GRE Quantitative Comparison

What a Quantitative Comparison question asks

Quantitative Comparison (QC) presents Quantity A and Quantity B, along with information or conditions. You select one of four standard relationships: A is greater; B is greater; the quantities are equal; or the relationship cannot be determined from the information. The method is different from a normal solve-for-x problem. You need to know what relationship is always supported, and often you can stop without finding exact values.

A common mistake is to calculate one convenient example and assume it proves the relationship. If a variable can take multiple values, a single case cannot establish an always-true claim. Another mistake is to compare expressions without checking restrictions such as positive, integer, nonzero, or geometric conditions. The given restrictions determine which test values are legitimate.

A dependable four-step method

1. Read all stated conditions and rewrite the variable restrictions in plain language.
2. Simplify each quantity or compare their difference if that makes the relationship clearer.
3. Test useful values, especially zero, negative numbers, fractions, and boundary values when allowed.
4. Choose the relationship that is true for every permitted case, not just the first case you tried.

Testing values is not a shortcut that replaces algebra. It is a way to disprove a relationship or discover whether the answer varies. If two test cases produce different relationships, then neither ‘A greater,’ ‘B greater,’ nor ‘equal’ is always correct; choose the cannot-be-determined option. If all cases suggest one answer, use algebra or the stated constraints to confirm it.

Worked example: unrestricted variable

This example illustrates why negative values and fractions matter. Many candidates test only positive integers and miss a counterexample. Before using test values, ask what the problem says about the variable. If it only says ‘real number,’ negative and fractional values are allowed. You should not silently assume that x is a positive integer.

Worked example: a restriction changes the result

When the algebra establishes a relationship for all permitted values, trust it. QC is not always a trap requiring exotic numbers. The skill is deciding whether a test case is enough, whether a restriction proves a general result, or whether the result varies.

Boundary values and zero

Zero is a useful test value when it is allowed. Expressions with denominators, square roots, or logarithms may restrict whether zero is valid, so check the domain first. For an inequality involving x² and x, zero can expose equality. For a fraction, a denominator cannot be zero. For a positive variable, zero is not permitted, but values close to zero may still be informative.

Boundary cases also matter with inequalities. If a variable is at least 3, test 3 and a larger value. If it is between 0 and 1, test a fraction such as one-half. If it is a positive integer, one may be the smallest allowed value. These cases often reveal whether a claim changes at a cutoff.

Worked example: absolute value

The phrase ‘any real number’ includes positive, zero, and negative values. If the prompt instead specified x < 0, the relationship would be determined. Read the condition before applying familiar identities.

Use algebra without over-solving

Sometimes subtracting Quantity B from Quantity A makes the comparison simple. If A − B is positive for every allowed value, A is greater; if negative, B is greater; if always zero, they are equal. If the sign depends on a variable whose range is unknown, the relationship may not be determined. Factor expressions carefully and preserve the restrictions.

For example, if A = (x + 1)² and B = x² + 2x, then A − B equals 1, so A is always greater for every real x. There is no need to expand both values for specific x. A short algebraic comparison can be more reliable than repeated substitution.

Geometry and diagrams

If a QC item includes a figure, use the stated measurements and relationships rather than assuming it is drawn to scale. A diagram may clarify which lengths or angles are related, but visual appearance does not establish exact equality. If the figure indicates a triangle but gives no side or angle constraint, do not assume it is isosceles because two sides look similar.

For geometric variables, list the actual restrictions. A length is positive; an angle may have a specified range; triangle side lengths must satisfy triangle inequalities. Those constraints can make a relationship determinate even when algebraic variables appear free. Conversely, missing information may leave several shapes possible.

A more complex worked example

The condition ‘positive’ excludes negative and zero, but it does not imply that x is at least 1. Testing one fraction, the boundary value, and one value above the boundary reveals all three relationships. A candidate who tests only x = 2 would select the wrong strict relationship.

Time management

The current GRE has 12 questions in its first Quantitative section with 21 minutes and 15 in the second with 26 minutes. QC items are included among the quantitative questions, but ETS does not promise a fixed number of QC questions. Practice the format without assuming an exact section composition.

Set up the comparison, test a small number of high-value cases, and move on when the relationship is clear. Do not calculate both quantities completely if a difference or counterexample resolves the problem. If the relationship cannot be determined, make sure it truly varies among allowed values; do not choose that option merely because the algebra looks unfamiliar.

Common mistakes

• Assuming an unstated variable is positive or an integer.
• Testing only one value and treating it as proof.
• Forgetting that equal can occur for one value while the overall relationship still varies.
• Treating a diagram as precisely drawn when the prompt does not say so.
• Ignoring domain restrictions such as nonzero denominators or positive side lengths.
• Continuing to compute after a counterexample already shows the relationship is not fixed.

Practice routine

After each QC problem, write the condition, test cases considered, and proof or counterexample. Include at least one case outside the obvious positive integer examples. Review whether you selected the answer from a general argument or from intuition. As your accuracy improves, add section timing and mixed quantitative questions so you practice switching methods.

The central habit is to ask what must be true for every allowed value. Sometimes a simple subtraction proves the answer. Sometimes a boundary value settles it. Sometimes two valid test cases prove that the relationship changes. Applying that habit makes QC a structured comparison instead of a guessing game.

A final worked example: do not overread a diagram

This example uses a geometric relationship rather than a picture’s appearance. A candidate can compare the sides and angles from the stated facts. If the drawing makes the largest angle look only slightly above 60 degrees, that visual impression is irrelevant. Conversely, if the prompt gave only a triangle outline without side or angle facts, the diagram would not establish the requested comparison.

Check response instructions and partial work

Some quantitative items ask for one answer, while multiple-select items may require every correct choice. QC has its own four relationship options. Read the directions each time instead of assuming that all quantitative questions use the same response format. If you make a scratch calculation, label which quantity it represents; a correct arithmetic result can still lead to the wrong choice if it is assigned to Quantity A or B incorrectly.

Use constraints before choosing test values

A variable restriction changes which examples are valid. If the prompt says n is an integer, fractions are not permitted. If n is a positive integer, zero and negative values are excluded. If a variable appears in a denominator, values that make the denominator zero are excluded even if the prompt does not repeat the restriction. Write these limits before testing values so a counterexample is valid.

The reciprocal example shows why ‘positive’ is not enough to decide whether a number exceeds its reciprocal. The boundary x = 1 separates the cases. Testing one value on each side and the boundary quickly exposes the changing relationship. If the question had specified x > 1, then x would be greater than 1/x.

These problems reward a short proof after the test cases. A counterexample can show that a proposed fixed comparison is wrong; it cannot by itself prove that one relationship always holds. To establish a fixed answer, use algebra or the stated restrictions to cover every permitted value. That distinction keeps substitution useful without letting a few examples replace reasoning.