Quantitative Comparison
Reasoning
Master the art of comparing two quantities without always solving: the fastest strategy for GRE Quantitative success.
GRE Quantitative
What Is a Quantitative Comparison?
Every QC question presents two quantities: Quantity A and Quantity B. Your job is not to solve for a number: it is to determine which quantity is greater, whether they are equal, or whether the relationship cannot be determined.
The Four Answer Choices
(A)
Quantity A is greater
( B )
Quantity B is greater
(C)
The two quantities are equal
(D)
The relationship cannot be determined
Anatomy of a QC Question
Some questions include given information (constraints) above the two columns. Others present the two quantities with no extra context.
The Core Strategy Framework
1
Simplify both sides. Add, subtract, multiply, or divide both quantities by the same positive value to reduce complexity.
2
Plug in numbers. Try easy values (0, 1, −1, fractions) to test the relationship. If two different sets of values give different outcomes, the answer is (D).
3
Estimate, don't calculate. If one quantity is obviously larger, you do not need an exact number.
4
Watch for constraints. Variables could be negative, zero, or fractions: always check edge cases.
Simplify Both Sides
You can perform the same operation on both Quantity A and Quantity B without changing the comparison: as long as you add/subtract anything, or multiply/divide by a positive number.
Quantity A
vs
Quantity B
\( 3x + 7 \)
vs
\( 3x + 4 \)
Cannot Be Determined vs Equal
The trickiest distinction on the GRE: knowing when the answer is (C) Equal versus (D) Cannot be determined.
(C) Equal
The relationship holds for every possible value of the variable. No matter what you plug in, \( A = B \).
(D) Cannot Determine
You found at least two cases that give different results: one where \( A > B \), and one where \( A < B \) (or \( A = B \)).
The Power of Plugging In
When variables are present without tight constraints, strategic number picking is your best friend. Always test these categories:
Positive integers
Try \( x = 1, 2, 10 \)
Negatives
Try \( x = -1, -3 \)
Zero
Try \( x = 0 \)
Fractions
Try \( x = \frac{1}{2}, -\frac{1}{3} \)
Step-by-Step Approach
1
Read the given information. Note any constraints on variables (e.g., \( x > 0 \), integer values only).
2
Simplify if possible. Cancel common terms, factor expressions, or rearrange to isolate the comparison.
3
Try to determine the relationship algebraically. If you can prove A > B (or A = B) for all allowed values, you are done.
4
If unsure, plug in numbers. Test positives, negatives, zero, and fractions. Two conflicting results means (D).
5
Choose your answer. Do not over-calculate: confidence in the comparison is enough.
Example 1: Algebra Comparison
Quantity A
vs
Quantity B
\( (x+1)^2 \)
vs
\( x^2 + 1 \)
Example 2: With Constraints
Quantity A
vs
Quantity B
\( \dfrac{x}{y} \)
vs
\( \dfrac{x^2}{y^2} \)
Example 3: Geometry Context
Quantity A
vs
Quantity B
Area of the circle
vs
\( 75 \)
Example 4: When They Are Equal
Quantity A
vs
Quantity B
\( (a - b)^2 \)
vs
\( a^2 - 2ab + b^2 \)
Common Mistakes
✕ Testing Only Positives
If there is no constraint, variables could be negative, zero, or fractions. Testing only positive integers misses critical cases.
✕ Multiplying by a Variable
Multiplying both sides by a variable that could be negative flips the inequality. Only multiply by values you know are positive.
✕ Over-Calculating
Spending 3 minutes computing exact answers. QC questions reward estimation and comparison, not precision.
Strategy Summary
| Scenario | Best Strategy | Likely Answer |
|---|---|---|
| No variables | Compute directly | (A), (B), or (C) |
| Constrained variable | Simplify algebraically | Any of (A), (B), (C), or (D) |
| Unconstrained variable | Plug in edge cases | Often (D) |
| Identical expressions | Simplify both sides | (C) |
| Geometry with fixed values | Estimate with formulas | (A), (B), or (C) |
Question 1 of 5
Quantity A
vs
Quantity B
\( \sqrt{144} + \sqrt{25} \)
vs
\( \sqrt{169} \)
Compare the two quantities above.
\( \sqrt{144} = 12 \) and \( \sqrt{25} = 5 \), so Quantity A = \( 12 + 5 = 17 \). Meanwhile \( \sqrt{169} = 13 \). Since \( 17 > 13 \), Quantity A is greater. Note: \( \sqrt{a} + \sqrt{b} \neq \sqrt{a+b} \) in general!
Question 2 of 5
Given: \( x \) is a positive integer.
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Quantity A
vs
Quantity B
\( 2x + 3 \)
vs
\( 3x + 1 \)
Compare the two quantities.
Subtract \( 2x + 1 \) from both sides: you get \( 2 \) vs \( x \). When \( x = 1 \): \( 2 > 1 \), so \( A > B \). When \( x = 2 \): \( 2 = 2 \), so they are equal. When \( x = 3 \): \( 2 < 3 \), so \( B > A \). The relationship changes, so the answer is (D).
Question 3 of 5
Quantity A
vs
Quantity B
\( (-3)^4 \)
vs
\( 3^4 \)
Compare the two quantities.
\( (-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 81 \). And \( 3^4 = 81 \). Raising a negative number to an even power always gives a positive result. Both quantities equal 81, so they are equal.
Question 4 of 5
Given: The perimeter of a rectangle is 40.
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Quantity A
vs
Quantity B
Area of the rectangle
vs
\( 100 \)
Compare the two quantities.
If the perimeter is 40, then \( 2(l + w) = 40 \), so \( l + w = 20 \). If it is a \( 10 \times 10 \) square: area = \( 100 \), so \( A = B \). If it is \( 15 \times 5 \): area = \( 75 \), so \( B > A \). Two different outcomes mean the answer is (D).
Question 5 of 5
Given: \( y < 0 \)
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Quantity A
vs
Quantity B
\( y^3 \)
vs
\( y^2 \)
Compare the two quantities.
Since \( y < 0 \), \( y^2 \) is always positive (squaring a negative gives a positive). And \( y^3 \) is always negative (cubing a negative stays negative). A positive number is always greater than a negative number, so Quantity B is always greater: answer is (B).