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, then

Question

BlackTom AI · Accepted Answer

B ≤ 0.5(A + B + C) because no more than 50% of the total investment A + B + C can be in company B.

1) Relevant concepts
- Percent constraint: "no more than 50%" means B is at most half of the total A + B + C.
- Algebraic manipulation of inequalities: expand, combine like terms, and reorder to standard linear-inequality form.

2) Step-by-step reasoning and intermediate steps
Start from the percentage statement:
B ≤ 0.5(A + B + C)

Bring the right-hand side to the left:
B − 0.5(A + B + C) ≤ 0

Expand the parentheses:
B − 0.5A − 0.5B − 0.5C ≤ 0

Combine the B terms (1B − 0.5B = 0.5B):
0.5B − 0.5A − 0.5C ≤ 0

Reorder terms to match the answer format:
−0.5A + 0.5B − 0.5C ≤ 0

(Optionally, multiply both sides by 2 to remove decimals: −A + B − C ≤ 0, which is algebraically equivalent.)

3) Why the correct answer is correct
The inequality −0.5A + 0.5B − 0.5C ≤ 0 is exactly the result of translating "B is at most 50% of the total" into an algebraic inequality and simplifying. It enforces that B cannot exceed half of A + B + C.

4) Why the other options are incorrect
- 0.5A − B − 0.5C ≤ 0 is different: it would rearrange to B ≥ 0.5A − 0.5C, not enforcing the 50%-of-total upper bound on B.
- A − 0.5B + C ≤ 0 rearranges to 0.5B ≥ A + C, which wrongly requires B to be at least twice A + C rather than at most half the total.
- B ≤ 0.5 is not equivalent because it bounds B by the fixed number 0.5 rather than by half of the total investment A + B + C.

Conclusion: The correct inequality 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, then单项选择题

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