Quantitative comparison: strategy for the four fixed choices

The format

Quantitative Comparison is one of the question types in GRE Quantitative Reasoning. Each question shows two quantities, Quantity A and Quantity B, sometimes with information above them that applies to both. You do not calculate an answer; you decide how the two quantities compare. The choices are always the same four:

Choice	Means
Quantity A is greater.	A is greater for every situation the information allows
Quantity B is greater.	B is greater for every situation allowed
The two quantities are equal.	they are equal for every situation allowed
The relationship cannot be determined from the information given.	different allowed situations give different relationships

The word "every" in the first three rows is the whole game. "A is greater" is wrong if even one allowed value makes the quantities equal.

When "cannot be determined" is impossible

If both quantities are fixed numbers, with no variables and no unknown figures, one number is greater than, less than or equal to the other, and that never changes. So the fourth choice is out the moment you see two fixed numbers. That leaves a three-way choice, and often a quick estimate settles it.

When variables are involved, the fourth choice needs proof: two allowed cases that give different relationships. One case can never show it, and one case can never rule it out either.

Strategy 1: simplify both sides together

Think of an unknown sign, $?$, between Quantity A and Quantity B, and change both sides in the same way, as you would solve an inequality. These moves keep the comparison intact:

• add or subtract the same quantity on both sides;
• multiply or divide both sides by the same positive quantity;
• square both sides, when both are known to be non-negative.

These moves are not safe: multiplying or dividing by a quantity that could be negative or zero (a negative multiplier flips the comparison, and zero erases it), and squaring when the signs are unknown ($-3 < 2$, but $9 > 4$).

For example, with Quantity A $\frac{x}{3} + 5$ and Quantity B $\frac{x}{3} + 4$, subtract $\frac{x}{3}$ from both sides: 5 against 4. Quantity A is greater for every $x$, without knowing anything about $x$.

Strategy 2: test values on purpose

When variables remain and simplifying stalls, try values, but choose them deliberately. Values must satisfy every condition given. Then cover the kinds of number that behave unusually:

Try	Because
0	products and powers become 0
1 and $-1$	powers do not change size (or only change sign)
a negative number	multiplying by it reverses order; odd powers stay negative
a fraction between 0 and 1	squaring makes it smaller
a large number	shows which side grows faster

If two ordinary values give the same relationship, try an unusual one before you commit. The common mistake is testing only positive whole numbers greater than 1, which hide every one of these behaviours.

Strategy 3: estimate instead of computing

With fixed numbers, you need only the comparison, not the values. To compare $\frac{17}{35}$ with $\frac{1}{2}$, note that half of 35 is 17.5, and $17 < 17.5$, so $\frac{17}{35} < \frac{1}{2}$. No division needed. Compare against easy benchmarks (½, 1, a round number) and decide whether the benchmark separates the two quantities. When an estimate is too close to call, compute exactly. The on-screen calculator is there for that.

Figures

When a question comes with a figure, base the comparison on what is given in words and numbers, and on what geometry forces, not on how the drawing looks. Two lengths that look equal may not be stated to be equal, and an angle that looks like a right angle may not be one.

A routine

1. Read the information above the quantities and note every condition (integer? positive? range?).
2. Are both quantities fixed numbers? If so, rule out "cannot be determined".
3. Simplify both sides together, with safe moves only.
4. If variables remain, test at least two contrasting allowed values, including unusual ones.
5. Choose. If two cases disagree, the answer is "cannot be determined"; if every case you tried agrees and the algebra explains why, choose that relationship.