**GRE Quantitative Comparison: Plugging In, Simplifying, & Geometric Reasoning**

GRE Quantitative Comparison: Plugging In, Simplifying, & Geometric Reasoning

A Premium Study Guide for Serious GRE Candidates

What This Is

Quantitative Comparison (QC) questions make up ~35% of the GRE Quant section (7–8 questions per test). They ask you to compare two quantities (Column A vs. Column B) and determine which is larger—or if the relationship cannot be determined. Mastering QC is the fastest way to boost your Quant score because: - Speed: QC questions are shorter than Problem Solving questions, so efficiency here frees up time for harder problems.
- Accuracy: The GRE tests conceptual understanding (not just calculations), so plugging in numbers, simplifying expressions, and geometric reasoning are critical.
- Traps: ETS designs QC questions to exploit common misconceptions (e.g., assuming variables are positive, ignoring zero, or misapplying geometric properties).

Example of a Real GRE-Style QC Question:
| Column A | Column B | |----------|----------| | ( x^2 - 5x + 6 ) | ( 0 ) |

Quantity A: ( x = 2 ) Quantity B: ( 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 from the information given.

(We’ll solve this later using the step-by-step strategy.)

Key Concepts & Techniques

1. Plugging In Numbers (PIN)

What it is: Substitute specific values for variables to test the relationship between Column A and Column B.
When to use it:
- When the question involves variables (e.g., ( x, y, n )) or undefined expressions (e.g., ( \frac{a}{b} )).
- When the answer choices include D (Cannot be determined)—PIN helps confirm if the relationship is consistent or variable.
- Always test at least 3 values:
- A positive integer (e.g., ( x = 2 )) - A negative number (e.g., ( x = -1 )) - Zero (e.g., ( x = 0 )) - Bonus: Test fractions (e.g., ( x = \frac{1}{2} )) or extreme values (e.g., ( x = 100 )) if the relationship isn’t clear.

2. Simplifying Expressions

What it is: Algebraically manipulate Column A and Column B to make them easier to compare.
When to use it:
- When both columns are algebraic expressions (e.g., ( 3x + 5 ) vs. ( 2x + 10 )).
- When the question involves inequalities (e.g., ( x > 2 ))—simplify to isolate the variable.
- Key moves:
- Subtract/add the same term to both columns.
- Multiply/divide both columns by a positive number (but never by a variable unless you’re certain it’s positive).
- Factor or expand expressions (e.g., ( x^2 - 4 ) → ( (x+2)(x-2) )).

3. Geometric Reasoning

What it is: Use properties of shapes, angles, and the coordinate plane to compare quantities.
When to use it:
- When the question involves figures (e.g., triangles, circles, rectangles).
- When lengths, areas, or angles are compared (e.g., "Is the area of Triangle A greater than the area of Triangle B?").
- Key properties to remember:
- Triangles: The sum of any two sides must be greater than the third side.
- Circles: The longest chord is the diameter; inscribed angles are half the central angle.
- Coordinate Geometry: Use the distance formula (( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} )) or slope (( \frac{y_2-y_1}{x_2-x_1} )).

4. The "D" Trap (Cannot Be Determined)

What it is: Answer choice D is correct when the relationship changes depending on the value of the variable(s).
When to suspect D:
- If plugging in different numbers gives different results (e.g., Column A > Column B for ( x = 1 ), but Column A < Column B for ( x = -1 )).
- If the question lacks sufficient constraints (e.g., "( x ) is a real number" vs. "( x ) is a positive integer").
- Warning: If the question provides specific constraints (e.g., "( x > 0 )"), D is rarely correct.

5. Cross-Multiplication (For Fractions/Inequalities)

What it is: Compare two fractions by cross-multiplying to avoid dealing with denominators.
When to use it:
- When both columns are fractions (e.g., ( \frac{a}{b} ) vs. ( \frac{c}{d} )).
- When the question involves inequalities with fractions (e.g., ( \frac{x}{y} > \frac{y}{x} )).
- Rule:
- If ( \frac{a}{b} > \frac{c}{d} ), then ( ad > bc ) (assuming ( b ) and ( d ) are positive).
- Never cross-multiply if denominators could be zero or negative!

6. Testing Edge Cases

What it is: Plug in values that break assumptions (e.g., zero, negatives, fractions, or extreme numbers).
When to use it:
- When the question seems too straightforward (ETS often hides traps in edge cases).
- When the answer choices include D—edge cases help confirm if the relationship is consistent.
- Common edge cases:
- ( x = 0 ) (tests if zero is allowed).
- ( x = 1 ) and ( x = -1 ) (tests symmetry).
- ( x = \frac{1}{2} ) (tests fractions).
- ( x = 100 ) (tests large numbers).

Step-by-Step Strategy (Follow This Every Time)

Step 1: Understand the Question

• Read carefully: Note any constraints (e.g., "( x ) is a positive integer") or diagrams.
• Identify variables: Are there unknowns? If yes, PIN is likely needed.
• Simplify first: If both columns are algebraic, try simplifying before plugging in.

Step 2: Plug In Numbers (If Variables Exist)

• Test 3+ values: Positive, negative, zero (and fractions if needed).
• Record results: For each value, note whether Column A >, <, or = Column B.
• If results vary → Answer is D.
• If results are consistent → Proceed to Step 3.

Step 3: Simplify Algebraically (If Possible)

• Subtract/add the same term to both columns to isolate variables.
• Factor or expand expressions to make them comparable.
• Cross-multiply if dealing with fractions (but check denominators first!).
• Example: Compare ( 3x + 5 ) vs. ( 2x + 10 ).
• Subtract ( 2x ) from both: ( x + 5 ) vs. ( 10 ).
• Subtract 5 from both: ( x ) vs. ( 5 ).
• Now it’s clear: If ( x > 5 ), A > B; if ( x < 5 ), A < B; if ( x = 5 ), A = B.

Step 4: Apply Geometric Reasoning (If Applicable)

• Draw the figure (if not provided) and label all given information.
• Use properties:
• For triangles: Sum of angles = 180°, side lengths must satisfy triangle inequality.
• For circles: Radius = distance from center to any point on the circle.
• For coordinate geometry: Use distance formula or slope.
• Example: Compare the area of a square with side ( s ) vs. a circle with diameter ( s ).
• Square area: ( s^2 ).
• Circle area: ( \pi (\frac{s}{2})^2 = \frac{\pi s^2}{4} ).
• Since ( \pi \approx 3.14 ), ( \frac{\pi}{4} \approx 0.785 ), so ( s^2 > \frac{\pi s^2}{4} ).

Step 5: Check for Traps

• Did you assume variables are positive? (ETS loves negative numbers.)
• Did you consider zero? (Many students forget this.)
• Did you cross-multiply without checking denominators? (Dangerous if denominators could be zero or negative.)
• Does the answer change with different values? (If yes, D is likely correct.)

Step 6: Select the Answer

• A: Column A is always greater.
• B: Column B is always greater.
• C: The columns are always equal.
• D: The relationship cannot be determined (varies with different values).

Fully Worked Example (Using the Strategy)

Question:
| Column A | Column B | |----------|----------| | ( x^2 - 5x + 6 ) | ( 0 ) |

Quantity A: ( x = 2 ) Quantity B: ( 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 from the information given.

Step 1: Understand the Question

• We’re comparing ( x^2 - 5x + 6 ) to ( 0 ) for two different ( x )-values.
• No constraints are given (e.g., ( x ) could be any real number).

Step 2: Plug In Numbers

Since ( x ) is a variable, we’ll test multiple values.

1. Test ( x = 2 ):
2. Column A: ( (2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0 ).
3. Column B: ( 0 ).

4. Result: A = B.

5. Test ( x = 3 ):

6. Column A: ( (3)^2 - 5(3) + 6 = 9 - 15 + 6 = 0 ).
7. Column B: ( 0 ).

8. Result: A = B.

9. Test ( x = 0 ):

10. Column A: ( (0)^2 - 5(0) + 6 = 6 ).
11. Column B: ( 0 ).

12. Result: A > B.

13. Test ( x = 1 ):

14. Column A: ( (1)^2 - 5(1) + 6 = 1 - 5 + 6 = 2 ).
15. Column B: ( 0 ).

16. Result: A > B.

17. Test ( x = 4 ):

18. Column A: ( (4)^2 - 5(4) + 6 = 16 - 20 + 6 = 2 ).
19. Column B: ( 0 ).

20. Result: A > B.

21. Test ( x = -1 ):

22. Column A: ( (-1)^2 - 5(-1) + 6 = 1 + 5 + 6 = 12 ).
23. Column B: ( 0 ).
24. Result: A > B.

Observation: For ( x = 2 ) and ( x = 3 ), A = B. For all other values, A > B.
But wait! The question asks for the relationship in general, not just for ( x = 2 ) or ( x = 3 ). Since the relationship changes, the answer is D.

But let’s double-check with Step 3.

Step 3: Simplify Algebraically

• Column A: ( x^2 - 5x + 6 ).
• Column B: ( 0 ).
• Factor Column A: ( (x-2)(x-3) ).
• Now, compare ( (x-2)(x-3) ) to ( 0 ).
• If ( x < 2 ) or ( x > 3 ), ( (x-2)(x-3) > 0 ) (A > B).
• If ( 2 < x < 3 ), ( (x-2)(x-3) < 0 ) (A < B).
• If ( x = 2 ) or ( x = 3 ), ( (x-2)(x-3) = 0 ) (A = B).

Conclusion: The relationship depends on ( x ), so the answer is D.

Common Mistakes

Mistake 1: Assuming Variables Are Positive

Why it happens: Students forget that ( x ) could be negative or zero.
Example:
| Column A | Column B | |----------|----------| | ( x^2 ) | ( x ) |

Wrong approach: Assume ( x > 0 ), so ( x^2 > x ) (A > B).
Correct approach: Test ( x = -1 ): ( (-1)^2 = 1 > -1 ) (A > B). Test ( x = 0.5 ): ( 0.25 < 0.5 ) (A < B). Answer: D.

Mistake 2: Ignoring Zero

Why it happens: Students forget to test ( x = 0 ), which often changes the relationship.
Example:
| Column A | Column B | |----------|----------| | ( \frac{1}{x} ) | ( x ) |

Wrong approach: Test ( x = 1 ) (A = B) and ( x = 2 ) (A < B), conclude D.
Correct approach: Test ( x = 0 ): ( \frac{1}{0} ) is undefined. Answer: D (since ( x = 0 ) is invalid).

Mistake 3: Cross-Multiplying Without Checking Denominators

Why it happens: Students cross-multiply fractions without ensuring denominators are positive.
Example:
| Column A | Column B | |----------|----------| | ( \frac{1}{x} ) | ( \frac{1}{x+1} ) |

Wrong approach: Cross-multiply to get ( x + 1 > x ), so A > B.
Correct approach: If ( x = -2 ), denominators are negative, and the inequality flips. Answer: D.

Mistake 4: Overlooking Geometric Constraints

Why it happens: Students misapply geometric properties (e.g., assuming a triangle exists when side lengths don’t satisfy the triangle inequality).
Example:
| Column A | Column B | |----------|----------| | Area of Triangle with sides 3, 4, 8 | 0 |

Wrong approach: Assume the triangle exists and calculate area.
Correct approach: Check triangle inequality: ( 3 + 4 = 7 \not> 8 ). Triangle doesn’t exist, so area = 0. Answer: C.

Mistake 5: Stopping After One Plug-In

Why it happens: Students test one value and assume the relationship holds for all values.
Example:
| Column A | Column B | |----------|----------| | ( x^3 ) | ( x^2 ) |

Wrong approach: Test ( x = 2 ): ( 8 > 4 ), so A > B.
Correct approach: Test ( x = -1 ): ( -1 < 1 ) (A < B). Answer: D.

GRE Traps & Timing

Trap 1: The "Always" Trap

What it is: ETS makes one column seem always greater by giving a specific value (e.g., "( x = 2 )"), but the question asks for the general case.
How to avoid it: Ignore the given values unless the question explicitly restricts ( x ) to those values.

Trap 2: The "Figure Not Drawn to Scale" Trap

What it is: Diagrams are often misleading (e.g., a right angle looks acute).
How to avoid it: Never trust the diagram. Use given measurements and geometric properties instead.

Trap 3: The "D is Rarely Correct" Trap

What it is: Students avoid D because it "feels" wrong, but ETS includes it when the relationship is truly variable.
How to avoid it: If plugging in different values gives different results, D is correct.

Timing

• Average time per QC question: 75–90 seconds.
• If stuck after 2 minutes: Guess and move on (don’t waste time on one question).
• Prioritize: Do the easiest QC questions first (they’re worth the same as hard ones).

Quick Practice

Question 1:

Answer: C (The two quantities are equal. Simplify Column A: ( \frac{y + x}{xy} = \frac{x + y}{xy} ).)

Question 2:

In the coordinate plane, point ( A ) is at ( (0, 0) ) and point ( B ) is at ( (4, 3) ).

Answer: C (Use distance formula: ( \sqrt{(4-0)^2 + (3-0)^2} = 5 ).)

Last-Minute Cram Sheet

1. PIN Rule: Always test positive, negative, zero (and fractions if needed).
2. Simplify First: Subtract/add the same term to both columns before comparing.
3. Never assume variables are positive—test negatives!
4. Zero is a trap—always check if ( x = 0 ) is allowed.
5. Cross-multiply only if denominators are positive (or flip the inequality if negative).
6. D is correct if the relationship changes with different values.
7. Triangle inequality: Sum of any two sides > third side.
8. Circle properties: Diameter = longest chord; inscribed angle = half central angle.
9. Coordinate geometry: Distance formula = ( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} ).
10. Time per QC: 75–90 seconds—don’t overthink!

Final Tip: On test day, write down your plug-in values to avoid careless mistakes. QC is about speed + accuracy—master this, and you’ll gain 10+ points on Quant.