GRE Quantitative Comparison Question Strategy

Mastering Quantitative Comparison questions is a non-negotiable skill for a high GRE score. These problems test your logical reasoning and number sense more than brute calculation, and a systematic approach can turn them from time-consuming puzzles into quick, confident points. Learning to efficiently compare two quantities, often containing variables, is central to maximizing your performance on the math section.

Understanding the Battlefield and the Rules

The format of a Quantitative Comparison question is always the same. You are given two quantities: Quantity A and Quantity B. Your task is not to solve for a specific value, but to determine the relationship between them. You must choose from four fixed answer choices: A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

These choices are always the same, so you should memorize them. The key to success lies in recognizing that your goal is to compare, not necessarily to calculate. Often, you can determine the relationship with minimal arithmetic if you understand the underlying mathematical principles. The fourth choice, "cannot be determined," is correct when the relationship between the quantities changes depending on what numbers you plug in for the variables. Spotting this possibility early is a critical strategy.

Strategy 1: The Power of Plugging In Numbers

The most effective and versatile strategy is to plug in numbers. This involves selecting simple, permissible values for any variables and calculating the resulting values for Quantity A and Quantity B. The process must be systematic.

1. Identify the Constraints: First, note any restrictions on variables (e.g., $x>0$, $y$ is an integer). Your chosen numbers must obey these rules.
2. Start Simple and Standard: Begin with easy, "nice" numbers like 0, 1, 2, -1, or $\frac{1}{2}$. Calculate both quantities.
3. Analyze the Outcome: If, after one plug-in, Quantity A is greater, your tentative answer is A. However, you are not done. You must now...
4. Try a Second Set of Numbers designed to try to break the initial relationship. If you used a positive number, try a negative one (if allowed). If you used an integer, try a fraction. If you used a small number, try a large one. The goal is to see if you can make Quantity B become greater, or the two quantities equal.

If you can find two different sets of numbers that produce two different relationships (e.g., A > B in one case, B > A in another), then you have proven that the answer is D: The relationship cannot be determined. If every permissible number you try yields the same consistent relationship, then that relationship is the answer.

Example: Compare $x^2$ (Quantity A) and $x^3$ (Quantity B), where $x$ is a real number.

• Plug in $x=2$: A = 4, B = 8. B is greater.
• Now try to break that. Plug in $x=-1$: A = 1, B = -1. A is greater.
• Since the relationship changed, the answer is D.

Strategy 2: Simplify Before You Compare

Before you start plugging in numbers, always look to simplify one or both quantities algebraically. Often, the quantities share common terms that can be canceled, or they can be factored or rewritten in a comparable form. This simplification can reveal the relationship instantly or make the plug-in process much simpler.

The guiding principle is: Perform the same operation on both quantities. You can add, subtract, multiply, or divide both quantities by the same positive number without changing their relationship. (Multiplying or dividing by a negative number reverses the inequality, which is risky; avoid it by testing numbers instead).

Example: Compare $\frac{4x+8}{2}$ (Quantity A) and $2x+4$ (Quantity B). Instead of plugging in, simplify Quantity A: $\frac{4x+8}{2}=2x+4$. You now see Quantity A and Quantity B are identical expressions. Therefore, they are equal for any value of $x$, and the answer is C.

Strategy 3: The Critical Role of Edge Cases

Certain numbers have special properties that can reverse or nullify standard mathematical relationships. Systematically testing these edge cases is how you reliably identify "D" answers and avoid traps. The most common edge cases to test are:

• Zero: A multiplier of zero can make entire quantities vanish.
• One and Negative One: These numbers behave uniquely with exponents ($1^n=1$, $(-1{)}^{even}=1$, $(-1{)}^{odd}=-1$).
• Fractions between 0 and 1: For positive numbers, $x^2<x$ when $0<x<1$. This is a classic trap.
• Extremes: Very large positive numbers and very small negative numbers.

A disciplined test of edge cases often provides the proof you need. For instance, if a variable is defined as $x>0$, you should test $x=\frac{1}{2}$ and $x=10$ to see if the relationship between $x$ and $x^2$ holds consistently.

Common Pitfalls

Pitfall 1: Assuming variables represent positive integers. The GRE loves to test your awareness that numbers can be zero, negative, or fractions. If no constraints are given, a variable could be any real number. Always consider non-integer and negative possibilities when testing.

Pitfall 2: Stopping after one successful test case. Finding that Quantity A is greater after plugging in $x=5$ does not mean the answer is A. You must ask, "Can I find a permissible number that would make B greater or equal?" If you can conceive of such a number, you must test it.

Pitfall 3: Overlooking shared information. Some questions have a centered piece of information (like a geometric diagram or an equation) above the two quantities. This information applies to both quantities equally. You must use it in your evaluation of each quantity.

Pitfall 4: Wasting time solving completely. Remember, you rarely need a precise value. If you can see that Quantity A simplifies to $5x+10$ and Quantity B simplifies to $5x+9$, you instantly know A is greater by 1, regardless of $x$. No further calculation is needed.

Summary

• Your goal is to compare, not calculate. Memorize the four answer choices and focus on determining the relationship.
• Systematically plug in numbers, starting with simple values and then intentionally trying to break the initial result by testing different types of numbers (negative, zero, fraction).
• Always simplify both quantities algebraically before doing any other work; they may become directly comparable.
• Make testing edge cases (0, 1, -1, fractions) a standard part of your process to expose "cannot be determined" answers.
• Choice D is correct when the relationship changes based on different permissible inputs. Your job is to actively look for this possibility.
• Avoid assumptions about numbers being positive, integers, or large. Let the constraints in the problem guide your testing.