Question 1201131
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Formulate a system of equations for the situation below and solve.
A private investment club has $400,000 earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high-risk, medium-risk, and low-risk. Management estimates that high-risk stocks will have a rate of return of 15%/year; medium-risk stocks, 10%/year; and low-risk stocks, 6%/year. The members have decided that the investment in low-risk stocks should be equal to the sum of the investments in the stocks of the other two categories. Determine how much the club should invest in each type of stock if the investment goal is to have a return of $40,000/year on the total investment. (Assume that all the money available for investment is invested.)
high-risk stocks $
medium-risk stocks $
low-risk stocks $
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        Probably, you will be very amazed if I tell you that this problem can be easily solved

        using only one unknown and one equation instead of using three unknowns and three equations.


        But I will show it right now, in the next 3 minutes.



<pre>
As the problem says, the total $400,000 is split in two equal parts: low-risk stock and the sum  
the investments in the stocks of the other two categories.


So, $200,000 go to the low risk stock at 6%, and another $200,000 go to the other two categories
at 10% and 15%.


Let 'x' be the amount invested at 15%.
Then the amount invested at 10% is (200000-x) dollars.


Write the total interest equation

    0.15x + 0.1*(200000-x) + 0.06*200000 = 40000.


Simplify and find x

    0.15x + 20000 - 0.1x + 12000 = 40000

    0.15x - 0.1x = 40000 - 20000 - 12000

         0.05x   =          8000

             x   =          8000/0.05 = 160000


Thus $160,000 invested at 15% (high-risk);  $200,000 - $160,000 = $40,000 invested at 10% (medium-risk);
and  $200,000 invested at 6%  (low-risk).
</pre>

At this point, the problem is solved completely using only one unknown and one equation.


It's immeasurably simpler, isn't it?


So, there are two ways to solve this problem: one way is to pretend that you are a stupid person
and follow blindly the problem's stupid instruction.  Or learn the advanced way and reduce calculations.