sat practice tests difficulty

• Skip to content

SAT Prep Courses & Tutoring

Free SAT Practice Questions

Doesn't everyone love the word "free"? This could not be truer—especially in the world of test prep. Practice makes perfect—or at least that's how the old saying goes—and free SAT practice means the world to ambitious students with high-scoring goals. If you're looking for a thorough breakdown of how to solve particular SAT question types, then we've got the free practice that could be a real game-changer in your course of study. Our step-by-step explanations illustrate for you what to expect from what each SAT question, revealing question-specific hurdles and common test-related traps.

Each of our free 60 practice questions is accompanied with a detailed explanation to clarify why a particular answer is correct, but even taking it one step further and entailing why the others are incorrect. We believe a well-rounded approach to understanding the SAT is crucial for success, which is why our breakdowns of important sample questions facilitate more than one way to arrive at the right choice. In the end, the only way to find out your trouble spots and successes is to practice, practice, practice—so why not get cracking on what we have to offer so you can ensure your destiny with a high score.

You have not answered any question so far. You can answer all questions in a row (click on "All Questions") or only all questions of a particular section (click on that Section) or a single selected question (click on that Question).

SAT is a registered trademark of the College Board, which is unaffiliated with and does not endorse this website.

The Most Difficult College Board Practice SAT Test

Finding The Hardest SAT Practice Test And Why It Matters

There are 8 free practice SAT exams offered through the College Board’s official website . These tests present the opportunity to simulate what it will be like to take the SAT on test day, and will also allow you to calculate your score to see where you are relative to your goals.

When you’re preparing for the SAT, taking advantage of every single resource you have is key, especially when the College Board is releasing official practice tests.

But, just like with the real SAT, you might wonder if one practice exam is going to be harder than the other. When it comes to the real thing, this also leads to a host of follow-up questions: what if my SAT is harder than usual? Will it hurt my chances of getting a good score? How do they make sure the test is fair?

Don’t worry — the College Board makes sure all of their SAT exams, whether it’s the October exam or the May exam, are as fair as possible for all test takers by equating scores. Read more to find out what equating does and to see which College Board practice test is the hardest.

What Does “Equating” The SAT Mean?

The College Board uses what it calls “Equating” to adjust for different difficulty levels between exams. This involves adjusting the score a student receives for getting a certain number of incorrect answers.

The variations between tests are not big, so it’s important to remember that your score will always fall in a similar range for a given number of wrong answers.

For example, if you wanted to get a perfect score on the 2021 April SAT, you would have needed to get every answer correct on all sections. But for the May 2021 SAT, students who missed one question on the reading section could still get a perfect score!

Is Equating The Same As Curving?

It’s important to remember that equating is not the same thing as curving. Curving is a retroactive process that adjusts an individual student’s performance based on the performance of other students on the same test.

Equating standards (the conversion table of right/wrong answers to score) is designed with the test itself, and does not change based on student performance on the exam.

No matter how other students perform on the test, you’ll still get the same score!

What Makes Some SAT Practice Tests Harder Than Others?

Some SATs might have to equate scales that allow students fewer wrong answers to get their desired score but with easier questions. Other SATs might be scaled more generously, but only because they have harder questions that the designers of the test expect students to struggle with.

It’s also important to remember that the test is broken up between both Math and  English  sections. Some exams have more generous Math scales and harder English scales. Similarly, tests might have slightly harder Math sections and slightly easier English sections. Whichever subject a student most struggles with will determine which test seems toughest.

It’s also extremely important to realize that what makes a practice test hard for you can be completely unique to your strengths and weaknesses. For this reason, it’s the most helpful to focus on practicing as much as possible during your SAT prep period to understand your test score before taking the actual SAT. If you see you’re consistently missing math section questions, focus on learning from your mistakes, studying SAT math concepts, and taking more practice math sections.

Regardless of a test’s level of difficulty, there are a few things you can always control. When you’re taking the actual SAT, make sure to never leave a question blank . The same advice applies to your practice SAT exams, but make sure you mark which ones you guessed on; if you guess correctly but didn’t make a note, you could skip over a question that you need help with.

The more practice tests you take, the more familiarity you gain with not just the structure of the exam, but also the curriculum they test you on. You might also begin to pick up on certain patterns or types of answer choices the test-makers like to include to confuse students!

Hardest Math SAT Practice Test

Remember that a more generous equating scale means a harder test. So for students who struggle with math, SAT Practice Test #3 is probably going to be the most difficult. That is because the equating scale looks like this compared to other practice tests:

Getting the same number of math questions wrong on Practice Test 3 leads to a score that is 20-40 points higher than on other practice tests. This is because the math questions are thought to be more difficult, and so it is harder to get that many correct answers.

If you’re looking for a math challenge, Test 3 is the way to go.

Hardest English SAT Practice Test

We can also do the same thing for the English sections of the SAT Practice Tests to see which exam is designed to be a little more difficult.

For students who struggle with SAT Reading, it turns out Practice Test #3 also has the toughest English scale too! Remember that the  Writing  and Reading scales are slightly different here, so for this table, we’re assuming our wrong answers were evenly divided between the Reading and Writing Sections.

If you get the same number of incorrect answers on Practice Test 3 as you do on Practice Test 10, the College Board thinks that should be worth a 50-60 point increase in your scaled score.

So for both Math and English, the College Board thinks students won’t get as many answers right with Practice Test 3 as they will on other tests.

Which Official SAT Practice Test Is The Hardest?

Since it has the hardest scale for both sections of the test, it should come as no surprise that we think Practice Test #3 is clearly the toughest official SAT practice test .

In online forums, students also tend to identify Test #3 as the hardest of the official practice tests, so there truly seems to be a clear consensus.

It’s important to remember though that a hard practice test can actually be a great resource! Students can work through the test untimed, as they know even with plenty of time the problems will still be quite challenging. They can also practice the test with a tutor or parent, who can help them with the toughest questions. Having a strong SAT score to include in your college applications can only help, not hurt.

Why You Should Take Practice Test 3

You might make more mistakes than you usually do if you take Practice Test 3. While it seems counterintuitive, mistakes can actually be super helpful when you’re preparing for your SAT because you can study the concepts you missed and learn from your mistakes.

Taking the hardest available practice test will mean you’ve done everything you possibly can in order to prepare for test day, which can also help you relieve some test anxiety because you know you’ve challenged yourself.

The SAT is obviously a very important part of the college admissions process, so making sure you have a good grasp of the SAT concepts can even help push you in the right direction if you’re looking for admission into a top school, especially one in the Ivy League.

Juggling a high school schedule isn’t easy, and preparing for the SAT on top of everything else can get extremely stressful. But, if you can find time to practice for the actual test, it will definitely pay off in the long run.

How Can I Use Practice Tests To Boost My SAT Score?

Taking full-length practice tests is a guaranteed way to help you score higher on the SAT. But, you shouldn’t be taking them on autopilot. Here are a few tips to help you utilize SAT practice tests to the maximum:

• Look over every single question you answered incorrectly or skipped after grading your practice test
• Follow the time limits noted for each section to help mimic test day conditions
• Read the answer explanations — it will help you understand where you went wrong and see how the test makers think
• Revisit old tests and redo some of the questions you got incorrect the first time — if you got it incorrect the first time but correct the second time, it shows you’re learning the curriculum and probably won’t make that same mistake again
• When you’re reviewing difficult math problems that you missed, make sure to solve out each step and understand the rationale behind every single one before moving on
• After grading your practice test, reread the reading passages to see if there is a way to identify and remember important information as you read — a smart way to do this is to underline and circle phrases or sentences you might need to quickly find again when you answer the questions
• Make a “study guide” listing all of the different topics or concepts you need to review and study before taking another practice test. For example, it can include concepts that you need to remember for the exam (like how to solve for slope)
• See what types of questions you miss and focus on how you can become better. If you’re missing main idea questions, look for online SAT resources that specifically focus on main idea questions for tips and tricks
• Understand your score percentile ! College admissions officers will be looking at how well you scored compared to your peers at school, so knowing where you stand will give you a good sense of how much more practice you might need

Even if you don’t have access to test prep books, you can use the free College Board practice tests and free online resources to practice for the SAT!

Most students like to take their SATs as juniors so they have some time before sending out college applications, so you should be thinking about your SAT prep schedule and when you plan to be done practicing so you can take the exam. Especially since sending colleges your SAT and ACT scores can take some time, you don’t want to be in a stressful rush. Start taking practice tests and completing test prep as early as you can!

All in all, if you’re the type of student who’s reading through this article to find the hardest SAT practice test, you’re probably already doing a great job with test preparation. Try not to get too stressed and keep working hard — good luck!

SoFlo Tutoring

We get it — taking standardized tests like the SAT is hard, time-consuming, and stressful. SoFlo Tutors are from top colleges across the states, have all gone through the same process, and scored in the top percentiles, whether it was for the SAT exam, ACT test, or even the PSAT.

SoFlo Tutors  are ready to help with expert knowledge and personalized programming that can fit the needs of any student.

Call or visit our website today to learn more about how SoFlo can help you reach your full potential.

• All official College Board practice SAT tests: https://satsuite.collegeboard.org/sat/practice-preparation/practice-tests

How To Determine How Many AP Classes To Take

How to get into mit: sat scores, act scores, and gpa requirements 2022, you may also like.

Should I Take The ACT Or The Digital SAT? | DSAT vs ACT

10 Most Difficult SAT Math Topics: Part 1 | Overview Of Matrices, Probability, Word Problems

Comments are closed.

We are exceptional online SAT/ACT tutors and college counselors that develop a personalized strategy tailored to your child’s strengths and weaknesses. Our founder Adam Shlomi had an 800 in Reading and 770 in Math on the SAT — good for the 99th percentile on both sections, went to Georgetown University, and has been tutoring for five years. We also offer free tutoring to those in need; money shouldn’t stop someone from achieving their dreams.

Watch Michael’s Story

Free Course: A Complete Guide to the SAT

In this free online class we share our #1 SoFlo secret for boosting SAT scores. Plus we teach tricks for every section of the test: Reading, Writing and Math. If you want to increase your score, you NEED this free class.

• 954-654-9777
• SAT/ACT Prep
• Math Tutoring to Raise Grades and Confidence
• Scholarship
• Media and Press

Latest Posts

Writing authentic essays for my stanford application | tutor feature, combining cs and business at umass amherst | soflo tutor feature, soflo digital sat platform vs bluebook | comparison of practice tests and study tools, the road to georgetown | soflo tutor feature.

Home > College Admissions > SAT > SAT Practice Test

SAT Practice Test

Exam summary.

0 of 10 Questions completed

Information

You have already completed the exam before. Hence you can not start it again.

Exam is loading…

You must sign in or sign up to start the exam.

You must first complete the following:

Exam complete. Results are being recorded.

Time has elapsed

You have reached 0 of 0 point(s), ( 0 )

Earned Point(s): 0 of 0 , ( 0 ) 0 Essay(s) Pending (Possible Point(s): 0 )

• SAT Algebra 0%
• SAT Geometry & Trigonometry 0%
• SAT Problem Solving & Data Analysis 0%

Congrats on taking our SAT Sample Quiz. Take one of our full-length SAT practice tests or one of our study sets. Everything is 100% free!

1 . Question

What percentage does 90 represent in relation to 360?

2 . Question

7 added to 4 times a quantity y is equal to 62. Which equation represents this condition?

• a. 7(4y) = 62
• b. 7 – 4y = 62
• c. 7 + 4y = 62
• d. 4y – 7 = 62

3 . Question

For a cost of $57, how many pounds of avocados were bought at a rate of $19 per pound?

4 . Question

The table displays three sets of values for x and their corresponding f(x) values for a linear function f. What equation accurately represents the definition of f(x)?

• a. f(x) = 6x + 65
• b. f(x) = 8x + 65
• c. f(x) = 65x + 73
• d. f(x) = 73x + 81

5 . Question

If \frac{x}{15}=20 , what is the value of \frac{15}{x} ?

6 . Question

What is the equation that defines line p in the xy-plane, given that it passes through the point (-4, 6) and has a slope of 6?

• a. y = 6x + 30
• b. y = 6x – 30
• c. y = 6 + 30x
• d. y = 6x – 30x

7 . Question

2.5b + 5r = 80

The provided equation expresses the connection between the quantity of biscuits (b) and raisins (r) that a reviewer business can handle in a single day. If the business is tending to 16 raisins on a specific day, what is the capacity for biscuits that it can manage on the same day?

8 . Question

In a specific rectangular area, the length-to-width ratio is 45 : 15. If the width of the region grows by 8 units, how should the length be adjusted to uphold this ratio?

• a. It must decrease by 24.5 units.
• b. It must increase by 24.5 units.
• c. It must decrease by 24 units.
• d. It must increase by 24 units.

9 . Question

In a circle with center O, the arc XY measures 105°. What is the degree measure of the angle XOY that corresponds to this arc?

10 . Question

In a right triangle, the side lengths are 3\sqrt3 , 2\sqrt3 ,and \sqrt{70} units. What is the area of the triangle in square units?

*Enter in just the number for your answer.

All SAT Practice Tests

If you want some more in-depth prep, use a free SAT practice test listed below.

• Practice Exams = Timed and Full-Length
• Practice Sets = Not Timed and Smaller Sets of Questions

SAT Practice Exam #1

SAT Practice Exam #2

SAT Practice Exam #3

SAT Practice Exam #4

SAT Math Practice Sets

SAT Reading & Writing Practice Sets

If you want some additional help with prepping for the SAT, consider using  SAT prep courses .

Official SAT PDF Practice Exams

If you want to study using PDFs, use the links below. These are full-length practice exams provided by the CollegeBoard.

Overview of the SAT

The SAT is an entrance exam used by colleges and universities to help make decisions about admissions.

The SAT is administered by the CollegeBoard and is given 7 times per year. The exam is a timed, mainly multiple-choice exam, taken by students in high school.

Beginning in the spring of 2024, the SAT will be going fully digital. The digital SAT (DSAT) will have some formatting and content changes to it, along with some other changes. You can review those changes below.

New SAT (Digital SAT)

This exam will be given to students starting in spring of 2024.

Old SAT (Written Exam)

This exam will be given to students up until spring of 2024.

The biggest difference between the old SAT and the digital SAT is that the digital SAT is taken completely online and has combined some sections to streamline the exam.

Scoring of the SAT

You will receive a score for math and a score for reading/writing. Each of those scores will be between 200 and 800 .

Your total score will be the sum of those two scores. You can receive a total score between 400 and 1600 on the SAT.

The average SAT score is 1060. Learn more about good SAT scores .

Registering for the SAT and Test Dates

You can register for the SAT online via the CollegeBoard site .

You will need to do the following when registering for the SAT:

• Have a Valid Photo ID
• Upload a Photo of Yourself When Registering Online
• Pay Registration Fees
• Print Your Admission Ticket

It will cost your $60 to take the SAT. There may be some additional fees like canceling, changing test center, etc..

The SAT is typically given 7 times per year. The exam is given on Saturdays. You can expect the exam to be given in these months:

• Early March
• Late August
• Early October
• Early November
• Early December

You can find exact dates on the official website .

Steps for Using SAT Practice

When using our free SAT practice tests, we recommend the following steps to get the most out of your time:

• Take 1 practice exam from each subject.
• Determine which subject you struggled the most with.
• Focus on that 1 subject moving forward. Take additional practice exams in that subject, study important concepts, and invest in a SAT prep course if need be.
• Take a practice exam on that subject again and see where you stand. Continue to focus on that subject if you need more help or move on to another subject if you did well on this practice exam.
• Repeat steps 3 and 4 for other subjects.

Benefits of Using SAT Practice Questions

There are many benefits to using SAT questions during your prep process. Some of those benefits include:

Help With Timing

The SAT exam is a timed test. Keeping a steady pace is critical to achieving a high score.

You can improve your decision making and your time by taking practice exams.

Test Familiarity

All standardized tests, including the SAT, have their own unique way of presenting questions and answer choices.

You will gain more familiarity and comfort with the SAT question style as you take more practice quizzes. On the real exam day, there will be no surprises.

Efficient Studying

When you take many practice exams, you will get a sense of your test strengths and weaknesses.

Many students mistakenly spend time working on their strengths while ignoring their weaknesses.

Knowing which subjects you struggle with will help you focus your study time.

Work On Problem Solving

Tests like the SAT measure your ability to solve problems, not just memorize information. It is critical to have strong problem-solving abilities.

The answer explanations provided in our score reports can help you understand how to solve problems that you may be struggling with.

Frequently Asked Questions

When is the sat going to the digital format.

The SAT will be switching to the digital format in the spring of 2024.

How many questions are on the SAT?

There are 154 questions on the old SAT (written version).

There are 98 questions on the new SAT (digital version).

How much time do you have to take the SAT?

You will have 3 hours and 15 minutes to take the old SAT (written version).

You will have 2 hours and 14 minutes to take the new SAT (digital version).

Can I use a calculator on the math section of the SAT?

There will be 1 math section in which you can use a calculator and 1 math section in which you cannot use a calculator on the old SAT (written version).

You will be permitted to use a calculator on all math modules on the new SAT (digital version).

What is a good way to practice for the SAT?

We recommend taking 1 practice exam for each subject. You can then determine which subject gave you the most trouble and focus your studies on that subject.

• Authored By: Adam Groden
• Last Updated: January 19, 2024

SAT Practice Test

Prepare for your test with realistic questions.

The SAT is considered a college readiness test, assessing the knowledge and skills necessary to succeed in an undergraduate program. Many universities and colleges across the country require the SAT as a prerequisite.

Take a free SAT practice test by clicking above!

SAT Practice Tests by Subject

If you need some extra practice in a specific subject, click one of the subjects below to get started on a subject-specific SAT practice test.

SAT Math Practice Test SAT Reading Practice Test

SAT Writing Practice Test Printable SAT Practice Tests

What’s on the SAT?

The SAT includes a reading and writing test and a math test. There are a total of 98 questions on the SAT, most of which are multiple-choice questions, and you are given 134 minutes to finish the test.

Here’s a quick breakdown of the SAT:

Check Out Mometrix's SAT Study Guide

Get practice questions, video tutorials, and detailed study lessons

How to Study for the SAT

How to study effectively.

Your success on SAT test day depends not only on how many hours you put into preparing but also on whether you prepared the right way. It’s good to check along the way to see whether your studying is paying off. One of the most effective ways to do this is by taking SAT practice tests to evaluate your progress. Practice tests are useful because they show exactly where you need to improve. Every time you take a free SAT practice test, pay special attention to these three groups of questions:

• The questions you got wrong
• The ones you had to guess on, even if you guessed right
• The ones you found difficult or slow to work through

This will show you exactly what your weak areas are and where you need to devote more study time. Ask yourself why each of these questions gave you trouble. Was it because you didn’t understand the material? Was it because you didn’t remember the vocabulary? Do you need more repetitions on this type of question to build speed and confidence? Dig into those questions and figure out how you can strengthen your weak areas as you go back to review the material.

Answer Explanations

Additionally, many SAT practice tests have a section explaining the answer choices. It can be tempting to read the explanation and think that you now have a good understanding of the concept. However, an explanation likely only covers part of the question’s broader context. Even if the explanation makes sense, go back and investigate every concept related to the question until you’re positive you have a thorough understanding.

Comprehend Each Topic

As you go along, keep in mind that the SAT practice test is just that: practice. Memorizing these questions and answers will not be very helpful on the actual test because it is unlikely to have any of the same exact questions. If you only know the right answers to the sample questions, you won’t be prepared for the real thing. Study the concepts until you understand them fully, and then you’ll be able to answer any question that shows up on the test.

Strategy for SAT Practice

When you’re ready to start taking practice tests, follow this strategy:

• Remove Limitations. Take the first test with no time constraints and with your notes and SAT study guide handy. Take your time and focus on applying the strategies you’ve learned.
• Time Yourself. Take the second practice test “open book” as well, but set a timer and practice pacing yourself to finish in time.
• Simulate Test Day. Take any other practice tests as if it were test day. Set a timer and put away your study materials. Sit at a table or desk in a quiet room, imagine yourself at the testing center, and answer questions as quickly and accurately as possible.
• Keep Practicing. Keep taking practice tests on a regular basis until you run out of practice tests or it’s time for the actual test. Your mind will be ready for the schedule and stress of test day, and you’ll be able to focus on recalling the material you’ve learned.

Check Out Mometrix's SAT Flashcards

Get complex subjects broken down into easily understandable concepts

By Peter Rench

Peter Rench joined Mometrix in 2009 and serves as Vice President of Product Development, responsible for overseeing all new product development and quality improvements. Mr. Rench, a National Merit Scholar, graduated magna cum laude with a Bachelor of Science in Mechanical Engineering and a minor in mathematics from Texas A&M University.

SAT Test – Home

by Mometrix Test Preparation | This Page Last Updated: March 22, 2024

On this page:

• SAT BootCamp
• SAT MasterClass
• SAT Private Tutoring
• SAT Proctored Practice Test
• ACT Private Tutoring
• Academic Subjects
• College Essay Workshop
• Academic Writing Workshop
• AP English FRQ BootCamp
• 1:1 College Essay Help
• Online Instruction
• Free Resources

Hardest SAT Math Problems (updated for Digital SAT)

Bonus Material: The Hardest SAT Math Problems Quiz

Aiming for a really great score on the SAT? Wondering if your math skills are up to the challenge of the hardest problems?

If you want to be able to get a perfect score, you have to be able to solve the hardest SAT math problems.

We used our extensive test-prep experience to find the questions that many students miss. The examples below are real problems from past official SATs. 

Give each of these 16 hard math problems a try, then read our step-by-step explanations to see if you’re solving them correctly.

If you’re thinking about getting SAT tutoring to help you tackle problems like these on the real SAT, be sure to check out our list of the 15 best SAT Tutoring Services

Then, download this quiz with 20 more of the hardest real SAT problems ever to see if you’re on track for a perfect score! 

Bonus Material: 20 of the All-Time Hardest SAT Math Problems

Math on the SAT

Math accounts for half of your Total SAT Score, regardless of whether you’re taking the old paper SAT or the new Digital SAT.

On the traditional, paper SAT (which will be phased out in early 2024), the Math section comprises section 3, which contains 20 questions, is 25 minutes long and does not allow you to use a calculator; and section 4, which contains 38 questions, is 55 minutes long and does allow a calculator.

On the upcoming digital SAT (which will come into place in spring of 2024), the format is considerably different. You’ll be given two 35-minute “modules” with 22 questions in each, with the difficulty level of the second one depending on your performance on the first one. In other words, if you do really poorly on the first set of 22 questions, the second set will be easier–but your overall math score will be negatively affected. You can use your calculator on both.

Every SAT covers the following math material:

Heart of Algebra: 33% of test . Linear equations and inequalities and their graphs and systems.

Problem Solving and Data Analysis: 29% of test . Ratios, proportions, percentages, and units; analyzing graphical data, probabilities, and statistics.

Passport to Advanced Math: 28% of test . Identifying and creating equivalent expressions; quadratic and nonlinear equations/functions and their graphs.

Additional Topics in Math: 10% of test . A wide variety of topics, including geometry, trigonometry, radians and the unit circle, and complex numbers.

On the old SAT , open-ended questions came at the end of each Math section. Many students find them harder because you can’t guess or work backwards from multiple-choice options.

However, what many students don’t know is that the first 1–3 of these grid-in questions will actually be easier than the last few multiple-choice questions. 

That’s because the math questions on the SAT get increasingly difficult over the course of each section, but the difficulty level starts over again with the grid-in questions.

The savvy student will know this and skip the harder multiple-choice questions to go answer the easier grid-in questions first. Of course, if you’re aiming for a perfect score, (on most tests) you’ll have to answer every question correctly . 

But on the new Digital SAT, these open-ended questions will pop up at different points throughout both modules. You may see them in the beginning, the middle, or the end: there’s no set place for these to appear. Nor is there a set difficulty: generally, we’ve seen these questions be slightly on the easier side, but this varies significantly from test to test.

Because there’s obviously no bubble sheet on the digital SAT, you’ll simply type your answer into the text box. Be sure to look for instructions in the question about how they want the answer formatted!

To work with us for one-on-one tutoring or for our group SAT classes, schedule a free consultation with our team .

Why these problems are essential if you’re aiming at a top school

A perfect score on the SAT Math is 800. The only way to get this score is to answer every question correctly . 

In order to score a 750, you can only miss 2 or 3 questions across both math sections .

A 750 Math SAT may sound like a very high score—and it is! It’s a very high score.

But at the very best schools in the US, three quarters of the students scored a 750 Math or better.

In fact, at the Ivy League and other top schools, at least a quarter of the students had a perfect score!

The average math scores are even higher at the top engineering schools. Three quarters of the students at CalTech had a 790 or 800, and three quarters of the students at MIT had at least a 780.

In order to be a competitive applicant to these schools, your SAT Math score should be within the “middle 50%” of the students at that school—in other words, more or less an average score for that school.

So if you’re aiming at an Ivy or one of the other top schools, you can only miss 2 or 3 questions out of the 58 math questions on the whole SAT.

If that’s your goal, make sure that you understand the problems explained below, and then try our quiz of 20 more real SAT questions that rank among the hardest questions ever.

SAT Problem #1

At first glance, this looks like a geometry question, since it talks about planes and lines and points . But this is actually an algebra question, dressed up with some geometric trappings. 

The key is to realize: 

1) We don’t need to solve for p and r individually. We just need to solve for (r/p) . 

2) The points themselves (p,r) and (2p, 5r) represent X and Y values on the line itself. (For example if p = 2 and r = 3 then that’s the same thing as an x-coordinate of 2 and a y-coordinate of 3.)

So let’s take a look at it. 

First, let’s plug in the p and r points for the x and y values to see what equations we end up with. 

y = x + b becomes r = p + b

y = 2x + b becomes 5r = 2(2p) + b or 5r = 4p + b

At this point we might get a little anxious because we have three variables. 

But we have to remember we don’t need to get the value of the individual letters, just the value of the relationship between r and p. 

That’s where b actually becomes helpful. Because we can now set both equations equal to b , plug in, and then see if we can manipulate the r and p to get them to express the same relationship we want. 

So, first set both equations equal to b to get: 

b = r – p

b = 5r – 4p

And since, obviously b = b … 

r – p = 5r – 4p

Let’s now use some basic algebra to put the like variables together, so:

Now we’re nearly home. All we have to do is manipulate the problem so r/p .

So, divide both sides by  3p :

4r / 3p = 1

Then multiply both sides by 3: 

And finally divide by 4, which gives us: 

CHOICE B  

Download the Hard SAT Math Problems quiz

SAT Problem #2

This is a question that can cause all sorts of problems if you forget your exponent rules—but it’s otherwise very straightforward. 

So let’s go over a few of those rules, just to get comfortable . . . and notice a pattern. I’ve included three below:

Two things to pay attention to:

First, when we divide variables with exponents, we keep the base and subtract the exponent. When we multiply variables with exponents, we keep the base and add the exponents. When we take a variable with an exponent to an additional power, we multiply the exponents. 

Second, in order to use the first two of these rules, the two numbers must have the same base . 

There is a base x on both the top and bottom of that fraction or the left and right side of that multiplication sign. 

So how does that help us here? 

Let’s forget the first half of the problem and look at the second:

We might look back at these exponent rules and throw our hands up—the top and bottom parts of this fraction don’t have the same base, so what am I supposed to do here? 

Except… 

8 and 2 actually DO have the same base. Base 2. 

Isn’t 2^3 equal to 8? 

So if we re-write the problem, plugging in 2^3 for 8, and thinking about that third exponent rule I gave you above, the equation will look like this: 

Now let’s go back to our exponent rules once more, and look at the first one. 

Because that tells us that… 

Well, hold on a second! 

We know the value of 3x – y . 

The problem tells us it’s 12.  

So we just plug in and get our answer… 

Which is CHOICE A. 

Keep up the practice! If you’d like help honing your skills, reach out to us for a free test prep consultation. All of our tutors are top 1% scorers who attended top-tier schools like Harvard and Princeton. That makes them uniquely qualified to help high-scoring students improve.

SAT Problem #3

A question like this confuses a lot of students because they either forget how minimums and maximums work or find it hard to keep track of which numbers they are plugging in and where. 

In order to solve it, it’s helpful to think of a function as a machine . We enter an input into the machine (an x value)—it acts on it—and then it gives us an output (a y value). 

Let’s also remember that when we’re talking about minimum and maximums we’re talking about the y value when the function is at its highest and lowest point . 

With these two facts in mind, the problem is going to be much simpler, so let’s take it on in parts…

Since the question is asking us for g(k) and k represents the maximum value of f , it’s going to be helpful to first… 

Find k .  

So what is the maximum value of f , the graphed function? Well, the maximum value (as we realized earlier) is the y value when the function is at its highest. 

Looking at the graph, it looks the function is at highest when x = 4 , and more importantly, when 

Therefore, k = 3 .

Now let’s consider our functions as machines. 

When the problem asks us for g(k) , it’s telling us that k is going to act as the input (the x value for the function). So g(k) , the value after the machine acts upon the function, is going to be the output , or the y value . 

So, g(k) is the same as g(x) , except we’re plugging in our value of k , which is 3, for our x value. 

The rest is very simple. 

We go to the table and find where x = 3 , then move our finger across to see the output for that value, which is 6.

CHOICE B. 

Test your SAT math knowledge with our quiz

SAT Problem #4

A version of this question has appeared on the SAT multiple times in recent years, and it often stumps students!

Here we have something that resembles a rotated version of the logo from Star Trek, and we’re asked to find the value of a degree inside the circle, between two points of the pointed figure.

We’re given a point that represents the center of the circle, along with two degree measurements inside the triangle-like figure. 

Generally, when we’re given a figure that looks unfamiliar to us—like the figure inside the circle— it can be extremely helpful to find a way to fix it (or cut it up) so that it’s made up of parts of shapes that are more familiar . 

So looking inside this circle, how might we “fix” this figure so that it becomes a little friendlier. 

Well, if we draw a line to the center of the circle ( P ) from the edge of the circle ( A ), then this unfamiliar figure suddenly becomes two triangles. 

And with triangles, unlike the figure we were originally given, we can apply some rules . 

Rules, for example, that dictate opposite sides of the triangle that have the same length will have the same opposite angles. 

And if we look at our drawing we see that two sides of our triangle are the same length because they’re both the radius …

And so we also know that the opposite angles of those sides will be the same… 

And we’ve been given one of those angles! 

Therefore, angles ⦣ABP and ⦣PAB will be the same—both 20 degrees. Let’s fill that in. 

Now again—because we have a triangle—we can apply another rule as well. 

We know that degrees of a triangle will add up to 180 degrees. 

So if we know one of the inner degrees of the triangle is 20, and the other is 20—the remaining angle has to be 140 degrees. (Because 180 – 40 = 140. )

We have two of these triangles, so we know the larger inner angles of both add up to 280. 

Because a circle is 360 degrees, the number of degrees “left over” when 280 is subtracted from 360 is 80. 

So X equals 80. 

There is actually a second clever way to solve this problem, involving arc measures. Can you spot it? (If not, don’t worry! Ask us how we did it here .)

SAT Problem #5

Here we have a problem that looks quite complicated—and one I find students often waste a lot of time on. They either try to plug in answers and work backwards… 

…or they waste time trying to combine the two terms on the right side of the equation and simplifying. 

It turns out the easiest way to solve this problem is by polynomial division , because we’ve already been given the answer! It’s the right-hand side of the equation: (-8x – 3) – (53 / (ax – 2)) .

That means that this is our answer to when (24x^2 + 25x – 47) is divided by ax – 2 .

So how does that help us get a value for a ? 

Well, let’s set this up as a polynomial division problem.

We’d write it as follows: 

(I’m not putting the second half of the right side of the equation on top because that’s going to be our remainder.) 

So now we have a simple question. What number divided into 24 , gives me -8 ? 

Well, that’s easy. It’s -3 , right? 

Because -3 * -8 gives me 24 . 

So a equals -3 ,   CHOICE B .

Now, you could spend time plugging in -3 for a and dividing through the rest of the problem to make sure your answer matches the one on the exam—but generally on a timed test you really shouldn’t do more work than necessary. 

In fact, by setting this up as a polynomial division problem, we’ve saved time precisely because we don’t have to complete all the work . . . just enough to get us our answer. 

Try the quiz

SAT Problem #6

Because the SAT is a timed test, “difficult” includes not only questions that are hard to solve, but also those that—if a few wrong decisions are made—take a long time to solve. 

Sure, you may get the right answer, but those extra seconds or minutes wasted will inevitably cost you on other questions later on the exam.

Generally speaking, you should be able to answer each question in about a minute. If you spend more than 60 seconds on a single question, you should put down your best guess and move on (and hope that you have extra time at the end to return to this question).

To that end, let’s look at this question. You’re asked to find the value of 3x – 2 , and you’re given this equation:

(⅔)(9x – 6) – 4 = (9x – 6)

Many students will immediately think: “This is totally straightforward: Solve for x and plug it back into the equation.” 

They’ll distribute the ⅔ and end up with something like this: 

6x – 4 – 4 = 9x – 6

and then go through all the algebra from there, to get… 3x = -2 . 

These students will then find that x = (-⅔) . 

A few unlucky students will then forget that they have to plug in, and they’ll choose the trap answer C. 

The lucky ones will plug the (-⅔) back into 3x – 2 and get the correct answer, -4 , A . 

However, it turns out there is actually a much quicker way to solve this problem! 

We can solve it without ever having to plug into a second equation. 

If we simply subtract (⅔)(9x-6) from both sides, we end up with… 

-4 = (⅓)(9x-6) . 

We can realize that (⅓) of 9x-6 is the same as 3x-2 . 

And, what do you know… 

-4 = 3x – 2 . 

Ready to try some of these problems on your own? Try our quiz with 20 more of the hardest real SAT problems ever to see if you could get a perfect score on the SAT Math!

SAT Problem #7

This is a question you could muscle through, but it’s going to be a lot easier if we find a few shortcuts and work from there. Remember, a hard question isn’t necessarily difficult because of the conceptual and mathematical effort it asks from you but also because of the time it might require.

So how do we save ourselves some time? 

First, let’s notice that in the answer choices none of these numbers repeat . There are eight distinct numbers in the answer choices. Therefore, if we were pressed for time we only really have to find one of the values of c , choose the corresponding answer choice, and then move on. 

Second, let’s look at the other piece of information this problem gives us besides the quadratic. 

It tell us that a + b = 8.

This should be especially helpful because we know from FOIL (and what the rest of the problem gives us) that a * b = 15 , because abx^2 is going to be equal to 15x^2. 

Because a + b = 8 and ab = 15 , we know that the values of a and b are going to be 3 and 5. 

(We don’t know which one is which, and that’s precisely why this problem has two possible values for c .)

At this point we’ve done most of the “hard” work to save time in this problem, and it hasn’t even been particularly hard!

Now all we have to do is assign one of 3 or 5 to a , assign the other to b , FOIL out the problem, and pick whichever choice corresponds to one of the values of c . 

Let’s say a = 3 and b = 5 .

It will work like this: 

(3x + 2)(5x + 7) = 15x^2 + 21x + 10x + 14 .

Which simplifies to… 

15x^2 + 31x + 14 .

Which means c = 31 .

31 only appears once in our answer choices, so the answer must be CHOICE D.  

SAT Problem #8

When you’re faced with one of these more difficult system-of-equations problems—specifically the ones that ask you for no solutions or infinite solutions —it’s going to be much, much easier to think about the problems geometrically. 

In other words, as two line equations. 

So what does it mean for two lines to have no solutions ?

Well, for two lines to have no solutions, they’d have to never intersect , correct? 

(Just like if one of these problems asks you about two lines with infinite solutions , they’re saying that the lines are the same . They’re laid on top of each other. )

In other words, they’d have to be… parallel lines . 

And parallel lines have the same… slope! 

So this question is asking you to find the correct value for the variable that gives these lines the equivalent slope . 

Obviously, the first step is to put both of these equations in slope-intercept form. We’d end up with:

y = (-a/2)x + 2

Now the rest is very simple. All we need is a value of a that makes the slopes equal, so that it solves the equation (-a/2) = 3 .

With some basic algebra, we end up with -a = 6 . This is the same as a = -6 .

So the answer is CHOICE A, -6. 

Are these problems feeling super hard for you? Want to work on more similar problems? Check out our one-on-one tutoring with Ivy-League instructors. A great experienced tutor can help you focus on the concepts that are the hardest for you until you understand them thoroughly.

SAT Problem 9

This is another type of problem that students often have conceptual difficulty with, causing them to waste much more time than they should. 

(Remember, basically every problem in the SAT math section is designed to be solved in a minute and half or less. If you’re taking three or four minutes on a math problem, you’ve probably made a mistake!)

Some students will see that (u-t) is defined but not u or t individually, so they’ll try either solving for u in terms of t (or vice versa), or they’ll try squaring (u-t) to get a solution. (Which is closer to the correct way to solve the problem, but still incorrect). 

Instead, to solve this problem we need to remember the difference of squares . 

Remember, that the difference of squares states the following… 

(x+y)(x-y) = x^2 – xy + xy – y^2 .

Which means… 

(x+y)(x-y) = x^2 – y^2 .

And doesn’t that look awfully familiar to… u^2 – t^2 ?

In fact, we can now replace  u^2 – t^2 with (u + t)(u – t) .

So the whole problem would now read: (u + t)(u – t)(u – t) . Since we know the value of (u + t) and (u – t) , this would simply be the same as (2)(5)(2) .

Which equals our answer… 

Ready to see how you’re doing? Download the quiz

SAT Problem #10

What makes this question confusing is that students often get thrown off by the repetition of the (⅓). 

They forget that when the ⅓ gets factored out of the parenthesis like that, it means it’s going to apply to the whole equation: both the  x^2 AND the -2 . 

Once we remember that, we can solve this problem by difference of squares . This will save us the time of having to brute force the answer choices and FOIL each one through for the different values of k. 

We’ll simply square k and subtract it from the  x^2 for each choice. 

That will give us the following four choices: 

(⅓)(x^2  – 4)

(⅓)(x^2 – 36)

(⅓)(x^2 – 2)

(⅓) (x^2 – 6)

A student might rush to choose the third answer choice, since it appears to look like the expression at the beginning of the problem, but remember what I told you at the beginning: 

We’re going to apply that ⅓ to both the x^2 AND the k ! 

If we multiply that ⅓ through, the choices suddenly look like this…  

(⅓)(x^2) – (4/3)

(⅓)(x^2) –  (12)

(⅓)(x^2) – (⅔)

(⅓)(x^2) – (2)

. . . and so the correct answer is actually the fourth choice, CHOICE D . 

Ready to try more hard problems on your own? Download our free quiz to try 20 more of the hardest ever (real) SAT problems.

SAT Problem #11

There are not many problems on the SAT that involve knowing the equation for a circle—in fact, circle equation problems don’t show up on every test—but that’s precisely why students often find a problem like this more difficult. 

First, let’s do a quick refresher on what the numbers in the equation of a circle mean. 

Any equation for a circle is going to be in this form: 

(x – h)^2 + (y – k)^2 = r^2

Where h and k represent the coordinates of the center and r is the radius. 

Let’s apply that to our problem here… 

(x + 3)^2 + (y – 1)^2 = 25.

Remember: because in the form of the circle equation the numbers inside the parenthesis are subtracted from x and y , when they appear inside the parenthesis as positives , that indicates the coordinate point will be negative. 

Therefore the center of this circle is at point (-3, 1) .

Because the radius is expressed as r^2 , then the 25 indicates the radius will be 5 . 

So we have a circle centered on the point (-3,1) and with a radius of 5 . 

So… now what? 

How do we figure out which of these points is not inside the circle? 

First, let’s draw the circle itself and look at it. On the SAT itself, you won’t have graph paper, so just draw a rough sketch!

Of course if we’re truly flummoxed we could graph the points, eliminate what we can . . . and guess. 

But that’s not ideal, obviously! 

Instead, let’s think about what the radius means. 

The radius demarcates the boundaries of the circle from the center. 

In other words, any points with a distance less-than-the-radius away from the center will lie within the circle. 

And any points more-than-the-radius distance from the center will lie outside of it. 

(Any points exactly-the-radius distance from the center will lie on the circle itself.) 

So all we have to do is find the point that is more than 5 units away from our center, and that will be our answer. 

To do this requires the distance formula. 

Remember, the distance formula is

A quick note: if you ever forget the distance formula, simply plot the two points on a graph, make a triangle with the distance between the two points and the hypotenuse, and use the Pythagorean Theorem to find the length of the hypotenuse, like this: 

Going back to our problem, let’s plug each of the points in along with our radius to the equation. (I’ll include the second point here, although since that’s our center we need not actually bother with it when we’re going through the problem.) We end up with: 

√(-3 – (-7))^2 +(1-(3))^2) = √20

√(-3 – (-3))^2 +(1-(1))^2) = √0

√(-3 – (0))^2 +(1-(0))^2) = √10

√(-3 – (3))^2 +(1-(2))^2) = √37

Only the square root of 37—choice D—is an answer that is larger than five. 

So that’s our correct choice, D . 

Want more hand-picked problems? Get the quiz

SAT Problem #12

More circles! Let’s recall how the equation for a circle looked. It’s… 

(x – h)^2 + (y – k)^2 = r^2 

What the problem gives us, unfortunately, does not resemble that equation… 

…so our goal is to get the equation in the problem to look like a normal equation for a circle. 

Once we do this, we’ll just have to take the square root of whatever is on the right side of the equation, and that will give us our answer.

But how? 

We need to do something called completing the square . 

For the SAT, this concept is slightly obscure—it’s one you may see only once (or not at all) on a given exam. It makes the question a bit more difficult. 

Completing the square is normally a process reserved for solving a quadratic equation, but if you look closely at the way this problem is set up – 

2x^2 – 6x + 2y^2 + 2y = 45

we see that what we really have here are two quadratic equations, so we just have to complete the square twice. 

First we have to get rid of the coefficient in front of the x and y squared, so we have to divide through by 2 . 

This gives us  x^2 – 3x + y^2 + y = 22.5 .

Now we’re reading to complete the square!

Let’s deal with the x terms first. We have to think of what number, if we had it here in the equation, would allow us to factor x^2 – 3x into something of the form (x – z)^2 , where z is a constant. If we think about it, we realize that z has to be half of b . In this case, that means half of -3 , so -1.5 .

When we pop that into our setup, we get (x – 1.5)^2 . If we FOIL this out, however, we see that we get x^2 – 3x + 2.25 .

So it turns out that in order to be able to rewrite our expression in the form we want, we need to add 2.25 to our equation. As always in algebra, we do the same thing to both sides, so now we have:

x^2 – 3x + 2.25 + y^2 + y = 22.5 + 2.25.

Now we do the same thing for the y terms! Again, we need to add something to the equation so that we could rewrite the y part of the expression in the form (y – z)^2 . To get this number, we take half of the b term and square it: 1 divided by 2 , then squared, so 0.5^2 or 0.25.

Again, we have to add this number to both sides of the equation. Now we’ve got:

x^2 – 3x + 2.25 + y^2 + y + 0.25 = 22.5 + 2.25 + 0.25.

We can factor and rewrite this like:

( x – 1.5)^2 + (y + 0.5)^2 = 25.

Alright, now this is finally in the right format for the equation for a circle!

The final step is to use this equation to find the radius.

We know that the equation for a circle is (x – h)^2 + (y – k)^2 = r^2  . Fortunately this works out really nicely, since 25 is just 5^2. The radius must be 5, CHOICE A .

We’ve tutored thousands of students and used that experience to assemble a list of 20 more problems that students frequently miss. Can you answer them correctly? Download the quiz now to find out!

SAT Problem #13

This question involves a number of moving parts and thus can be a little overwhelming for students to follow. 

It asks us to find, based on the rotation of the first gear, the rotation of the third. 

I find many students trip up on this problem by making two errors that are simple to fix, but relatively common. They fail to take the problem step by step… and they fail to write down their work as they track through the material. 

With that in mind, let’s work through the problem. 

Because gears A and C do not connect directly, but instead through gear B, we should first try to figure out the rotational relationship between A and B (at 100 rpm) before applying that to B and C. 

Because B is larger than A (and has more gears), A is going to rotate fully multiple times before B rotates once. 

How many times? Here it’s helpful to consider a ratio. 

A has 20 gears. 

B has 60 gears. 

So A is going to have to rotate three times before B rotates once . (20 goes into 60 three times.) 

Therefore, the ratio of rotation between A and B is 3 : 1 . 

Let’s write that down and then apply the same method to figure out the ratio between B and C. 

C has 10 gears. 

Here B only has to rotate a sixth of its distance for C to rotate once, so the ratio of rotation between B and C is 1 : 6 .

Now we take the number of RPMs the problem gives us, start with the gear on the left and multiply through with our ratios. 

So if Gear A rotates 100 times RPMS per minute, Gear B will rotate a third of that distance… 

So we divide 100 by 3. 

Because we know Gear C rotates six times as fast as Gear B, we then take our answer and multiply it by 6. 

So we get (100)(⅓)(6) . 

Which gives us 200 rpm. 

CHOICE C. 

SAT Problem #14

This question appears complicated—and students often get tripped up trying to either plug in numbers (which can be time consuming) or by searching for an equation that explains the relationship between the surface area and perimeter of the cube itself. 

This is especially tempting because while the question gives us the equation for the entire surface area of the cube, it only asks for the perimeter of one of the cube’s faces. 

However… 

If we think about the properties of a cube, this question actually becomes quite simple. 

First, let’s draw a cube.  

Again, the equation the problem gives us is for the entire surface area of the cube: 6(a/4)^2 .

But when we look at the cube, we may notice that it has, in fact, six faces. 

Therefore, each face would have one sixth of the surface area of the entire cube. 

So by dividing the equation by six, we get the surface area for one face of the cube, which is: 

But the question asks for the perimeter of one face of the cube. 

Let’s examine the drawing of the cube one more time. 

What shape is each cube face? It’s a square. 

And because each side of a square (let’s call each side x ) is equal to the other, the area of the square is going to be x^2, or the length of the side times itself. 

Well, wait a moment… 

If we go back to our equation for the surface area of ONE face of the cube, (a/4)^2 , we might notice that it’s in the same form as the equation for area of the square, except instead of x being squared, it’s (a/4) . 

And if we replace the x with (a/4) , we find that each side of the square is equivalent to (a/4) . 

Which makes finding the perimeter of this square quite simple, because it has four sides. 

So we merely add the four sides together: 

(a/4) + (a/4) + (a/4) + (a/4) . . .

which equals a . 

Which in this case is CHOICE B . 

Want more practice? We collected 20 more of the hardest SAT math problems. Download the quiz and take it with a 25-minute timer to mimic the real test!

SAT Problem #15

We have a lot of variables in this question, so it’s easiest to try to incorporate the extra piece of information we’re given, b = c – (½) , as best we can and then try to simplify the problem and solve from there. 

So how can we do that? 

The problem tells us b = c – (½) , which can also be expressed as b – c = -(½) .

(Once we put the b and c together on one side, it becomes easier to replace them together with a number). 

So what’s the best way to manipulate these two equations so that we’ll have b – c , which we can then replace with the (-½) and be left with x and y ? 

Because let’s remember that the problem does not ask us to solve for x and y individually. 

Just their relationship. 

So once we’re left with x and y as our only two variables, we should be able to make good progress. 

Anyhow, looking back over these two equations it seems the easiest way to be left with b – c is to… 

…subtract the bottom equation from the top one. 

When we do so, we’re left with the following: 

(3x – 3y) + (b – c) = (5x – 5y) + (-7 – (-7))

We replace b – c with -½  

And then combine like terms to get… 

(-½) = (5x – 3x) – (5y + 3y)

(-½) = 2x – 2y

Divide through by 2 … 

-¼ = x – y

Or x = y – (¼)

So our an answer is x = y – ¼ , CHOICE A .

Download 20 more of the hardest problems ever

SAT Problem #16

There are a few ways to solve this problem. The easiest one is simply to know the “remainder theorem.”

I don’t want to get too sidetracked with details, but remainder theorem states that when polynomial g(x) is divided by (x – a) , the remainder is g(a) .

In other words, when p(x) is divided by  (x-3) here, the remainder would be p(3) , which, according to the information we’re given, is -2 . 

That leads to CHOICE D . 

But what if, like many students, you don’t know the remainder theorem? (It’s pretty obscure and there’s a good chance you won’t see a problem about it on the entire exam.) 

Let’s look at an alternative way to solve the problem. 

If p(3) equals -2 , let’s imagine a function where that might be the case. 

We could do as simple one, like y = 3x – 11 , or a more complex one, like y = x^2 + 3x – 20 .

Either way, if I plug 3 into either of these functions for x , I get -2 as a y value. 

I should also notice immediately that (x – 5) , (x – 2) , and (x + 2) are not factors of either of them.

Clearly choices A, B, and C are not things that must be true. 

This also, by process of elimination, leads to CHOICE D. 

But just to check, let’s divide x – 3 into one of these functions – say 3x – 11 – and see what happens: 

The x goes into 3x three times – and three times (x-3) equals 3x – 9 .

When I subtract 3x – 9 from 3x – 11 , I get -2 , which is my remainder. 

Which points us, again, to CHOICE D .

Test your knowledge with 20 more problems

If these problems feel really hard, don’t panic—you can still do well on the SAT without answering every question correctly. 

The average SAT Math score for US students in 2022 was 52 8, and you have to answer about 32 out of 58 math questions correctly to get this score. That’s only a little over half of the questions!

However, if you want a high score—or a perfect score—you’ll have to be able to answer tough questions like these. You’ll need a very high score to be a competitive applicant for Harvard, Stanford, MIT, or other highly competitive schools.

The good news is that it’s very possible to raise your math score! 

In fact, it’s typically easier to improve your SAT Math score than your Reading & Writing score. Good preparation (on your own or with a tutor ) will fill in the knowledge gaps for any concepts that might be shaky and then practice the most common problem types until they feel easy.

We’ve worked with students who were able to see a 200-point increase on the Math section alone, through lots of hard work and practice.

To see how your math skills stack up against the toughest parts of the SAT, download our quiz with 20 more of the hardest SAT math questions, taken from real tests administered in recent years.

Once you know where you stand, keep up the practice!

If you’re interested in customized one-on-one tutoring support from an expert SAT tutor who can help you understand these tough problems, schedule a free consultation with Jessica or one of our founders . Our Ivy-League tutors are top scorers themselves who can help you with these more advanced concepts and strategies.

Bonus Material: Quiz: 20 of the All-Time Hardest SAT Math Problems

Emily graduated  summa cum laude  from Princeton University and holds an MA from the University of Notre Dame. She was a National Merit Scholar and has won numerous academic prizes and fellowships. A veteran of the publishing industry, she has helped professors at Harvard, Yale, and Princeton revise their books and articles. Over the last decade, Emily has successfully mentored hundreds of students in all aspects of the college admissions process, including the SAT, ACT, and college application essay. 

CHECK OUT THESE RELATED POSTS

Should You Transfer Colleges?

April 4, 2024

If you’re thinking about transferring colleges, there’s a lot you should think about. Not all reasons for transferring are great ones: read on to learn about good reasons to transfer, bad reasons to transfer, and…

How to Request your High School Transcript

You may need to get a copy of your high school transcript for college applications, scholarship competitions, or just for your own records. Learn the most efficient way to receive your high school transcript and …

13 Best SSAT Tutoring Services for 2024 (36 Services Reviewed)

March 29, 2024

A list of the 13 best online SSAT tutoring services, reviewed and ranked. Compare prices & quality. Best overall: PrepMaven. Best self-guided: Test Innovators. Best on a budget...

15 Best Online SAT Tutoring Services for 2024 (75 Tutoring Services Reviewed)

A list of the 15 best online SAT tutoring services, reviewed and ranked. Compare prices & instructor qualifications. Best overall: PrepMaven’s SAT tutoring ($79–349/hr). Best on a budget: Wyzant ($20–600/hr). Best…

14 Best College Essay Services for 2024 (40 Services Reviewed)

Working on college application essays? A list of the 14 best online college essay coaching and editing services, reviewed and ranked. Compare prices & essay consultant qualifications.

How ADHD Tutoring Can Help

ADHD can make staying on top of schoolwork and assignments particularly difficult. While tutoring can’t directly address the causes of ADHD, a good tutor can help students with organization, executive functioning, and …

SAT Accommodations for Students with ADHD

College Board offers many accommodation options for students with ADHD taking the SAT. We break down how to qualify for SAT accommodations, what accommodation options you’ll have, and how to make the most of these accommodations on the SAT.

How to Write a College Transfer Essay

At college and thinking about transferring to a different university? The transfer essay plays a key role in your chances of admission, and this post will break down how to write …

Letter of Recommendation for College Application (Guide and Sample)

Recommendation letters for college applications play a significant role in your chances of admission, but many students submit sub-par recommendation letters. Learn what you need to do to get a stellar…

Khan Academy Review: Is It The Best SAT Prep Service?

March 26, 2024

Considering using Khan Academy for SAT Prep? Read our review first to see if it’s a good choice!

Privacy Preference Center

Privacy preferences.

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Maximize your score with free Official Digital SAT ® Prep

Get ready for test day by practicing with purpose.

• Take a full-length practice test on College Board’s site
• Learn and practice your skills on Khan Academy

The inside scoop on the new digital SAT

For educators, for parents, the best way to practice for the digital sat is free for everyone..

18 Free SAT Practice Tests

Reviewed by:

Former Admissions Committee Member, Columbia University

Reviewed: 11/29/23

Are you considering incorporating an SAT practice test in your preparation? This guide will shed light on the significance of utilizing practice tests to boost your readiness for the SAT.

When preparing for the SAT , practice tests are invaluable tools. They offer you the opportunity to time yourself effectively, become acquainted with the types of questions you'll encounter, and experiment with various strategies for each section.

While practice tests are essential, it's crucial to understand that they serve a unique purpose. SAT practice tests are designed to help you refine your test-taking skills and approach each section methodically in order to achieve a higher score. They are not replicas of the SAT itself; they are structured differently.

In this guide, we will delve into some SAT practice tests and how you can make the most of them in your SAT preparation journey.

16 Free SAT Practice Tests

These free SAT practice tests are here to help you get ready for the SAT and improve yor college admissions chances. As you work through them, don’t forget to rely on SAT tips and strategies to boost your test-taking skills and maximize your potential for success on exam day. Now, let's dive in together and work toward SAT success. 

Practice Test 1 (2016-Present)

This free SAT practice test, available from 2016 to the present, serves as an excellent starting point for SAT preparation. It includes sections for questions , answers , and detailed answer explanations . It's a valuable resource for assessing your initial skill level. 

Practice Test 2 (Removed By College Board)

Unfortunately, Practice Test 2 has been removed by the College Board. However, you can still work your way through this SAT practice test with answers . Along with a long list of questions , you can find in-depth answer explanations .

Practice Test 3 (2016-Present)

Similar to Practice Test 1, this test offers another opportunity to practice and refine your SAT skills. As you work on the questions , and compare your work to the provided answers , make sure to utilize the provided answer explanations to understand where you might need improvement. 

Practice Test 4 (Removed By College Board)

Like Practice Test 2, Practice Test 4 has been removed by the College Board. While its questions , answers and explanations can still be helpful - ensure you focus on the available official practice tests to align with the most current SAT content. 

Practice Test 5 (2016-Present)

This practice test continues to offer valuable SAT practice questions , answers , and explanations . Consistent practice is key to SAT success.

Practice Test 6 (2016-Present)

Practice Test 6 is another resource to aid in your SAT preparation journey. Regularly incorporating practice tests into your study routine can help build confidence and familiarity with the test format . So, do your best to work through these questions , take a look at the answers , and familiarize yourself with the answer explanations . 

Practice Test 7 (2016-Present)

As you progress in your preparation, Practice Test 7 can be a useful benchmark to gauge your improvement. While you work on the questions and answers , try to analyze your performance and use answer explanations to target weak areas. 

Practice Test 8 (2016-Present)

Similar to the previous tests, Practice Test 8 offers additional chances to refine your SAT skills. Be sure to time yourself while you work through the questions to replicate test-day conditions accurately. Afterward, review the answers and answer explanations to understand where you can improve and develop effective strategies.

Practice Test 9 (2016-Present)

By this point in your preparation, you should be well-versed in SAT question types and formats. Practice Test 9 can help solidify your knowledge and strategies by providing you with a set of SAT questions , answers , and answer explanations .

Practice Test 10 (2016-Present)

As your test date approaches, think of Practice Test 10 as your last practice before the big day. Take a close look at how you performed, go over the questions and your answers , and make sure to check the provided answer explanations . This thorough approach will give you the confidence and readiness you need for the real SAT.

Official SAT Printable Practice Test (2013-14)

This official SAT practice test from 2013-14 offers additional practice questions and answers . While it may not be the most up-to-date resource, it can still be beneficial for extra practice.

Official SAT Printable Practice Test (2012-13)

Similarly, the SAT practice test from 2012-13 provides a chance to work on your skills with real SAT questions and compare your work to real answers .

Official SAT Printable Practice Test (2007-08)

For those seeking more practice, this test from 2007-08 offers an array of SAT questions and answers .

Official SAT Printable Practice Test (2004-05)

Another official practice test from 2004-05 provides historical SAT questions and answers to further enhance your preparation.

Official SAT Test (2004-05)

This official SAT test from 2004-05, with its included questions and answers , can be a valuable resource for test takers looking to practice on authentic SAT material.

Official SAT Test (2003-04)

Much like the previous test, this official SAT exam from 2003-04 is a valuable resource to strengthen your skills. It includes useful practice questions and answers to assist you in your preparation.

Official SAT Test (2002)

This official SAT test from 2002 serves as an additional resource to bolster your SAT readiness.

Official SAT Test (2001)

For those interested in more historical SAT questions , this official test from 2001 can be a valuable asset.

Free Digital SAT Practice Tests

Don't forget to explore the College Board website for free digital SAT practice tests. These resources offer the convenience of online practice and can closely mimic the actual test-taking experience.

Are SAT Practice Tests Harder or Easier Than the Real SAT?

The official SAT is neither harder nor easier than official practice tests since every SAT is scored differently. The official SAT does have a curve that determines how they will grade each section, whether it is harsh or lenient. 

Their levels rank SAT practice tests’ difficulty; the higher the level, the more difficult it is. It all depends on how you study with the practice tests; they grade the results the same as the actual SAT, so your score can remain the same when you take the official SAT. Some students claim that the questions were harder on the official SAT or at the same level as their practice ones.

So to put it simply, it is neither harder nor easier. It all depends on how prepared you are for the official test and how well you do while taking a test.

Practice Test Scores vs. Real Scores

Your practice SAT scores can predict what type of score you will get on the official test. However, if you scored lower on the SAT than you did on the practice test, that is not uncommon. There can be many reasons, such as test anxiety, less time to read and understand each question, or just not being as confident as you were with the practice tests. 

Again, this is purely subjective; each test-taker is different, along with their studying methods and ways of learning. The point of doing practice SATs is to find out your strengths and weaknesses and work on them to get an optimal score on the real SAT. 

It is always helpful to refer to the SAT scoring chart and track your practice test scores to see how you can improve your raw score. There are many tips and tricks on how to do better on your SAT practice tests so you can improve your chances of passing the official test with flying colors.

Why Practice Tests and the Real SAT Aren’t the Same

Even though they offer a preview of what you will face on the official test, practice tests are not 100% accurate compared to the real thing. The content and how you take the test can be vastly different, which is why some students are shocked to learn that their scores are vastly different from what they scored on the practice test. 

There are two major reasons why the practice SAT isn’t the same as the official one.

SAT Practice Tests aren’t taken in realistic settings. When doing the practice SAT, you are usually in the comfort of your own home, on your laptop or desktop, accessing the practice portals whenever you like. The guide to the answers is put in front of you, and you can take breaks whenever you like before getting to the difficult sections of the test. 

When the real test day comes, you wake up early to go to an unfamiliar test center where you can be subjected to uneasiness and limited breaks. You are then given a time limit on each section and may find yourself rushing to complete each section before time runs out. This is vastly different from the homemade open-book test you have been referencing all this time.

The second reason is that students tend to revert to their old habits under massive stress. The current SAT may have new content in its sections, and if you have taken prep courses and tutoring alongside your practice tests, you may have been taught new methods for approaching each question. 

For example, you may have been taught a certain way to answer a math problem when it comes to math. When it comes to SAT math, however, you were taught a new quicker method to approach the math questions for the sake of time conservation. 

When put in a new test set, you can forget the new methods and stick with what you are more familiar with. This can greatly hinder your SAT score , and your results can be lower than you anticipated. 

According to College Panda , the sections may have different question types as content unfamiliar with when you used the practice SATs. These patterns can throw you off guard when taking the official test.

This chart categorizes the differences between the official SAT by section: 

The reading sections are difficult to compare because there is no sure-fire way to quantify each passage’s difficulty level appropriately, and the questions can not be specifically categorized. 

Rest assured, though, the reading section may be tweaked as well to fit the current year’s SAT protocol.

How to Make the Most of Practice Tests

Even if the practice tests are not a carbon copy of the real thing, they are not useless. You need to understand how to use them to get the best possible score on the official test. Here are some tips on using the practice tests to your benefit.

Practice as a Real Simulation

To avoid a culture shock when you first enter the testing center on your test day:

• Do a practice run in your own house.
• Turn off your phone, get rid of snacks or drinks, and do other things that would provide a distraction to get an accurate exam room.
• Have short breaks, and use scrap paper for your test booklet. This will allow you to get used to the environment and note any discomfort you have had during this run.

That way, you can make adjustments and prepare for an actual day.

Practice Your Test-Taking Strategies

When you are studying, you develop certain exam-taking strategies. Perhaps you prefer dealing with easier questions first, then go back to the harder questions. Whatever method you have, gives you autonomy over the exam, but you have to know exactly which methods work in your favor. 

That way, you will be prepared for the official SAT and won’t be thrown off guard by the questions. Practice tests are a great way to keep track of your strategies, as they will be a very important part of your preparation process.

Practice with a Bubble Sheet

Bubble sheets are part of the SAT and are usually the most trivial instrument for students during the exam. You may get the answer right on the booklet but get it wrong on your bubble sheet. 

By practicing with a bubble sheet, you are giving yourself the advantage of keeping track of your answers when you take the real test. It can also prevent you from circling the wrong answer on a different line on the test (this is especially for the Reading section).

Find Your Area of Focus

Out of all the sections of the SAT, which one do you struggle with the most? Math? Reading? 

Are you more concerned with the time limits you have for each section? Are you susceptible to losing your energy halfway through an exam? 

When using practice SATs, you can pinpoint where you need to focus and how you can resolve those issues so you are relaxed and ready for test day. You cannot fix something you are not aware of. 

Review Practice Test Answers and Explanations

When you have taken a five-hour practice SAT, it would probably be exhausting to review the answers and the explanations. You must take a well-deserved break so you can check out the explanations with a refreshed and active mind. 

You should then identify which questions you answered incorrectly and why. Take note of the explanations and refer to any relevant material to be prepared for any similar questions in the future. 

Studying with SAT experts can also help you review questions and concepts efficiently. Expert tutors can help you pinpoint areas for improvement and sharpen up your test-taking skills! If you live in NYC, we can even provide specialized NYC SAT tutoring to help you ace your exams.

FAQs: SAT Practice Tests 

Still, have questions regarding the practice SAT and the official one? Here are some FAQs that give a better insight into the matter. 

1. How Can I Score as High on the Real SAT as I Did on the Practice SAT?

The best advice would be to practice! Do the practice tests, review your incorrect answers, find your strengths and weaknesses, and work on any weaknesses you noticed in practice SAT. These are your best options when it comes to wanting a great score. 

2. How Many Practice SATs Should I Take?

It is advised to take 3 to 4 mock tests about four to six weeks before the official test date. That way, you can practice in mock test environments and get used to them. Then, it would not hurt to take 2-3 more mock tests a month before the test day. It is important to time these tests, too to score in your desired range.

3. If the Practice Tests Aren’t Accurate, Why Should I Do Them?

Just because they are not 100% accurate does not mean they are useless to you. Practice SATs are more so a guide to give you a preview of what the SAT is like and allow you to analyze your strengths and test-taking skills. It can also help you work on any issues regarding the SAT so you are prepared once the official test date comes. 

4. What Are the Hardest SAT Practice Tests?

It depends on the student, but a consensus claims that Practice Test #3 could be the most difficult for a first-time SAT taker. It would help if you went over each practice test and ranked which one is hardest, so you can spend more time dealing with those before heading to the easier ones. 

5. What Are Some of the Best Free Sat Practice Tests?

You are always free to search online for practice tests to help you study. You can also ask your school about resources and information on affordable practice tests and prep courses. There is usually a list of the top best programs that offer Practice SATs.

6. I Have Test Anxiety. How Can I Take the Real SAT?

Being anxious is a normal response to something as stressful as the SAT. Aside from practicing, you can also take some breathing exercises and relaxation techniques during your study time to learn not to panic on the actual test day. These methods can be very effective in calming you down.

7. How Long Should I Study for the SAT?

It depends on you and your daily schedule. A solid schedule would advise you to study 10 to 20 hours a week, approximately two to three months before the test date. If you feel like you need more than that, you can adjust the hours of your own will.

Final Thoughts

To sum it up, these sample SAT practice tests are your key to SAT success. They provide the practice you need to excel on the actual test. With dedication and consistent practice, you'll be well-prepared for the SAT. So, don't wait, start using these resources today, and set yourself up for your best SAT score!

Take on 5 free practice SAT questions here

Get A Free Consultation

You may also like.

What is the Common College Application?

22 Best Free Summer Programs for High School Students in 2024

Recently viewed courses

Recently viewed.

Find Your Dream School

This site uses various technologies, as described in our Privacy Policy, for personalization, measuring website use/performance, and targeted advertising, which may include storing and sharing information about your site visit with third parties. By continuing to use this website you consent to our Privacy Policy and Terms of Use .

   COVID-19 Update: To help students through this crisis, The Princeton Review will continue our "Enroll with Confidence" refund policies. For full details, please click here.

• Digital SAT Home
• 1400+ Course
• 1500+ Tutoring
• Essentials Course
• Summer Camp
• Free Tests and Events
• College Advice
• Who Teaches
• Digital SAT FAQ
• SAT Test Dates

Free Digital SAT Practice Tests & Events

Sign up today to find out where you stand and how you can improve your SAT score.

Free Full-Length Digital SAT Practice Test

Mirror the exact format and question types found on the official College Board's Bluebook App.

SAT, ACT or Both?

Find out which test is right for you at our live event.

Free Digital SAT Model

Check out how the new digital test will look and feel on test day.

Get Advice From Our Experts

No Credit Card Required!

SAT Self-Paced 2-Week Free Trial

Take a sneak peek at our score-raising drills, video lessons, and tips.

Register Free >

Live College Admissions Webinars

Real admissions tips from our college admissions experts

View Options >

• Advice Articles

Need-to-know info that every future college student should have.

Check out our free resources and events!

• Practice Tests
• Strategy Sessions
• Other Free Classes
• Free Practice Tests, Events & Seminars

Our free webinars cover a wide variety of topics including test-taking strategies from our expert instructors that will help you get a better score.

SAT 1400+ Course

Score a 1400+ on the SAT or add 150 points depending on your starting score. Guaranteed, or your money back.*

*Please see our full terms and conditions for details.

Have a question?

Talk to a Princeton Review Enrollment Advisor. Call Now!

1-888-580-8378

Enrollment Advisor

1-800-2REVIEW (800-273-8439) ext. 1

1-877-LEARN-30

Mon-Fri 9AM-10PM ET

Sat-Sun 9AM-8PM ET

Student Support

1-800-2REVIEW (800-273-8439) ext. 2

Mon-Fri 9AM-9PM ET

Sat-Sun 8:30AM-5PM ET

Partnerships

• Teach or Tutor for Us

College Readiness

International

Advertising

Affiliate/Other

• Enrollment Terms & Conditions
• Accessibility
• Cigna Medical Transparency in Coverage

Register Book

Local Offices: Mon-Fri 9AM-6PM

• SAT Subject Tests

Academic Subjects

• Social Studies

Find the Right College

• College Rankings
• Applying to College
• Financial Aid

School & District Partnerships

• Professional Development
• Private Tutoring
• Mobile Apps
• Local Offices
• International Offices
• Work for Us
• Affiliate Program
• Partner with Us
• Advertise with Us
• International Partnerships
• Our Guarantees
• Accessibility – Canada

Privacy Policy | CA Privacy Notice | Do Not Sell or Share My Personal Information | Your Opt-Out Rights | Terms of Use | Site Map

©2024 TPR Education IP Holdings, LLC. All Rights Reserved. The Princeton Review is not affiliated with Princeton University

TPR Education, LLC (doing business as “The Princeton Review”) is controlled by Primavera Holdings Limited, a firm owned by Chinese nationals with a principal place of business in Hong Kong, China.

The SAT is coming back at some colleges. It’s stressing everyone out.

A patchwork of admissions test policies is wreaking havoc on students, parents and college admissions consultants.

A California mother drove 80 miles this month to find an SAT testing center with an open seat where her high school junior could take the exam. During college tours this spring, a teen recalled hearing some would-be applicants groan when admissions staffers announced they could not guarantee test-optional policies would continue.

And across the country, college counselors are fielding questions from teenagers alarmed, encouraged or simply confused by what seems like the return of the standardized test in admissions — maybe? Sort of? In some places, but not in others?

“You could be expecting and preparing for a certain way to apply to a college and present yourself — but then they change it mid-application process,” said Kai Talbert, a 17-year-old high school junior in Pennsylvania. “That’s really confusing. It can set back a lot of people.”

Colleges nationwide have been updating their coronavirus-era policies on standardized testing, which many dropped when the pandemic shut down in-person testing centers. Some of the most selective schools are declaring they will require tests again — including, across the last two months, Dartmouth College and Yale and Brown universities. Others, such as the University of Chicago and Columbia University, won’t. And still others have not yet picked a permanent policy: Princeton, Stanford, Cornell and the University of Pennsylvania have said they will remain test-optional for another year or two, and Harvard University plans to keep its test-optional policy at least through the 2025-26 application cycle.

Public universities have veered in different directions, too: The University of Tennessee system requires tests. The University of Michigan will be test-optional. The University of California system is test-blind, meaning schools refuse to consider SAT or ACT scores for admissions.

The patchwork of policies is wreaking havoc on applicants, parents and college admissions consultants nationwide, who are being forced to recalculate where and how they are willing to apply — or what to tell anxious teenagers about whether to test, retest or skip testing entirely — as decisions keep rolling out in real time.

Laurie Kopp Weingarten, founder of One-Stop College Counseling in New Jersey, said she has a new response whenever a student gives her a list of their school targets. She starts by going down the list, school by school, to review each institution’s testing rules and whether those seem likely to change.

Taking a breath, Weingarten rattled off a summary of the different testing requirements in place at every Ivy League school. It took her three minutes.

“Even just saying it, it sounds like insanity to me, and then we’re expecting kids to understand this?” Weingarten said. “Colleges should really analyze the data, come up with a decision and stop changing their mind.”

The shifting testing expectations are among many changes roiling college admissions this year. Colleges are still grappling with the fallout from the landmark Supreme Court ruling that ended the use of race-based affirmative action in admissions. Many are undertaking an array of experiments in response to the decision in a bid to maintain diverse admitted classes — ending legacy preferences in some cases, adding essay prompts on adversity or identity in others, or increasing outreach in low-income areas.

And the disastrous rollout of a federal financial aid form that was supposed to simplify the notoriously difficult process has left students, parents and schools scrambling .

This is the most hectic and distressing admissions cycle in recent memory, said Jennifer Nuechterlein, a college and career counselor at a New Jersey high school. She laid special blame on schools that reinstated testing mandates in the past two months, some of which affect the high school juniors who will begin applying in the fall. This class of teens will have to take the SAT or ACT, should they decide to do so, within the next six months.

“Students can’t just test overnight,” Nuechterlein said. “There are students who want to prep, there are students who are not math- or English-ready. ... Students are going to be unprepared.”

For the most ambitious, high-achieving students, the tests are another stressful hurdle to clear as they apply to the most selective colleges. And for many other students, the test scores — even if not required for admission — are mandatory if they want to qualify for some financial aid programs or, on some campuses, certain degree programs.

Critics of standardized tests have argued that they mirror, or exacerbate, societal inequities, in part because students from unstable homes or with limited resources cannot afford SAT or ACT tutors or testing preparation classes, or may not know of free resources such as Khan Academy . Even before the pandemic, some schools had moved to make the scores optional to avoid creating another barrier for students.

Then the pandemic hit, spurring a crisis response when students literally could not access spaces in which to take standardized tests, said Dominique J. Baker, a University of Delaware associate professor of education and public policy who studies admissions policies.

“There were a number of institutions that never would have chosen to have gone test-optional except the pandemic made them,” she said. “Those institutions, by and large, are going back to requiring test scores.”

MIT, Georgetown University and the University of Florida are among schools that quickly chose to reinstate the requirements, with MIT announcing the change in 2022. Many others have spent the years since the virus arrived studying what effect going test-optional had on their admitted classes.

At Brown, Yale and Dartmouth, officials said they had found something surprising: Considering test scores would help them identify more promising applicants from disadvantaged backgrounds, not fewer. After looking at their own data, leaders at the three Ivy League schools say they concluded that SAT and ACT scores are highly predictive of students’ academic performance in college, more so than high school grades. They also found that some less-advantaged students withheld their scores when sharing them would have boosted their chances.

Depriving admissions officers of SAT and ACT scores meant they were less able to evaluate an applicant’s chances of thriving at Brown, Provost Francis J. Doyle III said in an interview this month .

“Our analysis suggested our admissions could be more effective if we brought back testing as an instrument,” Doyle said.

The University of Texas at Austin is also choosing to require testing again, the school announced earlier this month. Jay Hartzell, the school’s president, said he and others worried the cost and preparation associated with the tests could keep students from applying. But about 90 percent of UT Austin applicants in the latest round took the SAT even though it was optional, Hartzell said. And the school found that students who declined to submit scores were less successful once enrolled.

John Friedman, a professor of economics and international and public affairs at Brown, said he wouldn’t be surprised if more of those highly selective schools reinstate a testing requirement. He was one of the authors of the study from Opportunity Insights, a nonprofit at Harvard University, on standardized test scores and student performance at a dozen “Ivy-plus” universities.

“It’s not just about the test scores being a good predictor,” he said. “We show in the paper that students who attend a school, having been admitted without a test score, perform at the bottom of the distribution.” He said schools should look at their own data to determine their policy.

Nonetheless, most schools nationwide will probably remain test-optional, predicted Angel Pérez, the chief executive of the National Association for College Admission Counseling. For many institutions, he said, the policy has been a huge success, bumping up the number of applicants and diversifying admitted classes.

He added that most American high-schoolers are applying to schools that admit nearly all applicants, to public schools or to colleges close to home, he said: “So the majority of students aren’t going to be impacted.”

Morehouse College is among those maintaining a test-optional policy, which the historically Black college adopted in 2020. Since going test-optional, Morehouse has seen an increased number of applicants and an increased acceptance rate from admitted students, said Michael Gumm, Morehouse’s director of admissions and recruitment.

The majority of Morehouse applicants choose not to submit scores, Gumm said, and more students are completing their applications than in the past. He said Morehouse is looking for leaders, so essays and letters of recommendation carry a lot of weight.

Gumm said he often preaches to students: “Your test scores do not make you who you are.”

But for some students, the tests remain a priority. Alina Bunch, a 16-year-old high school junior in Texas, said that even when she saw schools dropping test requirements, she never altered her plan to take the ACT. The exam, she says, is a way to demonstrate determination and academic rigor.

She thinks it’s generally a good thing that schools are bringing back testing requirements, because they can function as a mechanism of standardization in a sometimes subjective admissions process. She does fear the effects of reinstating test requirements for students who cannot afford tutoring.

But for herself — after taking a summer course to prepare for the ACT and scoring high on the exam — she has no real worries. “It was never a question for me, of whether I should do it or not,” Alina said.

Many students pursued similar strategies, continuing to take standardized tests throughout the test-optional trend. After a dramatic drop in 2020 spurred by the sudden closure of test sites, the number of students taking the SAT nationally has risen every year since, per the College Board, and reached 1.9 million for the class of 2023. That’s about 300,000 short of the last pre-pandemic total, when 2.2 million members of the class of 2019 sat for the exam — the largest-ever group to do so.

Joan Koven, who heads college consulting company Academic Access in Pennsylvania, said she never expected standardized testing to suffer a real drop in popularity.

“The ACTs and the SATs are Burger King and McDonald’s,” she said. “They’re not going away.”

But in some places, counselors wish they would. Priscilla Grijalva, a high school counselor in California’s San Jacinto Unified School District, said the elimination of test requirements in the UC system and California State University campuses was a godsend for the nearly 300 students she works with every year, a mix of White, Black and Latino teens, most of whom are socioeconomically disadvantaged.

In the past, many of her students applied only to community colleges. But now she has seen a sharp rise in those willing to aim for state universities.

“It has changed our students’ mindsets,” Grijalva said. “Now it’s like, ‘Hey, I can do this.’ They’re more confident in their leadership and their grades.”

But the flurry of recent announcements from schools altering their testing rules has proved alarming, she said. Her students “do feel the pressure coming back,” she said. “They’re starting to talk.”

Claire Elkin, 16, overheard some of this nervous chatter when she was touring colleges this spring with her family — making visits to places including the University of Virginia, Vanderbilt University and the University of North Carolina at Chapel Hill. At every school, recalled Claire, who took the ACT and intends to submit her scores, admissions tour leaders said something like: “Yeah, we’re test-optional now, but we can’t guarantee anything for you.”

Every time, the crowds of hopefuls around Claire broke into murmurs that ranged from anguished to angry, she said. She remembered one family whose daughters seemed especially upset, spurring the mother to jump into emergency action trying to calm the girls as the admissions presentation continued.

“A lot of kids my age can’t set a path right now for what they should be prioritizing when they’re applying for schools,” Claire said. “So there is definitely more panic.”

An earlier version of this article included incorrect information about the rise in the number of students who have taken the SAT since a drop sparked by the pandemic. The number of students taking the test reached 1.9 million for the class of 2023, about 300,000 fewer than for the class of 2019. The article has been corrected.

• Share full article

Advertisement

Supported by

Guest Essay

How the SAT Changed My Life

By Emi Nietfeld

Ms. Nietfeld is the author of the memoir “Acceptance.”

This month, the University of Texas, Austin, joined the wave of selective schools reversing Covid-era test-optional admissions policies, once again requiring applicants to submit ACT or SAT scores.

Many colleges have embraced the test-optional rule under the assumption that it bolsters equity and diversity, since higher scores are correlated with privilege. But it turns out that these policies harmed the teenagers they were supposed to help. Many low-income and minority applicants withheld scores that could have gotten them in, wrongly assuming that their scores were too low, according to an analysis by Dartmouth. More top universities are sure to join the reversal. This is a good thing.

I was one of the disadvantaged youths who are often failed by test-optional policies, striving to get into college while in foster care and homeless. We hear a lot about the efforts of these elite schools to attract diverse student bodies and about debates around the best way to assemble a class. What these conversations overlook is the hope these tests offer students who are in difficult situations.

For many of us, standardized tests provided our one shot to prove our potential, despite the obstacles in our lives or the untidy pasts we had. We found solace in the objectivity of a hard number and a process that — unlike many things in our lives — we could control. I will always feel tenderness toward the Scantron sheets that unlocked higher education and a better life.

Growing up, I fantasized about escaping the chaos of my family for the peace of a grassy quad. Both my parents had mental health issues. My adolescence was its own mess. Over two years I took a dozen psychiatric drugs while attending four different high school programs. At 14, I was sent to a locked facility where my education consisted of work sheets and reading aloud in an on-site classroom. In a life skills class, we learned how to get our G.E.D.s. My college dreams began to seem like delusions.

Then one afternoon a staff member handed me a library copy of “Barron’s Guide to the ACT .” I leafed through the onionskin pages and felt a thunderclap of possibility. I couldn’t go to the bathroom without permission, let alone take Advanced Placement Latin or play water polo or do something else that would impress elite colleges. But I could teach myself the years of math I’d missed while switching schools and improve my life in this one specific way.

After nine months in the institution, I entered foster care. I started my sophomore year at yet another high school, only to have my foster parents shuffle my course load at midyear, when they decided Advanced Placement classes were bad for me. In part because of academic instability like this, only 3 percent to 4 percent of former foster youth get a four-year college degree.

Later I bounced between friends’ sofas and the back seat of my rusty Corolla, using my new-to-me SAT prep book as a pillow. I had no idea when I’d next shower, but I could crack open practice problems and dip into a meditative trance. For those moments, everything was still, the terror of my daily life softened by the fantasy that my efforts might land me in a dorm room of my own, with endless hot water and an extra-long twin bed.

Standardized tests allowed me to look forward, even as every other part of college applications focused on the past. The song and dance of personal statements required me to demonstrate all the obstacles I’d overcome while I was still in the middle of them. When shilling my trauma left me gutted and raw, researching answer elimination strategies was a balm. I could focus on equations and readings, like the scholar I wanted to be, rather than the desperate teenager that I was.

Test-optional policies would have confounded me, but in the 2009-10 admissions cycle, I had to submit my scores; my fellow hopefuls and I were all in this together, slogging through multiple-choice questions until our backs ached and our eyes crossed.

The hope these exams instilled in me wasn’t abstract: It manifested in hundreds of glossy brochures. After I took the PSAT in my junior year, universities that had received my score flooded me with letters urging me to apply. For once, I felt wanted. These marketing materials informed me that the top universities offered generous financial aid that would allow me to attend free. I set my sights higher, despite my guidance counselor’s lack of faith.

When I took the actual SAT, I was ashamed of my score. Had submitting it been optional, I most likely wouldn’t have done it, because I suspected my score was lower than the prep-school applicants I was up against (exactly what Dartmouth found in the analysis that led it to reinstate testing requirements). When you grow up the way I did, it’s difficult to believe that you are ever good enough.

When I got into Harvard, it felt like a miracle splitting my life into a before and after. My exam preparation paid off on campus — it was the only reason I knew geometry or grammar — and it motivated me to tackle new, difficult topics. I majored in computer science, having never written a line of code. Though a career as a software engineer seemed far-fetched, I used my SAT study strategies to prepare for technical interviews (in which you’re given one or more problems to solve) that landed me the stable, lucrative Google job that catapulted me out of financial insecurity.

I’m not the only one who feels affection for these tests. At Harvard, I met other students who saw these exams as the one door they could unlock that opened into a new future. I was lucky that the tests offered me hope all along, that I could cling to the promise that one day I could bubble in a test form and find myself transported into a better life — the one I lead today.

Emi Nietfeld is the author of the memoir “ Acceptance .” Previously, she was a software engineer at Google and Facebook.

The Times is committed to publishing a diversity of letters to the editor. We’d like to hear what you think about this or any of our articles. Here are some tips . And here’s our email: [email protected] .

Follow The New York Times Opinion section on Facebook , Instagram , TikTok , WhatsApp , X and Threads .

Choose Your Test

Sat / act prep online guides and tips, complete list of free sat math practice problems.

Need to study for the SAT Math section but don't know where to start? On a budget, too? Not to worry! In this article, we've put together a comprehensive list of all the best (and free!) SAT Math study materials and guides currently available online.

Note: For info on the best SAT study materials you have to pay for, check out our picks for top SAT prep books.

Feature Image: Alan O'Rourke /Flickr

The Importance of Using High-Quality SAT Math Materials

It is incredibly important to get your study materials from the right places. The SAT is a very specific kind of test, and there are many (many!) websites and programs out there that, sadly, offer only poor-quality SAT resources.

If you study using mainly inferior SAT materials, you won't get a clear sense as to how you'll do on test day. Test prep is about assessing and improving upon your current level, but you won't be able to do this if you get a false sense of your skills based on poor study materials. Ultimately, products and test questions that don't accurately reflect the SAT are a waste of time.

Therefore, it's best to prioritize official SAT materials. The SAT is designed by the College Board, and their free materials are by far the best place to start studying. From here, you can branch off into programs and materials that use (or closely resemble) official SAT material.

Now, let's look at the best resources you can use for free official SAT Math practice questions.

Top 5 Free Resources for Quality SAT Math Practice

In this section, we go over the top five free resources you can use for quality SAT Math prep.

#1: The College Board

As the creator and issuer of the SAT, the College Board is the best place to begin for collecting your free SAT Math study materials. The official website offers full-length SAT practice tests and practice questions for all sections.

The closest you can get to taking the actual SAT is the free, full-length  practice exams in Bluebook , College Board’s online testing platform.  The six available practice tests provide you with a sneak peek of the interface, format, and scoring of the online test, so they’re a useful tool as you prepare for the digital format.

In addition to full-length exams, Bluebook also offers previews of individual questions from the actual SAT. Even these are useful as you’re trying out the digital interface and learning what kinds of math problems to expect.

Note that if you have testing accommodations that allow you to take the test on paper, you'll want to practice on the  the official, printable practice tests  created for the digital SAT. 

Old Official Full-Length SAT Practice Tests

While the newest practice tests will be most useful for your prep, you can also use these old (pre-digital) SAT practice tests, though keep in mind there will be differences 

There are 10 official practice tests from the 2016-2023 version of the SAT. Each test comes with an answer key and in-depth answer explanations to help you understand why you got questions wrong.

Although the general format of these tests is quite different, the individual math questions will be fairly similar to what you'll see on the digital SAT. Just keep in mind that they should be used for practicing skills and concepts rather than estimating your score.

#2: Khan Academy

Khan Academy is a nonprofit and partner of the College Board that offers a free online SAT prep program and practice questions.

While most of the questions come from official SAT practice materials (such as those linked above), others have been created or adapted with approval from or in tandem with the College Board itself.

Sign up for a free Khan Academy account to get access to SAT study questions and materials, complete with guides and explanation videos to aid your studying.

#3: PrepScholar SAT Blog

Here at PrepScholar, we offer tons of free resources, including detailed guides on every SAT math topic and a complete collection of our best SAT Math articles . All our guides include definitions and explanations, examples of how you'll see the topic on the test, and real SAT practice questions with detailed answer explanations.

Browse our SAT Math guides below to get started! All guides are arranged by topic.

Integers (Basic)

Integers (Advanced)

Fractions, Ratios, and Proportions

Statistics (Mean, Median, Mode)

Probability

Single Variable Equations

Systems of Equations

• Completing the Square

Coordinate Geometry

Lines and Slopes

Reflections, Translations, and Rotations

Plane and Solid Geometry

Lines and Angles

Solid Geometry

• Trigonometry

The 28 Critical SAT Math Formulas You MUST Know

In addition, we have strategy guides that will help you solve numerous SAT Math problems across the board:

Plugging in Answers

Plugging in Numbers

If you find our guides and quizzes helpful, you can also sign up for a free five-day trial to our test prep program. Our program assesses your current strengths and weaknesses and adapts based on your progress. It gives you practice questions tailored to your areas of needed improvement and provides answer explanations for all questions.

All our questions are based off real SAT test questions and, though the full program is not free, we guarantee your money back if you do not improve by 160 points.

#4: CrackSAT.net

CrackSAT is a free, unofficial SAT website offering PDFs of official SAT practice tests and an array of both official and unofficial SAT Math questions.

Many of the questions on CrackSAT come from SAT prep books by well-known test-prep companies including The Princeton Review and Kaplan. Though generally solid, unofficial SAT questions can vary wildly in quality, so don't rely on only these to give you a sense of your math abilities. Rather, it's best to use this website after you've exhausted all the official SAT resources above.

What's especially nice about this website is its organization: math questions are available in both multiple-choice and grid-in categories to give you the full range of math practice. You can also browse problems by concept (e.g., linear functions , exponents and radicals , etc.).

You've gathered all your study material ... so now what?

Ready to go beyond just reading about the SAT? Then you'll love the free five-day trial for our SAT Complete Prep program . Designed and written by PrepScholar SAT experts , our SAT program customizes to your skill level in over 40 subskills so that you can focus your studying on what will get you the biggest score gains.

Click on the button below to try it out!

How to Use SAT Math Study Materials Effectively: 6 Tips

It's just as important to know how to best utilize your SAT study materials as it is to know how to access them. These six tips will help you achieve your highest SAT Math score on test day.

#1: Take a Full Practice Test in One Sitting

You're probably most concerned about your SAT Math score, but it's still important to take a full SAT practice test so you can see how you'll fare on test day. Although answering one or two Math questions might not be too much of a challenge for you, in truth the SAT is a marathon—and if you aren't prepared, you'll likely be exhausted by the end of it. And exhaustion can lead anyone to make mistakes!

So before you dedicate your focus to the SAT Math section, take an official SAT practice test on the Bluebook app to see how your Math score fits into the larger test-taking picture. Make sure to take the whole test in one sitting; this will help you build endurance for test day.

#2: Use Proper Timing and Pace Yourself

As you take SAT practice tests and any Math-specific sections, be sure to use the same time limits you'll have on test day.

The chart below shows how much time you'll have on each SAT section as well as how much time you should (approximately) spend per question. Note that each section is split into two modules of equal lengths.

You'll have about 9 5 seconds per SAT Math question. Some questions might take you less or more time, but keep practicing so that you're not consistently spending too long on a single math question.

Remember that accuracy is only half the battle—you have to actually finish the questions within the time limits if you want to get a high Math score!

That said, don't worry if you run out of time while taking your first SAT practice test. This is completely normal and gives you a starting point from which you can later improve.

#3: Review Your Mistakes

It's one thing to take an SAT Math test, but you also need to look over your results and identify any patterns in your correct and incorrect answers. Do you tend to get the last few questions wrong on each Math section? Do function questions just seem to throw you for a loop?

First, figure out what went wrong. Then, see how you can pick up some easy points and diligently target those areas in your prep. (Remember that each Math question, no matter its difficulty, is worth the same number of points!)

#4: Target Your Biggest Weaknesses

Once you've pinpointed patterns in the errors you make, it's time to dig deeper and examine your biggest weaknesses. Your ultimate goal will be to tailor your studying so that you can focus on improving the areas you struggle with the most.

Since your overall objective is to answer as many questions as accurately as possible, you'll want to dedicate the majority of your study time to improving your weaknesses, rather than to reviewing material you already know well.

Refreshing your known material again and again might make you feel productive and confident, but it doesn't do a whole lot to improve your score. Your best bet is to focus on your problem areas by practicing and expanding your knowledge base.

#5: Set a Study Schedule

Don't be tempted to cram right before the test . You might have a busy schedule and feel as if your only option is to cram for the SAT, but improvement really happens over time. We generally advise giving yourself at least three to six months to study for the SAT.

If you don't have a lot of time left before test day, however, don't give up! No matter how much time you've got for studying, it is possible to come up with a balanced SAT study plan .

First, calculate the amount of time you have before your test date . (For example, say you've got a month before your test .)

Next, assign yourself at least three full practice tests (in addition to your normal study materials) and spread them out over the course of your study schedule. If you only have a week or so before test day, this probably won't be possible. In this case, try to fit in at least one or two practice tests wherever you can.

Nothing else replicates the SAT experience like a full practice test does, so don't underestimate its importance in your prep!

#6: Not Seeing Any Improvement? Get Additional Prep Help

Some students prefer to do all of their SAT studying by themselves, but others work better with a little guidance.

If the SAT subject matter just isn't making sense to you or you need help prioritizing your time, an SAT tutor or prep program (or both!) are excellent options to try. Both can give you that final push you need to stay focused and maximize your SAT Math score.

At PrepScholar, we offer expert SAT tutoring services in addition to a fully customizable SAT prep program . For more info, read our guides on how to find the best SAT tutor for you and how much you should spend on tutoring .

Targeted and careful practice, rest, and confidence: the perfect recipe for SAT success.

What's Next?

Don't know where to start? Take a look at our list of all math topics on the SAT Math section , and target your weaknesses with our individual SAT Math guides .

Want to know what the hardest SAT Math questions are? We've compiled a list of the 13 most challenging SAT Math questions , along with answer explanations for each.

Bitten by the procrastination bug? Time running out until test day? Check out how to beat the urge to procrastinate , and learn how to balance time for your studies.

Looking to get a perfect SAT Math score? Dig into our guide to getting a perfect 800 on SAT Math , written by a perfect scorer!

Want to improve your SAT score by 160 points?

Check out our best-in-class online SAT prep program . We guarantee your money back if you don't improve your SAT score by 160 points or more.

Our program is entirely online, and it customizes what you study to your strengths and weaknesses . If you liked this Math strategy guide, you'll love our program. Along with more detailed lessons, you'll get thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step program to follow so you'll never be confused about what to study next.

Check out our 5-day free trial:

Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. She is passionate about bringing education and the tools to succeed to students from all backgrounds and walks of life, as she believes open education is one of the great societal equalizers. She has years of tutoring experience and writes creative works in her free time.

Student and Parent Forum

Our new student and parent forum, at ExpertHub.PrepScholar.com , allow you to interact with your peers and the PrepScholar staff. See how other students and parents are navigating high school, college, and the college admissions process. Ask questions; get answers.

Ask a Question Below

Have any questions about this article or other topics? Ask below and we'll reply!

Improve With Our Famous Guides

• For All Students

The 5 Strategies You Must Be Using to Improve 160+ SAT Points

How to Get a Perfect 1600, by a Perfect Scorer

Series: How to Get 800 on Each SAT Section:

Score 800 on SAT Math

Score 800 on SAT Reading

Score 800 on SAT Writing

Series: How to Get to 600 on Each SAT Section:

Score 600 on SAT Math

Score 600 on SAT Reading

Score 600 on SAT Writing

Free Complete Official SAT Practice Tests

What SAT Target Score Should You Be Aiming For?

15 Strategies to Improve Your SAT Essay

The 5 Strategies You Must Be Using to Improve 4+ ACT Points

How to Get a Perfect 36 ACT, by a Perfect Scorer

Series: How to Get 36 on Each ACT Section:

36 on ACT English

36 on ACT Math

36 on ACT Reading

36 on ACT Science

Series: How to Get to 24 on Each ACT Section:

24 on ACT English

24 on ACT Math

24 on ACT Reading

24 on ACT Science

What ACT target score should you be aiming for?

ACT Vocabulary You Must Know

ACT Writing: 15 Tips to Raise Your Essay Score

How to Get Into Harvard and the Ivy League

How to Get a Perfect 4.0 GPA

How to Write an Amazing College Essay

What Exactly Are Colleges Looking For?

Is the ACT easier than the SAT? A Comprehensive Guide

Should you retake your SAT or ACT?

When should you take the SAT or ACT?

Stay Informed

Get the latest articles and test prep tips!

Looking for Graduate School Test Prep?

Check out our top-rated graduate blogs here:

GRE Online Prep Blog

GMAT Online Prep Blog

TOEFL Online Prep Blog

Holly R. "I am absolutely overjoyed and cannot thank you enough for helping me!”