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Assume that [tex]A[/tex], [tex]B[/tex], and [tex]C[/tex] are the dollars invested in three different common stocks from the New York Stock Exchange, Stock [tex]A[/tex], [tex]B[/tex], and [tex]C[/tex] respectively. In order to diversify the investments, the investing company requires that more than [tex]60\%[/tex] of the dollars invested can be in Stock [tex]A[/tex]. The constraint for this requirement can be written as:

a. [tex]A \geq 0.60(A + B + C)[/tex]
b. No given choice is correct.
c. [tex]A \geq 0.60[/tex]
d. [tex]0.40A - 0.60B - 0.60C \leq 0[/tex]
e. [tex]0.40A - 0.60B - 0.60C \geq 0[/tex]
f. [tex]A + B + C \leq 0.6A[/tex]



Answer :

GinnyAnswer
Let’s break down the constraint that says more than 60% of the total dollars invested should be in Stock A.

Given:
- [tex]$A$[/tex] is the amount invested in Stock A.
- [tex]$B$[/tex] is the amount invested in Stock B.
- [tex]$C$[/tex] is the amount invested in Stock C.

The total amount invested in all three stocks is [tex]$A + B + C$[/tex].

The constraint states that more than 60% of this total amount should be invested in Stock A. Mathematically, this can be written as:

[tex]\[ A \geq 0.60 \times (A + B + C) \][/tex]

Now, let's consider each given choice:

a. [tex]\(A \geq 0.60(A + B + C)\)[/tex]

This correctly represents the constraint given. It means that the amount invested in Stock A should be at least 60% of the total investment in all stocks.

b. No given choice is correct.

This cannot be correct because we can see that choice (a) correctly matches the given constraint.

c. [tex]\(A \geq 0.60 \)[/tex]

This does not make sense because it doesn't incorporate the amounts invested in Stocks B and C. It only takes into account the investment in Stock A in isolation.

d. [tex]\(0.40 A - 0.60 B - 0.60 C \leq 0\)[/tex]

Let's rewrite this inequality:

[tex]\[ 0.40A - 0.60B - 0.60C \leq 0 \][/tex]

Rearranging terms, we get:

[tex]\[ 0.40A \leq 0.60B + 0.60C \][/tex]

Which can be further simplified to:

[tex]\[ A \leq 1.5B + 1.5C \][/tex]

This inequality is not the same as our original constraint.

e. [tex]\(0.40 A - 0.60 B - 0.60 C \geq 0\)[/tex]

Again, let's rewrite and simplify:

[tex]\[ 0.40A - 0.60B - 0.60C \geq 0 \][/tex]

This can be rearranged to:

[tex]\[ 0.40A \geq 0.60B + 0.60C \][/tex]

Or:

[tex]\[ A \geq 1.5B + 1.5C \][/tex]

This also is not the same as our original constraint.

f. [tex]\(A + B + C \leq 0.6A\)[/tex]

This simplifies to:

[tex]\[ B + C \leq -0.4A \][/tex]

Which clearly does not make sense since investments should not result in negative values.

From all the choices, the correct representation of the constraint is indeed given in option (a):

[tex]\[ A \geq 0.60(A + B + C) \][/tex]

Therefore, the correct choice is:
a. [tex]\(A \geq 0.60(A + B + C)\)[/tex]

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