Logical reasoning — Costs and materials (choose the statement that must be true) Facts: Eyeglass frames cost between $35 and $350. Some eyeglass frames are made of titanium. Some eyeglass frames are made of plastic. Which statements must also be facts? I: Titanium frames cost more than plastic frames. II: Expensive frames last longer than cheap frames. III: Only a few frames cost less than $35.

Introduction / Context:The problem asks us to separate known facts from assumptions about price and durability. The given facts specify a price range and the existence of two materials; they do not give relative costs or performance differences for those materials.

Given Data / Assumptions:

• All eyeglass frames are priced in the interval [$35, $350].
• There exist titanium frames and plastic frames.
• No further relation between price and material is stated.

Concept / Approach:To be a “must,” a statement must hold in every interpretation compatible with the facts. If we can imagine any consistent scenario where a statement fails, it is not a necessary fact.

Step-by-Step Solution:

Verification / Alternative check:Pick any valid price examples (e.g., titanium at $80, plastic at $200; or swap). All facts hold, but I fails in at least one scenario. Similarly, II and III are independent of given data or contradict it.

Why Other Options Are Wrong:

• I only / II only / II and III only / I and III only: Each asserts information not secured by the premises.

Common Pitfalls:Importing real-world stereotypes (e.g., “titanium is pricier” or “expensive lasts longer”) when the logical task forbids unstated assumptions.

Final Answer:None of the statements is a known fact.