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Difficulty: MediumPercentages vs. Numbers Flaws

Agricultural Commissioner: Over the past five years, the proportion of our region's total farmed acreage dedicated to growing the new drought-resistant sorghum has increased from 10 percent to 30 percent. Since farmers typically only expand the cultivation of a crop when it proves highly profitable, the absolute number of acres dedicated to growing drought-resistant sorghum in our region must be exactly three times what it was five years ago.

Which of the following statements must be true in order for the Agricultural Commissioner's conclusion to be logically drawn? Select all that apply.

  1. The total farmed acreage in the region is exactly the same today as it was five years ago.Answer
  2. The total number of acres dedicated to crops other than drought-resistant sorghum has decreased over the past five years.Answer
  3. C
    The absolute number of farmers growing drought-resistant sorghum in the region has tripled over the past five years.
  4. D
    Farmers in the region no longer allocate any acreage to crops that are not drought-resistant.
  5. E
    The profits generated per acre of drought-resistant sorghum are three times higher today than they were five years ago.

Answer

The statements that must be true are that the total farmed acreage is exactly the same today as it was five years ago, and that the total acreage dedicated to other crops has decreased.
The argument's conclusion requires the total acreage to remain unchanged. If total acreage changed, 30% of the new total would not equal exactly three times 10% of the old total. Consequently, if the total acreage is unchanged and sorghum's share grew by 20 percentage points, the share of all other crops must have shrunk by 20 percentage points, meaning their absolute acreage decreased.

Step-by-Step Solution

1
Analyze the premises and the conclusion.
Premise: Sorghum acreage increased from 10% to 30% of total acreage. Conclusion: Absolute sorghum acreage is exactly three times what it was.
Identifying the structural components highlights the leap between proportional evidence and an absolute numerical claim.
2
Evaluate the mathematical relationship required to make the conclusion valid.
For an absolute number to triple when its corresponding percentage also triples, the baseline total must remain constant.
If the total acreage had grown, the absolute acreage of sorghum would have more than tripled. If it shrank, it would have less than tripled.
3
Determine the necessary mathematical consequences of a constant total.
If the total stays the same and sorghum takes up a larger slice (from 10% to 30%), the slice for all other crops must shrink (from 90% to 70%).
This confirms that both the constant total acreage and the resulting decrease in other acreage are logically necessary assumptions.

Key Concept

To validly infer an absolute numerical change from a percentage change, the underlying total (the baseline) must be known and accounted for.
Estimated Time:1m 30s
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Topics

Percentages vs. Numbers Flaws

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