GRE Prep: Quantitative Comparison Questions (Strategy: Avoid Calculation, Simplify, Plug Numbers)

GRE – Quantitative Comparison Questions (Strategy: Avoid Calculation, Simplify, Plug Numbers)

GRE Quantitative Comparison – Strategy: Avoid Calculation, Simplify, Plug Numbers

What This Is

Quantitative Comparison (QC) items ask you to decide which of two given expressions (Quantity A and Quantity B) is larger, or whether they are equal or incomparable. The goal is not to compute exact values but to determine the relationship quickly by simplifying, using algebraic tricks, or testing a single convenient number. For example:

Quantity A: (\displaystyle \frac{3x+5}{x+2})  Quantity B: 3.
You must decide whether A > B, A < B, A = B, or the relationship cannot be determined without exhaustive calculation.

Key Terms & Rules

• Quantity A / Quantity B: The two expressions you compare; answer choices are (A) A > B, (B) A < B, (C) A = B, (D) cannot be determined, (E) both A > B and A < B are possible.
• Plug‑in Method: Choose a value that satisfies all constraints (often a small integer) and substitute it to test the inequality.
• Simplify First: Cancel common factors, combine like terms, or rewrite fractions before plugging numbers.
• Monotonicity: Recognize when a function is always increasing or decreasing (e.g., (x^2) for (x\ge0)) to infer the direction of the inequality.
• Extreme‑Value Test: If a variable is bounded, test the smallest and largest permissible values; the relationship that holds at both extremes is the answer.
• Cross‑Multiplication (Positive Denominators): When denominators are known to be positive, you may multiply both sides without flipping the inequality sign.
• Common‑Factor Cancellation: If both quantities share a factor that is always positive, cancel it to reduce the comparison.
• Sign‑Analysis: Determine the sign of each expression (positive, negative, zero) to eliminate impossible answer choices.
• Variable‑Free Comparison: Sometimes the expressions reduce to a constant independent of the variable; then the answer is immediately known.
• “Cannot be Determined” Trap: Occurs when the variable’s domain allows the inequality to flip; watch for hidden sign changes or division by zero.

Step‑by‑Step Process Flow

1. Read the stem & note constraints – Identify any restrictions on the variable (e.g., “(x) is a positive integer”).
2. Simplify each quantity – Cancel factors, combine fractions, or rewrite using algebraic identities.
3. Check monotonicity or sign – Decide if the expression is always increasing/decreasing or always positive/negative in the given domain.
4. Plug a convenient number – Choose the simplest admissible value (often 0, 1, or the bound) and substitute into the simplified forms.
5. Compare the results – If the inequality holds for the test value and monotonicity guarantees it cannot reverse, select the corresponding answer choice; otherwise, answer “cannot be determined.”

Common Mistakes

• Mistake: Assuming a denominator is positive without checking the domain.
  Correction: Verify the sign of every denominator; if it could be negative, cross‑multiplication may flip the inequality.

• Mistake: Plugging a number that violates a hidden restriction (e.g., (x=0) when the expression contains (\frac{1}{x})).
  Correction: Always respect the stated domain; choose a value that keeps every term defined.

• Mistake: Cancelling a factor that could be zero.
  Correction: Only cancel factors that are guaranteed non‑zero in the given domain; otherwise you may lose a critical case.

• Mistake: Concluding “cannot be determined” because the test value gave equality, without checking other values.
  Correction: Test at least two distinct admissible values (or use monotonicity) to see if the relationship can change.

• Mistake: Doing full arithmetic instead of simplifying first, leading to time pressure.
  Correction: Prioritize algebraic reduction; a simplified form often reveals the answer without any calculation.

Exam Insights

1. Most‑tested trick: Expressions that look messy often reduce to a constant after cancellation—look for common factors early.
2. Typical distractor: Answer choice (D) “cannot be determined” appears when a variable can be both positive and negative; the test will hide a sign‑flip in a denominator or square root.
3. Bounded‑variable questions frequently ask you to compare a linear expression to a quadratic one; the extreme‑value test (plug the smallest and largest allowed values) usually settles the answer.
4. “Plug‑in” is a time‑saver for problems where the variable appears only once in each quantity; a single well‑chosen integer often decides the comparison.

Quick Check Questions

1. Quantity A: (\displaystyle \frac{2n+4}{n+2})  Quantity B: 2  ((n) is a positive integer)
   Answer: (C) Quantity A = Quantity B.
   Explanation: Simplify (\frac{2n+4}{n+2}=2) after factoring a 2 from the numerator; the ratio is exactly 2 for any (n\neq -2).

2. Quantity A: (\displaystyle 5 - \frac{1}{x})  Quantity B: 4  ((x>0))
   Answer: (A) Quantity A > Quantity B.
   Explanation: Since (\frac{1}{x}>0), (5-\frac{1}{x}) is always greater than (5-1=4).

3. Quantity A: (\displaystyle \frac{3k}{k+1})  Quantity B: 2  ((k) is a positive integer)
   Answer: (D) Cannot be determined.
   Explanation: For (k=1), A = 1.5 < 2; for (k=10), A ≈ 2.73 > 2. The inequality flips, so the relationship is indeterminate.

Last‑Minute Cram Sheet

1. Cross‑multiply only when denominators are positive; otherwise flip the inequality sign. ⚠️
2. Cancel a factor only if you know it’s never zero in the given domain. ⚠️
3. If both quantities simplify to the same expression, answer (C) – they are equal.
4. Plug‑in 1 (or the smallest allowed integer) first; many GRE problems are built around that value.
5. When a variable appears inside a square root, remember the expression is non‑negative; use this to eliminate sign‑flip traps.
6. Monotonic functions (e.g., (x^2) for (x\ge0)) keep the order of inequalities unchanged.
7. If a quantity is a constant plus a fraction with a positive denominator, the fraction’s sign decides the comparison.
8. Extreme‑value test: evaluate at the lower and upper bounds; if the relationship holds at both, it holds everywhere.
9. “Cannot be determined” appears when the variable can make the denominator zero or change sign. ⚠️
10. Always write down the domain constraints before simplifying; they often prevent illegal cancellations.