Let A, B, and C be the amounts invested in companies A, B, and C. If no more than 50% of the total investment can be in company B, how can you express that as a constraint?

Question

BlackTom AI · Accepted Answer

B ≤ 0.5(A + B + C) because at most 50% of the total investment (A + B + C) may be in B.

1) Relevant knowledge points and concepts
- "No more than 50%" translates to an inequality with ≤ and a 0.5 factor.
- Total investment = A + B + C.
- Algebraic manipulation (adding/subtracting terms and scaling by a positive constant) produces equivalent linear constraints.

2) Step-by-step reasoning
Start with the verbal constraint:
B ≤ 0.5(A + B + C)

Multiply both sides by 2 (positive factor, inequality direction preserved):
2B ≤ A + B + C

Bring all terms to the left side:
2B − A − B − C ≤ 0

Combine like terms (2B − B = B):
B − A − C ≤ 0

Multiply this inequality by 0.5 (again a positive scalar, so it remains equivalent):
0.5(B − A − C) ≤ 0

Distribute 0.5:
−0.5A + 0.5B − 0.5C ≤ 0

3) Intermediate calculations shown
B ≤ 0.5(A + B + C)
→ 2B ≤ A + B + C
→ B − A − C ≤ 0
→ 0.5(B − A − C) ≤ 0
→ −0.5A + 0.5B − 0.5C ≤ 0

4) Why the correct answer is correct
The inequality −0.5A + 0.5B − 0.5C ≤ 0 is algebraically equivalent to B ≤ 0.5(A + B + C). Each transformation used only addition/subtraction and multiplication by positive scalars, which preserve equivalence. Thus this expression precisely enforces that B is at most half of the total investment.

5) Why the other options are incorrect
- B ≤ 0.5: Fixes B relative to the number 0.5, not relative to the total A + B + C, so it's unrelated.
- A − 0.5B + C ≤ 0: Rearranged gives 2A − B + 2C ≤ 0 (after multiplying by 2), which is not equivalent to the required relation.
- 0.5A − B − 0.5C ≤ 0: Equivalent to A − 2B − C ≤ 0, not the same as B ≤ 0.5(A + B + C).
- 0.5A − 0.5B + 0.5C ≤ 0: Equivalent to A − B + C ≤ 0, also not equivalent.

6) Conclusion confirming the answer
The correct constraint expressed in the given options is −0.5A + 0.5B − 0.5C ≤ 0, which matches the provided answer.

Let A, B, and C be the amounts invested in companies A, B, and C. If no more than 50% of the total investment can be in company B, how can you express that as a constraint?Single choice

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