Study guide
The Quantitative Reasoning measure is delivered in two sections, 12 questions in 21 minutes followed by 15 questions in 26 minutes, drawing on arithmetic, algebra, geometry, and data analysis across four formats: Quantitative Comparison, Select-One, Select-One-or-More, and Numeric Entry. This chapter covers the arithmetic and algebra content most tested in that mix, uses ASCII math throughout, and, per the double-check standard, recomputes every worked solution by two independent methods so you can see the cross-check habit modeled directly.
Integer Properties, Percentages, and Ratios
Integer properties questions test even and odd rules, factors and multiples, and prime factorization. A reliable rule set: even times even is even, odd times odd is odd, even times odd is even; even plus even is even, odd plus odd is even, even plus odd is odd. Percentage problems are best handled by converting a percent to a decimal or fraction immediately, since 15 percent of 240 is easier as 0.15 x 240 or as (15/100) x 240. Method one: 0.15 x 240 = 36. Method two: 240 x 15 = 3600, and 3600 / 100 = 36. Both methods agree, so 15 percent of 240 is 36. Percent change problems require dividing by the original value, not the new one: if a price rises from 80 to 92, the increase is 12, and the percent increase is 12/80 = 0.15, or 15 percent. Recheck: 80 x 1.15 = 80 + 80(0.15) = 80 + 12 = 92, confirming the 15 percent figure independently. Ratio problems convert a part-to-part ratio into a part-to-whole fraction; if apples to oranges is 3 to 5 in a fruit bowl of 40 pieces total, the parts sum to 3 + 5 = 8, so apples equal (3/8) x 40 = 15. Recheck: oranges equal (5/8) x 40 = 25, and 15 + 25 = 40, matching the given total. When ratios involve three quantities, such as 2 to 3 to 5, treat the sum, here 10, as the whole and apply the same part-to-whole method to each term.
Exponents, Roots, and Estimation
Exponent rules to know cold: a^m x a^n = a^(m+n); a^m / a^n = a^(m-n); (a^m)^n = a^(m x n); a^0 = 1 for nonzero a; a^(-n) = 1/(a^n). For roots, sqrt(a) x sqrt(b) = sqrt(a x b), and sqrt(a/b) = sqrt(a)/sqrt(b) for nonnegative a and positive b. Example: simplify sqrt(72). Method one: 72 = 36 x 2, so sqrt(72) = sqrt(36) x sqrt(2) = 6 x sqrt(2). Method two: 72 = 4 x 18 = 4 x 9 x 2, so sqrt(72) = sqrt(4) x sqrt(9) x sqrt(2) = 2 x 3 x sqrt(2) = 6 x sqrt(2). Both factorizations agree at 6 x sqrt(2). Estimation matters because the GRE often rewards a fast approximate answer over a slow exact one, especially under Select-One choices spread far apart. To estimate 3.98 x 51.2, round to 4 x 51 = 204, which is close enough to select among answer choices like 198, 204, 210, and 350; the exact value, 3.98 x 51.2 = 203.776, confirms 204 as the nearest choice; note that a coarser rounding such as 4 x 50 = 200 confirms the answer is near 200 but falls between 198 and 204, so when choices are close together, keep one factor unrounded, as in 4 x 51.2 = 204.8, to separate them. Negative exponents and fractional bases are common traps: (1/2)^(-3) equals 2^3, which is 8, not 1/8; recheck by definition, (1/2)^(-3) = 1 / (1/2)^3 = 1 / (1/8) = 8, confirming the same result twice.
Linear and Quadratic Equations, Inequalities
For linear equations, isolate the variable using inverse operations in reverse order of operations. Solve 3x + 7 = 22: subtract 7 from both sides to get 3x = 15, then divide by 3 to get x = 5. Recheck by substitution: 3(5) + 7 = 15 + 7 = 22, matching the original equation. For quadratics, factor when possible or apply the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Solve x^2 - 5x + 6 = 0. Method one, factoring: look for two numbers multiplying to 6 and summing to -5, which are -2 and -3, giving (x - 2)(x - 3) = 0, so x = 2 or x = 3. Method two, the quadratic formula with a = 1, b = -5, c = 6: x = (5 +/- sqrt(25 - 24)) / 2 = (5 +/- 1) / 2, giving x = 3 or x = 2. Both methods agree on {2, 3}. For inequalities, the crucial rule is that multiplying or dividing both sides by a negative number reverses the inequality sign. Solve -2x + 4 > 10: subtract 4 to get -2x > 6, then divide by -2 and flip the sign to get x < -3. Recheck with a test value, x = -4: -2(-4) + 4 = 8 + 4 = 12, and 12 > 10 is true, confirming values below -3 satisfy the inequality, while x = -2 gives -2(-2) + 4 = 8, and 8 > 10 is false, confirming values above -3 do not.
Coordinate Geometry Basics and Numeric Entry
Coordinate geometry on this measure typically covers slope, the distance formula, and the midpoint formula. Slope between points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1). Find the slope between (2, 3) and (6, 11). Method one, direct substitution: (11 - 3) / (6 - 2) = 8 / 4 = 2. Method two, reverse the point order to check sign consistency: (3 - 11) / (2 - 6) = (-8) / (-4) = 2, the same result, confirming slope 2. The midpoint formula is ((x1+x2)/2, (y1+y2)/2); the midpoint of (2, 3) and (6, 11) is ((2+6)/2, (3+11)/2) = (4, 7). Recheck by confirming (4,7) is equidistant in coordinate steps from both endpoints: from (2,3) to (4,7) is a step of (2, 4), and from (4,7) to (6,11) is also a step of (2, 4), confirming the midpoint. Numeric Entry questions require typing an exact numeric answer into a box rather than choosing from options, so there are no wrong-answer choices to eliminate for a sanity check; instead, always verify your numeric entry answer by plugging it back into the original relationship, exactly as shown throughout this chapter, since a small arithmetic slip has no answer-choice pattern to catch it. When a Numeric Entry question allows a fraction, reduce it fully unless the instructions state otherwise, and pay close attention to units and rounding instructions, since these are common sources of an otherwise correct answer being marked wrong.
Quantitative Comparison Strategy
Quantitative Comparison questions present Quantity A and Quantity B and ask you to choose among four fixed options: Quantity A is greater; Quantity B is greater; the two quantities are equal; or the relationship cannot be determined from the given information. Because the choices never change, memorize them so you can spend all your time on the math itself. The most efficient method is often to avoid full computation and instead simplify both quantities by the same legal operation, such as subtracting a common term from both sides, since only the relationship matters, not the exact values. Example: Quantity A is 17^2 and Quantity B is 16 x 18. Method one, direct computation: 17^2 = 289, and 16 x 18 = 288, so Quantity A is greater by 1. Method two, algebraic shortcut: 16 x 18 = (17-1)(17+1) = 17^2 - 1, so Quantity B always equals Quantity A minus 1, confirming Quantity A is greater without needing the multiplication at all. A critical trap in Quantitative Comparison is assuming a variable is positive when the problem never says so; if x is unrestricted and Quantity A is x^2 while Quantity B is x, testing x = 2 gives A greater (4 versus 2), but testing x = 0.5 gives B greater (0.25 versus 0.5), and testing x = -1 gives A greater again (1 versus -1); because the relationship changes depending on which permitted value of x you pick, the correct answer is that the relationship cannot be determined. Always test at least one fractional value and one negative value, when permitted, before concluding a fixed relationship holds.
Key terms
- Quantitative Comparison
- — A question format asking whether Quantity A is greater, Quantity B is greater, the quantities are equal, or the relationship cannot be determined.
- Numeric Entry
- — A question format requiring the test-taker to type an exact numeric or fractional answer rather than select from choices.
- Select-One-or-More (Quant)
- — A quantitative format in which any number of listed choices, from one to all, may be correct, with no partial credit.
- Percent change
- — The change in a value divided by its original value, not its new value, expressed as a percentage.
- Prime factorization
- — Expressing an integer as a product of prime numbers, used to simplify roots, fractions, and ratio problems.
- Slope
- — The ratio of vertical change to horizontal change between two points, calculated as (y2-y1)/(x2-x1).
- Midpoint formula
- — The formula ((x1+x2)/2, (y1+y2)/2) giving the point exactly between two coordinate points.
- Quadratic formula
- — The formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a) used to solve equations of the form ax^2 + bx + c = 0.
- Inequality sign reversal
- — The rule that multiplying or dividing both sides of an inequality by a negative number reverses the inequality's direction.
- Estimation strategy
- — Rounding values to quickly approximate a computation when exact precision is unnecessary to select the correct choice.
Exam tips
- Memorize the four fixed Quantitative Comparison answer choices so you never spend test time rereading the options themselves.
- In Quantitative Comparison, simplify both quantities by the same operation instead of computing full values whenever possible; it is usually faster and just as certain.
- When a variable's sign or range is not restricted, test a positive integer, a fraction between 0 and 1, and a negative number before concluding the relationship is fixed.
- Recompute every algebra answer by substitution back into the original equation; it is the fastest way to catch a sign or arithmetic slip.
- On Numeric Entry, re-verify your final number against the original relationship, since there are no answer choices to signal an error.