An investor has up to $650,000 to invest in two types of investments. Type A pays 13% annually and type B pays 7% annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth is to be allocated to type B investments. What is the optimal amount that should be invested in each type of investment?
If the amounts in A and B are x and y, then (suppressing all the extra zeros),
maximize p=.13x+.07y subject to
x >= 65/2
y >= 65/4
x+y <= 65
Now use your favorite tool to solve. A good one is at
http://www.zweigmedia.com/RealWorld/LPGrapher/lpg.html
To find the optimal amounts to invest in each type of investment, we can set up a system of equations based on the given conditions.
Let's assume the amount invested in Type A is x, and the amount invested in Type B is y.
We know that the total amount available to invest is $650,000, so we have the equation:
x + y = 650,000
Since at least half of the total portfolio is allocated to Type A investments, we have the condition:
x >= (1/2)(x + y)
Similarly, since at least one-fourth is allocated to Type B investments, we have the condition:
y >= (1/4)(x + y)
Therefore, the system of equations is:
x + y = 650,000
x >= (1/2)(x + y)
y >= (1/4)(x + y)
To solve this system of equations, we can use substitution or elimination method. Let's use substitution here:
From the second equation, we can rewrite it as:
2x >= x + y
2x - x >= y
x >= y
From the third equation, we can rewrite it as:
4y >= x + y
4y - y >= x
3y >= x
Now we have the following equations:
x + y = 650,000
x >= y
3y >= x
To find the optimal amount to invest in each type of investment, we need to find a solution that satisfies all the equations simultaneously. We can start by setting x=y, and then substitute it into the first equation:
2y = 650,000
y = 325,000
Substituting y back into the first equation, we can solve for x:
x + 325,000 = 650,000
x = 650,000 - 325,000
x = 325,000
Therefore, the optimal amount to invest is $325,000 in Type A and $325,000 in Type B.
To find the optimal amount to invest in each type of investment, we need to set up a system of inequalities based on the given conditions.
Let's assume the amount to be invested in type A be x, and the amount to be invested in type B be y.
Based on the first condition, at least one-half of the total portfolio should be allocated to type A investments. So, x ≥ 1/2(x + y).
Based on the second condition, at least one-fourth should be allocated to type B investments. So, y ≥ 1/4(x + y).
Next, we are given the total amount available for investment, which is $650,000. So, the sum of the amounts invested in type A and type B should be equal to $650,000. This can be represented as x + y = $650,000.
Now, we have a system of inequalities:
1) x ≥ 1/2(x + y)
2) y ≥ 1/4(x + y)
3) x + y = $650,000
We can solve this system of inequalities to find the optimal amounts to be invested in type A and type B.