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Sagot :
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]
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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