A fund manager must decide how much money to invest in Kumoto stock (K) and how much to invest in Takayama stock (T). One requirement is that at least 40% of the money invested in the two companies' stock must be in Kumoto stock.

A correct modelling of this requirement is:

Question

A fund manager must decide how much money to invest in Kumoto stock (K) and how much to invest in Takayama stock (T). One requirement is that at least 40% of the money invested in the two companies' stock must be in Kumoto stock.

A correct modelling of this requirement is:

BlackTom AI · Accepted Answer

Since at least 40% of the total invested in the two stocks must be in Kumoto, express the requirement as a fraction: K/(K + T) ≥ 0.40.

1) Relevant knowledge points and concepts
- A percentage requirement on a part of a total is expressed as part/total ≥ percent.
- Multiply both sides by the positive total (K + T) to obtain a linear inequality.
- Standard algebraic manipulation to isolate terms yields an equivalent linear constraint.

2) Complete reasoning process, step by step
Start with the fraction form:
K/(K + T) ≥ 0.40, assuming K + T > 0 (nonzero total investment).
Multiply both sides by (K + T):
K ≥ 0.40(K + T)

Distribute 0.40 on the right:
K ≥ 0.40K + 0.40T

Subtract 0.40K from both sides:
K − 0.40K ≥ 0.40T
0.60K ≥ 0.40T

Bring all terms to the left to form a standard inequality:
0.60K − 0.40T ≥ 0

3) Intermediate steps shown (already displayed above)
K/(K+T) ≥ 0.4
→ K ≥ 0.4K + 0.4T
→ 0.6K ≥ 0.4T
→ 0.6K − 0.4T ≥ 0

You can further simplify by dividing by 0.2 (positive), giving 3K ≥ 2T, which is equivalent.

4) Why the correct answer is correct
0.6K − 0.4T ≥ 0 is exactly the algebraic rearrangement of the statement "K is at least 40% of K+T." It enforces that the Kumoto portion is no less than 40% of the combined investment.

5) Why the other options are incorrect
- 0.4K − 0.6T ≥ 0 implies 0.4K ≥ 0.6T or K ≥ 1.5T, a much stronger and different condition.
- 0.4K − T ≤ 0 is 0.4K ≤ T or K ≤ 2.5T, which does not enforce K being at least 40% of the total.
- 0.4K − T ≥ 0 is 0.4K ≥ T or K ≥ 2.5T, again not equivalent to the 40% requirement.

6) Conclusion
The correct linear inequality modeling the requirement is 0.6K − 0.4T ≥ 0, which matches the provided answer.

A fund manager must decide how much money to invest in Kumoto stock (K) and how much to invest in Takayama stock (T). One requirement is that at least 40% of the money invested in the two companies' stock must be in Kumoto stock. A correct modelling of this requirement is:单项选择题

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