Introduction
Whether you’re a math whiz or whether math is your least favorite subject, you can get a 36 on the ACT® Math section.
Anyone can get a perfect score, and if you use this guide to study effectively, then you can too. It’s not about how hard you study, it’s about how strategically you study.
This guide is broken into four sections.
I. All About the ACT®
II. 11 Must-Know Study Tips
III. How to Take the ACT® Math Section
IV. Practice Question Walkthrough
We’ll discuss how the ACT® Math section is structured, how you can maximize your study time, how you can take advantage of hidden tips and tricks for a higher score, and how you can tackle each question successfully.
Are you ready to learn how to get a 36 on ACT® Math? Sharpen your pencil and let’s get started!
I. All About the ACT®
What’s the Test Format?
Before you can ace the ACT® Math test, you need to know how it’s structured. Luckily, although the questions change from test to test, the format and the general content never change. This guide will show you how to use that format to your advantage!
Time limit
You have 60 minutes to answer 60 questions on the ACT® Math test, which breaks down to 1 minute per question. However, as we’ll cover later in this guide, a smart test-taker will use that time a little differently.
Placement
The ACT® Math section comes right after the 45-minute English section.
Reference sheet
Unlike the SAT®, the ACT® does not give you a formula reference sheet. However, if you prepare correctly, you won’t need one; it would just slow you down.
Allowed materials
In addition to several sharp number #2 pencils, you’re allowed to have a calculator on the ACT®. You can have a four-function calculator, scientific calculator or graphing calculator, except for:
- Texas Instrument calculators beginning with TI-89, TI-92 and TI-Nspire CAS (non-CAS TI-Nspire is allowed);
- Hewlett Packard calculators beginning with HP 48GII, HP 40G, HP 49G, and HP 50G;
- Casio calculators beginning with CFX-9970G, Algebra fx 2.0, ClassPad300, and ClassPad 330;
- Laptops and phone calculators.
When in doubt, check the official ACT® calculator regulations.
What’s ON the Test?
You don’t need to brush up on your BC Calculus in order to ace the math section. In fact, you don’t need to know anything past Algebra II. Here’s the breakdown:
- Pre-Algebra: 20-25%
- Algebra I: 15-20%
- Plane Geometry: 20-25%
- Coordinate Geometry: 15-20%
- Trigonometry: 5-10%
- Algebra II: 15-20%
The order of the test
The ACT® Math questions aren’t thrown together in a random order, which works to your advantage. Knowing where to find the easiest, least challenging questions will help you to maximize your time for the highest score.
As a general trend, the math questions get more difficult over the course of the exam. Although “difficulty” is relative, you’ll be able to see a clear difference between question 1 and question 60. As the exam goes on, questions get more complicated, necessitate more steps, and require a mixture of algebra and geometry.
Don’t be intimidated. The good news is that easy questions count for just as many points as difficult ones, and there is no penalty for guessing.
Questions 1-20 are easy. Questions 21-40 are medium. Questions 41-60 are difficult.
Later on in this guide, we’ll discuss how to properly divide your time among different levels of questions in order to ensure that you answer them all – with time to spare.
Content Order
Algebra and geometry will be spread out all through the exam. However, as a general rule, questions 1-30 tend to involve more algebra, and questions 31-60 are heavier on geometry and trigonometry.
How Well do You Need to Score?
Of course you would love to get a 36, but how does that work? Do you have to get every question right? Can you miss 5?
The number of questions that you get correct is your “raw score.” When the ACT® converts that number into a number out of 36, the result is your “scaled score.”
Let’s look at the breakdown to see what raw school you need in order to get the scaled score that you want.
| Scaled Score | Raw Score |
| 36 | 59-60 |
| 35 | 57-58 |
| 34 | 55-56 |
| 33 | 54 |
| 32 | 53 |
| 31 | 52 |
| 30 | 50-51 |
| 29 | 49 |
| 28 | 47-48 |
| 27 | 45-46 |
| 26 | 43-44 |
| 25 | 41-42 |
| 24 | 38-40 |
| 23 | 36-37 |
| 22 | 34-35 |
| 21 | 33 |
| 20 | 31-32 |
| 19 | 29-30 |
| 18 | 27-28 |
| 17 | 24-26 |
| 16 | 19-23 |
| 15 | 15-18 |
| 14 | 12-14 |
| 13 | 10-11 |
| 12 | 8-9 |
| 11 | 6-7 |
| 10 | 5 |
| 9 | 4 |
| 8 | –- |
| 7 | 3 |
| 6 | –- |
| 5 | 2 |
| 4 | –- |
| 3 | 1 |
| 2 | –- |
| 1 | 0 |
This chart will help you keep track of your progress during your ACT® review.
II. Must-Know Study Tips for the ACT® Math Section
Now that you know how the test is set up, you’re ready to crack the books!
How to Approach Studying
You can’t study for the ACT® Math section the same way you would study for, say, an AP® History exam or an SAT® Language test. Yes, there are some formulas to memorize, but rote memorization isn’t enough to get a perfect score. You will also need speed, practice and a targeted study plan.
There are thousands of practice ACT® questions on the Internet. One way to study is to do all of the practice questions – but that would take months, if not years, and you’re probably planning to take the ACT® within the next decade. You’ve also got the ACT® Science, English and Reading sections to study for as well.
Instead, use these eleven study strategies to approach the exam like a pro.
What you’ll need for your ACT® studying:
- a number two pencil, or several
- a calculator if you plan to use one on the official ACT®
- an official practice exam
- a watch or timer
- several clean sheets of paper for after the exam
- a deep breath
Got everything? Then let’s get started!
Study Tip #1: Determine your baseline by taking an official ACT® practice exam.
The only way to really predict how you will do on the ACT® is to take it. The ACT® website has five exams available for practice, for free. Make sure you print it out, so that you can get used to working with a hard copy like you will on the official exam.
It’s important that you get an official practice exam, or an exam that has a very similar range of questions as the official ACT®. If you get an exam that is too easy or too hard, you will get a false sense of your preparedness; you either won’t study enough or you’ll focus on the wrong material.
Now, when taking this practice test, you’ll want to recreate the actual testing conditions as much as possible. Even if playing music helps you to focus, you won’t be able to have your iPod in your testing room, so turn it off.
Go to a nice, quiet place. Set your timer for 60 minutes.
Why are you timing it? Because it’s not enough to simply know all the material on the ACT® Math section. If you know how to do every problem perfectly, but it takes you longer than 60 minutes, then you won’t do much better than the student who doesn’t know how to do the difficult problems. In fact, you may do worse, because at least that student can use effective guessing strategies.
Take the exam. But – and this is very important – when time runs out, don’t put down your pencil. If you still have unanswered questions, keep doing them, but mark them so that you know which they are.
Note: When you’re taking the practice exam, don’t take blind guesses. If you get the answer correct, it won’t accurately reflect your knowledge. The point here isn’t to score as high as you can, but to measure what you know.
Study Tip #2: Identify your weakness: speed or accuracy?
There are two very important components of the ACT® Math section: speed (whether you can complete all of the questions in time) and accuracy (how many questions you get correct).
Everyone could improve in both areas, but there’s a good chance that you are a little weaker in one area than another, so that’s what you should focus on first.
How do you know your weakness?
First, use the answer key to mark which questions you got right and which you got wrong. If you have a highlighter, highlight the ones that you got wrong so that they’ll be easier to find later. Also mark the questions that you answered after time ran out.
Using the chart above, calculate your scaled score from your raw score. DO NOT count the questions that you answered after time ran out, even if you got them correct.
Write down your scaled score so that you know your starting point. You’ll be measuring your progress regularly.
Write down the number of questions that you got correct out of the questions that you answered within the time limit, and the number of questions that you got correct out of the questions you answered after time was up.
If you got 40 or fewer questions correct out of all the questions, and you completed 45 or more questions within the time limit, then you have an ACCURACY weakness.
If you completed 44 or fewer questions within the time limit, but you got 41 or more questions correct out of all the questions, then you have a SPEED weakness.
If you completed 44 or fewer questions within the time limit and you got 40 or fewer questions correct out of all the questions, then you will want to work on both your SPEED and your ACCURACY. Work on accuracy first, to make sure that you have the necessary knowledge base in order to increase your speed.
If you got 41 or more questions correct out of all the questions and you completed 45 or more questions within the time limit, then you are on track to score at least a 25. How many questions did you fail to complete within the time limit? How many questions did you get incorrect? If the former is more than the latter, work on your SPEED first. Otherwise, work on your ACCURACY.
Tips #3-6 focus on accuracy, while tips # 7-8 focus on speed.
Tip #3: Pinpoint your content weak spots.
Okay, now that you’ve taken the test, you have a pretty reliable indicator of which content areas (algebra, geometry, etc.) that you excel in, and which could use a little more work.
Get out two clean sheets of paper. And you may want to resharpen your pencil. We’re going to figure out your true content weak spots.
It’s not enough to glance at a question that you missed, say, “Oh, that’s geometry,” and move on. Instead, using the answer key to your exam, rework the question until you get it right. By reworking the question, you’ll realize the exact parts of the question that trip you up. For example, maybe you knew the correct formulas for a trigonometry question, but because the question involved radicals and exponents, you got stuck. In that case, your weak spot isn’t trigonometry, it’s radicals and exponents.
On the first clean sheet of paper, for each question, write down the general concept where you are having trouble. On the second sheet of paper, write down the specific formula or rule that is giving you trouble.
For example, if you’re having trouble with multiplying exponents, then write “rules of exponents” on the first sheet of paper. On the second sheet, write “multiplying exponents.”
The first sheet will make sure that you cover all the rules of a particular concept. If you’re having trouble multiplying exponents, then you’ll likely have trouble dividing exponents as well. By reviewing all of the rules, you’ll cover your bases. This is your “key concepts” sheet. The second sheet will make sure that you tackle the very specific rule that is giving you trouble – it’s the “bare minimum” sheet. The next time you take an ACT® practice exam, even if you don’t know all the rules of exponents, at the bare minimum you should know how to multiply them.
You’re going to need both lists for the next step.
Tip #4: Make a cheat sheet.
Even if you’re not a visual person, you will benefit from this next exercise. Yes, you’re going to need another clean sheet of paper. You may also grab crayons and colored pencils. This should be fun.
Group the items on your “key concept” sheet into five groups, based on general topics. For example, say that you missed 20 questions, so there’s a lot of concept overlap on your “key concept” sheet. You’ve written down “area of triangles” twice, and “rules of right triangles” once. Then make one of your groups “trigonometry.”
Each group is called a Key Content Area. Dividing your “key concept” sheet into five groups will 1) help you organize your areas of weakness so that they’re easier to tackle, and 2) show you that you’re closer to an ACT® score of 36 than you think – you’re only five content areas away!
On the sheet of paper, create a detailed (and perhaps colorful) cheat sheet for all of the Key Content Areas. Draw diagrams. If you’re working with surface area or coordinate geometry, draw pictures. Create mnemonic devices. Use boxes and arrows.
Check out Albert’s 9 Must-Know ACT® Math Topics for help.
The more effort you put into the cheat sheet, the more that these formulas will stick with you, and the fewer times that you will have to review them. This active approach to studying is much more effective than simply reviewing concepts straight out of a textbook. Your brain will be forced to make numerical, lexical and visual connections, which still strengthen the concepts in your brain.
Plus, you’ll be using this cheat sheet as you go through practice problems. You may as well make it nice to look at.
On the back of the sheet, use your “bare minimum” sheet to make a list of all the specific formulas that you don’t know. For example, using the exponents example from earlier, you would write, “To multiply exponents with a common base, add the exponents,” with an example underneath. This doesn’t have to be as detailed as the front page (which will include the formulas from your “bare minimum” sheet as well). The point of this page is to have a raw, running list of formulas to keep in mind.
Tip #5: Drill, drill, drill!
Now that you’ve created your cheat sheet, put it into action! You can’t learn how to use math equations simply by memorizing them from a sheet. However, reviewing the information on your sheet and having the sheet on hand during your ACT® review will help.
For each of your Key Content Areas, find practice problems. A lot of them. Albert has an extensive selection. Instead of doing multiple practice tests, focus on questions in the areas where you struggle.
Every time you study (which should be several times a week), do at least twenty-five questions in each Key Content Area, for a total of one hundred twenty-five questions. You don’t need to time yourself.
If you come across particular formulas and concepts that you struggle with while you’re practicing, add them to your cheat sheet. The back of your cheat sheet should be a growing list.
Tip #6: Be ruthless with your mistakes.
There are two types of studiers in the world.
There are those who do practice problems, check their answers, and use the answer key to get a vague sense of why they missed the problem and how to solve it correctly.
Then there are those who actually want to score a 36.
Be relentless with the problems that you got wrong. If you missed a problem, tear it apart. Underline key words. Work it through once with the answer key, following step by step. Then work it through again without the answer key to make sure that the concept sticks.
Even on the questions that you get correct, be vigilant. If you’re even slightly unsure about a question while you’re doing practice exams, mark the question for review and do it again later.
Similarly, remember to mark the questions that you guessed on. This will help you determine whether you’re using effective guessing strategies. It will ensure that you study these concepts in the future so that you don’t have to guess next time.
On a sheet of paper each day, keep track of how many questions you get incorrect for each of your Key Concept Areas. Over time, as you review concepts, keep track of new formulas on your cheat sheet and learn the rhythm of ACT® Math questions, you should be getting fewer and fewer questions incorrect.
If your number doesn’t change, then you’re not studying quite the right concepts. For example, maybe it’s not the question itself, it’s the format of the question that’s tripping you up – go to Tip #8 for specific help with word problems.
Doing all of these practice problems is a lot of work, but practice is the way to get a 36 on ACT® Math. And doing practice problems is less work than having to study for the entire ACT® all over again in the next cycle.
Tip #7: To save time, isolate your weak spots.
This part of the guide is for anyone who would like to focus on raising their speed. We’ll go into general test-taking efficiency strategies in the next section of the guide, but here we’ll look at strategies designed to improve your speed on the exam. If you already have a masterful level of the content on the ACT® Math section, then speed is your only obstacle.
First, take another ACT® Math practice exam. Don’t put yourself on a time limit, but do keep track of your time as you go along:
- If a question takes you about 30 seconds, put a checkmark beside it.
- If a question takes you about a minute, put a slash beside it.
- If a question takes you between one minute and two minutes, put an X.
- If a question takes you over two minutes, put an exclamation mark.
Checkmarks and slashes are good. That means you’re on track to complete the test in under 60 minutes. Xs are okay – if you can do the easy questions quickly, then you’ll have between one and two minutes for the more difficult questions, although it’s always better to stay closer to one minute. Exclamation marks aren’t good. If you’re taking more than two minutes on a question, it indicates that you’re not familiar with the material. We’ll discuss how you should budget your time in the next section.
Now that you’ve marked all of your questions, get out another sheet of paper. Look at the questions that have Xs and exclamation marks. What type of questions are they? Trigonometry? Word problems? Probability? Surface area? On the sheet of paper, make a running list of the concepts that are taking you more time. We’ll call them Time Drains.
If you’re working on your speed, then that means you know the material that the Time Drains ask, so you’re getting the questions correct. Therefore, you just need to practice your strategy. To do this, get practice questions corresponding to your Time Drains.
Every day that you study, do 2 sets of 10 practice problems for each Time Drain without timing yourself. Then do 1 set of 10 practice problems while timing yourself to see how long it takes you. Again, mark Xs and exclamation marks accordingly.
At the end of each week, compare your times to see how much time you’ve shaved off of each section. Then do a full ACT® Math practice exam at the end of each week. Through pure practice, your time should get faster and faster.
Tip #8: Master time-saving tricks.
In addition to rote practice, you can employ a few tricks to shave some seconds off your time. These extra seconds could be the difference that gets you to a 36.
Identify questions that will take a long time, and skip them. After a lot of practice marking how much time each question takes you, you’ll start to get a feel for what questions will take more time than others. When you encounter these questions on the ACT®, star them and skip over them to make sure that you answer the questions you’re fastest at first. Since every question counts for the same amount of points, it’s better to do three questions you’re speedy at than to complete one question that takes more than two minutes. Come back to the difficult questions at the end of the test.
Learn when to move on. Sometimes, you think a question will be easy, but as soon as you start working it out, it becomes a sticky situation that you’re not sure how to escape from. If you find yourself getting stuck for longer than a minute, make your best guess, bubble that in lightly, and come back to this question at the end of the exam. Taking space will help you refresh, and may give you fresh perspectives on the question when you return.
Bubble at the end. Instead of doing a question, taking the time to find the correct slot on your answer key, bubbling it in, and then returning to the next question, you can save a minute or two by bubbling page by page. While you’re doing the problems, circle the answer in your book. At the end of each page – or even after every ten or fifteen questions or so – bubble all of your answers. (Just be sure to bubble the correct ones.)
Plug in the answer. If you’re stuck on a question, try working backwards, plugging the answer choices into the problem.
Tip #9: Tackle word problems as their own beasts.
ACT® Math word problems are a separate beast unto themselves. Even if you know a concept perfectly, you may trip over it when it’s broken up and hidden inside of a tricky word problem. Study word problems across a variety of topics to ensure that you’re prepared for whatever this exam throws at you.
Do five word problems a day. Or more. Do word problems in concepts that you feel confident in, and concepts that you’re struggling with. The practice will help you get a feel for how word problems are set up.
Draw a picture. Turning a word problem into a diagram or a picture will help you visualize it. If the problem gives you the dimensions of a shape, then draw the shape and label the sides. Then you can solve it like you would solve any other geometry problem. If the problem requires algebra, then create a picture and jot down the formulas.
Underline key information as you read. Word problems are tricky, so underlining the most important information will help you focus on the key parts.
Read the word problem twice. Read the problem once, underlining words, and then set up your diagram and equations. Read the problem again to make sure that you haven’t missed any information.
Write out the formula. Don’t immediately plug numbers into an equation. To make sure that you’re setting up the formula properly, write out the formula first, and then write it out with the numbers plugged into it. For example, if the question asks you to solve a right triangle, write out the Pythagorean theorem beside the question. Then, beneath that, write the equation with the numbers plugged in.
Check to see whether the question is asking for an equation or a solution. Some word problems will ask you, “What equation would you use to…?” The answer choice will be an equation. Pay attention to these types of questions to make sure that you don’t waste time solving the equation if the question does not ask for that.
Tip #10: Test regularly.
Once a week or once every two weeks, depending on how much time you’ve allotted to study, take an ACT® Math practice test as a diagnostic. Adjust your study plan as necessary. For example, if trigonometry used to be one of your Key Content Areas but you’ve mastered it after a week, feel free to replace it with a concept you’re less sure about. The cheat sheet isn’t a bible, it’s a guide to help you find the right path to a 36.
When you take an ACT® Math practice test, try to recreate the testing conditions as much as possible. Use the correct calculator, take out your earbuds and sharpen a pencil.
In the next section, we’ll discuss effective test-taking tips. Very important note – don’t wait until exam day to implement the test-taking tips. Practice them during your ACT® review so that 1) you get used to them, 2) you’re not scrambling to remember them on the exam, and 3) you can see if they work for you. Every tip doesn’t work for everyone, so you will need time to find your perfect techniques.
At least two weeks before the ACT®, begin taking the entire ACT® once a week for practice. This will give you a more accurate reflection of where you stand on the entire exam, and will help you refine your study skills to focus more on certain sections.
Now that we’ve covered the eleven most important study tips that you will need, let’s look at how you can maximize your time and effectiveness when you actually sit for the test!
Tip #11: If you have limited study time, use it wisely.
Hopefully, you’re coming to this guide with at least a month before you take the ACT®. Spreading your ACT® studying out over a long time will aid your retention, keep you from burning out, and will give you time to really sharpen your Key Content Areas and your speedy test-taking skills.
However, if you have less time, you can still get a 36. You will just have to be extremely self-disciplined and make sure that you are studying not just hard, but effectively.
If you only have seven days of before the exam, here’s what to do:
- Day One: Take a practice test to identify where you are weak on content. Create a cheat sheet for your Key Content Areas, as detailed in Tip #4.
- Days Two – Three: Each day, do fifteen practice questions in each of your Key Content Areas and five word problems. Note: practice questions only count when you can do them without looking at your cheat sheet.
- Day Four: Take another practice test to determine where you are still not scoring as high as you could on content. Rewrite your cheat sheet if you need to in order to focus on different concepts. Use Tip #8 to save time.
- Days Five and Six: Each day, do ten practice questions in each of your Key Content Areas, five word problems, and twenty-five random practice questions.
- Day Seven: Rest before the big exam tomorrow!
III. How to Take the ACT® Math Section
How you take the exam is just as important as how you study for it. In fact, it may be more important – all of the preparation in the world won’t help you if you freeze once you get to the testing center.
Don’t panic. Even if you’re not naturally a strong test-taker, these tips will help you concentrate, make the most of your time, and get the score that you’ve studied so hard for.
Take Deep Breaths.
You may be nervous. After all, the ACT® is important, right? It will determine your entire college future, right?
Wrong. Colleges will also take your essays, grades, recommendation letters and extracurricular activities into account, sometimes even more than your ACT®. Some universities don’t even consider standardized testing scores, and if you’re applying for an arts program, then your art samples will matter much more than your scores.
The ACT® is important, but you’ve studied. You’ve prepared. If you’ve followed the advice in this guide, then you’re on track for a high score, so don’t let nerves tell you otherwise.
Take deep breaths and eat a good meal before the exam. You’re going to do fine!
Know when to Use a Calculator.
Many students fall into the trap of, “I have a calculator, so I must use it.” They second-guess their own mental math skills. Unfortunately, a calculator can work to your disadvantage – in some cases, it’s slower than doing the math by hand, and it’s easy to make typing errors that you don’t notice.
Basic operations
Chances are, you don’t need a calculator for very simple addition, subtraction, multiplication and division. “Very simple” means one-digit, or working with numbers that are multiples of five or ten.
Fractions
Be careful about using your calculator when working with fractions, because many calculators automatically convert fractions to decimals. If the answer choices are in fractions, then you’ll have to convert them into decimals too, which just wastes time.
Logs
When doing logarithms, it is much faster and more accurate to use your calculator.
Pi
When you see pi in a math problem, check the answer choices. If the answer choices contain pi, then do the question by hand. If the answer choices don’t contain pi, then use the “pi” function on your calculator to get the quickest and most accurate result.
Radicals and Exponents
At first glance, exponents and radicals may send you running for your calculator, because they can get complicated quickly – who really knows how to take the 6th root by hand? However, check the answer choices. Many times, the answer choices will contain radicals, which means that you don’t need to use your calculator for any tricky, decimal-based math.
Know how to Use a Calculator.
Some calculators are tricky. There are some functions on your graphic calculator that you will never, ever use. You will only need a very small fraction of your calculator’s functions on the ACT®; if you can graph, use exponents and radicals, and work with logarithms, then you’re off to a great start.
While you’re studying, make sure that you pay close attention to the calculator functions that you need, and add them to your cheat sheet.
However, be careful when you’re taking the exam: it’s easy to slip up and input the wrong numbers by mistake. Check your input twice before pressing “enter.”
When you’re putting in equations, also pay close attention to the order of operations, which is the automatic order in which your calculator will perform functions. You’ve probably learned this order in school, because it’s the basis of algebra and multi-step problems.
PERMDAS
- Parentheses
- Exponents/Radicals
- Multiplication/Division
- Addition/Subtraction
When inputting your equations, pay close attention to how you’re setting up your equation. If there’s a particular part of the equation that you need your calculator to perform first, put that in parentheses.
Allot Time Wisely.
Sixty minutes for sixty questions doesn’t necessarily mean a minute per question. In fact, if you spend a minute on each of the easy first twenty questions, then you’re likely to run out of time in the final third of the exam.
Allot approximately 30 seconds each for questions 1-20, 1 minute for questions 21-40, and 1 minute 30 seconds for questions 41-60. This comes out to about 15 minutes for the first 20 questions, 20 for the second, and 25 for the last.
You should almost never spend two minutes or more on a question. If you find yourself stuck or working in circles, then guess.
How to Guess
Guessing on the ACT® Math section isn’t a game of chance, it’s a skill. And you can sharpen that skill.
Always answer. The ACT® does not take off points for incorrect answers, so you shouldn’t leave any questions blank. By guessing, you have a 20% (or more) chance of getting it right. If you leave it blank, you have 0.
Know when to guess. If you’re completely stuck on a problem and the answers aren’t offering any hints on how to solve the problem, guess. If time is running out and you don’t have time to do the problem properly, guess. If you’ve eliminated some answers but still can’t figure out the correct one, guess.
Pick a letter. Before the ACT® starts, pick a letter between A and E. This will be your Guessing Letter. This tip doesn’t apply when you are eliminating answer choices, but when you’re unable to eliminate any answers, bubble in your Guessing Letter. Why? Because statistically, your Guessing Letter should be correct 20% of the time, so you’re almost guaranteed to get 20% of those questions correct. If you choose a letter at random for every question, you actually have lower odds of choosing the correct one.
Eliminate, eliminate, eliminate. If you guess out of five answers, you have a 20% chance at choosing the correct answer. If you can eliminate even one answer and guess out of four, then your odds shoot up to 25%. If you can eliminate one more, your odds shoot up to 33%. Eliminate wherever you can.
You might be wondering, “How can I eliminate incorrect answers if I don’t know what the correct one is?” You know more than you think. You might not know what the correct answer is, but you can get some idea of what it isn’t.
For example, if it’s clear from the equation that the answer will be a negative number, then cross out answers that have positive numbers. If four out of the five of the answers are fractions, then the correct answer is probably not the one with the decimal point.
Pick the round number. You can use a calculator on the ACT® exam, but the exam is designed so that you don’t have to. Therefore, the ACT® will never give you a problem that is too difficult to be solved by hand. When in doubt, narrow down your options to the round numbers (multiples of 5, for example), because those are easier for people without calculators to solve.
Approximate. Round up to numbers that are easier to work with, see if you can solve the question, and choose the answer that is closest to your approximated answer.
Eyeball it. The ACT® claims that geometric shapes aren’t drawn to scale, but they approximately are. If you’re running out of time or having trouble remembering the correct formulas, use the pictures as a guide.
Beware the traps. If you’re working in questions 41-60 and one answer seems too simple and straightforward, then it probably is. Remember, the last twenty questions are designed to be difficult.
Take Two Passes Over the Exam.
Since you’ve done so many practice tests, you have a good sense of which questions you’re a master at, and which take a little more time and concentration.
When you go through the exam the first time, do all of the questions that you’re confident about. If you stumble across a question you’re unsure about or that you know will take a lot of time, mark it.
After you finish the first pass, go back through to the questions you marked. Of these, focus on the ones that you think you can figure out, instead of trying to crack the ones you can’t. When in doubt, use elimination to narrow down your choices, and then make an educated guess.
Relax.
When the ACT® Math section is over, exhale a sigh of relief. You’re done! No matter what your scaled score is, you should be proud that you worked so hard.
IV. Walkthrough
Let’s walk through a few practice questions using the test-taking techniques that we discussed above. These are from Albert.
This is a word problem, so we will need to pay close attention to make sure we don’t miss any key information.
Read through it once, underlining the key information.
A sphere and a cone have the same volume.
The radius of the base of the cone is equal to its height.
If the surface area of a sphere is 616 square units, what is the approximate height of the cone?
It may help you to circle “height of the cone,” since that is what the question is asking you to find.
Keep in mind that the question says “approximate,” so it’s okay to round your decimals.
Consider drawing a picture of the cone and sphere, and labeling their bases and heights so that you know what you’re searching for.
Define your variables. For example:
The volume of a sphere = a
The volume of a cone = b
The radius of a cone’s base = c
The surface area of a sphere = s
The radius of a sphere = d
The height of the cone = h
Read through the question one last time to ensure that you didn’t miss anything.
Write out the formulas that you will need:
Volume of a sphere:
\dfrac {4 }{ 3 } \pi r^{ 3 }
Volume of a cone:
\dfrac {1 }{ 3 } \pi r^{ 2 }\ast h
Surface area of a sphere:
4\pi r^2
Then, set up equations from the information provided by the word problem.
A sphere and a cone have the same volume.
Therefore:
a = b
The radius of the base of the cone is equal to its height.
c = h
If the surface area of a sphere is 616 square units.
s = 616
What is the approximate height of the cone?
h = ?
To solve this problem, you need to work backwards. Start with the given number – in this case, it’s 616. We have set the radius of our sphere to be d.
s = 616
s = 4\pi d^2=616
\pi d^2=154
d^2 = 49
d = 7
The radius of the sphere is 7. Now that you know the radius of the sphere, you can find its volume. We are still using dfor the sphere’s radius.
a = \dfrac {4 }{ 3 } \pi d^{ 3 }
a = \dfrac {4 }{ 3 } \pi 7^{ 3 }
a = \dfrac {4 }{ 3 } \pi \ast 343
a = 457 \pi
a = 1436.65
The volume of the sphere is 1436.65. That is the same volume as the cone b. Remember that the radius of the cone’s base r is equal to its height h.
a = 1436.65 = b
b = 1436.65
b = \dfrac {1 }{ 3 } \pi r^{ 2 }\ast h =1436.65
\dfrac {1 }{ 3 } \pi r^{ 2 }\ast r =1436.65
\dfrac {1 }{ 3 } \pi r^{ 3 } =1436.65
\pi r^{3} =4309.95
r^{3} = 1371
r=11.11
The radius of the cone’s base r is approximately 11.11. Therefore, the height is also approximately 11.11.
The answer is E, 11.11.
There are two ways to solve this question. Think critically about which way you would like to use.
But before you do any calculations, make sure you understand every part of the problem, and underline any keywords. The keyword in this problem is integer.
From your studying, you should know that “integer” is just a technical word for whole number. So you can go ahead and eliminate any answer that doesn’t conform to that definition. Goodbye, C!
You also know that:
x^{ \dfrac {1}{2} } = \sqrt {x}
Therefore, for each number in that equation, you could find the square root, or you could plug the equation into your calculator as is. It is faster to take the square root of each number – you can still use your calculator.
\dfrac {\sqrt {32} + \sqrt {18}} {\sqrt{8}}
\dfrac {\sqrt {32} + \sqrt {18}} {\sqrt{8}} = 3.5
Compare your answer with the answers offered. 4 is the smallest integer that is greater than 3.5, so the answer is B, 4.
On the surface, percentages can seem confusing, because you’re working with a lot of decimals and fractions, but this question boils down to one equation.
Anna has averaged 72% (.72) on her first 3 tests.
3(.72)
She has 6 more tests to take, and she’s not sure what she needs to make on them yet. She knows that she will make a number x out of 100.
3(.72) + 6(.01x)
She wants her average of the 9 tests to be 80% (.8).
\dfrac {3(.72) + 6(.01x)} {9}= .8
Now it’s a simple solve-for-x problem!
\dfrac {3(.72) + 6(.01x)} {9}= .8
3(.72) + 6(.01x) = 7.2
2.16 + .06x = 7.2
.06x = 5.04
x = 84
For a question like this, if you’re running out of time, you could also use the “plug in the answer” method that we discussed briefly under Tip #8: Master Time-Saving Tricks. You can plug each answer into the equation, and find the smallest number that will give you an average of at least 80.
The answer is C, 84.
V. Conclusion
Now you know how to get a 36 on ACT® Math – you’re prepared for anything this section will throw at you. Keep in mind the studying and the test-taking tips that you’ve just learned, and you’re well on your way to a 36!
Good luck!
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