ACT Math Study Guide 2026 | Exam Predictor

Part 145 questions, 50 minutes

How the ACT Math Section Works

45 self-contained, four-choice questions on the math of courses up to the start of grade 12, with nine scores reported.

ACT Math assesses the skills students typically acquire in courses taken up to the beginning of grade 12, with emphasis on what you need for entry-level college math. Every question is self-contained and has four answer choices. ACT assumes knowledge of basic formulas and computational skills, but says recall of complex formulas and extensive computation are not required.

The Directions You Will See

The Math directions tell you not to linger over problems that take too long, and list four assumptions that hold unless a question says otherwise:

Illustrative figures are not necessarily drawn to scale.
Geometric figures lie in a plane.
The word "line" indicates a straight line.
The word "average" indicates arithmetic mean.

The directions do not include a formula sheet, so the formulas in Unit 7 of this guide need to be in your head.

Nine Scores From One Section

Your report shows a Math scale score from 1 to 36 and eight reporting category scores: Preparing for Higher Math, its five subcategories, Integrating Essential Skills, and Modeling. Math also feeds the STEM score if you take Science.

The Reporting Categories

Category	Approximate share
Preparing for Higher Math	80%
Number and Quantity	10 to 12%
Algebra	17 to 20%
Functions	17 to 20%
Geometry	17 to 20%
Statistics and Probability	12 to 15%
Integrating Essential Skills	20%
Modeling	Counted across the categories above

Algebra, Functions and Geometry carry equal weight, so do not let one become a weak spot. Number and Quantity is the smallest category, but it holds the topics students forget: complex numbers, matrices and vectors.

The Top of the Scale Is Tight

In the conversion table for ACT's 2025/2026 Practice Test 1, 41 and 40 correct both convert to 36, 39 converts to 35, 36 to 33, 30 to 28 and 21 to 20. Conversions vary by form, but missing a few hard questions at the end costs far less than careless slips on easy ones.

Key Concepts

Preparing for Higher MathAbout 80% of Math: Number and Quantity, Algebra, Functions, Geometry, and Statistics and Probability.

Integrating Essential SkillsAbout 20% of Math: multi-step, non-routine problems built from rates, percentages, proportions, area and volume, averages and medians.

ModelingEvery question that involves producing, interpreting, evaluating or improving a model. Each is also counted in another category.

Part 2Math section only

The ACT Calculator Policy

What you can bring, what is banned, and how to use a calculator without losing time.

Calculators are allowed only on the mathematics section, and ACT says all problems can be solved without one. You must bring your own; testing staff do not provide them, and sharing is not permitted.

What Is Allowed

Four-function, scientific and graphing calculators, unless the model is prohibited.
Some calculators need modification: remove documents and any programs with computer algebra system (CAS) functionality, remove paper tape, turn off sound, cover an infrared data port with heavy opaque material, and remove power cords.
The non-CAS TI-Nspire is permitted.
Online testers have the Desmos graphing calculator built into the testing platform during Math.

What Is Prohibited

Any calculator with built-in or downloaded CAS functionality, including all models beginning TI-89 or TI-92, the TI-Nspire CAS, the HP Prime, HP 48GII, all HP 40G, 49G and 50G models, the Casio fx-CP400 (ClassPad 400), ClassPad 300 and m330, Algebra fx 2.0 and all CFX-9970G models.
Laptops, tablets, phones, pen-input devices and calculators with QWERTY keyboards.

Breaking the policy is treated as prohibited behavior and leads to dismissal. ACT updates its list, so check act.org/calculator-policy before test day.

Using It Well

ACT's own advice is that many problems are best done without a calculator, and to solve simple calculations by hand. Reach for it for messy arithmetic, checking a graph's intercepts, or testing a value. If you test online, practice with Desmos in advance: graphing both sides of an equation and reading the intersection can solve many algebra questions quickly.

Exam tip: Practice with the calculator you will use on test day, as ACT recommends. Bring fresh batteries or a backup, and know where the functions you need (fractions, square roots, the table feature) are before the clock starts.

Part 310 to 12% of Math

Number and Quantity

Real and complex numbers, exponents and radicals, vectors and matrices.

ACT describes this subcategory as knowledge of real and complex number systems and reasoning with quantities in many forms, including integer and rational exponents, vectors and matrices.

Exponent rules: multiply means add exponents, divide means subtract, power of a power means multiply. A rational exponent is a root.
Complex numbers: i squared is -1, and powers of i cycle every four. To divide, multiply by the conjugate.
Vectors: add component by component; magnitude comes from the Pythagorean theorem.
Matrices: add entry by entry; multiply row by column, which requires the inner dimensions to match.

Worked Example: Rational Exponents and Complex Division

\begin{aligned} 27^{2/3} &= \left(\sqrt[3]{27}\right)^2 = 3^2 = 9 \\[6pt] \frac{5}{2+i} &= \frac{5}{2+i}\cdot\frac{2-i}{2-i} = \frac{5(2-i)}{4 - i^2} = \frac{5(2-i)}{5} = 2 - i \end{aligned}

Worked Example: Matrices and Vectors

\begin{aligned} \begin{bmatrix} 1 & 2 \\ 3 & 0 \end{bmatrix}\begin{bmatrix} 4 \\ 5 \end{bmatrix} &= \begin{bmatrix} 1\cdot4 + 2\cdot5 \\ 3\cdot4 + 0\cdot5 \end{bmatrix} = \begin{bmatrix} 14 \\ 12 \end{bmatrix} \\[8pt] \mathbf{u} = \langle 3, -1 \rangle,\ \mathbf{v} = \langle 1, 5 \rangle &\Rightarrow \mathbf{u}+\mathbf{v} = \langle 4, 4 \rangle,\quad \lVert \mathbf{u}+\mathbf{v} \rVert = \sqrt{4^2+4^2} = 4\sqrt{2} \end{aligned}

Common mistake: Treating a negative exponent as a negative number. A negative exponent means a reciprocal: 2 to the power -3 is 1/8, not -8.

Part 417 to 20% of Math

Algebra

Linear, polynomial, radical and exponential relationships, systems of equations, and applying results to real contexts.

ACT lists solving, graphing and modeling many kinds of expressions; interpreting linear, polynomial, radical and exponential relationships; and solving systems, even when written as a simple matrix equation.

Linear equations: slope from two points, slope-intercept and point-slope forms, parallel lines share a slope, perpendicular slopes are negative reciprocals.
Systems: elimination when coefficients line up, substitution when one variable is already isolated.
Quadratics: factor first; use the quadratic formula when factoring fails. The discriminant tells you how many real solutions there are.
Radical equations: square both sides, then check every solution in the original, because squaring can create extraneous ones.
Inequalities: flip the sign when multiplying or dividing by a negative.

Worked Example: A Radical Equation

\begin{aligned} \sqrt{x+7} &= x - 5 \\ x + 7 &= x^2 - 10x + 25 \\ 0 &= x^2 - 11x + 18 = (x-2)(x-9) \\ \text{Check } x = 9:&\ \sqrt{16} = 4 = 9 - 5\ \checkmark \qquad \text{Check } x = 2:\ \sqrt{9} = 3 \ne -3\ \times \end{aligned}

Only x = 9 works. An answer choice that lists both 2 and 9 is the trap.

Exam tip: When a question asks for an expression such as x + y or 2a, look for a way to get it directly. Adding or subtracting the equations of a system often gives the requested combination without solving for each variable.

Part 517 to 20% of Math

Functions

Notation, linear, radical, piecewise, polynomial, exponential and logarithmic functions, transformations and graph features.

ACT expects you to know the definition, notation, representation and application of functions, to manipulate and translate them, and to interpret important features of graphs.

Notation and composition: f(g(x)) means apply g first, then f.
Transformations: f(x) + k shifts up, f(x - h) shifts right, -f(x) reflects over the x-axis, a f(x) stretches vertically.
Graph features: intercepts, zeros, maximum and minimum points, domain and range, increasing and decreasing intervals, and asymptotes.
Piecewise functions: check which interval the input falls in before you substitute.
Exponential functions: a times b to the x grows when b is greater than 1 and decays when b is between 0 and 1. The initial value is a.
Logarithms: a log is an exponent. Convert to exponential form to solve.

Worked Example: Logs and Exponential Growth

\begin{aligned} \log_2(x - 1) = 5 &\iff x - 1 = 2^5 = 32 \iff x = 33 \\[6pt] P(t) = 1200(1.05)^t:&\quad \text{initial value } 1200,\ \text{growth rate } 5\%\ \text{per period} \\[4pt] \log_3 81 - \log_3 9 &= \log_3 \tfrac{81}{9} = \log_3 9 = 2 \end{aligned}

Common mistake: Reading f(x - 3) as a shift left. Inside the parentheses the shift goes the opposite way to the sign: f(x - 3) moves the graph 3 units right.

Part 6Geometry 17 to 20%; Statistics and Probability 12 to 15%

Geometry, Statistics and Probability

Shapes and solids, triangles and circles, trigonometric ratios and conic sections; then center, spread, data and probability.

Geometry

ACT lists congruence and similarity, surface area and volume, composite objects, missing values in triangles, circles and other figures, trigonometric ratios, and equations of conic sections.

Similar triangles: corresponding sides are in the same ratio; areas scale by the ratio squared, volumes by the ratio cubed.
Special right triangles: 30-60-90 sides are in the ratio 1 : root 3 : 2; 45-45-90 sides are 1 : 1 : root 2.
Circles: complete the square to turn a general equation into center-radius form.
Composite figures: break the shape into rectangles, triangles and sectors, or subtract a cut-out from a whole.

Statistics and Probability

ACT lists describing center and spread, analyzing data collection methods, modeling bivariate data, and calculating probabilities from sample spaces.

The median resists outliers; the mean does not. An outlier also increases range and standard deviation.
Random sampling lets results generalize to the population; random assignment in an experiment supports cause and effect.
Scatterplots: the slope of a line of best fit is the predicted change in y per unit of x.
Without replacement, the second probability uses one fewer item in the numerator and denominator.

Worked Example: Completing the Square for a Circle

\begin{aligned} x^2 + y^2 + 8x - 2y + 8 &= 0 \\ (x^2 + 8x + 16) + (y^2 - 2y + 1) &= -8 + 16 + 1 \\ (x + 4)^2 + (y - 1)^2 &= 9 \quad\Rightarrow\quad \text{center } (-4, 1),\ r = 3 \end{aligned}

Exam tip: Figures are not necessarily drawn to scale, so never measure by eye when the question gives numbers. But a rough sketch of your own, with the given lengths labeled, catches most setup errors in geometry questions.

Part 7Integrating Essential Skills about 20% of Math

Integrating Essential Skills, Modeling and the Formulas to Know

Multi-step problems from core skills, what modeling questions ask, and the formula list to memorize.

Integrating Essential Skills questions ask about rates and percentages, proportional relationships, area, surface area and volume, average and median, and expressing numbers in different ways. ACT describes them as non-routine problems that chain several steps together. The math is familiar; the challenge is keeping track.

Worked Example: Percent Change in Two Steps

A store raises a price by 25%, then puts the item on sale at 20% off the new price. What is the overall change?

p \times 1.25 \times 0.80 = p \times 1.00 \quad\Rightarrow\quad \text{no overall change (not } {+}5\%\text{)}

Modeling Questions

Modeling questions ask you to produce, interpret, evaluate or improve a model: write the equation for a situation, explain what a slope or intercept means in context, or judge whether a model is reasonable. For a linear model, the slope is the rate and the intercept is the starting value. For an exponential model, the base tells you the percent change per period. Always attach units.

The Formulas to Know

ACT assumes basic formulas without giving them to you. Memorize these:

\begin{array}{ll} \textbf{Slope, line:} & m = \dfrac{y_2 - y_1}{x_2 - x_1},\quad y = mx + b,\quad y - y_1 = m(x - x_1) \\[10pt] \textbf{Distance, midpoint:} & d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2},\quad \left(\tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2}\right) \\[10pt] \textbf{Quadratic formula:} & x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a},\quad \text{vertex at } x = -\tfrac{b}{2a} \\[10pt] \textbf{Exponents, logs:} & a^m a^n = a^{m+n},\ (a^m)^n = a^{mn},\ a^{-n} = \tfrac{1}{a^n},\ \log_b x = y \iff b^y = x \\[10pt] \textbf{Areas:} & \text{triangle } \tfrac12 bh,\ \text{trapezoid } \tfrac12(b_1+b_2)h,\ \text{circle } \pi r^2,\ \text{circumference } 2\pi r \\[10pt] \textbf{Volumes:} & \text{prism } Bh,\ \text{cylinder } \pi r^2 h,\ \text{cone } \tfrac13\pi r^2 h,\ \text{sphere } \tfrac43\pi r^3 \\[10pt] \textbf{Triangles:} & a^2 + b^2 = c^2,\ \sin\theta = \tfrac{\text{opp}}{\text{hyp}},\ \cos\theta = \tfrac{\text{adj}}{\text{hyp}},\ \tan\theta = \tfrac{\text{opp}}{\text{adj}},\ \sin^2\theta + \cos^2\theta = 1 \\[10pt] \textbf{Circles, arcs:} & (x-h)^2 + (y-k)^2 = r^2,\quad \text{arc} = \tfrac{\theta}{360^\circ}\cdot 2\pi r,\quad \text{sector} = \tfrac{\theta}{360^\circ}\cdot \pi r^2 \\[10pt] \textbf{Sequences:} & a_n = a_1 + (n-1)d,\quad a_n = a_1 r^{\,n-1} \\[10pt] \textbf{Probability, mean:} & P(A) = \tfrac{\text{favorable}}{\text{total}},\ P(A \text{ and } B) = P(A)P(B \mid A),\ \bar{x} = \tfrac{\text{sum}}{n} \end{array}

Exam tip: ACT warns that an answer choice is often an intermediate result. In multi-step problems, reread the last line of the question before you choose: the total, not the increase; the radius, not the diameter; the percent, not the amount.

Common mistake: Adding percent changes. A 25% increase followed by a 20% decrease is not a 5% increase; multiply the factors (1.25 times 0.80) instead.

Part 8About 67 seconds per question

Pacing and Problem-Solving Habits

How to spend 50 minutes on 45 questions, and the checks ACT itself recommends.

Do not linger. The directions say to solve as many as you can, then return to the others. Guess, flag and move on after about two minutes on any one question. There is no guessing penalty.
Solve first, then match. ACT notes that working backwards from all four choices can take a lot of time. Backsolve only when testing a choice or two is clearly faster than the algebra.
If your answer is not there, reread the problem, and check whether your expression can be rewritten into one of the choices.
Check reasonableness. A negative length or a probability above 1 means a setup error.
Use spare time to check the questions you were least sure about.

Exam prediction: This topic frequently appears on the ACT Math exam. See our full ACT Math predictions

Part 910 ACT-style questions

ACT Math Practice Quiz

10 original ACT-style math questions across every category. Select an answer to see why every choice is right or wrong.

ACT Math Practice Quiz

Original questions written in the style of the enhanced ACT Math section, with four choices each. They are not official ACT questions. As on the ACT, odd questions use A to D and even questions use F to J.

10 Questions

Number and Quantity · Exponents

1What is the value of

(2^{3})^{2} \cdot 2^{- 4}

?

Choose 1 answer:

Algebra · Systems of equations

2If

2 x + y = 11

and

x - y = 1

, what is the value of

x + y

?

Choose 1 answer:

Functions · Composition

3Given

f (x) = x^{2} - 3

and

g (x) = 2 x + 1

, what is

f (g (2))

?

Choose 1 answer:

Geometry · Right triangle trigonometry

In right triangle ABC, angle C is the right angle, BC = 6, AC = 8 and AB = 10.

4What is

sin A

?

Choose 1 answer:

Statistics and Probability · Mean and median

A data set contains the values 3, 7, 7, 8 and 15. The value 40 is added to the set.

5Which statement about the mean and median is true?

Choose 1 answer:

Integrating Essential Skills · Percent change

6A jacket costs 80 dollars. Its price is raised by 20%, and then the new price is reduced by 20%. What is the final price, in dollars?

Choose 1 answer:

Modeling · Interpreting a linear model

A seedling's height, in centimeters,

w

weeks after planting is modeled by

h (w) = 4 + 1.5 w

.

7What does the number 1.5 represent in this model?

Choose 1 answer:

Number and Quantity · Complex numbers

8Which of the following is equal to

i^{23}

, where

i^{2} = - 1

?

Choose 1 answer:

Statistics and Probability · Without replacement

A bag holds 4 red marbles and 6 blue marbles. Two marbles are drawn at random without replacement.

9What is the probability that both marbles are red?

Choose 1 answer:

Geometry · Conic sections

10What are the center and radius of the circle

x^{2} + y^{2} - 6 x + 4 y - 12 = 0

?

Choose 1 answer:

ACT Math Study Guide

How the ACT Math Section Works

The Directions You Will See

Nine Scores From One Section

The Reporting Categories

The Top of the Scale Is Tight

The ACT Calculator Policy

What Is Allowed

What Is Prohibited

Using It Well

Number and Quantity

Worked Example: Rational Exponents and Complex Division

Worked Example: Matrices and Vectors

Algebra

Worked Example: A Radical Equation

Functions

Worked Example: Logs and Exponential Growth

Geometry, Statistics and Probability

Geometry

Statistics and Probability

Worked Example: Completing the Square for a Circle

Integrating Essential Skills, Modeling and the Formulas to Know

Worked Example: Percent Change in Two Steps

Modeling Questions

The Formulas to Know

Pacing and Problem-Solving Habits

ACT Math Practice Quiz

ACT Math Practice Quiz

Practise on a predicted ACT Math paper

Sources