DAT Quantitative Reasoning: Contradiction Means Cannot Be Determined

DAT High-Yield Concept

The Contradiction Rule: Finding just two valid test values that yield opposite comparison outcomes proves conclusively that the relationship cannot be determined.
Instant Stop: Stop testing additional values the instant a single contradiction occurs—wasting time on further calculations is unnecessary.
Strict Constraint Compliance: Every test value selected must strictly satisfy all stated problem constraints (e.g., $x > 0$ , $x$ is an integer).
Empirical Fallacy: Repeated agreement across multiple test values suggests a consistent relationship but never proves it; you must verify algebraically before concluding a fixed outcome.

The Single-Contradiction Rule

Core Principle: In Quantitative Comparison questions, if one allowed scenario makes Quantity A greater than Quantity B, but another allowed scenario makes Quantity B greater than Quantity A (or makes them equal), the mathematical relationship depends on the specific variable chosen. Therefore, the relationship cannot be determined from the information given.

When testing numbers in Quantitative Comparison questions on the DAT, your primary strategic goal is to disprove a fixed outcome by looking for a contradiction. If two permissible values yield opposing relationships, the answer is immediately established as "the relationship cannot be determined."

One contradiction ends the question; agreement does not — Two permitted test values that give opposite comparisons prove that no fixed relationship exists, so the answer is that the relationship cannot be determined. With x greater than zero, one half makes one over x the larger quantity while 2 makes x the larger, and the question is finished at that moment. The logic runs one way only: a single contradiction is a proof, but agreement across several values is merely evidence and must be confirmed algebraically. Test values must also obey every restriction in the problem, since narrowing the range to x greater than 1 changes the answer from cannot be determined to B.

Worked Example 1: Finding a Contradiction

Problem: Given that $x > 0$ , compare Quantity A: $x^2$ and Quantity B: $x$ .

Step-by-Step Reasoning:

Select Case 1: Choose an integer satisfying $x > 0$ , such as $x = 2$ . Here, Quantity A $= 2^2 = 4$ and Quantity B $= 2$ . Since $4 > 2$ , Quantity A is greater.
Select Case 2: Search for a value that might reverse the result, such as a fraction between $0$ and $1$ . Choose $x = \frac{1}{2}$ (valid because $\frac{1}{2} > 0$ ). Here, Quantity A $= \left(\frac{1}{2}\right)^2 = \frac{1}{4}$ and Quantity B $= \frac{1}{2}$ . Since $\frac{1}{4} < \frac{1}{2}$ , Quantity B is greater.
Conclusion: Because Case 1 yields Quantity A > Quantity B while Case 2 yields Quantity A < Quantity B, a contradiction exists. Stop testing immediately; the relationship cannot be determined.

Common Exam Trap: Testing only whole positive integers (e.g., $x = 2$ and $x = 3$ ) leads to false agreement (Quantity A > Quantity B), causing students to select Quantity A as greater while ignoring positive proper fractions.

Honoring Stated Domain Restrictions

Test values are only valid if they honor every domain restriction declared in the problem stimulus. Violating restrictions leads to false contradictions and incorrect answers.

Mnemonic — C.A.P. Strategy for Value Testing:

C - Constraints First: Check domain rules ( $x > 0$ , integers only, $a \neq b$ ) before picking values.
A - Alternate Classes: Test distinct number classes allowed by the rule (positives, negatives, fractions, zero).
P - Prove or Stop: Stop instantly on contradiction; otherwise prove algebraically.

For instance, if a problem specifies that $x$ is a positive integer ( $x \in \{1, 2, 3, \dots\}$ ), you cannot test $x = 0$ , negative numbers, or fractions such as $x = \frac{1}{2}$ . Choosing illegal test values invalidates your comparison.

Empirical Agreement vs. Algebraic Proof

A common pitfall on the DAT is assuming that because several test values yield the same outcome, the relationship is guaranteed to hold universally. Empirical testing can only prove that a relationship cannot be determined (via contradiction); it can never rigorously prove that one quantity is always greater or equal.

Worked Example 2: Confirming Agreement Algebraically

Problem: Given that $x$ is an integer such that $x \ge 2$ , compare Quantity A: $x^2 + 1$ and Quantity B: $2x$ .

Step-by-Step Reasoning:

Test Values:
- If $x = 2$ : Quantity A $= 2^2 + 1 = 5$ ; Quantity B $= 2 \times 2 = 4$ . (Quantity A > Quantity B)
- If $x = 3$ : Quantity A $= 3^2 + 1 = 10$ ; Quantity B $= 2 \times 3 = 6$ . (Quantity A > Quantity B)
Recognize Need for Proof: Both test values yield Quantity A > Quantity B. However, this empirical agreement does not constitute mathematical proof.
Algebraic Confirmation: Compare the expressions by evaluating their difference: $(x^2 + 1) - 2x = x^2 - 2x + 1 = (x - 1)^2$ For any integer $x \ge 2$ , the expression $x - 1 \ge 1$ , so $(x - 1)^2 \ge 1^2 = 1 > 0$ . Since $(x^2 + 1) - 2x > 0$ , Quantity A is strictly greater than Quantity B for all allowed domain values.

Strategic Dimension	Contradiction Testing	Algebraic Proof
Primary Objective	Disprove a fixed relationship by finding 2 conflicting cases	Establish a universal truth for all valid domain values
Required Outcome	1 contradiction (e.g., A > B in Case 1, A < B in Case 2)	Algebraic inequality holding across all domain conditions
When to Stop	Immediately upon establishing the first contradiction	When the algebraic difference or simplification is conclusive
Conclusive Answer	The relationship cannot be determined	Quantity A is greater, Quantity B is greater, or Equal

Critical Thinking Questions

If testing $x = 1$ yields Quantity A = Quantity B, and testing $x = 2$ yields Quantity A > Quantity B, why is this sufficient to choose "cannot be determined" without testing negative numbers?
Suppose a prompt states $y < 0$ . What distinct classes of numbers must you test to thoroughly hunt for a potential contradiction?
Why does getting Quantity A > Quantity B for five consecutive test values fail to prove that Quantity A is greater for all valid inputs?

References

Sources returned by Google Search grounding during lesson verification.

DAT Quantitative Reasoning Quiz: Quantitative Comparison - Varsity Tutors — varsitytutors.com
Quantitative Comparison: 'The Relationship Cannot Be Determined from the Information Given' Answer Choice — magoosh.com

Contradiction Means Cannot Be Determined

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