Dividing Investments for Maximum Return

Date: 09/26/1999 at 18:21:07
From: Andrew
Subject: Algebra II with Trigonometry

A woman wants to invest $60,000 in mutual funds and municipal bonds. 
She does not want to invest more than 50%, or less than 20%, of her 
money in mutual funds. The minimum investment for municipal bonds is 
$10,000, and they are only guaranteed up to $40,000, so she will not 
invest more than $40,000 in municipal bonds. The mutual funds should 
produce a return of 10%, and the municipal bonds a return of 12%. How 
much should she invest in each to maximize her income? What is her 
maximum income?

I honestly have never really understood interest problems, so I need 
help approaching this, setting up the equations, and then solving it.

Thank you.

Date: 09/29/1999 at 11:17:46
From: Doctor Rick
Subject: Re: Algebra II with Trigonometry

Hi, Andrew,

This problem has a lot of little pieces of information, and a lot of 
steps in order to make use of that information. Each step is pretty 
simple, but all the little steps add up.

As in most algebra problems, we should start by defining some 
variables. How about these:

     F = dollars invested in mutual funds
     B = dollars invested in municipal bonds
     I = total interest in dollars

Since all of the $60,000 will be invested in one or the other, we know 
that

    [1] F + B = 60,000

Before we get to the interest, we are told some things that limit the 
ranges of F and B. I will deal with two of these four things:

1. The woman does not want to invest more than 50% of the $60,000 in 
   mutual funds. This tells us that

          F <= 0.5 * 60,000
  
   That <= is supposed to be a "greater than or equal to" sign. Work 
   out the arithmetic:

     [2] F <= 30,000

2. The minimum investment for municipal bonds is $10,000. This tells 
   us that

     [3] B >= 10,000

   I leave the maximum F and the minimum B for you to work out.

We will want to pick one variable to work with, since equation [1] 
tells us how to find the other. Let's work with F. Solve equation [1] 
for B:

     [4] B = 60,000 - F

What does equation [3] tell us about the range of our main variable, 
F? Substitute equation [4] in equation [3]:

          60,000 - F >= 10,000

Solve this inequality for F:

                   60,000 >= 10,000 + F

          60,000 - 10,000 >= F

     [5]           50,000 >= F

Compare equations [2] and [5]. If F <= 30,000 then F is certainly less 
than 50,000, so equation [5] doesn't add any new information, and we 
can forget it.

Remember those other two conditions on the range of the variables. You 
will be able to set a lower bound for F (that is, F >= something) 
based on those two conditions.

Now that you know the range of F, you can start looking at the 
interest earned on the money. The mutual funds produce a return of 
10%, and the municipal bonds a return of 12%. This means that the 
interest on F is 10% of F, and the interest on B is 12% of B. As an 
equation,

     [6]  I = 0.1 F + 0.12 B

You can use equation [4] to get an equation in I and F only. Then it 
might help you to graph the equation. Make F the x-axis, and I the 
y-axis. Add the range of F to your graph: something <= F <= 30,000. 
What value of F within this range gives the greatest value of I? When 
you know this, you can use your equation in I and F to find the 
maximum interest.

I hope this helps you.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/