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Any infrastructure project that utilizes synthetic materials must pass a secondary safety inspection. Furthermore, no project that passes a secondary safety inspection is completed within its original budget. Every project managed by the transit authority utilizes synthetic materials, and some projects managed by the transit authority are funded by state bonds. Finally, only projects that incorporate sustainable design are funded by state bonds.

If the statements above are true, which one of the following must also be true?

  1. At least one project that incorporates sustainable design fails to be completed within its original budget.Answer
  2. B
    Any project that incorporates sustainable design receives funding from state bonds.
  3. C
    If an infrastructure project is not managed by the transit authority, it will be completed within its original budget.
  4. D
    Infrastructure projects funded by state bonds face more stringent safety standards than other infrastructure projects.
  5. E
    There is at least one project that passes a secondary safety inspection but is not funded by state bonds.

Answer

The correct answer states that at least one project that incorporates sustainable design fails to be completed within its original budget.
The correct option is provable by connecting multiple formal premises. We know that some transit authority projects are funded by state bonds, and because ONLY projects with sustainable design receive state bonds, those specific transit projects must incorporate sustainable design. Furthermore, every transit project utilizes synthetic materials and thus passes a secondary safety inspection, which guarantees they are not completed within budget. Therefore, there exists at least one project (the state-bond-funded transit project) that incorporates sustainable design and is not completed within its original budget.

Step-by-Step Solution

1
Map the universal conditional chain starting from the transit authority.
Transit Authority -> Synthetic Materials -> Secondary Safety Inspection -> NOT Completed Within Budget.
This establishes that every single project managed by the transit authority ultimately fails to be completed within its original budget.
2
Translate the 'only' statement into a standard conditional.
The phrase 'only projects that incorporate sustainable design are funded by state bonds' translates to: State Bonds -> Sustainable Design.
The word 'only' modifies the necessary condition, meaning sustainable design is required for state bond funding.
3
Link the 'some' statement to the established conditions.
We are told: Transit Authority <-some-> State Bonds. Using the translation from Step 2, this means: Transit Authority <-some-> Sustainable Design.
Because some transit projects are funded by state bonds, and all state bond projects have sustainable design, at least some transit projects must have sustainable design.
4
Synthesize the overlapping groups to find the final necessary truth.
From Step 3, there is an intersecting group of projects that are both Transit Authority AND Sustainable Design. From Step 1, ALL Transit Authority projects go over budget. Therefore, this intersecting group of Sustainable Design projects must also go over budget.
Any characteristic that applies to ALL transit authority projects must also apply to the subset of transit authority projects that incorporate sustainable design.

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