1.2 Quantitative Comparison (QC) Core Strategies

1.2 Quantitative Comparison (QC) Core Strategies

Quick Answer: Quantitative Comparison (QC) questions constitute roughly 30% to 40% of the GRE Quant section. Each item presents Quantity A and Quantity B alongside optional centered information, requiring you to determine which quantity is larger or if the relationship cannot be determined. Mastering QC relies on four core strategies: understanding the 4 fixed answer options, using the FROZEN number-plugging framework, performing simultaneous algebraic simplifications, and avoiding geometric scale traps.

Quantitative Comparison Question Anatomy

QC items feature a standardized format that never changes on test day. You are presented with two quantities—Quantity A and Quantity B—and must choose one of four constant choices:

• Choice A: Quantity A is greater.
• Choice B: Quantity B is greater.
• Choice C: The two quantities are equal.
• Choice D: The relationship cannot be determined from the information given.

+-------------------------------------------------------------+
| Given: x is an integer such that x > 1.                      |
|                                                             |
| Quantity A: x^2 + 1                                         |
| Quantity B: x^3                                             |
|                                                             |
| (A) Quantity A is greater.                                  |
| (B) Quantity B is greater.                                  |
| (C) The two quantities are equal.                           |
| (D) The relationship cannot be determined from information. |
+-------------------------------------------------------------+

The Fundamental Deterministic Rule (Choice D Trap)

A critical rule of GRE QC questions is: If a problem contains only fixed numerical constants and no variables, Choice D can NEVER be correct. For example, comparing $\sqrt{50}$ and $7$ yields a fixed mathematical truth ($7 = \sqrt{49} < \sqrt{50}$, so Quantity A is greater). Choice D only applies when variables or unspecified geometric dimensions allow multiple contradictory relationships (e.g., Quantity A > Quantity B in Case 1, but Quantity A < Quantity B in Case 2).

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The FROZEN Framework for Testing Values

When variables are present without tight constraints, testing numbers is one of the fastest ways to evaluate or eliminate choices. To thoroughly test for edge cases, memorize the FROZEN mnemonic:

The Two-Case Goal for Choice D

When plugging in numbers, your primary goal is to try to prove Choice D.

• Test Case 1: Pick $x = 2$. Evaluate quantities. Suppose Quantity A > Quantity B.
• Test Case 2: Pick an edge-case number from FROZEN (e.g., $x = 1/2$ or $x = 0$). If Quantity A < Quantity B or Quantity A = Quantity B, you have proven that multiple relationships exist. The answer is instantly Choice D.

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Simultaneous Algebraic Simplification

Instead of plugging in numbers, you can treat Quantity A and Quantity B like two sides of an inequality scale and simplify them simultaneously.

Permitted Operations (Preserve Comparison Direction):

1. Add or Subtract the same number or expression from both quantities.
2. Multiply or Divide both quantities by a POSITIVE number.
3. Simplify expressions on either side independently (e.g., factoring $x^2 - y^2$ to $(x-y)(x+y)$).

Forbidden Operations (Invalid / Dangerous):

1. DO NOT multiply or divide by a variable unless you know with 100% certainty that the variable is strictly positive. (If negative, the inequality sign flips!).
2. DO NOT square both sides unless both quantities are guaranteed to be positive. (For instance, $-5 < 2$, but squaring gives $25 > 4$).

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Worked Step-by-Step Math Example: Algebraic QC

Problem:

Given that $a > 0$ and $b > 0$, compare:

• Quantity A: $\frac{a^2 + 2ab + b^2}{a + b}$
• Quantity B: $\sqrt{a^2 + b^2}$

Step-by-Step Solution:

1. Simplify Quantity A Algebraically: Notice that the numerator of Quantity A is a perfect square trinomial: $a^2 + 2ab + b^2 = (a + b)^2$ Substitute this into Quantity A: $\text{Quantity A} = \frac{(a + b)^2}{a + b} = a + b$

2. Set Up the Simplified Comparison: Now compare:

   • Quantity A: $a + b$
   • Quantity B: $\sqrt{a^2 + b^2}$

3. Square Both Quantities (Valid since $a>0, b>0 \implies a+b > 0$):

   • $(\text{Quantity A})^2 = (a + b)^2 = a^2 + 2ab + b^2$
   • $(\text{Quantity B})^2 = (\sqrt{a^2 + b^2})^2 = a^2 + b^2$

4. Subtract $a^2 + b^2$ from Both Quantities:

   • $(\text{Quantity A})^2 - (a^2 + b^2) = 2ab$
   • $(\text{Quantity B})^2 - (a^2 + b^2) = 0$

5. Evaluate the Result: Since $a > 0$ and $b > 0$, the product $2ab$ must be strictly positive ($2ab > 0$). Therefore, $(\text{Quantity A})^2 > (\text{Quantity B})^2$, which means Quantity A is greater.

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Geometric Figure Traps: Figures Are NOT Drawn to Scale

One of the most frequent traps on GRE Quantitative Comparison involves geometric diagrams.

Official ETS Rule: Diagrams accompanying Quantitative Comparison questions are NOT drawn to scale unless explicitly stated. You cannot infer angle sizes, line segment lengths, or relative areas from visual appearance alone.

       B
      /|
     / |
    /  |
   /   |
  /____|
 A      C
 (Diagram shows triangle ABC appearing to be a right triangle with AB > BC)

If the prompt does NOT explicitly state that $\angle C = 90^\circ$, you cannot assume it is a right triangle. If it does not provide side measurements, you cannot assume $AB > BC$ merely because it looks longer on the screen. Unless backed by geometric theorems or explicit labels, the answer to visual comparison questions is almost always Choice D.

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Avoiding Unnecessary Calculations

Do not waste time computing massive multi-digit products when comparing magnitude.

Comparison Example:

• Quantity A: $53 \times 98$
• Quantity B: $52 \times 99$

Strategic Benchmark Analysis:

Rewrite both products using a shared base:

• $\text{Quantity A} = (52 + 1) \times 98 = 52 \times 98 + 98$
• $\text{Quantity B} = 52 \times (98 + 1) = 52 \times 98 + 52$

Subtracting $52 \times 98$ from both sides leaves:

• Quantity A: $98$
• Quantity B: $52$

Since $98 > 52$, Quantity A is greater without ever performing long multiplication!