Chapter 7, Problem 51RE

To determine

Find the amount placed in each investment.

Answer to Problem 51RE

Investment amounts are: $16,000, $13,0000 and $11,000

Explanation of Solution

Given:

An investment of $40,000 yielding $3500 interest per year.

The interest rates for the three investments are 7%, 9% and 11%.

Second investment is $3000 less than the first.

Third investment is $5000 less than the first.

Let x, y and z be the amount in each of the investment scheme.

Total investment amount is $40,000.

The equation can be written as.

  $x+y+z=40,\mathrm{000.........}(1)$

Given the interest rates are 7, 9 and 11% for each and total interest in an year is $3500 and the equation can be written as:

  $0.07x+0.09y+0.11z=3500$

Multiplying equation (2) with 100,

  $7x+9y+11z=\mathrm{350000..........}(2)$

Second and third invested amount is less than first by $3000 and $5000 respectively

  $\begin{array}{l}y=x-3000\Rightarrow x=y+3000\hfill \\ z=x-5000\Rightarrow x=z+5000\hfill \end{array}$

Equating the right hand side of the above two equations:

  $\begin{array}{l}y+3000=z+5000\\ y-z=\mathrm{2000...........}(3)\end{array}$

The system of equations is:

  $\begin{array}{cc}x+y+z& =40,000\\ 7x+9y+11z& =350000\\ y-z& =2000\end{array}$

Dividing the above resultant equation with (2)

  $y+2z=35000$

The new system of equation is:

  $\begin{array}{cc}x+y+z& =40,000\\ y+2z& =35000\\ y-z& =2000\end{array}$

Adding -1 times equation (2) with equation (3),

  $\begin{array}{c}-y-2z=-35000\\ y-z=2000\\ -3z=-33000\\ z=11000\end{array}$

Now, the new system has become,

  $\begin{array}{cc}x+y+z& =40,000\\ y+2z& =35000\\ z& =11000\end{array}$

Back substituting $z=11000$ in equation (2),

  $\begin{array}{l}y+2z=35000\text{Write the equation}\hfill \\ y+2(11000)=35000\text{Substitute for z}\hfill \\ y=35000-22000\text{Combie like terms}\hfill \\ y=13000\text{Solve for y}\hfill \end{array}$

Back substituting values of y and z in equation (1),

  $\begin{array}{r}\hfill x+y+z=40,000\\ \hfill x+13000+11000=40,000\\ \hfill x+13000+11000=40,000\\ \hfill x=16000\end{array}$

So the investment amounts will be $16,000, $13,0000 and $11,000 into investment schemes that gives 7%, 9% and 11% interest rates respectively.