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Difficulty: HardIdentifying Must-Be-True Statements and Valid Deductions

In Metropolitan Area M, municipal agricultural policies dictate that every commercial urban farm utilizing hydroponic vertical farming methods receives a Tier-1 municipal water rate discount. Furthermore, any agricultural initiative in Metropolitan Area M receiving a Tier-1 municipal water rate discount is subject to an annual environmental audit conducted by the Department of Ecology. Finally, no agricultural initiative in Metropolitan Area M that is subject to an annual environmental audit conducted by the Department of Ecology is permitted to use synthetic chemical pesticides. During the last fiscal year, exactly 40 percent of all urban farms operating in Metropolitan Area M used synthetic chemical pesticides.

If the statements above are true, which of the following must also be true on the basis of them?

  1. A
    No urban farm in Metropolitan Area M that refrained from using synthetic chemical pesticides during the last fiscal year was subject to an annual environmental audit.
  2. B
    Every urban farm in Metropolitan Area M that was subject to an annual environmental audit during the last fiscal year utilized hydroponic vertical farming methods.
  3. At least 40 percent of the urban farms operating in Metropolitan Area M during the last fiscal year did not receive a Tier-1 municipal water rate discount.Answer
  4. D
    During the last fiscal year, more than half of all urban farms in Metropolitan Area M received a Tier-1 municipal water rate discount.
  5. E
    At least 40 percent of the urban farms operating in Metropolitan Area M during the last fiscal year were subject to an annual environmental audit.

Answer

At least 40 percent of the urban farms operating in Metropolitan Area M during the last fiscal year did not receive a Tier-1 municipal water rate discount.
The stimulus establishes a conditional chain: Hydroponic vertical farming implies a Tier-1 water discount, which implies an environmental audit, which in turn implies that synthetic chemical pesticides are NOT used. Thus, any farm receiving a Tier-1 water discount cannot use synthetic chemical pesticides (Tier-1 Discount → No Synthetic Pesticides). Taking the contrapositive reveals that if a farm uses synthetic chemical pesticides, it CANNOT receive a Tier-1 water discount (Synthetic Pesticides → No Tier-1 Discount). Since exactly 40 percent of urban farms used synthetic chemical pesticides, those specific 40 percent must not have received a Tier-1 discount. Therefore, it must be true that at least 40 percent did not receive the discount.

Step-by-Step Solution

1
Formalize the conditional statements given in the premises.
Hydroponic (H) → Tier-1 Discount (T) → Environmental Audit (E) → NO Synthetic Pesticides (~P).
Establishing the chain of necessary conditions allows us to trace deductions across all premises.
2
Deduce the transitive relation between Tier-1 Discount and Synthetic Pesticides, then find its contrapositive.
Direct chain: T → ~P. Contrapositive: P → ~T (If a farm uses synthetic pesticides, it does NOT receive a Tier-1 discount).
Taking the contrapositive provides a valid deduction connecting pesticide use directly to discount status.
3
Apply the given numerical fact to the contrapositive deduction.
Since 40% of farms used synthetic pesticides (P), those exact 40% cannot receive a Tier-1 discount (~T).
Combining the contrapositive rule with the explicit percentage proves that at least 40% of farms must not receive the discount.

Key Concept

Conditional Logic Chains and Contrapositive Inferences
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Topics

Identifying Must-Be-True Statements and Valid Deductions

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