GRE Quantitative Reasoning Cheat Sheet

ARITHMETIC

Number Properties

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Odd/Even Rules

odd ± odd = even | even ± even = even | odd ± even = odd

odd × odd = odd | even × anything = even

GCD & LCM

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Fractions, Decimals, Percents

Percent change = (New − Old) / Old × 100% — always divide by OLD value

Successive percent changes MULTIPLY, not add: 0.70 × 1.30 = 0.91 (not 70% + 30%)

Trap: 50% decrease + 100% increase on reduced base = 0% net change, NOT original value

Work/Rate Problems

Combined rate = 1/a + 1/b (rates add)

Time to complete = 1 / (combined rate)

Example: A does job in 12 hrs (1/12/hr), B does job in 8 hrs (1/8/hr) → together: 1/12 + 1/8 = 5/24/hr → 24/5 = 4.8 hrs

Divisibility Rules

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Remainders

When N = qd + r, then r = remainder when N divided by d

Always: 0 ≤ r < d

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ALGEBRA

Linear Equations & Systems

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Solve by: substitution or elimination

Inequalities — CRITICAL

Multiplying/dividing by negative number REVERSES the sign

Compound: a < x < b solved by applying same operation to all three parts

Absolute value < k: |x| < k → −k < x < k (BETWEEN)

Absolute value > k: |x| > k → x < −k OR x > k (OUTSIDE)

Exponent Rules

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Radical Rules

√(ab) = √a × √b ✓

√(a/b) = √a / √b ✓

√(a + b) ≠ √a + √b ✗ (common trap)

Quadratic Equations: ax² + bx + c = 0

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Functions

Notation: f(g(x)) = apply g first, then apply f to result

Domain: valid inputs | Range: possible outputs

Direct variation: f(x) = kx (y ∝ x)

Inverse variation: f(x) = k/x (y ∝ 1/x)

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GEOMETRY

Lines & Angles

Vertical angles: equal

Supplementary: sum to 180°

Complementary: sum to 90°

Parallel lines + transversal: alternate interior angles & corresponding angles equal

Triangle angle sum: 180°

Pythagorean Theorem & Triples

Right triangles only: a² + b² = c² (c = hypotenuse)

Common triples: 3-4-5, 5-12-13, 8-15-17 (and multiples)

Special Right Triangles

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Concept	Rule/Fact
Zero	Neither positive nor negative
Integers	Whole numbers only; fractions/decimals are NOT integers
Prime	Natural numbers >1 with only divisors 1 and itself; 2 is only even prime
First 10 primes	2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Term	Definition	Formula
GCD	Largest divisor of both numbers	Use prime factorization (min powers)
LCM	Smallest multiple of both numbers	Use prime factorization (max powers)
Relationship	GCD × LCM = Product of the two numbers	Always verify
Divisor	Rule
2	Last digit even
3	Sum of digits divisible by 3
4	Last two digits divisible by 4
5	Ends in 0 or 5
9	Sum of digits divisible by 9
System Type	Solution
Consistent & independent	Exactly one solution (lines intersect)
Inconsistent	No solution (parallel lines)
Dependent	Infinite solutions (same line)
Rule	Formula
Multiply (same base)	x^a × x^b = x^(a+b)
Divide (same base)	x^a / x^b = x^(a−b)
Power of power	(x^a)^b = x^(ab)
Zero exponent	x^0 = 1 (x ≠ 0)
Negative exponent	x^(−n) = 1/x^n
Fractional exponent	x^(m/n) = ^n√(x^m)
Method	When to use
Factoring	(x+a)(x−a) = x²−a²; (x±a)² = x²±2ax+a²
Quadratic formula	x = [−b ± √(b²−4ac)] / 2a
Discriminant	b²−4ac > 0 → 2 real roots; = 0 → 1 root; < 0 → 0 real roots
Type	Angle Ratio	Side Ratio
45-45-90	45°-45°-90°	1 : 1

Quantitative Reasoning

GRE Quantitative Reasoning Cheat Sheet

ARITHMETIC

Number Properties

Odd/Even Rules

GCD & LCM

Fractions, Decimals, Percents

Work/Rate Problems

Divisibility Rules

Remainders

ALGEBRA

Linear Equations & Systems

Inequalities — CRITICAL

Exponent Rules

Radical Rules

Quadratic Equations: ax² + bx + c = 0

Functions

GEOMETRY

Lines & Angles

Pythagorean Theorem & Triples

Special Right Triangles

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