How to Solve GRE Quantitative Comparison (QC) Questions: The Complete Guide

How to Solve GRE Quantitative Comparison (QC) Questions: The Complete Guide

(1,200+ words, actionable under timed conditions)

Introduction

"Quantitative Comparison appears 7–8 times on every GRE Quant section—master it, and you’ll gain 3–5 raw points, enough to jump from the 70th to the 90th percentile."

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The GRE isn’t testing your ability to compute; it’s testing your ability to compare under pressure. Specifically, it probes for:

1. Precision in comparison, not calculation – Can you determine which quantity is larger without solving for exact values?
2. Resistance to traps – Will you fall for common pitfalls (e.g., assuming variables are positive, ignoring zero, or misapplying formulas)?
3. Efficiency under constraints – Can you make the comparison in 60–90 seconds without overcomplicating?

ANATOMY OF THE QUESTION

Structure Breakdown

Representative Example

Question: Let x be a positive integer. Quantity A: ( x^2 + 5x + 6 ) Quantity B: ( (x + 2)(x + 3) )

Answer Choices: A) Quantity A is greater. B) Quantity B is greater. C) The two quantities are equal. D) The relationship cannot be determined.

(Correct answer: C)

THE DECISION FRAMEWORK (Step-by-Step)

Run this every time, in order. No skipping.

Step 1: Read the Stem and Circle Constraints

• Action: Underline or circle any conditions (e.g., "x > 0," "n is even").
• Why: These often determine whether D is correct.
• Example: In the question above, "x is a positive integer" rules out D (since the relationship holds for all such x).

Step 2: Simplify Both Quantities

• Action: Expand, factor, or rewrite both quantities to make them comparable.
• If algebraic: Factor, combine like terms, or express in terms of a common variable.
• If numerical: Compute or estimate.
• Why: Direct comparison is faster than solving separately.
• Example:
• Quantity A: ( x^2 + 5x + 6 )
• Quantity B: ( (x + 2)(x + 3) = x^2 + 5x + 6 ) → Both simplify to the same expression.

Step 3: Compare Directly

• Action: Ask: "Is A > B, A < B, A = B, or does it depend?"
• If algebraic: Subtract B from A. If the result is always positive/negative/zero, the answer is A/B/C.
• If numerical: Plug in numbers (see Step 4).
• Why: Avoids unnecessary computation.
• Example:
• ( A - B = (x^2 + 5x + 6) - (x^2 + 5x + 6) = 0 ) → A = B (Answer C).

Step 4: Test Edge Cases (If Needed)

• Action: If the relationship isn’t obvious, plug in extreme values (e.g., 0, 1, -1, large numbers).
• Rule: Test at least 2 values (e.g., x = 1 and x = 100).
• Why: Some relationships change based on input (e.g., ( x^2 ) vs. ( x )).
• Example:
• If Quantity A = ( x ) and Quantity B = ( x^2 ):
  • x = 0.5 → A > B
  • x = 2 → B > A → Answer D.

Step 5: Eliminate Wrong Answers

• Action: Cross out options that contradict your findings.
• If A > B for all tested values, eliminate B, C, and D.
• If A = B for all tested values, eliminate A, B, and D.
• Why: Narrows choices even if you’re unsure.

Step 6: Choose the Best Answer

• Action: Pick the remaining option.
• If you’re stuck between C and D, ask: "Does the relationship always hold, or does it depend on the variable?"
• Why: Forces a decision under time pressure.

Worked Examples

Example 1: Straightforward (Algebraic)

Question: Let x be a real number. Quantity A: ( 3x + 5 ) Quantity B: ( 2x + 10 )

Step 1: No constraints (x is any real number). Step 2: Subtract B from A: ( (3x + 5) - (2x + 10) = x - 5 ) Step 3: Compare ( x - 5 ) to 0: - If ( x > 5 ), A > B. - If ( x = 5 ), A = B. - If ( x < 5 ), A < B. Step 4: Since x can be any real number, the relationship depends on x. Answer: D

Example 2: Common Trap (Ignoring Zero)

Question: Let n be an integer. Quantity A: ( n^2 ) Quantity B: ( n^3 )

Step 1: n is an integer (no other constraints). Step 2: Test values: - n = 0 → A = 0, B = 0 → A = B - n = 1 → A = 1, B = 1 → A = B - n = 2 → A = 4, B = 8 → B > A - n = -1 → A = 1, B = -1 → A > B Step 3: The relationship changes. Trap: Students assume n is positive and pick B. Answer: D

Example 3: Hard Variant (Inequalities)

Question: Let ( 0 < x < 1 ). Quantity A: ( x^2 ) Quantity B: ( \sqrt{x} )

Step 1: Constraint: ( 0 < x < 1 ). Step 2: Test x = 0.25: - A = ( (0.25)^2 = 0.0625 ) - B = ( \sqrt{0.25} = 0.5 ) → B > A Step 3: Test x = 0.81: - A = ( (0.81)^2 = 0.6561 ) - B = ( \sqrt{0.81} = 0.9 ) → B > A Step 4: For all ( 0 < x < 1 ), ( \sqrt{x} > x^2 ). Answer: B

WRONG ANSWER PATTERNS

These are the exact traps the GRE uses. Memorize them.

Common Mistakes

TIME STRATEGY

• Target time: 60–90 seconds per question.
• When to skip: If you’re stuck after 2 minutes, flag and return.
• Minimum work needed:
• Simplify both quantities.
• Test 1–2 edge cases (if needed).
• Eliminate wrong answers.

Pro Tip: If you can’t simplify, plug in numbers immediately (Step 4).

BACKSOLVING AND SHORTCUTS

1. Number Substitution (Plugging In)

• When to use: When quantities are algebraic and the relationship isn’t obvious.
• How:
• Pick easy numbers (0, 1, -1, 10, 0.5).
• Test extreme values (very large/small numbers).
• Example:
• Quantity A: ( x^2 + 1 )
• Quantity B: ( 2x )
• Test x = 0 → A = 1, B = 0 → A > B
• Test x = 1 → A = 2, B = 2 → A = B
• Test x = 2 → A = 5, B = 4 → A > B → Relationship depends on x → D.

2. Elimination-First Strategy

• When to use: When you’re unsure but can rule out options.
• How:
• If A > B for one value, eliminate B and C.
• If A < B for another value, eliminate A and C.
• If A = B for any value, eliminate A and B.
• Example:
• Quantity A: ( x )
• Quantity B: ( x^2 )
• x = 0.5 → A > B (eliminate B, C)
• x = 2 → B > A (eliminate A) → Only D remains.

3. Avoid Exact Calculation

• When to use: When quantities are numerical but complex.
• How: Estimate or compare ratios.
• Example:
• Quantity A: ( \frac{1}{3} + \frac{1}{4} )
• Quantity B: ( \frac{1}{2} )
• ( \frac{1}{3} \approx 0.33 ), ( \frac{1}{4} = 0.25 )
• A ≈ 0.58, B = 0.5 → A > B.

1-Minute Recap

"Here’s your QC battle plan for test day:

1. Circle constraints first—they decide if D is correct.
2. Simplify both quantities—expand, factor, or rewrite to compare directly.
3. Test edge cases—plug in 0, 1, -1, and large numbers if the relationship isn’t obvious.
4. Eliminate wrong answers—cross out options that contradict your findings.
5. Pick the best remaining choice—if you’re stuck between C and D, ask: Does the relationship always hold, or does it depend?

Remember: QC isn’t about solving—it’s about comparing. Stay disciplined, and you’ll gain 3–5 points on Quant. Now go practice!

KEY TAKEAWAYS

1. Simplify first, plug in second.
2. Test edge cases to avoid traps.
3. D is only correct if the relationship truly depends on the variable.
4. Spend 60–90 seconds max per question.

Next Steps: - Drill 20–30 QC questions under timed conditions. - Review every mistake using the framework above. - Focus on why you got it wrong (e.g., ignored constraints, assumed positivity).

Master this, and you’ll consistently score in the 90th+ percentile on GRE Quant.