Quantitative Comparison

What Is Quantitative Comparison?

Compare Quantity A with Quantity B using the given conditions. Decide whether A is greater, B is greater, the quantities are equal, or the relationship cannot be determined. You need a valid relationship, not necessarily the individual values.

Detail	Study reference
Exam	GRE General Test
Response	Choose one of four relationships
Timing	Shared Quantitative section timer
Primary skill	Domain restrictions and valid proof

How It Is Scored

Choose one of the four standard relationship options. A single counterexample can disprove an assumed fixed relationship. Numerical testing supports a proof only when the tested cases exhaust the permitted possibilities.

Official Video

ETS introduces the skills assessed across the GRE. This is a general exam overview; use the worked examples below for this task.

Explore ETS preparation videos for Verbal Reasoning, Quantitative Reasoning and Analytical Writing.

Technical Foundations

Domains first

An unspecified variable is not automatically a positive integer. Test negative values, zero, proper fractions and values above one when allowed. Conditions such as x > 0 rule out some cases but still allow fractions.

Safe algebra

Adding the same expression preserves a comparison. Multiplying an inequality by a negative number reverses it; multiplying by an expression of unknown sign can be invalid. Division by zero is undefined.

Counterexamples

If one valid input makes A greater and another makes B greater or equal, the relationship is undetermined. Conversely, trying several values that agree does not prove the result for all values.

Step-by-Step Approach

Step	Action and purpose
1	List all restrictions on variables and quantities.
2	Simplify shared terms without unsafe division or squaring.
3	Try boundary values and contrasting permitted cases.
4	Choose a fixed relationship only with proof; otherwise exhibit differing valid outcomes.

Worked Examples

These are original teaching examples, not official ETS questions. Suggested practice timings are study targets, not extra test time.

Example 1

Given x > 0, compare A: x² with B: x.

Explanation / model: Cannot be determined. At x = 2, A = 4 > 2. At x = 1/2, A = 1/4 < 1/2. At x = 1 the quantities are equal. Positivity alone does not settle the comparison.

Example 2

Given x < 0, compare A: x² + 1 with B: x.

Explanation / model: A is greater. A is at least 1, while B is negative. No exact value of x is needed; the sign argument proves the relationship.

Common Mistakes to Avoid

Mistake	Why it fails	Correction
Assuming a diagram is to scale	Appearance is not a stated measurement.	Use labels and geometric facts.
Testing only integers	Fractions can reverse comparisons involving powers.	Include 1/2 when permitted.
Dividing by x without checking	x may be zero or negative.	Separate sign cases first.

Practice Routine

Practise ten comparisons, writing either a proof or two contrasting valid examples for every answer. Maintain a domain checklist: sign, zero, fraction, equality boundary, integer restriction. Retest each error with a new value rather than memorizing the original numerical example.

Quick Reference

Check	Reminder
Before answering	List all restrictions on variables and quantities.
Before submitting	Choose a fixed relationship only with proof; otherwise exhibit differing valid outcomes.
Review focus	Domain restrictions and valid proof

Sources and Format Check

Format checked on 22 October 2026 against ETS GRE guidance. Explanations, examples, routines and time allocations within tasks are teaching guidance. Follow the instructions and timer on your test screen.