GRE General Test Reference Sheet

How to revise from this sheet

1. 1Cover the right-hand column and say the relationship out loud before reading it. Recognising a formula is not the same as recalling it, and only recall survives a timed section.
2. 2Work down a single domain at a time, then answer a few questions in that domain immediately. Recall that is never used decays fastest.
3. 3When an entry has a condition attached, rehearse the condition as part of the statement. Most exam traps are built on applying a correct formula outside its hypothesis.

ETS publishes separate Quantitative Reasoning and Verbal Reasoning content overviews for the GRE General Test.

Quantitative reasoning

Arithmetic and algebra identities, the standard geometry facts, and the data-analysis definitions that decide comparison items.

Percent and change

$$\text{change} = \frac{\text{new} - \text{old}}{\text{old}}$$

percent change = (new - old)/old; an increase of p% then a decrease of p% does not return to the start; successive changes multiply as (1 + p1)(1 + p2)

Percent of and percent greater than are different questions; read which base is intended.

Ratios and proportions

$$\frac{a}{b} = \frac{c}{d} \iff ad = bc$$

a ratio a : b scales to ak : bk; a proportion is solved by cross-multiplication; combined rates add, so working together takes 1/(1/t1 + 1/t2)

Rate problems are proportion problems in disguise; write the unit ('per what?') before computing.

Exponent and radical rules

$$a^{m}a^{n} = a^{m+n}, \quad (a^{m})^{n} = a^{mn}, \quad a^{m/n} = \sqrt[n]{a^{m}}$$

same base: multiply adds exponents, divide subtracts, power of a power multiplies; a^(-n) = 1/a^n; a^(m/n) is the n-th root of a to the m-th power

There is no rule that simplifies a^m + a^n; only products and quotients combine.

Algebraic identities

$$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}, \quad r_1 + r_2 = -\frac{b}{a}, \quad r_1r_2 = \frac{c}{a}$$

(a + b)^2 = a^2 + 2ab + b^2; (a - b)^2 = a^2 - 2ab + b^2; a^2 - b^2 = (a+b)(a-b); quadratic roots are (-b +/- sqrt(b^2 - 4ac))/(2a) with sum -b/a and product c/a

The difference of squares is the single most useful factorisation on the test; look for it before expanding anything.

Lines and coordinates

$$m = \frac{y_2 - y_1}{x_2 - x_1}, \qquad d = \sqrt{(x_2-x_1)^{2} + (y_2-y_1)^{2}}$$

slope = rise/run; y = mx + b; parallel lines share a slope and perpendicular slopes multiply to -1; distance is the Pythagorean formula and the midpoint is the coordinate average

A vertical line has undefined slope, not zero slope.

Triangles

$$a^{2} + b^{2} = c^{2}, \qquad 1 : \sqrt{3} : 2, \qquad 1 : 1 : \sqrt{2}$$

angles sum to 180 degrees; area = base times height over 2; the Pythagorean theorem holds in right triangles; the 30-60-90 sides are in ratio 1 : sqrt(3) : 2 and the 45-45-90 sides are 1 : 1 : sqrt(2); any side is less than the sum of the other two

Similar triangles scale lengths by k, areas by k squared, and volumes by k cubed.

Circles and solids

$$A = \pi r^{2}, \quad V_{\text{cyl}} = \pi r^{2}h, \quad V_{\text{sph}} = \tfrac{4}{3}\pi r^{3}$$

circumference 2 pi r and area pi r^2; an arc of theta degrees is theta/360 of the circumference; a rectangular solid has volume lwh; a cylinder has pi r^2 h; a sphere has 4 pi r^3 / 3

An inscribed angle is half the central angle on the same arc, so an angle inscribed in a semicircle is a right angle.

Descriptive statistics

the mean uses every value, the median is the middle value, the mode is the most frequent; the range is max minus min and the interquartile range is Q3 minus Q1; standard deviation measures spread about the mean

Adding a constant to every value shifts the mean and median but leaves the standard deviation unchanged.

Normal distribution and z-scores

$$z = \frac{x - \mu}{\sigma}$$

z = (x - mean)/standard deviation; in a normal distribution about 68% of values lie within one standard deviation of the mean, about 95% within two

The percentages apply only to an approximately normal distribution, which the question must state or imply.

Counting and probability

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}, \qquad P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

ordered selections n!/(n-r)!, unordered n!/(r!(n-r)!); P(A or B) = P(A) + P(B) - P(A and B); P(not A) = 1 - P(A)

'At least one' problems are almost always faster through the complement.

Quantitative comparison discipline

compare rather than compute; test 0, 1, a fraction between 0 and 1, and a negative before choosing; answer D only when two tests give different orderings

If a variable's sign or size is unconstrained, the answer is usually that the relationship cannot be determined — but prove it with two cases, never assume it.

Verbal reasoning

Verbal recall is structural: what each question type rewards, and the connective words that fix a blank's direction.

Text completion method

predict the blank from the sentence's own evidence before reading the options, then match; in multi-blank items solve the most constrained blank first

Each blank is scored independently of the others in the same item, so a confident blank is worth filling even when another is unclear.

Sentence equivalence method

two options must produce sentences of the same meaning; a word with no synonym among the choices cannot be part of the answer

Pairing the options first is faster than evaluating six words one at a time.

Directional signals

continuation: moreover, indeed, in fact, similarly; contrast: although, yet, nevertheless, despite, paradoxically; cause: because, since, thus, consequently

The signal word sets the blank's sign. Getting the direction right eliminates more options than knowing the vocabulary does.

Reading comprehension question types

main idea, detail, inference, purpose of a phrase, structure of the argument, and strengthen or weaken

An inference must follow from the passage alone; an option that is true in the world but unsupported by the text is still wrong.

Argument anatomy

the conclusion is what the author wants you to accept; premises are the support; an assumption is the unstated link the argument needs

Negating a true assumption breaks the argument — the fastest test for an assumption option.

Vocabulary in context

the tested sense of a word is the one the sentence supports, which is often a secondary meaning

Words carry tone. Sorting an unfamiliar word into positive, negative, or neutral is frequently enough to eliminate three options.

Sources

Entries drawn from a cited source list it below. Standard results and numerical constants that no source in our registry covers are given as recall material without a citation rather than attributed to a source that does not support them.

1. GRE General Test Quantitative Reasoning Overview — ETS. Accessed 2026-08-02. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
2. GRE General Test Verbal Reasoning Overview — ETS. Accessed 2026-08-03. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
3. Prealgebra 2e, Section 6.5: Solve Proportions and Their Applications — OpenStax. Accessed 2026-07-11. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
4. Elementary Algebra 2e, Section 6.2: Use Multiplication Properties of Exponents — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
5. Elementary Algebra 2e, Section 6.7: Integer Exponents and Scientific Notation — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
6. Intermediate Algebra 2e, Section 8.3: Simplify Rational Exponents — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
7. Introductory Statistics 2e, Section 2.5: Measures of the Center of the Data — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
8. Introductory Statistics 2e, Section 2.7: Measures of the Spread of the Data — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
9. Introductory Statistics 2e, Section 6.1: The Standard Normal Distribution — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
10. Introductory Statistics 2e, Section 3.3: Two Basic Rules of Probability — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
11. Contemporary Mathematics, Section 7.3: Combinations — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.