Chapter 7.3, Problem 91E

Solution

To state: The amount that Sophia have in each investment if she receives $10,843 interest the first year.

She has $80,000 and she invests part of it in CDs earning 6.7% APY, part in bonds earning 9.3% APY, and remainder in a growth fund earning 15.6% APY. She invests 3 times as much in the growth funds as in the other two combined.

The amount invested in CDs is $14500, in bonds is $5500, and in growth funds are $60000.

Given information:

Sophia has $80,000 and she invests part of it in CDs earning 6.7% APY, part in bonds earning 9.3% APY, and remainder in a growth fund earning 15.6% APY. She invests 3 times as much in the growth funds as in the other two combined.

Explanation:

Let’s take x as the amount invested in CDs at 6.7%, y as the amount invested in bonds at 9.3%, and z as the amount invested in growth funds at 15.6%.

Now make equations using the given information.

Total amount invested is: $x+y+z=80,000$

Amount invested in the growth funds is 3 times as much in the other two combined.

  $\begin{array}{c}z=3\left(x+y\right)\\ z=3x+3y\\ 0=3x+3y-z\end{array}$

The total interest earned is $10,843:

  $0.067x+0.093y+0.156z=10,843$

The three equations are:

  $\begin{array}{c}x+y+z=80,000\\ 3x+3y-z=0\\ 0.067x+0.093y+0.156z=10,843\end{array}$

Let’s take,

  $A=\left[\begin{array}{ccc}1& 1& 1\\ 3& 3& -1\\ 0.067& 0.093& 0.156\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}80000\\ 0\\ 10843\end{array}\right]$

The system of equation can be written as: $A\cdot X=B$

Then $A\cdot X=\left[\begin{array}{ccc}1& 1& 1\\ 3& 3& -1\\ 0.067& 0.093& 0.156\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

Solve for X by multiplying both sides by $A^{-1}$ . Then $X=A^{-1}\cdot B$ .

  $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]={\left[\begin{array}{ccc}1& 1& 1\\ 3& 3& -1\\ 0.067& 0.093& 0.156\end{array}\right]}^{-1}\left[\begin{array}{c}80000\\ 0\\ 10843\end{array}\right]$

Use graphing calculator to find the value,

  

Therefore, the amount that Sophia has invested in CDs is $14500, in bonds is $5500, and in growth funds is $60000.