ACT Mathematics — Complete Study Guide
Read this first. It is written for you.
If maths has never quite clicked, this guide is built for exactly that. Nothing here assumes you already understand it. Every rule is explained in plain English before the formula appears, every symbol is spelled out in words, and every method comes with a worked example where every step is shown — nothing is left for you to "just know".
You do not have to be naturally good at maths to do well on this test. The ACT Mathematics section examines a fixed and published list of ideas. It is not trying to surprise you with new mathematics; it is checking whether you can use familiar mathematics accurately, at speed, on a problem worded in an unfamiliar way. That is a skill, and skills are built by practice.
The test changed in 2025. Make sure you are studying the right one.
This matters more than anything else on this page. In April 2025 ACT replaced its maths paper, and from spring 2026 the old one is not given at all. A great deal of the ACT practice material in circulation — including books still being sold — is for the paper that no longer exists.
| the old paper | the paper you will sit | |
|---|---|---|
| questions | 60 | 45 |
| minutes | 60 | 50 |
| time per question | 60 seconds | 67 seconds |
| answer choices | five (A–E / F–K) | four (A–D / F–J) |
| questions that count | 60 | 41 — four are unscored trials |
| calculator | allowed throughout | allowed throughout |
| formula sheet | none | none |
| score range | 1–36 | 1–36 |
ACT's own wording, from its guide for educators:
There are 44 fewer questions overall. […] The number of answer choices in math questions has been reduced from five to four.
How to tell instantly whether a practice test is out of date: look at the answer choices. If you see an E or a K, it is the old paper. Its mathematics is still perfectly good practice — the content did not change — but its shape, its timing and its pacing are wrong.
Four choices, not five — and why that is good news
Five choices meant a blind guess paid off one time in five. Four choices means it pays off one time in four. Combined with the extra seven seconds per question, a question you have to guess on is now worth more than it used to be.
There is no penalty for a wrong answer. None. A blank is worth exactly zero and a guess is worth a quarter of a mark on average. Never leave anything blank, even if you have to fill something in blindly in the last ten seconds.
The letters alternate, and that is deliberate
Odd-numbered questions are lettered A, B, C, D. Even-numbered questions are lettered F, G, H, J.
This is not decoration. It is a safety net: if you slip a row on the answer sheet, you will find yourself trying to shade a "B" on a row that only offers F, G, H and J, and you will catch the mistake immediately. Every question in this book is lettered the same way, so you build the habit here.
The one thing the ACT does that the SAT does not
The ACT gives you no formula sheet.
The SAT prints a page of formulas at the start of every maths module — areas, volumes, the special right triangles, the circle equation. The ACT prints nothing. Everything you need must be in your head when you sit down.
That single fact changes how you must prepare, and it is why this book has Chapter 8, which contains nothing but the formulas the ACT expects you to know, with the reason each one is true so it stays learned rather than memorised and forgotten.
The other ACT-only topics, none of which appear on the SAT at all:
- matrices — adding them, multiplying by a number, determinants
- complex numbers — the number , and its powers
- vectors — magnitude and components
- sequences — arithmetic and geometric, and the th term
- logarithms
- trigonometry beyond right triangles — the law of sines and the law of cosines
- radians, and the graphs of sine and cosine
If you are moving across from SAT preparation, that list is your gap.
What is on it, and how much of each
ACT publishes the blueprint. There are two reporting categories, and the first splits into five:
| category | share of the test | roughly, out of 45 |
|---|---|---|
| Preparing for Higher Math | 80% | 36 |
| — Algebra | 17–20% | 9 |
| — Functions | 17–20% | 8 |
| — Geometry | 17–20% | 8 |
| — Statistics & Probability | 12–15% | 6 |
| — Number & Quantity | 10–12% | 5 |
| Integrating Essential Skills | 20% | 9 |
Integrating Essential Skills is the category people ignore, because its mathematics is the easiest — rates, percentages, ratios, averages, area. It is also a fifth of the paper, which is more than Statistics and Number & Quantity put together. Chapter 1 is devoted to it, and it comes first on purpose.
Modeling is also reported, but it is not separate content: a question counts as modeling and as one of the categories above.
⚠ A warning about ACT's own free practice booklet
ACT publishes a free booklet called Preparing for the ACT. It is genuinely useful and you should do it. But its maths paper is not built to the blueprint above.
We counted it, question by question, from ACT's own scoring key. Of its 41 scored questions, 25 are Preparing for Higher Math and 16 are Integrating Essential Skills — a 61% / 39% split, not 80% / 20%. That is almost exactly the old paper's mix, which the three official pre-2020 forms confirm at 58% / 42%.
Independent analysis of four released enhanced forms puts the real tests at 80/20, matching the published blueprint.
What that means for you: the free booklet gives you about twice as much easy foundational material as the real test will, and correspondingly less algebra, functions and geometry. If you practise only on it, you will be over-prepared for the easy fifth and under-prepared for the rest. This book is built to 80/20.
How it is scored
You get one mark per correct answer out of the 41 that count. Nothing is subtracted for a wrong one. That total — your raw score — is converted to the familiar 1–36 scale score.
The conversion is not a percentage. From ACT's own table for the official enhanced form:
| raw score (out of 41) | scale score |
|---|---|
| 41 | 36 |
| 38 | 34 |
| 33 | 30 |
| 28 | 27 |
| 23 | 22 |
| 18 | 18 |
| 13 | 16 |
| 8 | 14 |
| 3 | 9 |
Read the middle of that table carefully. Around the middle of the range one extra correct answer is worth about one scale point. Five more right answers is roughly five points. That is a very large return for a modest amount of study, and it is concentrated in the questions you nearly get right — which is what the worked solutions in this book are for.
The mock papers in Chapter 9 are scored with this exact table.
Time, and how to spend it
50 minutes, 45 questions — 67 seconds each. That is the whole difficulty of the ACT in one number. The mathematics is not harder than the SAT's; there is simply less time per question.
Three consequences worth taking seriously:
- The questions get harder as you go. Not strictly, but strongly. The first ten are mostly one-step; the last ten are the ones that take real thought. So a minute spent stuck on question 34 costs you a question 12 you could have had. Go through in order, and skip anything that does not open up within about 30 seconds. Come back to it.
- Answer every question you skip anyway. Before you move on, put something in the box. If you come back, change it. If you run out of time, you have a one-in-four chance instead of nothing.
- A calculator is allowed on every question, so arithmetic speed is not what is being tested — setting the problem up correctly is. Most wrong options in this book are a perfectly correct calculation of something the question did not ask.
About the calculator
Bring one you already know. The permitted-calculator rules are ACT's, they change occasionally, and you must check them on ACT's own site before test day — a prohibited calculator can cost you the whole section.
What matters more than the model: a calculator will happily give you a precise answer to the wrong equation. Every "silly mistake" you have ever made was a setup mistake, not an arithmetic one. This book puts its effort there.
How to use this book
Every question here has been checked by machine. The answer to each one is solved independently, by computer algebra, from the same source that prints the options on the page — so the answer key cannot disagree with the question. And every wrong option is there because of a specific, named mistake, which the solution tells you. When you pick a wrong answer, you can find out exactly which slip you made.
A workable order:
- Chapter 0 — this guide. Once.
- Chapter 8 — the formulas. Start learning them now, a few at a time, and keep coming back. They are the single cheapest marks in the book, and the ACT will not give them to you.
- Chapter 1 — Integrating Essential Skills. A fifth of the test, and the most improvable part of it.
- Chapters 2 to 7 — the five higher-maths categories. In any order; do the one you are weakest at first, not last.
- Chapter 9 — the two full papers. Time them properly: 50 minutes, one sitting, no interruptions. An untimed mock paper measures your mathematics; a timed one measures your ACT score, and they are not the same number.
Do the second paper only when you have worked through what the first one showed you. A mock paper you have not learned from is a mock paper wasted.
Topics covered in this guide: the 2025 format change · four answer choices · alternating answer letters · no formula sheet · the published blueprint · the warning about ACT's own booklet · scoring and the conversion table · timing · the calculator · a study order