Maximizing income

Question

A financial consultant wishes to invest a total of $30,000 in two types of securities, one that yields 10% per year and another that yields 8% per year. Furthermore, she believes that the amount invested in the first security should be at most one-third of the amount invested in the second security. What investment program should the consultant pursue in order to maximize income? I just need the set of up the equations... not necessarily the answer, I’m stuck on the one-third part.

Answer 1

This is best set up as a matrix of 2 math expressions.

 

First one:

 

Let x = amount invested at 10%

Let y = amount invest at 8%

 

Make this into the first math statement:  x + y = 30,000

 

Second one:

 

The amount invested at 8% should be at least 3 times the amount invest at 10%

 

Make that into a statement:  x = 3y; putting it all to the left of the equal sign:  x - 3y = 0

 

Set this up into the following matrix:

 

1          1          =          30000

1         -3          =                 0

 

I want to get the R2, C1 value to equal 0, so I do the following row operation:

 

-3R1 + R2 = new R2; which gives us the new matrix of:

 

1          1          =          30000

0         -4          =        -90000

 

Next, I want to get the R1, C2 value to equal 0, so I do the following row operation:

 

R2 + 4R1 = new R1, which gives us the new matrix of:

 

4          0          =          30000

0         -4          =        -90000

 

Then, I want to do the two following row reductions:

 

1/4 R1 = new R1

-1/4 R2 = new R2

These give us the following updated matrix:

 

1          0         =          7500

0          1         =        22500

 

So, we now have:  x = $7,500 at 10%, y = $22,500 at 8%

 

This leaves us with two (2) checks to make:

 

(1)  x + y = 30,000; so:  $7,500 + $22,500 = $30,000  [Check]

(2)  3x - y = 0; so:  3 * $7,500 = $22,500 [Check]