Quantitative Comparison

The format

A Quantitative Comparison question shows two quantities, Quantity A and Quantity B, sometimes with information about them centred above. You decide how they compare. The four answer choices are always the same (ETS, Overview of the Quantitative Reasoning Measure):

1. Quantity A is greater.
2. Quantity B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

Because the choices never change, learn exactly what each one claims:

• "A is greater" means A is greater for every possible value allowed by the given information.
• "B is greater" means the same for B.
• "Equal" means equal in every allowed case.
• "Cannot be determined" means the relationship changes depending on which allowed values you pick.

So the whole question is often this: is the relationship the same in every case? If you find one case where A is greater and another where B is greater, or where they are equal, the answer is the fourth choice.

ETS adds a useful corollary: if both quantities can be fixed by computation (if they are just numbers), the answer can never be the fourth choice (ETS, Overview of the Quantitative Reasoning Measure).

Strategy 1: simplify the comparison

ETS suggests treating the comparison like an equation you are solving. Write A on the left, B on the right and an unknown relation between them, then simplify both sides step by step (ETS, Overview of the Quantitative Reasoning Measure). The rules are those for inequalities:

You may…	Effect on the comparison
add or subtract the same quantity on both sides	unchanged
multiply or divide both sides by the same positive number	unchanged
multiply or divide both sides by a negative number	reverses
multiply or divide by an expression whose sign you don't know	not allowed
square both sides	safe only when both sides are known to be nonnegative

Worked example 1.

Quantity A	Quantity B
$3(x+4) - 2x$	$x + 10$

Simplify A: $3x + 12 - 2x = x + 12$. Now compare $x + 12$ with $x + 10$. Subtract $x$ from both sides: $12$ against $10$. Quantity A is greater, for every $x$.

Worked example 2.

Quantity A	Quantity B
$\dfrac{7}{8}$	$\dfrac{8}{9}$

Multiply both sides by $72$ (positive, so the relation keeps its direction): $63$ against $64$. Quantity B is greater. Or notice that $\frac78 = 1 - \frac18$ and $\frac89 = 1 - \frac19$; subtracting the smaller fraction $\frac19$ leaves more.

Worked example 3.

Quantity A	Quantity B
$\sqrt{50} + \sqrt{18}$	$\sqrt{128}$

Simplify the roots: $\sqrt{50} = 5\sqrt2$, $\sqrt{18} = 3\sqrt2$, so A $= 8\sqrt2$; and $\sqrt{128} = \sqrt{64 \cdot 2} = 8\sqrt2$. The two quantities are equal. The calculator would give $7.07 + 4.24 = 11.31$ and $11.31$, but the exact route is faster and leaves no doubt about rounding.

Strategy 2: plug in numbers, and choose them wisely

When the quantities contain variables, try values. ETS's advice is to consider all kinds of numbers before answering: zero, positive and negative numbers, small and large numbers, fractions and decimals (ETS, Overview of the Quantitative Reasoning Measure). A handy checklist:

• 0 and 1, which behave specially under multiplication and powers;
• a negative number, which flips inequalities and changes the sign of odd powers;
• a fraction between 0 and 1, which gets smaller when squared;
• a large number, which shows long-run behaviour.

Only use values the given information allows. If $x > 0$, don't test $-1$.

Worked example 4. Given: $x > 0$.

Quantity A	Quantity B
$x^2$	$x^3$

Try $x = 2$: A $= 4$, B $= 8$, so B is greater. Try $x = \frac12$: A $= \frac14$, B $= \frac18$, so A is greater. Two different relationships: cannot be determined. (Testing only $x = 2$ and $x = 3$ would have misled you. This is why fractions are on the list.)

Worked example 5. Given: $x$ and $y$ are integers and $x > y$.

Quantity A	Quantity B
$x - y$	$1$

Because they are integers and $x > y$, the difference $x - y$ is at least $1$. With $x = 5, y = 4$ the quantities are equal; with $x = 5, y = 2$, A is greater. Cannot be determined. If the question had said B was $0$, the answer would be "A is greater," since $x - y \ge 1 > 0$ in every case.

Worked example 6. Given: $x^2 = 9$.

Quantity A	Quantity B
$x$	$3$

$x$ could be $3$ (equal) or $-3$ (B greater). Cannot be determined. Forgetting the negative root is one of the most common QC errors.

Worked example 7: reversing an inequality. Given: $x > y$.

Quantity A	Quantity B
$-3x$	$-3y$

Divide both sides by $-3$, which reverses the comparison: compare $x$ with $y$ but with the relation flipped. Since $x > y$, we get $-3x < -3y$. Quantity B is greater. Check: $x = 2, y = 1$ gives $-6$ against $-3$.

Strategy 3: estimate instead of computing

ETS advises against needless computation. Simplify, transform or estimate only as far as you need to compare (ETS, Overview of the Quantitative Reasoning Measure).

Worked example 8.

Quantity A	Quantity B
$19\%$ of $412$	$21\%$ of $388$

Exact arithmetic is possible, but think first. A has the smaller percent of the larger number, so estimates won't settle it quickly. Use the products instead: $19 \times 412 = 7{,}828$ and $21 \times 388 = 8{,}148$. B is greater. A neat check: $(20 - 1)(400 + 12)$ against $(20 + 1)(400 - 12)$, that is, $8{,}000 + 240 - 400 - 12$ against $8{,}000 - 240 + 400 - 12$. The second is larger. Quantity B is greater. Sometimes a quick exact calculation is the efficient route; the point is to choose it deliberately.

Strategy 4: geometry comparisons

Figures are not necessarily drawn to scale. You may assume that lines shown as straight are straight, that points on a line are in the order shown, and that objects are in the relative positions shown. You may not assume lengths or angle measures from appearance (ETS, Math Conventions). For a geometry QC, ETS suggests redrawing the figure: keep what is fixed by the given information and vary what isn't (ETS, Overview of the Quantitative Reasoning Measure).

Worked example 9. A triangle has two sides of length $5$ and $8$, and the third side has length $c$.

Quantity A	Quantity B
$c$	$13$

By the triangle inequality, any side is less than the sum of the other two: $c < 5 + 8 = 13$. Quantity B is greater, however the triangle is drawn.

Common traps

• Swapping the choices. You decide "B is greater" and click "A is greater." ETS specifically warns about reversing the first two choices (ETS, Overview of the Quantitative Reasoning Measure).
• Choosing "cannot be determined" for two fixed numbers. If both quantities can be computed, the relationship is determined.
• Testing only easy positive integers. Fractions, negatives and zero often change the answer.
• Assuming integers. Unless the question says integer, a variable can be any real number in its allowed range (ETS, Math Conventions).
• Dividing by a variable of unknown sign. That might reverse the inequality without your knowing it. Rearrange instead.
• Trusting the picture. Unless the figure is in a coordinate plane or is a data graph, it isn't drawn to scale (ETS, Math Conventions).

Check yourself

• What exactly does "Quantity A is greater" claim?
• Why can't the answer be "cannot be determined" when both quantities are numbers?
• Which operations reverse a comparison, and which are not allowed?
• Which kinds of numbers should you test, and why fractions in particular?
• What can and can't you assume from a geometry figure?