The GRE Quantitative Reasoning section tests your understanding of fundamental mathematical concepts, your ability to reason quantitatively, and your skill in solving problems with quantitative methods. While conceptual understanding and problem-solving strategies are paramount, knowing key formulas is crucial for efficiency and accuracy. This guide covers the essential formulas you must know to excel in the GRE Quant section, categorized for easier learning and recall.

Why Formulas Matter (and When They Don’t)

Formulas are shortcuts. They encapsulate mathematical relationships derived from core principles. Memorizing them allows you to quickly apply these principles without re-deriving them during the timed test. However, relying solely on memorization without understanding the underlying concepts can be detrimental. The GRE often tests your ability to adapt formulas or recognize when a conceptual shortcut or estimation is faster than plugging numbers into a formula.

Key Takeaway: Learn the formulas, understand their origins and limitations, and practice applying them in diverse problem contexts.

General Strategies for Using Formulas on the GRE

• Understand, Don’t Just Memorize: Know *why* a formula works. This helps you adapt it or recognize its applicability in non-standard problems.
• Know When to Use Them: Identify the problem type and the relevant formula quickly.
• Know When *Not* to Use Them: Sometimes estimation, logical reasoning, or picking numbers is faster or more appropriate, especially in Quantitative Comparison questions.
• Check Units: Ensure consistency in units before applying formulas (e.g., hours vs. minutes, meters vs. kilometers).
• Practice Application: Regularly solve practice problems that require these formulas to build speed and accuracy.
• Utilize the On-Screen Calculator Wisely: It’s useful for arithmetic but can be slow. Use it for complex calculations, not basic ones. Know its functions (square root, etc.).

I. Arithmetic Formulas

Arithmetic deals with numbers and basic operations. It forms the foundation for many GRE Quant problems.

1. Number Properties

• Even/Odd Rules:
  • Even ± Even = Even
  • Odd ± Odd = Even
  • Even ± Odd = Odd
  • Even × Even = Even
  • Odd × Odd = Odd
  • Even × Odd = Even
• Integer: Whole numbers (…, -2, -1, 0, 1, 2, …)
• Prime Number: A positive integer greater than 1 with only two distinct positive divisors: 1 and itself (e.g., 2, 3, 5, 7, 11, 13…). Note: 2 is the only even prime number. 1 is not prime.
• Divisibility Rules (Examples):
  • Divisible by 2: Last digit is even.
  • Divisible by 3: Sum of digits is divisible by 3.
  • Divisible by 4: Last two digits form a number divisible by 4.
  • Divisible by 5: Last digit is 0 or 5.
  • Divisible by 6: Divisible by both 2 and 3.
  • Divisible by 9: Sum of digits is divisible by 9.

2. Percentages

• Percent Definition: “Per hundred.” x% = x/100
• Percent Change: [(New Value – Old Value) / Old Value] × 100%
• Percentage Of: (Part / Whole) × 100%
• Finding a Percent of a Number: x% of y = (x/100) * y
• Simple Interest: Interest = Principal × Rate × Time (I = PRT). The rate (R) and time (T) must be in consistent units (e.g., annual rate and time in years).
• Compound Interest: Final Amount = Principal × (1 + Rate/n)^(n × Time). Where ‘n’ is the number of times interest is compounded per time period (e.g., n=1 for annually, n=2 for semi-annually, n=4 for quarterly, n=12 for monthly). The GRE often tests the *concept* rather than complex calculations. You might compare simple vs. compound interest or calculate interest compounded annually for a short period.

3. Ratios and Proportions

• Ratio: A comparison of two quantities, often expressed as a:b or a/b.
• Proportion: An equation stating that two ratios are equal: a/b = c/d. Cross-multiplication (ad = bc) is often used to solve proportions.
• Ratio Problems Strategy: If the ratio of A to B is x:y, the actual quantities can often be represented as xk and yk, where k is a constant multiplier.

4. Rates

• Fundamental Rate Formula: Rate = Quantity / Time (or Distance / Time, Work / Time)
• Distance, Rate, Time: Distance = Rate × Time (D=RT)
• Average Speed: Total Distance / Total Time. Important: This is generally *not* the average of the speeds unless the time spent traveling at each speed is equal.
• Work Rate: Work = Rate × Time. If multiple entities work together, their rates add up: Rate_Total = Rate₁ + Rate₂ + …
• Individual Work Rate: Often expressed as 1 / (Time to complete job alone). For example, if Person A takes 3 hours to do a job, their rate is 1/3 of the job per hour.

5. Sequences (Basic)

• Arithmetic Sequence: Each term is found by adding a constant difference (d) to the previous term. Formula for the nth term: a_n = a₁ + (n-1)d, where a₁ is the first term.
• Geometric Sequence: Each term is found by multiplying the previous term by a constant ratio (r). Formula for the nth term: a_n = a₁ * r^(n-1). (Less common, but good to recognize).

II. Algebra Formulas

Algebra involves variables, expressions, equations, and functions. It’s heavily tested on the GRE.

1. Exponents and Roots

• x^a * x^b = x^(a+b)
• x^a / x^b = x^(a-b)
• (x^a)^b = x^(a*b)
• (xy)^a = x^a * y^a
• (x/y)^a = x^a / y^a
• x⁰ = 1 (for x ≠ 0)
• x^-a = 1 / x^a
• x^(1/a) = ^a√x (a^th root of x)
• x^(a/b) = (^b√x)^a = ^b√(x^a)
• √x * √y = √(xy)
• √x / √y = √(x/y)

2. Equations and Inequalities

• Linear Equations: Equations of the form ax + b = c. Solved using basic algebraic manipulation (isolate the variable).
• Quadratic Equations: Equations of the form ax² + bx + c = 0. Can often be solved by:
  • Factoring: Expressing the quadratic as (px + q)(rx + s) = 0.
  • Quadratic Formula: x = [-b ± √(b² – 4ac)] / 2a. Use this when factoring is difficult or impossible. The term (b² – 4ac) is the discriminant, which tells you the nature of the roots (positive = 2 real roots, zero = 1 real root, negative = no real roots).
• Systems of Equations: Two or more equations with two or more variables. Common methods for solving:
  • Substitution: Solve one equation for one variable and substitute that expression into the other equation.
  • Elimination: Multiply equations by constants so that adding or subtracting them eliminates one variable.
• Inequalities: Similar rules to equations, BUT: Multiplying or dividing both sides by a negative number reverses the inequality sign.

3. Algebraic Identities (Important to Recognize)

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• a² – b² = (a + b)(a – b) (Difference of Squares)

4. Absolute Value

• Definition: |x| = distance of x from 0 on the number line. |x| = x if x ≥ 0, and |x| = -x if x < 0.
• Equations: If |x| = k (where k ≥ 0), then x = k or x = -k.
• Inequalities:
  • If |x| < k (where k > 0), then -k < x < k.
  • If |x| > k (where k > 0), then x > k or x < -k.

5. Functions

• Notation: f(x) represents a rule applied to the input x.
• Evaluation: To find f(a), substitute ‘a’ for ‘x’ in the function definition.
• Unusual Symbols: The GRE sometimes defines functions with odd symbols (e.g., x ♦ y = x² + y). Treat the symbol like a function name and apply the given rule.

III. Geometry Formulas

Geometry questions involve shapes, lines, angles, and coordinate planes.

1. Lines and Angles

• Sum of angles on a straight line = 180°
• Sum of angles around a point = 360°
• Vertical angles are equal.
• Parallel lines cut by a transversal: Corresponding angles are equal, alternate interior angles are equal, consecutive interior angles sum to 180°.

2. Triangles

• Sum of interior angles = 180°
• Area = (1/2) × Base × Height
• Perimeter = Sum of the lengths of the three sides.
• Triangle Inequality Theorem: The sum of the lengths of any two sides must be greater than the length of the third side. (|a – b| < c < a + b)
• Pythagorean Theorem (Right Triangles Only): a² + b² = c², where a and b are the lengths of the legs, and c is the length of the hypotenuse.
• Common Pythagorean Triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), and their multiples. Recognizing these saves calculation time.
• Special Right Triangles:
  • Isosceles Right Triangle (45°-45°-90°): Sides are in the ratio x : x : x√2.
  • 30°-60°-90° Triangle: Sides opposite the angles 30°, 60°, 90° are in the ratio x : x√3 : 2x.
• Properties of Isosceles triangles (two equal sides, two equal base angles) and Equilateral triangles (three equal sides, three 60° angles).

3. Quadrilaterals

• Sum of interior angles = 360°
• Parallelogram: Opposite sides parallel and equal, opposite angles equal, consecutive angles sum to 180°. Area = Base × Height.
• Rectangle: A parallelogram with four 90° angles. Area = Length × Width. Perimeter = 2(Length + Width). Diagonal = √(Length² + Width²).
• Square: A rectangle with four equal sides (s). Area = s². Perimeter = 4s. Diagonal = s√2.
• Trapezoid: Quadrilateral with exactly one pair of parallel sides (bases b₁, b₂). Area = [(b₁ + b₂) / 2] × Height.

4. Circles

• Radius (r), Diameter (d = 2r)
• Area = πr²
• Circumference = 2πr = πd
• Arc Length = (Central Angle / 360°) × Circumference
• Sector Area = (Central Angle / 360°) × Area

5. Coordinate Geometry

• Slope (m): (y₂ – y₁) / (x₂ – x₁) (“rise over run”)
• Slope-Intercept Form of a Line: y = mx + b (m = slope, b = y-intercept)
• Parallel Lines: Slopes are equal (m₁ = m₂).
• Perpendicular Lines: Slopes are negative reciprocals (m₁ * m₂ = -1).
• Distance Formula: √[(x₂ – x₁)² + (y₂ – y₁)²] (derived from Pythagorean theorem)
• Midpoint Formula: [(x₁ + x₂)/2, (y₁ + y₂)/2]

6. 3D Shapes (Solids)

• Rectangular Solid (Box):
  • Volume = Length × Width × Height
  • Surface Area = 2(LW + LH + WH)
  • Length of the main diagonal = √(L² + W² + H²)
• Cube (Special case of Rectangular Solid with L=W=H=s):
  • Volume = s³
  • Surface Area = 6s²
  • Length of the main diagonal = s√3
• Right Circular Cylinder:
  • Volume = Area of Base × Height = πr²h
  • Lateral Surface Area (side) = Circumference × Height = 2πrh
  • Total Surface Area = Lateral Surface Area + Area of 2 Bases = 2πrh + 2πr²

IV. Data Analysis Formulas

These questions involve statistics, probability, and interpretation of data presented in graphs and tables.

1. Statistics

• Average (Arithmetic Mean): Sum of terms / Number of terms
• Median: The middle value in an ordered set of numbers. If the set has an even number of terms, the median is the average of the two middle terms.
• Mode: The value that appears most frequently in a set. A set can have multiple modes or no mode.
• Range: Maximum value – Minimum value
• Standard Deviation (Concept): Measures the spread or dispersion of data points around the mean.
  • A low standard deviation indicates data points are close to the mean.
  • A high standard deviation indicates data points are spread out over a wider range.
  • You likely won’t need to *calculate* standard deviation using its complex formula, but you must understand what it represents and be able to compare the relative standard deviations of different data sets (e.g., {10, 20, 30} vs. {19, 20, 21}).
• Weighted Average: [(Value₁ × Weight₁) + (Value₂ × Weight₂) + …] / (Total Weight)

2. Probability

• Basic Probability: P(Event) = (Number of favorable outcomes) / (Total number of possible outcomes)
• Probability is always between 0 and 1 (inclusive). 0 means impossible, 1 means certain.
• Probability of an event NOT occurring: P(Not A) = 1 – P(A)
• Probability of Independent Events A and B both occurring: P(A and B) = P(A) × P(B) (Events are independent if the outcome of one does not affect the outcome of the other).
• Probability of Mutually Exclusive Events A or B occurring: P(A or B) = P(A) + P(B) (Events are mutually exclusive if they cannot both happen at the same time).
• Probability of Non-Mutually Exclusive Events A or B occurring: P(A or B) = P(A) + P(B) – P(A and B) (Subtract the overlap).

3. Combinatorics (Counting)

• Fundamental Counting Principle: If an event can occur in ‘m’ ways, and another independent event can occur in ‘n’ ways, then the two events can occur together in m × n ways.
• Permutations (Order Matters): The number of ways to arrange ‘k’ items selected from a set of ‘n’ distinct items. Formula: P(n, k) = n! / (n-k)! (where “!” denotes factorial, e.g., 5! = 5×4×3×2×1).
• Combinations (Order Does Not Matter): The number of ways to choose ‘k’ items from a set of ‘n’ distinct items, where the order of selection doesn’t matter. Formula: C(n, k) = n! / [k! * (n-k)!]. Also written as “n choose k” or ⁿC_k.
• Key Distinction: Use permutations when the arrangement or sequence is important (e.g., arranging letters in a word, assigning specific roles). Use combinations when selecting a group where the order doesn’t matter (e.g., choosing a committee, picking lottery numbers).

4. Data Interpretation

While not strictly formulas, you need to be proficient in reading and interpreting information from:

• Tables
• Bar graphs
• Line graphs
• Pie charts
• Scatter plots

Key skills include finding specific values, calculating percentages or ratios from the data, identifying trends, and making inferences. Pay close attention to titles, labels, units, and scales.

Integrating Formulas with Problem-Solving

Knowing formulas is only half the battle. The real skill lies in applying them correctly within the context of a GRE problem.

• Identify the Topic: Read the question carefully to determine if it’s about geometry, algebra, probability, etc.
• Extract Information: Note down the given values and what the question is asking for. Draw diagrams for geometry problems.
• Select the Right Formula(s): Choose the formula(s) that relate the given information to the desired unknown. Sometimes multiple steps or formulas are needed.
• Plug In and Solve Carefully: Substitute the known values into the formula. Perform calculations accurately, using the on-screen calculator if necessary for complex arithmetic.
• Check for Reasonableness: Does the answer make sense in the context of the problem? (e.g., A length can’t be negative, a probability can’t be greater than 1).

Conclusion

Mastering these essential formulas is a significant step towards achieving a high score on the GRE Quantitative Reasoning section. Remember that practice is key – the more you use these formulas to solve diverse problems, the more comfortable and efficient you will become. Combine formula knowledge with strong conceptual understanding and smart test-taking strategies (like estimation and process of elimination) for optimal results. Good luck with your preparation!