1 · The comparison question
Part of the Upper Level Quantitative Reasoning section shows two quantities, one in Column A and one in Column B, sometimes with information above them. The four choices are always the same:
(A) The quantity in Column A is greater. (B) The quantity in Column B is greater. (C) The two quantities are equal. (D) The relationship cannot be determined from the information given.
2 · Plain numbers: never D
If both columns are plain numbers with no letters, one is bigger or they are equal, so the answer is never D. Column A: 2¹⁰. Column B: 10³. 1,024 is more than 1,000: the answer is A.
3 · Compare without finishing the work
Column A: 49/99. Column B: 1/2. Half of 99 is 49.5, and 49 is less, so 49/99 is less than 1/2: the answer is B, with no dividing.
You can add, subtract, or multiply both columns by the same positive number to simplify. Never multiply both by a number that might be negative.
4 · Letters: try different kinds of numbers
Test numbers from each kind the information allows: a whole number above 1, 1, 0, a negative, and a fraction between 0 and 1.
Column A: x². Column B: x³. If x = 2, B is bigger. If x = ½, A is bigger. Two different results mean the answer is D.
5 · Read the information carefully
"0 < x < 1" means powers make x smaller: x³ < x² < x < √x. "x < 0" means x² is positive and x³ is negative.
"The average of a and b is 20, and a > 25" pins b down: a + b = 40, so b is less than 15.
6 · Common traps
−3² is −9, but (−3)² is 9. A 25% raise followed by a 20% cut ends exactly where it started (1.25 × 0.8 = 1), but a 20% raise then a 20% cut does not.
x² − y² = (x + y)(x − y), so if x + y = 9 and x − y = 3, then x² − y² = 27 without finding x or y.
7 · Moving toward each other and catching up
Two cars 210 miles apart drive toward each other at 45 and 60 mph: the gap closes at 105 mph, so they meet in 2 hours.
A car leaves at 9:00 at 50 mph; another follows at 10:00 at 70 mph. The first is 50 miles ahead, and the gap closes at 20 mph: 2.5 hours after 10:00, at 12:30.
8 · Work problems
Add what each worker does in one hour. If Al alone needs 6 hours and Bo alone needs 3, together they do 1/6 + 1/3 = 1/2 of the job per hour.
If Al works alone for 2 hours first, he finishes 1/3 of the job; the remaining 2/3 at 1/2 per hour takes 4/3 hours, or 80 minutes.
9 · Digits and clocks
A two-digit number with tens digit t and ones digit u is 10t + u; reversed, it is 10u + t. The difference is always 9 times the difference of the digits.
On a clock the minute hand moves 6° per minute and the hour hand ½° per minute. At 4:30 the minute hand is at 180° and the hour hand at 120° + 15° = 135°, so the angle is 45°, not 60°.
10 · Pacing
Comparison questions are usually quicker than word problems. If one is taking too long, try two kinds of numbers, make your best choice, and move on. There is no penalty for a wrong answer.
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