is $\$160000$ at 4%, $\$170000$ at $6.5\%$ , $\$320000$ at 9%.

Given Information:

Mitcham's Furniture Store borrowed $650,000 to expand its facilities and extend its product line. Some of the money was borrowed at 4%, some at 6.5%, and the rest at 9%.

Calculation:

Consider the given information,

Suppose that x is representing amount borrowed at 4%, y is representing amount borrowed at 6.5%, and z is representing amount borrowed at 9%.

Set up the equations. Mitcham's Furniture Store borrowed $650000.

  $x+y+z=650000$

The annual interest was $46250:

  $0.04x+0.065y+0.09z=46250$

The amount borrowed at 9% was twice the amount borrowed at 4%:

Write the system as a matrix equation  $AX=B$ .

  $A=\left[\begin{array}{ccc}1& 1& 1\\ 0.04& 0.065& 0.09\\ -2& 0& 1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],\text{ and }B=\left[\begin{array}{c}65000\\ 46250\\ 0\end{array}\right]$

Solve for  X  by multiplying both sides by  $A^{-1}$ . That is,  $X=A^{-1}BX$

. Using a graphing calculator (as used in the book), find the solution of the system as shown:

  

Therefore, the required amount is $\$160000$ at 4%, $\$170000$ at $6.5\%$ , $\$320000$ at 9% .