GRE Quantitative

GRE Quantitative Formula Reference Sheet

Every formula and concept you need for GRE Quantitative Reasoning. Covers arithmetic, algebra, coordinate geometry, statistics, probability, combinatorics, and set theory.

Updated for GRE revised format · 170-point scale · Quant Comparison + Problem Solving

Take a Free GRE Practice Exam →GRE Quant Guide GRE Top 100 Vocabulary

130–170

Score range

Each Quant section scored separately

27/section

Questions

Two Quant sections on the exam

47 min/section

Time

About 1.75 min per question

On-screen

Calculator

Basic 4-function; no graphing

2 types

Question types

QC and Problem Solving (PS)

Adaptive

Difficulty

Section 2 difficulty depends on section 1

Arithmetic & Number Theory

Divisibility rules, prime numbers, and integer properties tested in GRE Quant.

LCM (Least Common Multiple)

LCM(a, b) = (a × b) / GCD(a, b)

LCM × GCD = a × b

GCD (Greatest Common Divisor)

Use prime factorization: GCD = product of shared prime factors (lowest powers)

Also called HCF

Prime factorization

n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ

Every integer > 1 has a unique prime factorization

Number of factors

τ(n) = (a₁+1)(a₂+1)...(aₖ+1)

If n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ

Remainder theorem

a ≡ r (mod m) means a = qm + r

0 ≤ r < m; remainder when a is divided by m

Sum of first n integers

1 + 2 + ... + n = n(n+1) / 2

Arithmetic series with first term 1

Sum of first n odd integers

1 + 3 + 5 + ... + (2n−1) = n²

Sum of n odd integers starting from 1

Sum of consecutive integers

Sum = n × average = n × (first + last) / 2

n = count of terms

Divisibility by 3

Sum of digits divisible by 3

E.g., 273: 2+7+3=12, divisible by 3

Divisibility by 9

Sum of digits divisible by 9

E.g., 729: 7+2+9=18, divisible by 9

Algebra

Algebraic identities and factoring patterns essential for GRE Quant Comparison and Problem Solving.

FOIL

(a + b)(c + d) = ac + ad + bc + bd

First, Outer, Inner, Last

Difference of squares

a² − b² = (a + b)(a − b)

Frequently tested factoring identity

Sum of squares

(a + b)² = a² + 2ab + b²

Also (a − b)² = a² − 2ab + b²

Sum of cubes

a³ + b³ = (a + b)(a² − ab + b²)

Less common but appears in hard questions

Difference of cubes

a³ − b³ = (a − b)(a² + ab + b²)

Companion to sum of cubes

Completing the square

x² + bx = (x + b/2)² − (b/2)²

Used to find vertex of parabola

Quadratic formula

x = [−b ± √(b² − 4ac)] / 2a

Solves ax² + bx + c = 0

Inequality — sign flip rule

Multiply/divide by negative → flip the inequality sign

E.g., −2x > 4 → x < −2

Absolute value inequality

|x| < a → −a < x < a

|x| > a → x < −a or x > a

Exponent rules

aᵐ·aⁿ = aᵐ⁺ⁿ · aᵐ/aⁿ = aᵐ⁻ⁿ · (aᵐ)ⁿ = aᵐⁿ

Also a⁻ⁿ = 1/aⁿ and a⁰ = 1

Coordinate Geometry

Lines, parabolas, and circles in the coordinate plane.

Slope formula

m = (y₂ − y₁) / (x₂ − x₁)

Rate of change; rise over run

Slope-intercept form

y = mx + b

m = slope, b = y-intercept

Distance formula

d = √[(x₂−x₁)² + (y₂−y₁)²]

Euclidean distance between two points

Midpoint formula

M = ((x₁+x₂)/2 , (y₁+y₂)/2)

Coordinates of the midpoint

Standard form of a parabola

y = ax² + bx + c

Vertex x = −b/2a; opens up if a > 0

Vertex form of a parabola

y = a(x − h)² + k

Vertex at (h, k)

Equation of a circle

(x − h)² + (y − k)² = r²

Center (h, k), radius r

Parallel lines

Same slope: m₁ = m₂

Different y-intercepts; never intersect

Perpendicular lines

Slopes: m₁ × m₂ = −1

Slopes are negative reciprocals

Statistics

Descriptive statistics — the most heavily tested statistics content on the GRE.

Mean (arithmetic average)

x̄ = Σxᵢ / n

Sum of all values divided by count

Median

Middle value of sorted data

Mean of two middle values if n is even

Mode

Most frequently occurring value

A dataset can have multiple modes or none

Range

Range = max − min

Basic measure of spread

Standard deviation (concept)

SD measures how spread out values are around the mean

Larger SD = more spread; GRE does not require computing SD

Weighted average

x̄ = Σ(wᵢxᵢ) / Σwᵢ

Each value multiplied by its weight/frequency

Percent change

% change = (new − old) / old × 100

Positive = increase; negative = decrease

Percent of a number

x% of n = (x/100) × n

E.g., 35% of 80 = 0.35 × 80 = 28

Average speed

Avg speed = total distance / total time

Do NOT average speeds — use total distance and time

Probability

Probability rules and counting principles tested in GRE Problem Solving.

Basic probability

P(A) = favorable outcomes / total outcomes

Assumes equally likely outcomes; 0 ≤ P(A) ≤ 1

Complement rule

P(not A) = 1 − P(A)

Probability of event not happening

Addition rule (general)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Inclusion-exclusion; removes double-counting

Addition rule (mutually exclusive)

P(A ∪ B) = P(A) + P(B)

Events cannot both occur; P(A ∩ B) = 0

Multiplication rule (independent)

P(A ∩ B) = P(A) × P(B)

Events do not affect each other

Conditional probability

P(A | B) = P(A ∩ B) / P(B)

Probability of A given B has occurred

Expected value

E(X) = Σ[xᵢ · P(xᵢ)]

Weighted average of outcomes by probability

Combinatorics

Counting, permutations, and combinations.

Fundamental counting principle

If event A has m outcomes and B has n, together: m × n

Multiply independent choices

Permutations

P(n, r) = n! / (n − r)!

Ordered arrangements of r items from n

Combinations

C(n, r) = n! / [r!(n − r)!]

Unordered selections; also written ⁿCᵣ

Factorial

n! = n × (n−1) × (n−2) × ... × 2 × 1

0! = 1 by definition

Circular permutations

(n − 1)!

Arrangements of n objects in a circle

Arrangements with repeats

n! / (k₁! × k₂! × ... )

Divide by factorial of each repeated element count

Set Theory

Venn diagram and set intersection formulas — common in GRE Quantitative Comparison.

Union of two sets

|A ∪ B| = |A| + |B| − |A ∩ B|

Inclusion-exclusion for two sets

Union of three sets

|A ∪ B ∪ C| = |A| + |B| + |C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|

Extended inclusion-exclusion

Only in A

|A only| = |A| − |A ∩ B|

Elements in A but not B

Neither A nor B

|neither| = Total − |A ∪ B|

Elements in neither set

Ratios, Rates & Percents

Proportional reasoning — one of the most common GRE Quant topics.

Ratio scaling

If a : b = m, then a = mk and b = nk for some k

Use a common multiplier to find actual values

Direct variation

y = kx

k = constant; as x increases, y increases proportionally

Inverse variation

y = k / x → xy = k

As x increases, y decreases

Mixture problems

c₁v₁ + c₂v₂ = c₃(v₁ + v₂)

c = concentration, v = volume

Simple interest

I = Prt

P = principal, r = rate, t = time in years

Compound interest

A = P(1 + r/n)^(nt)

n = compounds per year

Work rate

1/A + 1/B = 1/T

A and B are individual times; T is combined time

GRE Quant strategy tips

Plug in numbers for QC

For Quantitative Comparison questions with variables, try 0, 1, −1, and a fraction. If the relationship changes, the answer is 'D — cannot be determined.'

Estimation on a basic calculator

The on-screen GRE calculator is basic. For complex calculations, round aggressively and estimate. Most answer choices are spread far enough apart.

Know when to use algebra vs. arithmetic

Some GRE problems can be solved faster by substituting numbers than by solving algebraically. Develop the skill to recognize which approach is faster.

Memorize statistics formulas cold

Statistics (mean, median, standard deviation concept, percentiles) appears in nearly every GRE and is heavily tested in Quantitative Comparison.

Apply these formulas on a real GRE

Take a full-length GRE Quantitative practice exam with authentic Quantitative Comparison and Problem Solving questions.

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