**GRE Quantitative Comparison: Geometry – Adding/Subtracting Equal Quantities from Both Columns**

GRE Quantitative Comparison: Geometry – Adding/Subtracting Equal Quantities from Both Columns

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What This Is

Quantitative Comparison (QC) questions involving geometry often test whether you can manipulate expressions by adding or subtracting the same quantity from both columns without changing the relationship between them. The GRE uses this tactic to disguise simple comparisons behind seemingly complex geometric setups. Mastering this technique lets you simplify problems quickly, avoid unnecessary calculations, and save 30–60 seconds per question—time you can reinvest in harder problems. Example:

GRE-Style Example:
In the figure, ABCD is a rectangle with AB = 6 and AD = 8. Point E lies on side BC such that BE = 3. Compare: - Quantity A: Area of triangle ABE - Quantity B: Area of triangle EDC

(Answer: The quantities are equal—this guide will show you why.)

Key Concepts & Techniques

1. Additive Invariance Principle
2. What it is: If you add or subtract the same value from both Quantity A and Quantity B, the relationship (A > B, A < B, or A = B) does not change.

3. When to use it: When both columns share a common term or area that can be canceled out (e.g., overlapping regions, shared sides, or identical sub-expressions).

4. Area/Perimeter Subtraction

5. What it is: For composite shapes, subtract identical areas or lengths from both columns to isolate the difference.

6. When to use it: When the question compares parts of a larger shape (e.g., triangles within a rectangle, shaded vs. unshaded regions).

7. Variable Elimination

8. What it is: If both columns contain the same variable (e.g., an unknown side length), subtract it from both sides to simplify.

9. When to use it: When the question involves variables in both columns (e.g., "x is a positive integer; compare 2x + 5 vs. x + 10").

10. Symmetry Exploitation

11. What it is: If a figure is symmetric, the quantities may be equal even if they look different.

12. When to use it: When the figure has reflective or rotational symmetry (e.g., rectangles, circles, isosceles triangles).

13. Plugging in Numbers (Last Resort)

14. What it is: Assign values to unknowns (e.g., side lengths) to test the relationship.

15. When to use it: Only if the question involves variables and the above methods fail. Never assume values unless the question allows it.

16. Common Overlaps

17. What it is: If two shapes share a region, subtract that region from both columns to compare the remaining parts.
18. When to use it: When the question involves overlapping areas (e.g., two circles intersecting, a triangle inside a square).

Step-by-Step Strategy

Follow these steps for every QC geometry question involving addition/subtraction:

1. Identify the Common Element

2. Look for a term, area, or length that appears in both Quantity A and Quantity B. Circle it.

3. Subtract the Common Element from Both Columns

4. Rewrite the comparison by subtracting the common element from both sides. This simplifies the problem.

5. Compare the Remaining Quantities

6. After subtraction, compare what’s left. If the remaining parts are identical, the quantities are equal. If not, determine which is larger.

7. Check for Hidden Symmetry or Constraints

8. Ask: Does the figure have symmetry? Are there unstated constraints (e.g., right angles, parallel lines)? These can make quantities equal even if they don’t look it.

9. Test Edge Cases (If Needed)

10. If the question involves variables, test extreme values (e.g., very large or very small numbers) to confirm the relationship holds.

11. Eliminate Impossible Answers

12. If the simplified comparison shows A = B, eliminate choices (A) and (B). If A > B, eliminate (B) and (C), etc.

Fully Worked GRE-Style Example

Question:
In the figure, ABCD is a square with side length 10. Points E and F lie on sides AB and AD, respectively, such that AE = AF = 6. Compare: - Quantity A: Area of triangle AEF - Quantity B: Area of quadrilateral EBCF

Step 1: Identify the Common Element
- Both quantities involve parts of the square ABCD. The total area of the square is common to both.

Step 2: Subtract the Common Element
- Quantity A = Area of triangle AEF - Quantity B = Area of quadrilateral EBCF = Total area of square – Area of triangle AEF - Subtract the area of triangle AEF from both quantities: - New Quantity A = 0 - New Quantity B = Total area of square – 2 × (Area of triangle AEF)

Step 3: Compare the Remaining Quantities
- The total area of the square is 10 × 10 = 100.
- Area of triangle AEF = (6 × 6)/2 = 18 (since AE = AF = 6 and angle A is 90°).
- New Quantity B = 100 – 2 × 18 = 64.
- New Quantity A = 0.
- Clearly, 64 > 0, so Quantity B > Quantity A.

But wait! This seems counterintuitive. Let’s re-examine Step 2.

Correction:
- Quantity B is the area of quadrilateral EBCF, which is Total area of square – Area of triangle AEF.
- So, Quantity A = Area of triangle AEF - Quantity B = Total area – Area of triangle AEF - Subtract Quantity A from both sides: - New Quantity A = 0 - New Quantity B = Total area – 2 × (Area of triangle AEF) - But this is incorrect because we’re not comparing the same thing. Instead, add the area of triangle AEF to both quantities: - Quantity A + Area of triangle AEF = 2 × (Area of triangle AEF) - Quantity B + Area of triangle AEF = Total area of square - Now compare: - 2 × 18 = 36 - Total area = 100 - 100 > 36, so Quantity B > Quantity A.

Final Answer: (B) Quantity B is greater.

Common Mistakes

1. Mistake: Forgetting to subtract the same quantity from both columns.
2. Why it happens: Students rush and subtract only from one column, changing the relationship.

3. Correct approach: Always perform the same operation on both columns.

4. Mistake: Assuming symmetry where none exists.

5. Why it happens: Students see a rectangle or square and assume quantities are equal without verifying.

6. Correct approach: Only assume symmetry if the figure or question explicitly states it.

7. Mistake: Overcomplicating with variables.

8. Why it happens: Students plug in numbers for variables when the question can be solved algebraically.

9. Correct approach: Use variables only if the question requires it (e.g., "x is a positive integer").

10. Mistake: Misidentifying the common element.

11. Why it happens: Students subtract the wrong area or length (e.g., subtracting a side length when they should subtract an area).
12. Correct approach: Double-check that the subtracted element is identical in both columns.

GRE Traps & Timing

1. Trap: Hidden Overlaps
2. What it is: The GRE may include overlapping regions in both columns, making it seem like the quantities are different when they’re not.

3. How to avoid: Always look for shared areas or lengths and subtract them.

4. Trap: Variable Constraints

5. What it is: The question may imply constraints (e.g., "x is a positive integer") that limit the relationship between quantities.

6. How to avoid: Read the question carefully for hidden constraints.

7. Timing:

8. Aim for 60–90 seconds per QC question. If you’re stuck after 90 seconds, guess and move on.

Quick Practice

Question:
In the figure, ABCD is a rectangle with AB = 4 and AD = 6. Point E lies on side BC such that BE = 2. Compare: - Quantity A: Area of triangle ABE - Quantity B: Area of triangle EDC

Answer: (C) The two quantities are equal.
Explanation: Subtract the area of triangle ABE from both quantities. Quantity A becomes 0, and Quantity B becomes the area of rectangle ABCD minus twice the area of triangle ABE. Since the areas of ABE and EDC are equal (both have base 4 and height 2), the quantities are equal.

Last-Minute Cram Sheet

1. Add/subtract the same thing from both columns → Relationship stays the same.
2. For areas: Subtract overlapping regions to isolate differences.
3. For lengths: Subtract shared sides to simplify comparisons.
4. Symmetry = equality (e.g., rectangles, isosceles triangles).
5. Never assume values unless the question allows it.
6. If both columns have a variable, subtract it to simplify.
7. Common trap: Overlapping regions in both columns—subtract them!
8. Time budget: 60–90 seconds per QC question.
9. If stuck, test edge cases (e.g., very large/small numbers).
10. Always check for hidden constraints (e.g., "x is a positive integer").

Final Tip: On test day, if a QC geometry question looks complex, look for what’s the same in both columns and subtract it first. This will simplify 90% of these questions.