Official Quantitative Syllabus • Complete Cheat Sheet

Complete GRE Math Formula Sheet & Cheat Sheet

All essential formulas required for GRE Quantitative Reasoning rendered in clear mathematical notation. Covers Arithmetic, Algebra, Geometry, and Data Analysis with test-day traps.

1. Arithmetic & Numbers 2. Algebra & Equations 3. Geometry 2D & 3D 4. Statistics & Probability

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1. Basic Arithmetic & Number Properties

Fundamental rules for operations, exponents, radicals, percentages, and divisibility.

Order of Operations (PEMDAS)

\text{P} \to \text{E} \to \text{M / D (left-to-right)} \to \text{A / S (left-to-right)}

Test Tip: Parentheses, Exponents, Multiplication & Division, Addition & Subtraction. Always evaluate left to right when operations share precedence.

Fraction Arithmetic (Add, Multiply, Divide)

\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}, \quad \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}, \quad \frac{a/b}{c/d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

Test Tip: To divide fractions, multiply by the reciprocal of the divisor.

Basic Exponent Laws

x^a \cdot x^b = x^{a+b}, \quad \frac{x^a}{x^b} = x^{a-b}, \quad (x^a)^b = x^{ab}, \quad (xy)^a = x^a y^a

Test Tip: Bases must be identical to combine exponents via addition or subtraction.

Zero & Negative Exponents

x^0 = 1 \quad (x \neq 0), \quad x^{-n} = \frac{1}{x^n}, \quad x^{1/n} = \sqrt[n]{x}

Test Tip: A negative exponent inverts the base to its reciprocal. Fractional exponents denote roots.

Radical Rules & Pitfalls

\sqrt{ab} = \sqrt{a}\sqrt{b}, \quad \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, \quad \sqrt{a+b} \neq \sqrt{a} + \sqrt{b}

Test Tip: Common Trap: Never split square roots across addition or subtraction!

Percent Change Formula

\text{Percent Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%

Test Tip: Always divide by the original (base) amount, never the new amount.

Simple Interest & Compound Interest

I = P \cdot r \cdot t, \quad A = P\left(1 + \frac{r}{n}\right)^{nt}

Test Tip: $P = \text{Principal}$ , $r = \text{annual rate as decimal}$ , $t = \text{years}$ , $n = \text{compounding periods per year}$ .

Product of GCF and LCM

\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b

Test Tip: Holds exclusively for two positive integers $a$ and $b$ .

Even / Odd Rules & Zero

\text{Even} \pm \text{Even} = \text{Even}, \quad \text{Odd} \pm \text{Odd} = \text{Even}, \quad \text{Even} \times \text{Any} = \text{Even}

Test Tip: Zero ( $0$ ) is an even integer (neither positive nor negative). Consecutive integers always alternate even and odd.

2. Basic Algebra, Equations & Rates

Quadratic identities, linear equations, work rates, distance, and coordinate plane formulas.

Core Quadratic Factoring

(a + b)^2 = a^2 + 2ab + b^2, \quad (a - b)^2 = a^2 - 2ab + b^2, \quad a^2 - b^2 = (a-b)(a+b)

Test Tip: Difference of squares ( $a^2 - b^2$ ) is one of the most tested algebraic shortcuts on the GRE.

Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \quad \text{for } ax^2 + bx + c = 0

Test Tip: Discriminant $b^2 - 4ac > 0 \implies 2$ real roots; $= 0 \implies 1$ real root; $< 0 \implies 0$ real roots.

Distance, Rate & Time

D = R \times T, \quad R = \frac{D}{T}, \quad T = \frac{D}{R}

Test Tip: Units must be consistent (e.g., convert minutes to hours if speed is in miles per hour).

Average Speed Formula

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

Test Tip: Common Trap: Never take the simple arithmetic mean of two speeds for equal distances!

Work Rate & Combined Efficiency

\frac{1}{t_1} + \frac{1}{t_2} = \frac{1}{t_{\text{together}}}, \quad \text{Work} = \text{Rate} \times \text{Time}

Test Tip: Convert individual completion times to reciprocal hourly rates before adding them together.

Slope & Linear Equations

m = \frac{y_2 - y_1}{x_2 - x_1}, \quad y = mx + b

Test Tip: Parallel lines have equal slopes: $m_1 = m_2$ . Perpendicular lines have negative reciprocal slopes: $m_1 \cdot m_2 = -1$ .

Coordinate Distance & Midpoint

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, \quad M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

Test Tip: The coordinate distance formula is directly derived from the Pythagorean theorem in the $xy$ -plane.

Inequality Sign Traps

a < b \implies -a > -b, \quad x^2 < x \iff 0 < x < 1

Test Tip: Multiplying or dividing both sides by a negative number flips the inequality symbol. Squaring numbers between 0 and 1 makes them smaller.

3. Basic Geometry (2D & 3D)

Polygons, triangles, circles, coordinate geometry, and 3D solid geometry.

Triangle Area & Angle Sum

A = \frac{1}{2} b h, \quad \angle A + \angle B + \angle C = 180^\circ

Test Tip: The height $h$ must always be perpendicular to base $b$ .

Pythagorean Theorem & Common Triples

a^2 + b^2 = c^2, \quad (3, 4, 5), \quad (5, 12, 13), \quad (8, 15, 17), \quad (7, 24, 25)

Test Tip: Applies exclusively to right triangles where $c$ is the hypotenuse.

Special Right Triangles

30^\circ-60^\circ-90^\circ: x : x\sqrt{3} : 2x \quad | \quad 45^\circ-45^\circ-90^\circ: x : x : x\sqrt{2}

Test Tip: Opposite $30^\circ$ is $x$ ; opposite $60^\circ$ is $x\sqrt{3}$ ; hypotenuse is $2x$ .

Triangle Third-Side Inequality

|a - b| < c < a + b

Test Tip: The third side of any triangle is strictly between the difference and sum of the other two sides.

Circle Area, Circumference & Arc Length

A = \pi r^2, \quad C = 2\pi r = \pi d, \quad \text{Arc} = \frac{\theta}{360^\circ} \cdot 2\pi r, \quad \text{Sector Area} = \frac{\theta}{360^\circ} \cdot \pi r^2

Test Tip: Diameter $d = 2r$ . Central angle $\theta$ is proportional to the fraction of $360^\circ$ .

Rectangular Solid (Box) & Space Diagonal

V = l \cdot w \cdot h, \quad SA = 2(lw + lh + wh), \quad \text{Space Diagonal} = \sqrt{l^2 + w^2 + h^2}

Test Tip: Space diagonal is the longest straight line between two opposite 3D vertices.

Cylinder Volume & Surface Area

V = \pi r^2 h, \quad \text{Lateral Area} = 2\pi r h, \quad \text{Total SA} = 2\pi r^2 + 2\pi rh

Test Tip: Volume equals base circle area $(\pi r^2)$ multiplied by height $h$ .

4. Statistics, Probability & Combinatorics

Averages, medians, normal distribution, permutations, combinations, and probability.

Arithmetic Mean (Average)

\text{Mean} = \frac{\text{Sum of all terms}}{n}, \quad \text{Sum} = \text{Mean} \times n

Test Tip: If you know the average and the number of elements, you can immediately find the total sum.

Complement Rule (At-Least-One)

P(\text{not } E) = 1 - P(E), \quad P(\text{at least 1}) = 1 - P(\text{none})

Test Tip: Calculating $1 - P(\text{none})$ is much faster than computing multiple compound success branches.

Combinations vs Permutations

nCr = \frac{n!}{r!(n-r)!} \quad (\text{Order does not matter}), \quad nPr = \frac{n!}{(n-r)!} \quad (\text{Order matters})

Test Tip: Select a committee = $nCr$ ; assign distinct positions (President, VP, Secretary) = $nPr$ .

Normal Distribution (Empirical Rule)

\pm 1\sigma \approx 68.2\%, \quad \pm 2\sigma \approx 95.4\%, \quad \pm 3\sigma \approx 99.7\%

Test Tip: A symmetric bell curve centered at the mean $\mu$ . Exactly 50% of the data lies above the mean.

Overlapping Sets (Two Groups)

\text{Total} = \text{Group A} + \text{Group B} - \text{Both} + \text{Neither}

Test Tip: Subtract "Both" once to avoid double-counting elements in the intersection of both groups.

Put These Formulas into Practice

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