The Mathematics Test and how to work it

What ETS says the test is

The GRE Subject Test in Mathematics is an achievement test in undergraduate mathematics. Here are the facts as ETS publishes them for the 2026–27 testing year, in our words:

Feature	What ETS says
Length	Approximately 66 multiple-choice questions (ETS, GRE Subject Test Content and Structure)
Time	2 hours 50 minutes in total, with no separately timed sections (ETS, Content and Structure; Practice Book, Overview)
Answer format	Five suggested answers for each question; choose the best one (Practice Book, test directions)
Scoring	The number answered correctly is converted to a scaled score on a 200–990 scale in 10-point steps; nothing is subtracted for wrong answers (ETS, Understanding Your GRE Subject Test Scores; Strategies and Tips)
Delivery	Computer-delivered at test centers worldwide, and at home on select dates in the testing windows (ETS, Content and Structure; Register for the GRE FAQ; Taking a GRE Subject Test At Home)
Calculator	None: no onscreen calculator for the Subject Tests, and personal calculators only as an approved accommodation (GRE Information Bulletin 2026–27)
Breaks	None during a Subject Test (Bulletin; Strategies and Tips)
2026–27 windows	September 14–27, 2026; October 19–November 1, 2026; April 19–May 2, 2027 (ETS, GRE Subject Tests for score users)

Two conventions from the practice book's directions are worth memorizing so they never cost you a second: a logarithm written without a base is the natural logarithm, and $\mathbb Z,\ \mathbb Q,\ \mathbb R,\ \mathbb C$ are the integers, rationals, reals and complex numbers (Practice Book, test directions).

What is on it

ETS's content specification has three parts. The percentages are ETS's estimates, and ETS says the actual shares differ somewhat from one edition of the test to the next (GRE Mathematics Test fact sheet).

Area	Share	What ETS lists
Calculus	about 50%	Differential and integral calculus of one and of several variables, with calculus-based applications and connections to coordinate geometry, trigonometry, differential equations and other branches
Algebra	about 25%	Elementary algebra; linear algebra (matrix algebra, linear systems, vector spaces, linear transformations, characteristic polynomials, eigenvalues and eigenvectors); abstract algebra and number theory (groups, rings and modules, fields, number theory)
Additional topics	about 25%	Introductory real analysis (sequences and series of numbers and functions, continuity, differentiability, integrability, elementary topology of $\mathbb R$ and $\mathbb R^n$); discrete mathematics (logic, set theory, combinatorics, graph theory, algorithms); other topics (general topology, geometry, complex variables, probability and statistics, numerical analysis)

So on a 66-question form you should expect roughly 33 calculus questions, 16 or 17 algebra questions and 16 or 17 from the additional topics. ETS adds two warnings worth taking seriously: the list is not exhaustive, and questions that need nothing beyond a solid precalculus background can be among the hardest on the test (fact sheet). That is why this course has a full lesson on elementary algebra.

The clock

$170$ minutes for $66$ questions is

$$\frac{170}{66}\approx 2.58 \text{ minutes} \approx 2 \text{ minutes } 35 \text{ seconds per question.}$$

That average hides the real shape of the test: many questions take under a minute if you know the fact being tested, and a few take five or more. ETS's own advice is to work quickly but carefully, and not to linger over questions you find extremely difficult or unfamiliar, because every question carries the same weight (Strategies and Tips).

A plan that works:

1. First pass (about 100–110 minutes). Answer every question you can do with confidence. If a question will clearly take long, choose your best guess, mark it with the mark-and-review tool, and move on. The testing software lets you mark questions, see a list of answered and marked questions, and change answers while time remains (Strategies and Tips).
2. Second pass (about 50 minutes). Return to marked questions, easiest first.
3. Last minutes. Make sure every question has an answer. Your score is the number right; a blank earns nothing and a wrong answer costs nothing (Strategies and Tips; Practice Book, Test-Taking Strategies).

Technique 1: try a special case

When a question says "for every function such that …" or "which of the following must be …", the answer must hold for the simplest object satisfying the hypotheses. Find that object.

Worked example. A differentiable function $f\colon\mathbb R\to\mathbb R$, not identically zero, satisfies $f(x+y)=f(x)f(y)$ for all $x,y$, and $f'(0)=3$. What is $f(2)$?

The exponential $e^{kx}$ satisfies the functional equation, and $f'(0)=k=3$, so $f(x)=e^{3x}$ and $f(2)=e^{6}$. The theory confirms it: $f(0)=f(0)^2$ and $f\ne 0$ force $f(0)=1$; then $f'(x)=\lim_{h\to0}\frac{f(x)f(h)-f(x)}{h}=f(x)f'(0)$, so $f'=3f$ and $f=e^{3x}$. On the test you need only the first line.

The trap: the special case tells you what the answer must be when the answer is unique. It cannot prove a "must be true" statement; it can only refute one.

Technique 2: Roman numerals and counterexamples

A common format lists statements I, II, III and asks which must be true. Decide each statement separately, then match. For each statement, ask "what is the nastiest object that satisfies the hypotheses?"

Worked example. Let $f$ be continuous on the open interval $(0,1)$. Which must be true? I. $f$ is bounded. II. $f$ attains a maximum. III. $f$ is uniformly continuous.

Take $f(x)=1/x$: it is unbounded, has no maximum, and is not uniformly continuous. One counterexample kills all three: the answer is "none". Had the interval been the closed interval $[0,1]$, all three would hold (extreme value theorem and the Heine–Cantor theorem; see the real analysis lesson). The whole question turns on one bracket, which is exactly the kind of detail to read slowly.

Elimination logic. If you are sure statement I is false, cross out every choice containing I. If you are sure II is true, cross out every choice lacking II. Often two decisions leave one choice.

Technique 3: test the choices

When the answer is a number and checking is easier than solving, check.

Worked example. For which $k$ does the system $x+2y=3,\ 2x+ky=6$ have infinitely many solutions? Choices: $1, 2, 3, 4, 6$.

The second equation must be a multiple of the first; with $k=4$ it is exactly twice the first. Checking takes seconds: only $k=4$ makes the coefficient matrix singular, and then the right-hand sides are also in ratio $2$.

Technique 4: limiting cases and dimensions

A formula offered as an answer must survive the cases you know.

Worked example. Which expression gives the volume of the cap cut from a ball of radius $R$ by a plane at height $h$ above the bottom ($0\le h\le 2R$)?

(A) $\tfrac{\pi h^2(3R-h)}{6}$ (B) $\tfrac{\pi h^2(3R-h)}{3}$ (C) $\tfrac{\pi h(3R-h)^2}{3}$ (D) $\pi h^2(R-h)$ (E) $\tfrac{\pi h^2(2R-h)}{3}$

At $h=2R$ the cap is the whole ball, $\tfrac43\pi R^3$. (A) gives $\tfrac23\pi R^3$, (B) gives $\tfrac{4}{3}\pi R^3$, (C) gives $\tfrac23\pi R^3$, (D) is negative, (E) gives $0$. Only (B) survives, and it also gives the hemisphere $\tfrac23\pi R^3$ at $h=R$. No integral needed.

Technique 5: estimate, because there is no calculator

Without a calculator, know a few anchors cold: $e\approx 2.718$, $\pi\approx3.1416$, $\ln 2\approx0.693$, $\ln 3\approx 1.099$, $\ln 10\approx 2.303$, $\sqrt2\approx1.414$, $\sqrt3\approx1.732$, $e^{\pi}\approx 23.14$.

Worked example. Which is larger, $e^\pi$ or $\pi^e$? Compare $\frac{\ln x}{x}$ at $x=e$ and $x=\pi$: the function $g(x)=\frac{\ln x}{x}$ has $g'(x)=\frac{1-\ln x}{x^2}$, so it is largest at $x=e$. Hence $\frac{\ln \pi}{\pi}<\frac{1}{e}$, i.e. $e\ln\pi<\pi$, i.e. $\pi^e<e^\pi$. The same idea settles every "$a^b$ versus $b^a$" comparison.

Common traps

• Hypotheses read too fast. Open versus closed interval, "differentiable" versus "continuously differentiable", "some" versus "every", real versus complex scalars. Underline them on your scratch paper.
• Answering a different question. You computed $f(2)$; the question asked for $f'(2)$, or for the sum of the roots, or for the number of solutions.
• Sign and constant slips from doing algebra in your head. Write the line down; the test is long and fatigue is real.
• Sinking five minutes into one question on the first pass. Mark it and leave.
• Leaving blanks. There is no penalty for a wrong answer.

How to study with this course

ETS notes that the questions measure knowledge built over a long time, that preparation in the weeks or months before the test can help to some extent, and that a general review of your college courses is probably the best preparation (Strategies and Tips; Practice Book, Preparing for the Test). This course is organized that way: one lesson per area of the specification, each with the theorems that the test uses, worked examples, traps, and original practice questions with full explanations. Finish with our full-length practice exam and with the free official practice book that ETS offers on its Prepare for a GRE Subject Test page, taken under real timing.

Check yourself

• How many questions, how many minutes, and how many minutes per question on average?
• What are the three content areas and their approximate weights?
• Why should no question be left blank?
• What does $\log x$ mean on this test?
• Give one function that refutes "every continuous function on $(0,1)$ is bounded".
• Which is larger, $2^{\pi}$ or $\pi^{2}$? (Hint: $g(x)=\ln x/x$, and $g(2)=g(4)$.)