15 Hardest SAT Math Questions in 2023-2024

July 1, 2023

college math problems hard

For some students, “math” is a scary word, particularly in the context of the SAT. While test takers can often utilize context clues to make an educated guess on reading-oriented questions, math problems can sometimes feel like they are written in a foreign language. In pursuit of a good SAT score , many students engage in SAT prep to build their knowledge, skills, and confidence. As part of that prep, some students may wish to challenge themselves by tackling the hardest SAT math questions. If that sounds familiar, this post is for you!

Below, we discuss some of the hardest SAT math questions, identifying what qualities make them difficult and strategies that will help you solve them. Whether you’re a math aficionado or a novice hoping to build your skills, this post will tell you what you need to know about hard SAT math questions to help you do your best.

SAT Math Basics

Before discussing the hardest SAT math questions, let’s go over the composition of the SAT math section. The Math section consists of 58 questions that students have 80 minutes to complete. These questions fall into two sections: calculator active and calculator inactive. The majority of questions are multiple-choice, though a small portion are “grid-in” questions, in which students write their answers. Below is a more detailed breakdown of the composition of the SAT Math section from the College Board .

SAT Math: Calculator Active

SAT Math: Calculator Inactive

What’s covered on the SAT Math test?

There are four categories of questions on the SAT Math test:

  • Heart of Algebra, 19 questions
  • Problem-Solving and Data Analysis, 17 questions
  • Passport to Advanced Math, 16 questions
  • Additional Topics in Math, 6 questions

Heart of Algebra questions focus on students’ knowledge of linear equations and systems. Questions may ask students to develop equations that represent a given situation or establish connections between different linear equations.

In comparison, Problem Solving and Data Analysis questions measure students’ quantitative literacy through concepts they’re likely to need in college courses and everyday life, including ratios, percentages, and proportional relationships. Students may address problems in real-world settings or describe relationships in graphs or statistics.

As its title suggests, Passport to Advanced Math tests students on the knowledge they’ll need to specialize in mathematically-oriented topics, such as STEM subjects or economics. These questions will also evaluate students on the skills they’ll need to excel in calculus and advanced statistics courses. As one might expect, this is a category that may produce some of the hardest SAT math questions.

Finally, Additional Topics in Math sounds like a catch-all, but students can reasonably expect to encounter questions focused on geometry, trigonometry, and complex numbers. This category may also include some hard SAT math questions, given students’ varying levels of familiarity with these subjects.

Preparing for the SAT Math Test

As you can see, the SAT Math test covers a wide variety of topics. While it might be tempting to jump straight to the hardest SAT math questions, it’s important to first establish a clear baseline by taking a practice test. Doing so will allow you to familiarize yourself with the structure of the SAT. Moreover, this practice test will provide you with an opportunity to reflect on your strengths and weaknesses so you can identify what topics warrant more practice. Once you know what your priorities are, you can start your SAT prep through the materials provided by College Board or an SAT prep manual.

15 Hardest SAT Math Questions

Now that we have that groundwork in place, we can discuss our selections for hard SAT math questions. We have opted to categorize questions around four common challenges students may experience, providing several examples of each. As you read our selections, bear in mind that difficulty is relative. We have selected questions that we believe are challenging due to their composition. However, this may not be the case for all students. Therefore, we recommend students identify their personal SAT prep goals to ensure they are being strategic in their studies. All questions are sourced from College Board’s practice tests.

Hard SAT Math Questions: Specialized or less familiar forms of math

Of all of the hard SAT math questions, perhaps none are more difficult than those that deal with more specialized mathematical subjects, such as trigonometry. Test takers have typically had less exposure to these subjects, which can make solving these problems more difficult. Therefore, it is important that students review a variety of mathematical concepts to ensure they are equipped to answer all types of questions. Here are a few examples:

1) Calculator Inactive, Grid-In

In a right triangle, one angle measures x°, where sin x° = ⅘ . What is cos(90° − x°)?

As this problem illustrates, students need a basic understanding of trigonometry functions to tackle this type of question. A complete solution for this problem is available on page 31 of the answer guide for SAT Practice Test 1.

2) Calculator Active, Grid-In

A group of friends decided to divide the $800 cost of a trip equally among themselves. When two of the friends decided not to go on the trip, those remaining still divided the $800 cost equally, but each friend’s share of the cost increased by $20. How many friends were in the group originally?

Solving this problem necessitates that students have the ability to utilize quadratic equations, which is a more advanced form of math relative to many of the concepts tested on the SAT Math test. A complete explanation is available on page 47 of the answer guide for SAT Practice Test 6.

3) Calculator Active, Multiple Choice

The world’s population has grown at an average rate of 1.9 percent per year since 1945. There were approximately 4 billion people in the world in 1975. Which of the following functions represents the world’s population P, in billions of people, t years since 1975?

college math problems hard

This problem engages students’ knowledge of exponential growth. However, rather than simply solving an equation, students must understand the logic of exponential functions well enough to translate the information provided into the correct equation. The complete solution for this problem is available on page 47 of the answer guide for SAT Practice Test 7.

4) Calculator Inactive, Grid-In

Triangle PQR has right angle Q . If sin R = ⅘, what is the value of tan P ?

This problem requires that students utilize trigonometry functions, as well as the Pythagorean theorem, to arrive at the correct answer. A complete solution for this problem is available on page 44 of the answer guide for SAT Practice Test 9.

Hard SAT Math Questions: Problems with multistep solutions

Many problems on the SAT Math test require students to complete multiple steps to arrive at an answer. While the math involved may not be difficult in itself, a multistep process creates opportunities for students to make mistakes. For this reason, students should practice solving problems with multistep solutions to avoid careless errors. Let’s look at a few examples:

5) Calculator Inactive, Multiple Choice

If (ax+2)(bx+7)=15x^2+ cx+14 for all values of x, and a+b=8, what are the two possible values for c?

B) 6 and 35

C) 10 and 21

D) 31 and 41

To answer this question, students must understand the logic of how these variables and equations relate to one another. Relevant skills students would need to solve this problem include mastery of algebra and the ability to use factoring. A complete explanation for this problem is available on page 30 of the answer guide for SAT Practice Test 1.

6) Calculator Active, Multiple Choice

A rectangle was altered by increasing its length by 10 percent and decreasing its width by p percent. If these alterations decreased the area of the rectangle by 12 percent, what is the value of p?

Concepts involved in this problem, including calculating area and percentages, are likely familiar to most students. However, students may stumble when completing the steps necessary to find the answer, which involves writing equations to represent the values of the original area of the rectangle, the altered values for the length and width, and the decreased area of the rectangle. This lengthy process leaves room for mistakes, making this problem deceptively challenging. A complete solution is available on page 35 of the answer guide for SAT Practice Test 3.

Hardest SAT Math Problems (Continued)

7) Calculator Active, Grid-In

If Ms. Simon starts her drive at 6:30 a.m., she can drive at her average driving speed with no traffic delay for each segment of the drive. If she starts her drive at 7:00 a.m., the travel time from the freeway entrance to the freeway exit increases by 33% due to slower traffic, but the travel time for each of the other two segments of her drive does not change. Based on the table, how many more minutes does Ms. Simon take to arrive at her workplace if she starts her drive at 7:00 a.m. than if she starts her drive at 6:30 a.m.? (Round your answer to the nearest minute.)

Again, if we judged this problem strictly on the math involved, it probably wouldn’t be considered one of the hardest SAT math questions. However, the multiple steps and calculations it requires make it easy for students to make mistakes. The complete solution is available on page 49 of the answer guide for SAT Practice Test 5. A similar example is available below.

8) Calculator Active, Grid-In

Number of Contestants by Score and Day

The same 20 contestants, on each of 3 days, answered 5 questions in order to win a prize. Each contestant received 1 point for each correct answer. The number of contestants receiving a given score on each day is shown in the table above.

What was the mean score of the contestants on Day 1?

The complete solution for this problem is available on page 47 of the answer guide for SAT Practice Test 7.

Hard SAT Math Questions: Problems that are difficult to comprehend

Although math involves numbers, having a firm grasp of reading comprehension and logic is often necessary to understand a problem. Looking at a block of text can sometimes be overwhelming, which is why it’s important to practice reading word problems so you can learn how to understand the variables involved and tackle these hard SAT math questions. Here are a few examples:

9) Calculator Active, Multiple Choice

A square field measures 10 meters by 10 meters. Ten students each mark off a randomly selected region of the field; each region is square and has side lengths of 1 meter, and no two regions overlap. The students count the earthworms contained in the soil to a depth of 5 centimeters beneath the ground’s surface in each region. The results are shown in the table below.

Which of the following is a reasonable approximation of the number of earthworms to a depth of 5 centimeters beneath the ground’s surface in the entire field?

Between the described 10×10 grid and the data chart, there is a lot to sift through in this question. While the math involved isn’t especially difficult (students primarily need to be comfortable with ratios to solve this problem), the sheer number of variables in the question could make it challenging to understand and, therefore, to solve. A complete explanation for this problem is available on page 40 of the answer guide for SAT Practice Test 1.

10) Calculator Active, Multiple Choice

Of the following four types of savings account plans, which option would yield exponential growth of the money in the account?

  • A) Each successive year, 2% of the initial savings is added to the value of the account.
  • B) Each successive year, 1.5% of the initial savings and $100 is added to the value of the account.
  • C) Each successive year, 1% of the current value is added to the value of the account.
  • D) Each successive year, $100 is added to the value of the account.

This problem has less to do with precise calculations and more to do with a student’s ability to translate the answers into mathematical concepts, specifically linear versus exponential growth. Therefore, the challenge is for students to consider the logic of each option to determine which would support exponential growth. A complete solution for this problem is available on page 34 of the answer guide for SAT Practice Test 3.

11) Calculator Active, Grid-In

The problem outlined below refers to the following information:

If shoppers enter a store at an average rate of r shoppers per minute and each stays in the store for an average time of T minutes, the average number of shoppers in the store, N, at any one time is given by the formula N = rT. This relationship is known as Little’s law.

The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little’s law to estimate that there are 45 shoppers in the store at any time.

Little’s law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spends an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store?

Because this problem has a contextual paragraph, there is a fair amount of text students have to work through. This quantity of information can easily obscure the relationships between the values discussed. However, by working through the question carefully, students can understand the logic of the problem. A complete solution is available on page 38 of the answer guide of the SAT Practice Test 3.

12) Calculator Active, Multiple Choice

The 22 students in a health class conducted an experiment in which they each recorded their pulse rates, in beats per minute, before and after completing a light exercise routine. The dot plots below display the results.

college math problems hard

Let s 1 and r 1 be the standard deviation and range, respectively, of the data before exercise, and let s 2 and r 2 be the standard deviation and range, respectively, of the data after exercise. Which of the following is true?

  • s 1 = s 2 and r 1 = r 2
  • s 1 < s 2 and r 1 < r 2
  • s 1 > s 2 and r 1 > r 2
  • s 1 ≠ s 2 and r 1 = r 2

This problem requires that students utilize their interpretative abilities to break down the provided charts and context to determine how the standard deviations compare. A complete solution for this problem is available on page 46 of the answer guide for SAT Practice Test 8.

Hard SAT Math Questions: Problems that test multiple concepts

Some questions on the SAT will require that students leverage multiple mathematical skills and concepts to arrive at an answer. For these questions, the threshold for achieving the correct answer is higher simply because they require mastery of multiple concepts. Let’s look at a few examples:

13) Calculator Inactive, Grid-In

At a lunch stand, each hamburger has 50 more calories than each order of fries. If 2 hamburgers and 3 orders of fries have a total of 1700 calories, how many calories does a hamburger have?

This problem looks simple enough and, in fact, the math involved really isn’t that hard. However, what makes this problem challenging is that it requires students to understand systems of equations well enough to write equations that represent the described situation. Students then have to utilize the system of equations they create to solve the problem using algebra. A complete explanation for this problem is available on page 26 of the answer guide for SAT Practice Test 3.

14) Calculator Inactive, Grid-In

In triangle ABC, the measure of ∠B is 90°, BC = 16, and AC = 20. Triangle DEF is similar to triangle ABC, where vertices D, E, and F correspond to vertices A, B, and C, respectively, and each side of triangle DEF is 1 3 the length of the corresponding side of triangle ABC. What is the value of sin F ?

This question requires that students be comfortable with basic trigonometry and the geometric concept of similarity. This, in turn, necessitates an understanding of ratios. Being able to layer these skills will ensure students arrive at the appropriate solution. A complete explanation of this problem is available on page 27 of the answer guide for SAT Practice Test 3. Below is another example of a question that layers these concepts.

15) Calculator Active, Grid-In

college math problems hard

In the figure above _ _ _ _   ¾. If _ _  +15  and _ _ = 4, what is the length of _ _?

A complete solution for this problem is available on page 47 of the answer guide for SAT Practice Test 7.

Final Thoughts: The Hardest SAT Math Problems

After working through these problems, take a moment to reflect. If you struggled or are feeling overwhelmed, that might be a sign you need to do a little more studying. Consider consulting College Board’s SAT Study Guide or our post on the most important SAT math formulas for assistance. If you breezed through these problems, congratulations! Math is clearly a strength of yours. Consider turning your attention to other areas, such as SAT vocabulary words. Happy studying and best of luck!

Got other SAT-related questions? Check out our other SAT resources:

  • Entering Class Statistics
  • Should I Apply Test Optional?
  • When Do SAT Scores Come Out?

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Emily Smith

Emily earned a BA in English and Communication Studies from UNC Chapel Hill and an MA in English from Wake Forest University. While at UNC and Wake Forest, she served as a tutor and graduate assistant in each school’s writing center, where she worked with undergraduate and graduate students from all academic backgrounds. She also worked as an editorial intern for the Wake Forest University Press as well as a visiting lecturer in the Department of English at WFU, and currently works as a writing center director in western North Carolina.

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You’ve studied and now you’re geared up for the ACT math section (whoo!). But are you ready to take on the most challenging math questions the ACT has to offer?

If you're up for the challenge, preparation starts by understanding the toughest question types you'll face on the ACT . So if you’ve got your heart set on that perfect score (or you’re just really curious to see what the most difficult questions will be), then this is the guide for you.

We’ve put together what we believe to be the most most difficult math question types the ACT has given to students in the past 10 years, complete with strategies and answer explanations for each . These are all real ACT math questions, so understanding and studying them is one of the best ways to improve your current ACT score and knock it out of the park on test day.

Let's dive in!

Brief Overview of the ACT Math Section

Like all topic sections on the ACT, the ACT math section is one complete section that you will take all at once. It will always be the second section on the test and you will have 60 minutes to completed 60 questions .

The ACT arranges its questions in order of ascending difficulty. As a general rule of thumb, questions 1-20 will be considered “easy,” questions 21-40 will be considered “medium-difficulty,” and questions 41-60 will be considered “difficult.”

The way the ACT classifies “easy” and “difficult” is by how long it takes the average student to solve a problem as well as the percentage of students who answer the question correctly. The faster and more accurately the average student solves a problem, the “easier” it is. The longer it takes to solve a problem and the fewer people who answer it correctly, the more “difficult” the problem.

(Note: we put the words “easy” and “difficult” in quotes for a reason—everyone has different areas of math strength and weakness, so not everyone will consider an “easy” question easy or a “difficult” question difficult. These categories are averaged across many students for a reason and not every student will fit into this exact mold.)

All that being said, with very few exceptions, the most difficult ACT math problems will be clustered in the far end of the test. Besides just their placement on the test, these questions share a few other commonalities. We'll take a look at example questions and how to solve them and at what these types of questions have in common, in just a moment.

But First: Should You Be Focusing on the Hardest Math Questions Right Now?

If you’re just getting started in your study prep, definitely stop and make some time to take a full practice test to gauge your current score level and percentile. The absolute best way to assess your current level is to simply take the ACT as if it were real , keeping strict timing and working straight through (we know—not the most thrilling way to spend four hours, but it will help tremendously in the long run). So print off one of the free ACT practice tests available online and then sit down to take it all at once.

Once you’ve got a good idea of your current level and percentile ranking, you can set milestones and goals for your ultimate ACT score. If you’re currently scoring in the 0-16 or 17-24 range , your best best is to first check out our guides on using the key math strategies of plugging in numbers and plugging in answers to help get your score up to where you want it to. Only once you've practiced and successfully improved your scores on questions 1-40 should you start in trying to tackle the most difficult math problems on the test.

If, however, you are already scoring a 25 or above and want to test your mettle for the real ACT, then definitely proceed to the rest of this guide . If you’re aiming for perfect (or close to) , then you’ll need to know what the most difficult ACT math questions look like and how to solve them. And luckily, that’s exactly what we’re here for.

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Hardest Types of ACT Math Questions

Now that you're positive that you should be trying out these difficult math questions, let’s get right to it! We've broken hard ACT questions into sections based on type. Additionally, the answers to these questions are in a separate section below, so you can go through them all at once without getting spoiled.

Type #1: Graph Interpretation

Knowing how to read and interpret graphs can be tricky, especially since that information can also require you to use other math skills, like finding points on a line or calculating angles.

To unpack a graph question, first ask yourself—what is the prompt actually asking me to do? Figuring out the core mathematical concept behind the graphed information will make it easier to answer the question.

Check out the two graph interpretation questions below. Notice that both are asking you to interpret the graph data by using other mathematical concepts! With that in mind, try solving them on your own to get practice solving these tough questions. (Remember: we've included the answers below!)

Graph Interpretation Questions

QUESTION #1:

In the figure below, line $q$ in the standard $(x,y)$ coordinate plane has equation $-2x+y=1$ and intersects line $r$, which is distinct from line $q$, at a point on the $x$-axis. The angles $∠a$ and $∠b$ formed by these lines and the $x$-axis are congruent. What is the slope of line $r$?

body-graph-1-1

G. $-{1}/2$

K. Cannot be determined

First, turn our given equation for line q into proper slope-intercept form .

Now, we are told that the angles the lines form are congruent. This means that the slopes of the lines will be opposites of one another [Note: perpendicular lines have opposite reciprocal slopes, so do NOT get these concepts confused!].

Since we have already established that the slope of line $q$ is 2, line $r$ must have a slope of -2.

Our final answer is F , -2

QUESTION #2:

The graph of the equation $h=-at^2 + bt + c$, which describes how the height, $h$, of a hit baseball changes over time, $t$, is shown below.

body-graph-2

If you alter only this equation's $c$ term, which gives the height at time $t=0$, the alteration has an effect on which of the following?

I. The $h$-intercept II. The maximum value of $h$ III. The $t$-intercept

H. III only

J. I and III only

K. I, II, and III

The equation we are given ($−at^2+bt+c$) is a parabola and we are told to describe what happens when we change $c$ (the $y$-intercept).

From what we know about functions and function translations , we know that changing the value of $c$ will shift the entire parabola upwards or downwards, which will change not only the $y$-intercept (in this case called the "$h$-intercept"), but also the maximum height of the parabola as well as its $x$-intercept (in this case called the $t$-intercept). You can see this in action when we raise the value of the $y$-intercept of our parabola.

body-question-2-graphs

Options I, II, and III are all correct.

Our final answer is K, I, II, and III

Type #2: Trigonometry Questions

Trigonometry questions are tough for a number of reasons. First, they require you to memorize formulas and values in order to properly interpret questions. There's also a lot of conversion involved: you have to know how (and when!) to convert radians to degrees and visa versa.

And don't forget functions! Trigonometry uses non-linear functions, which is tough because its graph isn't a line or part of a line. Because trigonometry combines so many advanced math concepts, they can trip you up.

Test your trig skills on the advanced questions below. Try to work the, without peeking at the answers!

Trigonometry Questions

The equations of the two graphs shown below are $y_1(t) = a_1\sin (b_1t)$ and $y_2(t)=a_2\cos(b_2t)$, where the constants $b_1$ and $b_2$ are both positive real numbers.

body-graph-3

Which of the following statements is true of the constants $a_1$ and $a_2$?

A. $0 < a_1 < a_2$

B. $0 < a_2 < a_1$

C. $a_1 < 0 < a_2$

D. $a_1 < a_2 < 0$

E. $a_2 < a_1 < 0$

The position of the a values (in front of the sine and cosine) means that they determine the amplitude (height) of the graphs. The larger the a value, the taller the amplitude.

Since each graph has a height larger than 0, we can eliminate answer choices C, D, and E.

Because $y_1$ is taller than $y_2$, it means that $y_1$ will have the larger amplitude. The $y_1$ graph has an amplitude of $a_1$ and the $y_2$ graph has an amplitude of $a_2$, which means that $a_1$ will be larger than $a_2$.

Our final answer is B , $0 < a_2 < a_1$.

For x such that $0 < x <π/2$, the expression ${√{1 – \cos^2 x}/{\sin x} + {√{1 – \sin^2 x}/{\cos x}$ is equivalent to:

J. $-\tan{x}$

K. $\sin{2x}$

If you remember your trigonometry shortcuts , you know that $1−{\cos^2}x+{\cos^2}x=1$. This means, then, that ${\sin^2}x=1−{\cos^2}x$ (and that ${\cos^2}x=1−{\sin^2}x$).

So we can replace our $1−{\cos^2}x$ in our first numerator with ${\sin^2}x$. We can also replace our $1−{\sin^2}x$ in our second numerator with ${\cos^2}x$. Now our expression will look like this:

${√{\sin^2 x}/{\sin x}+{√{\cos^2 x}/{\cos x }$

We also know that the square root of a value squared will cancel out to be the original value alone (for example,$√{2^2}=2$), so our expression will end up as:

$={\sin{x}}/{\sin{x}}+{\cos{x}}/{\cos{x}}$

Or, in other words:

Our final answer is H , 2.

Type #3: Logarithms

Logarithms can be hard because they require you to differentiate between exponential and logarithmic questions and understand logarithmic notation. If you can't read the question, you can't answer it, after all!

You'll also need to understand exponents to work with logarithms since the two concepts are intertwined. Working through the following equation can help you learn how to manage these tough problems.

Logarithm Questions

What is the real value of $x$ in the equation $\log_2{24} – \log_2{3} = \log_5{x}$?

ANSWER #1: If you’ve brushed up on your log basics, you know that $\log_b(m/n)=\log_b(m)−\log_b(n)$. This means that we can work this backwards and convert our first expression into:

$\log_2(24)-\log_2(3)=\log_2(24/3)$

$=\log_2(8)$

We also know that a log is essentially asking: "To what power does the base need to raised in order to achieve this certain value?" In this particular case, we are asking: "To which power must 2 be raised to equal 8?" The answer to this question is 3, since $(2^3=8)$, so $\log_2(8)=3$

Now this expression is equal to $\log_5(x)$, which means that we must also raise our 5 to the power of 3 in order to achieve $x$. So:

$3=\log_5(x)$

Our final answer is J , 125.

Type #4: Functions

Functions are one of those question types that are tough for some students but easier for others. That's because as functions get more complex, so does finding the correct answer. If you understand how functions work, you can break down more complex problems into simpler steps.;

Try your hand at the hard function question below to test your knowledge.

Function Questions

The functions $y = \sin x \and y = \sin(x + a) + b$, for constants $a$ and $b$, are graphed in the standard $(x,y)$ coordinate plane below. The functions have the same maximum value. One of the following statements about the values of $a$ and $b$ is true. Which statement is it?

body-graph-4

A. $a < 0$ and $b = 0$

B. $a < 0$ and $b > 0$

C. $a = 0$ and $b > 0$

D. $a > 0$ and $b< 0$

E. $a > 0$ and $b> 0$

The only difference between our function graphs is a horizontal shift , which means that our b value (which would determine the vertical shift of a sine graph) must be 0.

Just by using this information, we can eliminate every answer choice but A, as that is the only answer with $b=0$. For expediency's sake, we can stop here.

Our final answer is A , $a<0$ and $b=0$

Advanced ACT Math note : An important word in ACT Math questions is "must," as in "something must be true." If a question doesn't have this word, then the answer only has to be true for a particular instance (that is, it could be true.)

In this case, the majority of the time, for a graph to shift horizontally to the left requires $a>0$. However, because $\sin(x)$ is a periodic graph, $\sin(x+a)$ would shift horizontally to the left if $a=-π/2$, which means that for at least one value of the constant $a$ where $a<0$, answer A is true. In contrast, there are no circumstances under which the graphs could have the same maximum value (as stated in the question text) but have the constant $b≠0$.

As we state above, though, on the real ACT, once you reach the conclusion that $b=0$ and note that only one answer choice has that as part of it, you should stop there. Don't get distracted into wasting more time on this question by the bait of $a<0$!

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Whoo! You made it to the finish line—go you!

What Do the Hardest ACT Math Questions Have in Common?

Now, lastly, before we get to the questions themselves, it is important to understand what makes these hard questions “hard.” By doing so, you will be able to both understand and solve similar questions when you see them on test day, as well as have a better strategy for identifying and correcting your previous ACT math errors.

In this section, we will look at what these questions have in common and give examples for each type. In the next section, we will give you all 21 of the most difficult questions as well as answer explanations for each question, including the ones we use as examples here.

Some of the reasons why the hardest math questions are the hardest math questions are because the questions do the following:

#1: Test Several Mathematical Concepts at Once

EXAMPLE QUESTION:

Consider the functions $f(x) = √{x}$ and $g(x) = 7x + b$. In the standard $(x,y)$ coordinate plane, $y = f(g(x))$ passes through $(4,6)$. What is the value of $b$?

E. $4 – {7√6}$

As you can see, this question deals with a combination of functions and coordinate geometry points.

#2: Require Multiple Steps

Many of the most difficult ACT Math questions primarily test just one basic mathematical concept. What makes them difficult is that you have to work through multiple steps in order to solve the problem. (Remember: the more steps you need to take, the easier it is to mess up somewhere along the line!)

An integer from 100 through 999, inclusive, is to be chosen at random. What is the probability that the number chosen will have 0 as at least 1 digit?

A. $19/900$

B. $81/900$

C. $90/900$

D. $171/900$

E. $271/{1{,}000}$

Though it may sound like a simple probability question, you must run through a long list of numbers with 0 as a digit. This leaves room for calculation errors along the way.

#3: Use Concepts You're Less Familiar With

Another reason the questions we picked are so difficult for many students is that they focus on subjects you likely have limited familiarity with. For example, many students are less familiar with algebraic and/or trigonometric functions than they are with fractions and percentages, so most function questions are considered “high difficulty” problems.

The functions $y =$ sin$x$ and $y =$ sin$(x + a) + b$, for constants $a$ and $b$, are graphed in the standard $(x,y)$ coordinate plane below. The functions have the same maximum value. One of the following statements about the values of $a$ and $b$ is true. Which statement is it?

body-graph-5

D. $a > 0$ and $b < 0$

E. $a > 0$ and $b > 0$

Many students get intimidated with function problems because they lack familiarity with these types of questions.

#4: Give You Convoluted or Wordy Scenarios to Work Through

Some of the most difficult ACT questions are not so much mathematically difficult as they are simply tough to decode. Especially as you near the end of the math section, it can be easy to get tired and misread or misunderstand exactly what the question is even asking you to find.

In the complex plane, the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. The complex number $a + bi$ graphed in the complex plane is comparable to the point $(a,b)$ graphed in the standard $(x,y)$ coordinate plane.

The modulus of the complex number $a + bi$ is given by ${√{a^2+b^2}$ . Which of the complex numbers $z_1$ , $z_2$ , $z_3$ , $z_4$ , and $z_5$ below has the greatest modulus?

body-graph-6

This question presents students with a completely foreign mathematical concept and can eat up the limited available time.

#5: Appear Deceptively Easy

Remember—if a question is located at the very end of the math section, it means that a lot of students will likely make mistakes on it. Look out for these questions, which may give a false appearance of being easy in order to lure you into falling for bait answers. Be careful!

Which of the following number line graphs shows the solution set to the inequality $|x–5|<1$ ?

body-graph-7

This question may seem easy, but, because of how it is presented, many students will fall for one of the bait answers.

#6: Involve Multiple Variables or Hypotheticals

The more difficult ACT Math questions tend to use many different variables—both in the question and in the answer choices—or present hypotheticals. (Note: The best way to solve these types of questions—questions that use multiple integers in both the problem and in the answer choices—is to use the strategy of plugging in numbers .)

If $x$ and $y$ are real numbers such that $x > 1$ and $y < -1$, then which of the following inequalities must be true?

A. ${x/y} > 1$

B. ${|x|^2} > |y|$

C. ${x/3} – 5 > {y/3} – 5$

D. ${x^2} + 1 > {y^2} + 1$

E. ${x^{-2}} > {y^{-2}}$

Working with hypothetical scenarios and variables is almost always more challenging than working with numbers.

body_bakery

The Take-Aways

Taking the ACT is a long journey; the more you get acclimated to it ahead of time, the better you'll feel on test day. And knowing how to handle the hardest questions the test-makers have ever given will make taking your ACT seem a lot less daunting.

If you felt that these questions were easy , make sure not underestimate the effect of adrenaline and fatigue on your ability to solve your math problems. As you study, try to follow the timing guidelines (an average of one minute per ACT math question) and try to take full tests whenever possible. This is the best way to recreate the actual testing environment so that you can prepare for the real deal.

If you felt these questions were challenging , be sure to strengthen your math knowledge by checking out our individual math topic guides for the ACT . There, you'll see more detailed explanations of the topics in question as well as more detailed answer breakdowns.

What’s Next?

Felt that these questions were harder than you were expecting? Take a look at all the topics covered on the ACT math section and then note which sections you had particular difficulty in. Next, take a look at our individual math guides to help you strengthen any of those weak areas.

Running out of time on the ACT math section? Our guide to helping you beat the clock will help you finish those math questions on time.

Aiming for a perfect score? Check out our guide on how to get a perfect 36 on the ACT math section , written by a perfect-scorer.

Want to improve your ACT score by 4 points?

Check out our best-in-class online ACT prep classes . We guarantee your money back if you don't improve your ACT score by 4 points or more.

Our classes are entirely online, and they're taught by ACT experts . If you liked this article, you'll love our classes. Along with expert-led classes, you'll get personalized homework with thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step, custom program to follow so you'll never be confused about what to study next.

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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.

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College students are still struggling with basic math. Professors blame the pandemic

George Mason Term Instructor Ermias Kassaye, left, helps a student figure out an equation during a summer math boot camp on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. Researchers say math learning suffered during the pandemic for various reasons. An intensely hands-on subject, math was hard to translate to virtual classrooms. (AP Photo/Kevin Wolf)

George Mason Term Instructor Ermias Kassaye, left, helps a student figure out an equation during a summer math boot camp on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. Researchers say math learning suffered during the pandemic for various reasons. An intensely hands-on subject, math was hard to translate to virtual classrooms. (AP Photo/Kevin Wolf)

George Mason Term Instructor Ermias Kassaye, standing left, leads a summer math boot camp on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. Dozens opted to spend a week of summer break at the university brushing up on math lessons that didn’t stick during pandemic schooling. The northern Virginia school started Math Boot Camp because of alarming numbers of students arriving with gaps in their math skills. (AP Photo/Kevin Wolf)

A student uses his phone to copy the whiteboard at the end of a summer math boot camp session on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. (AP Photo/Kevin Wolf)

Students uses their bodies to plot their location on a graph based on the number they are holding during a summer math boot camp session on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. Researchers say math learning suffered during the pandemic for various reasons. An intensely hands-on subject, math was hard to translate to virtual classrooms. (AP Photo/Kevin Wolf)

Diego Fonseca, left, and his fellow students uses their bodies to plot their location on a graph based on the number they are holding during a summer math boot camp session on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. “I managed to use the knowledge of the boot camp, and I got into calculus,” Fonseca says. “I didn’t have any expectation I’d do that.” (AP Photo/Kevin Wolf)

Students take part in a summer math boot camp on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. (AP Photo/Kevin Wolf)

George Mason undergraduate math major David Wigginton writes an equation on a whiteboard as students Ethan Hill, right, and Diego Fonseca, center, take part in a summer math boot camp on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. (AP Photo/Kevin Wolf)

Rosa Sarmiento, second from left, and Alicia Davis, center, work together to solve the math equation written on a whiteboard during a summer math boot camp session on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. (AP Photo/Kevin Wolf)

George Mason Term grad student Aman D’Souza, right, calls on Alicia Davis, second from left, during summer math boot camp on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. (AP Photo/Kevin Wolf)

A student takes a break next to a whiteboard during a summer math boot camp on Thursday, Aug. 1, 2023 at George Mason University in Fairfax. Va. The pandemic disrupted all learning but caused an outsize impact in math. (AP Photo/Kevin Wolf)

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college math problems hard

FAIRFAX, Va. (AP) — Diego Fonseca looked at the computer and took a breath. It was his final attempt at the math placement test for his first year of college. His first three tries put him in pre-calculus, a blow for a student who aced honors physics and computer science in high school.

Functions and trigonometry came easily, but the basics gave him trouble . He struggled to understand algebra, a subject he studied only during a year of remote learning in high school.

“I didn’t have a hands-on, in-person class, and the information wasn’t really there,” said Fonseca, 19, of Ashburn, Virginia, a computer science major who hoped to get into calculus. “I really struggled when it came to higher-level algebra because I just didn’t know anything.”

Fonseca is among 100 students who opted to spend a week of summer break at George Mason University brushing up on math lessons that didn’t stick during pandemic schooling. The northern Virginia school started Math Boot Camp because of alarming numbers of students arriving with gaps in their math skills.

Giada Gambino, 10, left, becomes frustrated with a problem on a math worksheet from school as her mother helps her work through it at the dining room table in their home Wednesday, Aug. 23, 2023, in Spring, Texas. (AP Photo/Michael Wyke)

Colleges across the country are grappling with the same problem as academic setbacks from the pandemic follow students to campus. At many universities, engineering and biology majors are struggling to grasp fractions and exponents. More students are being placed into pre-college math, starting a semester or more behind for their majors, even if they get credit for the lower-level classes.

Colleges largely blame the disruptions of the pandemic, which had an outsize impact on math. Reading scores on the national test known as NAEP plummeted, but math scores fell further , by margins not seen in decades of testing. Other studies find that recovery has been slow.

At George Mason, fewer students are getting into calculus — the first college-level course for some majors — and more are failing. Students who fall behind often disengage, disappearing from class.

“This is a huge issue,” said Maria Emelianenko, chair of George Mason’s math department. “We’re talking about college-level pre-calculus and calculus classes, and students cannot even add one-half and one-third.”

For Jessica Babcock, a Temple University math professor, the magnitude of the problem hit home last year as she graded quizzes in her intermediate algebra class, the lowest option for STEM majors. The quiz, a softball at the start of the fall semester, asked students to subtract eight from negative six.

“I graded a whole bunch of papers in a row. No two papers had the same answer, and none of them were correct,” she said. “It was a striking moment of, like, wow — this is significant and deep.”

Before the pandemic, about 800 students per semester were placed into that class, the equivalent of ninth grade math. By 2021, it swelled to nearly 1,400.

“It’s not just that they’re unprepared, they’re almost damaged,” said Brian Rider, Temple’s math chair. “I hate to use that term, but they’re so behind.”

Researchers say math learning suffered for various reasons. An intensely hands-on subject, math was hard to translate to virtual classrooms . When students fell behind in areas like algebra, gaps could go unnoticed for a year or more as they moved to subjects such as geometry or trigonometry. And at home, parents are generally more comfortable helping with reading than math.

As with other learning setbacks, math issues are most pronounced among Black, Latino, low-income and other vulnerable students, said Katharine Strunk, who led a study on learning delays in Michigan and is now dean of the graduate school of education at the University of Pennsylvania.

“Those are the students who were most impacted by the pandemic, and they’re the ones who are going to suffer the longer-term consequences,” she said. “They’re not going to have the same access.”

Colleges say there’s no quick fix. Many are trying to identify gaps sooner, adopting placement tests that delve deeper into math skills. Some are adding summer camps like George Mason’s, which helped participants increase placement test scores by 59% on average.

In lieu of traditional remedial classes, which some research finds to be ineffective, more schools are offering “corequisite” classes that help students shore up on the basics while also taking higher courses like calculus.

Penn State tackled the problem by expanding peer tutoring. Professors report that students who participate have scored 20% higher on exams, said Tracy Langkilde, dean of Penn State’s College of Science.

What’s becoming a persistent problem at some colleges has been a blip for others. At Iowa State University, known for its engineering program, students entering in 2020 were far more likely to be placed in lower-level math classes, and grades fell. That group of students has had continued trouble, but numbers improved for the following year’s class, said Eric Weber, math department chair.

At Temple, there’s been no rebound. Professors tried small changes: expanded office hours, a new tutoring center, pared-down lessons focused on the essentials.

But students didn’t come for help, and they kept getting D’s and F’s. This year, Babcock is redesigning the algebra course. Instead of a traditional lecture, it’ll focus on active learning, an approach that demands more participation and expands students’ role in the learning process. Class will be more of a group discussion, with lots of problems worked in-class.

“We really want students to feel like they’re part of their learning,” Babcock said. “We can’t change their preparation coming in, but we can work to meet their needs in the best way possible.”

George Mason also is emphasizing active learning. Its new placement test helps students find gaps and fill them in before taking it again, with up to four attempts. During the school year, students struggling in math can switch to slower-paced versions that take two terms instead of one.

At math camp, Fonseca felt he was making up ground. He studied hard, doing practice problems on the train ride to camp. But when he got to the placement test’s algebra portion, he made the same mistakes. His final score again placed him in pre-calculus.

The setback would have meant spending at least one extra semester catching up on math at George Mason. In the end, Fonseca decided to start at Northern Virginia Community College. After two years, he plans to transfer to one of Virginia’s public four-year universities.

A couple weeks after camp, Fonseca again found himself taking a placement test, this time for the community college.

“I managed to use the knowledge of the boot camp, and I got into calculus,” he said. “I didn’t have any expectation I’d do that.”

The Education Reporting Collaborative, a coalition of eight newsrooms, is documenting the math crisis facing schools and highlighting progress. Members of the Collaborative are AL.com, The Associated Press, The Christian Science Monitor, The Dallas Morning News, The Hechinger Report, Idaho Education News, The Post and Courier in South Carolina, and The Seattle Times.

The Associated Press education team receives support from the Carnegie Corporation of New York. The AP is solely responsible for all content.

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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.

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! 

Download our quiz with 20 of the hardest SAT Math problems

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.

sample SAT math grid-in problems

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.

MIT campus

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.

US schools with the highest SAT math scores

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

college math problems hard

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. 

student practicing SAT math questions

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

college math problems hard

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:

key exponent rules

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:

solving the SAT math problem

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: 

solving the SAT math problem

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

Because that tells us that… 

solving the SAT math problem

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… 

solving the SAT math problem

Which is CHOICE A. 

studying for the SAT Math test

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

college math problems hard

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 .

student practicing SAT math problems on an iPad

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

college math problems hard

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. 

solving this hard SAT geometry problem

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. 

solving this hard SAT geometry problem

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. 

solving this hard SAT geometry problem

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

So X equals 80. 

student solving math problem

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

college math problems hard

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: 

solving the SAT math problem

(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. 

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SAT Problem #6

college math problems hard

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 . 

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SAT Problem #7

college math problems hard

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

college math problems hard

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. 

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SAT Problem 9

college math problems hard

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… 

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SAT Problem #10

college math problems hard

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 . 

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SAT Problem #11

college math problems hard

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!

graph of the circle equation

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

distance formula

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: 

distance formula graph

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 . 

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SAT Problem #12

college math problems hard

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 .

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SAT Problem #13

college math problems hard

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

college math problems hard

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.  

drawing of 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 . 

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SAT Problem #15

college math problems hard

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 .

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SAT Problem #16

college math problems hard

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 .

solving the hard SAT math problem

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!

Harvard campus

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

college math problems hard

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. 

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These Are the 10 Hardest Math Problems Ever Solved

They’re guaranteed to make your head spin.

pierre de fermat, math equation

On the surface, it seems easy. Can you think of the integers for x, y, and z so that x³+y³+z³=8? Sure. One answer is x = 1, y = -1, and z = 2. But what about the integers for x, y, and z so that x³+y³+z³=42?

That turned out to be much harder—as in, no one was able to solve for those integers for 65 years until a supercomputer finally came up with the solution to 42. (For the record: x = -80538738812075974, y = 80435758145817515, and z = 12602123297335631. Obviously.)

That’s the beauty of math : There’s always an answer for everything, even if takes years, decades, or even centuries to find it. So here are nine more brutally difficult math problems that once seemed impossible, until mathematicians found a breakthrough.

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The Poincaré Conjecture

the french astronomer and mathematician henri poincare at work in his office

In 2000, the Clay Mathematics Institute , a non-profit dedicated to “increasing and disseminating mathematical knowledge,” asked the world to solve seven math problems and offered $1,000,000 to anybody who could crack even one. Today, they’re all still unsolved, except for the Poincaré conjecture.

Henri Poincaré was a French mathematician who, around the turn of the 20th century, did foundational work in what we now call topology. Here’s the idea: Topologists want mathematical tools for distinguishing abstract shapes. For shapes in 3D space, like a ball or a donut, it wasn’t very hard to classify them all . In some significant sense, a ball is the simplest of these shapes.

Poincaré then went up to 4-dimensional stuff, and asked an equivalent question. After some revisions and developments, the conjecture took the form of “Every simply-connected, closed 3-manifold is homeomorphic to S^3,” which essentially says “the simplest 4D shape is the 4D equivalent of a sphere.”

Still with us?

A century later, in 2003, a Russian mathematician named Grigori Perelman posted a proof of Poincaré’s conjecture on the modern open math forum arXiv. Perelman’s proof had some small gaps, and drew directly from research by American mathematician Richard Hamilton. It was groundbreaking, yet modest.

After the math world spent a few years verifying the details of Perelman’s work, the awards began . Perelman was offered the million-dollar Millennium Prize, as well as the Fields Medal, often called the Nobel Prize of Math. Perelman rejected both. He said his work was for the benefit of mathematics, not personal gain, and also that Hamilton, who laid the foundations for his proof, was at least as deserving of the prizes.

Fermat’s Last Theorem

pierre de fermat

Pierre de Fermat was a 17th-century French lawyer and mathematician. Math was apparently more of a hobby for Fermat, and so one of history’s greatest math minds communicated many of his theorems through casual correspondence. He made claims without proving them, leaving them to be proven by other mathematicians decades, or even centuries, later. The most challenging of these has become known as Fermat’s Last Theorem.

It’s a simple one to write. There are many trios of integers (x,y,z) that satisfy x²+y²=z². These are known as the Pythagorean Triples, like (3,4,5) and (5,12,13). Now, do any trios (x,y,z) satisfy x³+y³=z³? The answer is no, and that’s Fermat’s Last Theorem.

Fermat famously wrote the Last Theorem by hand in the margin of a textbook, along with the comment that he had a proof, but could not fit it in the margin. For centuries, the math world has been left wondering if Fermat really had a valid proof in mind.

Flash forward 330 years after Fermat’s death to 1995, when British mathematician Sir Andrew Wiles finally cracked one of history’s oldest open problems . For his efforts, Wiles was knighted by Queen Elizabeth II and was awarded a unique honorary plaque in lieu of the Fields Medal, since he was just above the official age cutoff to receive a Fields Medal.

Wiles managed to combine new research in very different branches of math in order to solve Fermat’s classic number theory question. One of these topics, Elliptic Curves, was completely undiscovered in Fermat’s time, leading many to believe Fermat never really had a proof of his Last Theorem.

The Classification of Finite Simple Groups

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From solving Rubik’s Cube to proving a fact about body-swapping on Futurama , abstract algebra has a wide range of applications. Algebraic groups are sets that follow a few basic properties, like having an “identity element,” which works like adding 0.

Groups can be finite or infinite, and if you want to know what groups of a particular size n look like, it can get very complicated depending on your choice of n .

If n is 2 or 3, there’s only one way that group can look. When n hits 4, there are two possibilities. Naturally, mathematicians wanted a comprehensive list of all possible groups for any given size.

The complete list took decades to finish conclusively, because of the difficulties in being sure that it was indeed complete. It’s one thing to describe what infinitely many groups look like, but it’s even harder to be sure the list covers everything. Arguably the greatest mathematical project of the 20th century, the classification of finite simple groups was orchestrated by Harvard mathematician Daniel Gorenstein, who in 1972 laid out the immensely complicated plan.

By 1985, the work was nearly done, but spanned so many pages and publications that it was unthinkable for one person to peer review. Part by part, the many facets of the proof were eventually checked and the completeness of the classification was confirmed.

By the 1990s, the proof was widely accepted. Subsequent efforts were made to streamline the titanic proof to more manageable levels, and that project is still ongoing today .

The Four Color Theorem

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This one is as easy to state as it is hard to prove.

Grab any map and four crayons. It’s possible to color each state (or country) on the map, following one rule: No states that share a border get the same color.

The fact that any map can be colored with five colors—the Five Color Theorem —was proven in the 19th century. But getting that down to four took until 1976.

Two mathematicians at the University of Illinois, Urbana-Champaign, Kenneth Appel and Wolfgang Hakan, found a way to reduce the proof to a large, finite number of cases . With computer assistance, they exhaustively checked the nearly 2,000 cases, and ended up with an unprecedented style of proof.

Arguably controversial since it was partially conceived in the mind of a machine, Appel and Hakan’s proof was eventually accepted by most mathematicians. It has since become far more common for proofs to have computer-verified parts, but Appel and Hakan blazed the trail.

(The Independence of) The Continuum Hypothesis

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In the late 19th century, a German mathematician named Georg Cantor blew everyone’s minds by figuring out that infinities come in different sizes, called cardinalities. He proved the foundational theorems about cardinality, which modern day math majors tend to learn in their Discrete Math classes.

Cantor proved that the set of real numbers is larger than the set of natural numbers, which we write as |ℝ|>|ℕ|. It was easy to establish that the size of the natural numbers, |ℕ|, is the first infinite size; no infinite set is smaller than ℕ.

Now, the real numbers are larger, but are they the second infinite size? This turned out to be a much harder question, known as The Continuum Hypothesis (CH) .

If CH is true, then |ℝ| is the second infinite size, and no infinite sets are smaller than ℝ, yet larger than ℕ. And if CH is false, then there is at least one size in between.

So what’s the answer? This is where things take a turn.

CH has been proven independent, relative to the baseline axioms of math. It can be true, and no logical contradictions follow, but it can also be false, and no logical contradictions will follow.

It’s a weird state of affairs, but not completely uncommon in modern math. You may have heard of the Axiom of Choice, another independent statement. The proof of this outcome spanned decades and, naturally, split into two major parts: the proof that CH is consistent, and the proof that the negation of CH is consistent.

The first half is thanks to Kurt Gödel, the legendary Austro-Hungarian logician. His 1938 mathematical construction, known as Gödel’s Constructible Universe , proved CH compatible with the baseline axioms, and is still a cornerstone of Set Theory classes. The second half was pursued for two more decades until Paul Cohen, a mathematician at Stanford, solved it by inventing an entire method of proof in Model Theory known as “forcing.”

Gödel’s and Cohen’s halves of the proof each take a graduate level of Set Theory to approach, so it’s no wonder this unique story has been esoteric outside mathematical circles.

Gödel’s Incompleteness Theorems

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Gödel’s work in mathematical logic was totally next-level. On top of proving stuff, Gödel also liked to prove whether or not it was possible to prove stuff . His Incompleteness Theorems are often misunderstood, so here’s a perfect chance to clarify them.

Gödel’s First Incompleteness Theorem says that, in any proof language, there are always unprovable statements. There’s always something that’s true, that you can’t prove true. It’s possible to understand a (non-mathematically rigorous) version of Gödel’s argument, with some careful thinking. So buckle up, here it is: Consider the statement, “This statement cannot be proven true.”

Think through every case to see why this is an example of a true, but unprovable statement. If it’s false, then what it says is false, so then it can be proven true, which is contradictory, so this case is impossible. On the other extreme, if it did have a proof, then that proof would prove it true … making it true that it has no proof, which is contradictory, killing this case. So we’re logically left with the case that the statement is true, but has no proof. Yeah, our heads are spinning, too.

But follow that nearly-but-not-quite-paradoxical trick, and you’ve illustrated that Gödel’s First Incompleteness Theorem holds.

Gödel’s Second Incompleteness Theorem is similarly weird. It says that mathematical “formal systems” can’t prove themselves consistent. A consistent system is one that won’t give you any logical contradictions.

Here’s how you can think of that. Imagine Amanda and Bob each have a set of mathematical axioms—baseline math rules—in mind. If Amanda can use her axioms to prove that Bob’s axiom system is free of contradictions, then it’s impossible for Bob to use his axioms to prove Amanda’s system doesn’t yield contradictions.

So when mathematicians debate the best choices for the essential axioms of mathematics (it’s much more common than you might imagine) it’s crucial to be aware of this phenomenon.

The Prime Number Theorem

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There are plenty of theorems about prime numbers . One of the simplest facts—that there are infinitely many prime numbers—can even be adorably fit into haiku form .

The Prime Number Theorem is more subtle; it describes the distribution of prime numbers along the number line. More precisely, it says that, given a natural number N, the number of primes below N is approximately N/log(N) ... with the usual statistical subtleties to the word “approximately” there.

Drawing on mid-19th-century ideas, two mathematicians, Jacques Hadamard and Charles Jean de la Vallée Poussin, independently proved the Prime Number Theorem in 1898. Since then, the proof has been a popular target for rewrites, enjoying many cosmetic revisions and simplifications. But the impact of the theorem has only grown.

The usefulness of the Prime Number Theorem is huge. Modern computer programs that deal with prime numbers rely on it. It’s fundamental to primality testing methods, and all the cryptology that goes with that.

Solving Polynomials by Radicals

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Remember the quadratic formula ? Given ax²+bx+c=0, the solution is x=(-b±√(b^2-4ac))/(2a), which may have felt arduous to memorize in high school, but you have to admit is a conveniently closed-form solution.

Now, if we go up to ax³+bx²+cx+d=0, a closed form for “x=” is possible to find, although it’s much bulkier than the quadratic version. It’s also possible, yet ugly, to do this for degree 4 polynomials ax⁴+bx³+cx²+dx+f=0.

The goal of doing this for polynomials of any degree was noted as early as the 15th century. But from degree 5 on, a closed form is not possible. Writing the forms when they’re possible is one thing, but how did mathematicians prove it’s not possible from 5 up?

The world was only starting to comprehend the brilliance of French mathematician Evariste Galois when he died at the age of 20 in 1832. His life included months spent in prison, where he was punished for his political activism, writing ingenious, yet unrefined mathematics to scholars, and it ended in a fatal duel.

Galois’ ideas took decades after his death to be fully understood, but eventually they developed into an entire theory now called Galois Theory . A major theorem in this theory gives exact conditions for when a polynomial can be “solved by radicals,” meaning it has a closed form like the quadratic formula. All polynomials up to degree 4 satisfy these conditions, but starting at degree 5, some don’t, and so there’s no general form for a solution for any degree higher than 4.

Trisecting an Angle

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To finish, let’s go way back in history.

The Ancient Greeks wondered about constructing lines and shapes in various ratios, using the tools of an unmarked compass and straightedge . If someone draws an angle on some paper in front of you, and gives you an unmarked ruler, a basic compass, and a pen, it’s possible for you to draw the line that cuts that angle exactly in half. It’s a quick four steps, nicely illustrated like this , and the Greeks knew it two millennia ago.

What eluded them was cutting an angle in thirds. It stayed elusive for literally 15 centuries, with hundreds of attempts in vain to find a construction. It turns out such a construction is impossible.

Modern math students learn the angle trisection problem—and how to prove it’s not possible—in their Galois Theory classes. But, given the aforementioned period of time it took the math world to process Galois’ work, the first proof of the problem was due to another French mathematician, Pierre Wantzel . He published his work in 1837, 16 years after the death of Galois, but nine years before most of Galois’ work was published.

Either way, their insights are similar, casting the construction question into one about properties of certain representative polynomials. Many other ancient construction questions became approachable with these methods, closing off some of the oldest open math questions in history.

So if you ever time-travel to ancient Greece, you can tell them their attempts at the angle trisection problem are futile.

Headshot of Dave Linkletter

Dave Linkletter is a Ph.D. candidate in Pure Mathematics at the University of Nevada, Las Vegas. His research is in Large Cardinal Set Theory. He also teaches undergrad classes, and enjoys breaking down popular math topics for wide audiences.

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10 Hardest AP Calculus AB Practice Questions

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  • How Will AP Scores Impact Your Chances?

Overview of the AP Calculus AB Exam

  • Hardest AP Calc AB Practice Questions

With about 60% of students passing in 2020, the AP Calculus AB Exam is pretty tough. This test is one of the longer ones , and takes a total of 3 hours and 15 minutes. As with any math test, the key to this exam is practice! In this article, we’ll go over some of the harder questions you may encounter on the exam, along with detailed explanations of how to solve them.

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The AP Calculus AB exam will be offered both on paper and digitally in 2021.

The paper administration is held on May 4, 2021 and May 24, 2021:

  • No calculator: 30 questions (60 minutes)
  • Calculator: 15 questions (45 minutes)
  • Calculator: 2 questions (30 minutes)
  • No Calculator: 4 questions (60 minutes)

The digital administration is held on June 9, 2021:

  • 45 questions (1 hour 45 minutes), 50% of exam score
  • 6 questions (1 hour 30 minutes), 50% of exam score

For the digital exam, a calculator is allowed on all sections.

The AP Calculus AB course is organized into 8 units. The units are listed below, along with their weighting for the multiple choice section of the exam:

  • Limits and Continuity (10–12%)
  • Differentiation: Definition and Fundamental Properties (10–12%)
  • Differentiation: Composite, Implicit, and Inverse Functions (9–13%)
  • Contextual Applications of Differentiation (10–15%)
  • Analytical Applications of Differentiation (15–18%)
  • Integration and Accumulation of Change (17–20%)
  • Differential Equations (6–12%)
  • Applications of Integration (10–15%)

10 Hardest AP Calculus AB Questions

Here are some tough AP Calculus AB Questions for you to look over.

college math problems hard

You’ll definitely need to understand limits and their properties for the AP Calculus AB exam. For this particular question, we can start by trying to plug in \(\pi\) .

For the numerator, we get: \(\cos(\pi)+\sin(2\pi)+1=-1+0+1=0\).

For the denominator, we get: \(x^2-\pi^2=0\).

Since we have a 0 in both the numerator and denominator, we’re able to use L’Hospital’s rule, which means we’ll need to take the derivative of the numerator and denominator, separately.

Taking the derivative of the numerator yields: \(-\sin(x)+2\cos(2x)\).

Also, the derivative of the denominator is: \(2x\).

So, our limit now becomes: \(\lim_{x \to \pi} \frac{-\sin(x)+2\cos(2x)}{2x}=\frac{-\sin(\pi)+2\cos(2\pi)}{2\pi}=\frac{0+2(1)}{2\pi}=\frac{2}{2\pi}=\frac{1}{\pi}\) , which means our answer is B.

college math problems hard

When it comes to continuity, an easy rule of thumb is to check whether you can draw the graph without lifting your pencil. In this case, the graph only has one interruption, at \(x=0\). So, \(f\)  is continuous at all points besides \(x=0\).

Since \(f\)  is discontinuous at \(x=0\), answer choices B and D are incorrect (since the question asks where \(f\)  is continuous but isn’t differentiable).

So, either A or C is correct, which means we need to check differentiability at \(x=1\) and \(x=-2\).

At \(x=1\), we have a corner, so \(f\)  is not differentiable at \(x=1\).

Also, at \(x=-2\), we have a vertical tangent, and \(f\) is therefore not differentiable at \(x=-2\).

Then, answer choice C is correct.

college math problems hard

Questions involving slope fields tend to involve a lot of guess and check. For this question, we can start by looking at key \(x\) and \(y\) values.

First, if we look along the \(y\)-axis, we see that the slope is \(0\). So, regardless of our \(y\)-value, if \(x=0\), we should have that \(\frac{dy}{dx}=0\) .

For A, if we plug in \(x=0\), we get: \(\frac{dy}{dx}=0y+0=0\).

For B, if we plug in \(x=0\), we get: \(\frac{dy}{dx}=0y+y=y\).

For C, if we plug in \(x=0\), we get: \(\frac{dy}{dx}=y+1\) .

For D, if we plug in \(x=0\), we get: \(\frac{dy}{dx}=(0+1)^2=1\).

So, we see that the only equation which has tangent slopes of \(0\) along the \(y\)-axis is the one that corresponds to choice A.

college math problems hard

Recall that the average value of a function \(f\) on the interval \([a,b]\) is given by the formula: 

\(f_{avg}=\frac{1}{b-a} \int_{a}^b f(x)dx\).

So, we’ll need to compute the integral of \(f\)  over \([-4,4]\). Since we’re given a graph, we can do this by calculating the areas of different sections. We can divide up the graph into triangles and trapezoids:

Keep in mind that the value from \((-2,1)\) is negative since the function lies below the \(x\)-axis. To compute the integral, we can add up all our values:

\(\int_{-4}^4 f(x)dx=1-3+2+3/2=3/2\).

But, we’re not done yet! We still need to multiply by \(\frac{1}{4-(-4)}=1/8\).

So, the average value is \((1/8)(3/2)=3/16\).

college math problems hard

These questions are really easily missed when students fail to apply chain rule. When we find \(f'(x)\) , we’ll need to be careful to apply chain rule.

Let’s set \(F(x)=\int_{1}^x \frac{1}{1+\ln{t}}\) . Then, \(f(x)=F(x^3)\) .

So, \(f'(x)=F'(x^3)\) .

But, when we differentiate \(F(x^3)\) , we’ll need to apply chain rule and multiply by the derivative of \(x^3\) .

This means that \(F'(x^3)=(F(x^3))'(x^3)’\) . So, \(f'(x)=F'(x^3)=\frac{1}{1+\ln{x^3}}\cdot3x^2\).

Then, \(f'(2)=\frac{1}{1+\ln{2^3}}\cdot3(2)^2=\frac{12}{1+\ln{8}}\) .

college math problems hard

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college math problems hard

Recall the following formula used when converting integrals to limits:

\(\int_{a}^b f(x)dx=\lim_{n \to \infty}\sum_{k=1}^n f(a+(\frac{b-a}{n})k)\cdot\frac{b-a}{n}\).

So, in this case, we have that \(a=3\) , \(b=5\) , and \(f(x)=x^4\) . Also, \(b-a=2\) .

Then, \(\int_{3}^5 x^4dx=\lim_{n \to \infty}\sum_{k=1}^n (3+\frac{2k}{n})^4\cdot\frac{2}{n}\) , which corresponds to choice D.

college math problems hard

To solve the differential equation \(\frac{dy}{dt}=ky\) , we’ll first need to divide both sides by \(y\) and multiply both sides by \(dt\) .

This yields \(\frac{dy}{y}=k\:dt\) . Since we’ve separated our variables \(y\) and \(t\), we can now integrate:

\(\int \frac{dy}{y}=\int k\:dt\).

\(\ln{y}=kt+C\).

To isolate for \(y\), we’ll need to put both sides as a power of \(e\) :

\(e^{\ln{y}}=e^{kt+C}\)

\(y=e^{kt+C}=e^{kt}\cdot e^{C}=e^{kt}\cdot C=Ce^{kt}\).

We can now use a point from the table, \((0,4)\) to solve for \(C\):

\(4=Ce^{k(0)}\)

This means that \(y=4e^{kt}\).

To solve for \(k\), let’s use another point from the table, \((2,12)\):

\(12=4e^{k(2)}\)

\(3=e^{2k}\).

Let’s take the natural log of both sides:

\(\ln{3}=\ln{e^{2k}}\)

\(\ln{3}=2k\)

\(k=\frac{1}{2}\ln{3}\).

So, we get that \(y=f(t)=4e^{\frac{t}{2}\ln{3}}\) .

college math problems hard

You should expect to be asked to interpret information on the AP Calculus AB exam. For this question, since \(H(t)\) is the temperature of a room (in ºF) \(t\) minutes after a thermostat is adjusted, \(H'(t)\) would be the change in the temperature of the room per minute, \(t\) minutes after the thermostat is adjusted.

So, if \(H'(5)=2\) means that 5 minutes after the thermostat is adjusted, the change in temperature is 2 ºF per minute. Since 2 is positive, the temperature is increasing, and D is the correct answer choice.

You may be tempted to pick answer B, but it states that “the temperature of the room increases by 2 degrees,” which talks about a single event rather than the rate of change.

college math problems hard

Though this question allows the use of a calculator, we’ll still need to do quite a few calculations by hand. First, recall the relationship between position, velocity, and acceleration: \(x”(t)=v'(t)=a(t)\) . So, to get to the position function, we’ll need to integrate acceleration twice.

\(v(t)=\int a(t)\:dt=\int -6t^2-t\:dt=-2t^3-\frac{1}{2}t^2+C\).

From the problem, we know that at time \(t=0\) seconds, the velocity of the car is \(80\) meters per second. So, we can use that \(v(0)=80\) to solve for \(C\) .

\(v(0)=-2(0)^3-\frac{1}{2}(0)^2+C=80\Rightarrow C=80\).

This question is tricky since we aren’t given both our bounds. We know the time period starts at \(t=0\) and ends at the moment the race car stops.

To find the time that the race car stops, we’ll need to set \(v(t)=0\)  (since if the car is stopped, the velocity should be \(0\) meters per second).

We can do this by graphing the velocity function and finding the zeros. If we graph \(y=-2x^3-\frac{1}{2}x^2+80\)  we see that the zero is \(3.339\).

Now, using our calculators, we can integrate the absolute value of the velocity function to determine the distance travelled from \(t=0\) to \(t=3.339\):

\(\int_{0}^{3.339} |-2t^3-\frac{1}{2}t^2+80|\:dt=198.766\).

Note that we integrated the absolute value to determine the total distance travelled. Integrating just the velocity function gives us the displacement of the race car.

Question 10

college math problems hard

For related rates problems, it’s helpful to start with a familiar formula. In this case, since we’re given information about the volume of a sphere, let’s use that formula:

\(V=\frac{4}{3}\pi r^3\)

Now, we can differentiate with respect to time, \(t\):

\(\frac{dV}{dt}=4\pi r^2\cdot \frac{dr}{dt}\)

We know that \(\frac{dV}{dt}=2\pi\)  and \(r=5\) , so we can solve for \(\frac{dr}{dt}\) :

\(2\pi =4\pi (5)^2 \cdot \frac{dr}{dt} \Rightarrow \frac{dr}{dt}=1/50\).

Next, we’ll need to use the surface area formula:

\(S=4\pi r^2\)

Again, we differentiate with respect to time \(t\) to find the rate at which the surface area is decreasing when the radius is 5 meters:

\(\frac{dS}{dt}=8\pi r \cdot \frac{dr}{dt}\)

We can plug in the appropriate values of \(r\) and \(\frac{dr}{dt}\)  to find \(\frac{dS}{dt}\) .

\(\frac{dS}{dt}=8\pi (5)(1/50)=4\pi /5\).

Know your calculator!

Especially on the digital exam, you’ll be using your calculator a lot. Knowing your calculator well will help you get through questions much more quickly. For example, some calculus questions may be able to be solved without the use of a calculator, but there are many cases where using a quick calculator trick during intermediate steps will save you a significant amount of time.

Time yourself

Since the AP Calculus AB exam requires you to answer many questions in limited time, it’s imperative that you learn to properly pace yourself. So, when answering practice questions, try to time yourself in a format that’s similar to the exam (i.e. give yourself 1 hour and 45 minutes to answer 45 multiple choice questions).

This will help you practice your pacing and if you find that you’re struggling to finish on time, you can rethink your strategy. Since all the multiple choice questions carry equal weight, skipping difficult or time-consuming problems is more beneficial for you.

Check out these other articles as you prepare for your AP exams:

  • Ultimate Guide to the AP Calculus AB exam
  • 2021 AP Exam Schedule + Study Tips
  • How to Understand and Interpret Your AP Scores
  • How Long Is Each AP Exam? A Complete List

Related CollegeVine Blog Posts

college math problems hard

Practice Test

  • Introduction to Prerequisites
  • 1.1 Real Numbers: Algebra Essentials
  • 1.2 Exponents and Scientific Notation
  • 1.3 Radicals and Rational Exponents
  • 1.4 Polynomials
  • 1.5 Factoring Polynomials
  • 1.6 Rational Expressions
  • Key Equations
  • Key Concepts
  • Review Exercises
  • Introduction to Equations and Inequalities
  • 2.1 The Rectangular Coordinate Systems and Graphs
  • 2.2 Linear Equations in One Variable
  • 2.3 Models and Applications
  • 2.4 Complex Numbers
  • 2.5 Quadratic Equations
  • 2.6 Other Types of Equations
  • 2.7 Linear Inequalities and Absolute Value Inequalities
  • Introduction to Functions
  • 3.1 Functions and Function Notation
  • 3.2 Domain and Range
  • 3.3 Rates of Change and Behavior of Graphs
  • 3.4 Composition of Functions
  • 3.5 Transformation of Functions
  • 3.6 Absolute Value Functions
  • 3.7 Inverse Functions
  • Introduction to Linear Functions
  • 4.1 Linear Functions
  • 4.2 Modeling with Linear Functions
  • 4.3 Fitting Linear Models to Data
  • Introduction to Polynomial and Rational Functions
  • 5.1 Quadratic Functions
  • 5.2 Power Functions and Polynomial Functions
  • 5.3 Graphs of Polynomial Functions
  • 5.4 Dividing Polynomials
  • 5.5 Zeros of Polynomial Functions
  • 5.6 Rational Functions
  • 5.7 Inverses and Radical Functions
  • 5.8 Modeling Using Variation
  • Introduction to Exponential and Logarithmic Functions
  • 6.1 Exponential Functions
  • 6.2 Graphs of Exponential Functions
  • 6.3 Logarithmic Functions
  • 6.4 Graphs of Logarithmic Functions
  • 6.5 Logarithmic Properties
  • 6.6 Exponential and Logarithmic Equations
  • 6.7 Exponential and Logarithmic Models
  • 6.8 Fitting Exponential Models to Data
  • Introduction to Systems of Equations and Inequalities
  • 7.1 Systems of Linear Equations: Two Variables
  • 7.2 Systems of Linear Equations: Three Variables
  • 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
  • 7.4 Partial Fractions
  • 7.5 Matrices and Matrix Operations
  • 7.6 Solving Systems with Gaussian Elimination
  • 7.7 Solving Systems with Inverses
  • 7.8 Solving Systems with Cramer's Rule
  • Introduction to Analytic Geometry
  • 8.1 The Ellipse
  • 8.2 The Hyperbola
  • 8.3 The Parabola
  • 8.4 Rotation of Axes
  • 8.5 Conic Sections in Polar Coordinates
  • Introduction to Sequences, Probability and Counting Theory
  • 9.1 Sequences and Their Notations
  • 9.2 Arithmetic Sequences
  • 9.3 Geometric Sequences
  • 9.4 Series and Their Notations
  • 9.5 Counting Principles
  • 9.6 Binomial Theorem
  • 9.7 Probability

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

For the following exercises, evaluate the expression.

2 ( x + 3 ) − 12 ; x = 2 2 ( x + 3 ) − 12 ; x = 2

y ( 3 + 3 ) 2 − 26 ; y = 1 y ( 3 + 3 ) 2 − 26 ; y = 1

Write the number in standard notation: 3.1415 × 10 6 3.1415 × 10 6

Write the number in scientific notation: 0.0000000212.

For the following exercises, simplify the expression.

−2 ⋅ ( 2 + 3 ⋅ 2 ) 2 + 144 −2 ⋅ ( 2 + 3 ⋅ 2 ) 2 + 144

4 ( x + 3 ) − ( 6 x + 2 ) 4 ( x + 3 ) − ( 6 x + 2 )

3 5 ⋅ 3 −3 3 5 ⋅ 3 −3

( 2 3 ) 3 ( 2 3 ) 3

8 x 3 ( 2 x ) 2 8 x 3 ( 2 x ) 2

( 16 y 0 ) 2 y −2 ( 16 y 0 ) 2 y −2

9 x 16 9 x 16

121 b 2 1 + b 121 b 2 1 + b

6 24 + 7 54 − 12 6 6 24 + 7 54 − 12 6

−8 3 625 4 −8 3 625 4

( 13 q 3 + 2 q 2 − 3 ) − ( 6 q 2 + 5 q − 3 ) ( 13 q 3 + 2 q 2 − 3 ) − ( 6 q 2 + 5 q − 3 )

( 6 p 2 + 2 p + 1 ) + ( 9 p 2 −1 ) ( 6 p 2 + 2 p + 1 ) + ( 9 p 2 −1 )

( n − 2 ) ( n 2 − 4 n + 4 ) ( n − 2 ) ( n 2 − 4 n + 4 )

( a − 2 b ) ( 2 a + b ) ( a − 2 b ) ( 2 a + b )

For the following exercises, factor the polynomial.

16 x 2 − 81 16 x 2 − 81

y 2 + 12 y + 36 y 2 + 12 y + 36

27 c 3 − 1331 27 c 3 − 1331

3 x ( x − 6 ) − 1 4 + 2 ( x − 6 ) 3 4 3 x ( x − 6 ) − 1 4 + 2 ( x − 6 ) 3 4

2 z 2 + 7 z + 3 z 2 − 9 ⋅ 4 z 2 − 15 z + 9 4 z 2 − 1 2 z 2 + 7 z + 3 z 2 − 9 ⋅ 4 z 2 − 15 z + 9 4 z 2 − 1

x y + 2 x x y + 2 x

a 2 b − 2 b 9 a 3 a − 2 b 6 a a 2 b − 2 b 9 a 3 a − 2 b 6 a

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  • Book title: College Algebra
  • Publication date: Feb 13, 2015
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College Algebra Questions With Answers Sample 1

College algebra multiple choice questions, with answers, are presented. The solutions are at the bottom of the page. Also Detailed solutions with full explanations are included.

Question 1o

Question 11, question 12, question 13, question 14, question 15, question 16, question 17, question 18, question 19, question 20, answers to the above questions, more references and links.

College Algebra

Also known as "High School Algebra"

OK. So what are you going to learn here?

You will learn about Numbers, Polynomials, Inequalities, Sequences and Sums, many types of Functions, and how to solve them.

You will also gain a deeper insight into Mathematics, get to practice using your new skills with lots of examples and questions, and generally improve your mind.

With your new skills you will be able to put together mathematical models so you can find good quality solutions to many tricky real world situations.

Near the end of most pages is a "Your Turn" section ... do these! You need to balance your reading with doing . Answering questions helps you sort things out in your mind. And don't guess the answer: use pen and paper and try your best before seeing the solution.

So what is this thing called Mathematics? And how do you go about learning it?

  • Welcome to Mathematics
  • Learning Mathematics
  • The Language of Mathematics
  • Symbols in Algebra

Next, we need to think about mathematics in terms of sets .

  • Introduction to Sets

Now we know what a set is, let us look at different sets of numbers that are useful:

  • The Evolution of Numbers
  • Prime and Composite Numbers
  • Fundamental Theorem of Arithmetic
  • Whole Numbers and Integers
  • Rational Numbers
  • Using Rational Numbers
  • Irrational Numbers
  • 0.999... = 1
  • Real Numbers
  • Imaginary Numbers
  • Complex Numbers
  • Multiplying Complex Numbers
  • The Complex Plane
  • Common Number Sets

Inequalities

"Equal To" is nice but not always available. Maybe we only know that something is less than, or greater than. So let's learn about in equalities.

  • Introduction to Inequalities
  • Properties of Inequalities
  • Solving Inequalities
  • Solving Inequality Word Questions

We will be using exponents a lot, so let's get to know them well.

  • Variables with Exponents
  • Using Exponents in Algebra
  • Squares and Square Roots
  • Squares and Square Roots in Algebra
  • Fractional Exponents
  • Laws of Exponents
  • Exponents of Negative Numbers

Polynomials

Polynomials were some of the first things ever studied in Algebra. They are simple, yet powerful in their ability to model real world situations.

  • What is a Polynomial?
  • Adding And Subtracting Polynomials
  • Multiplying Polynomials
  • Polynomials - Long Multiplication
  • Dividing Polynomials
  • Polynomials - Long Division
  • Degree (of an Expression)
  • Special Binomial Products
  • Difference of Two Cubes
  • Factoring in Algebra
  • Solving Polynomials
  • Roots of Polynomials: Sums and Products
  • Rational Expressions
  • Using Rational Expressions
  • Fundamental Theorem of Algebra
  • Remainder Theorem and Factor Theorem
  • General Form of a Polynomial

Graphing Polynomials

  • How Polynomials Behave
  • Polynomials: The Rule of Signs
  • Polynomials: Bounds on Zeros

And, of course, we need to know about equations ... and how to solve them.

  • Equations and Formulas
  • Solving Equations
  • Solving Word Questions
  • Zero Product Property
  • Implication and Iff
  • Theorems, Corollaries, Lemmas

Graphs can save us! They are a great way to see what is going on and can help us solve many things. But we need to be careful, as they sometimes don't give the full story.

  • Cartesian Coordinates
  • Pythagoras' Theorem
  • Distance Between 2 Points
  • Graph of an Equation
  • Finding Intercepts From an Equation
  • Symmetry in Equations
  • Linear Equations

They are just equations for lines. But they come in many forms.

  • Equation of a Straight Line
  • Point-Slope Equation of a Line
  • General Form of Equation of a Line
  • Equation of a Line from 2 Points
  • Midpoint of a Line Segment
  • Parallel and Perpendicular Lines

A function relates an input to an output. But from that simple foundation many useful things can be built.

  • What is a Function?
  • Domain, Range and Codomain
  • Evaluating Functions
  • Increasing and Decreasing Functions
  • Maxima and Minima of Functions
  • Even and Odd Functions
  • Set-Builder Notation

Common Functions Reference :

  • Square Function
  • Square Root Function
  • Cube Function
  • Reciprocal Function
  • Absolute Value Function
  • Floor and Ceiling Function
  • Function Transformations
  • Equation Grapher
  • Operations with Functions
  • Composition of Functions
  • Inverse Functions

Equations of Second Degree

"Second degree" just means the variable has an exponent of 2, like x 2 . It is the next major step after linear equations (where the exponent is 1, like x).

  • Quadratic Equations
  • Factoring Quadratics
  • Completing the Square
  • Derivation of Quadratic Formula
  • Graphing Quadratic Equations
  • Quadratic Equations in the Real World
  • Circle Equations

We already have experience in solving, but now we can learn more!

  • Mathematical Models and Mathematical Models 2
  • Approximate Solutions
  • Intermediate Value Theorem
  • Solving Radical Equations
  • Change of Variables
  • Algebra Mistakes

We learned about inequalities above, now let's learn how to solve them.

  • Graphing Linear Inequalities
  • Inequality Graphing Tool
  • Solving Quadratic Inequalities
  • Solving Rational Inequalities
  • Absolute Value in Algebra

Exponents and Logarithms

We already know about exponents ... well logarithms just go the other way. And together they can be very powerful.

  • Introduction to Logarithms
  • Exponents, Roots and Logarithms
  • Working with Exponents and Logarithms
  • Exponential Function
  • Logarithmic Function
  • Exponential Growth and Decay
  • Systems of Linear Equations

What happens when we have two or more linear equations that work together? They can often be solved! It isn't very hard but can take a lot of calculations.

  • Types of Matrix
  • How to Multiply Matrices
  • Determinant of a Matrix
  • Inverse of a Matrix:
  • Using Elementary Row Operations (Gauss-Jordan)
  • Using Minors, Cofactors and Adjugate
  • Scalar, Vector, Matrix and Vectors
  • Matrix Calculator
  • More at Matrix Index
  • Solving Systems of Linear Equations Using Matrices
  • Systems of Linear and Quadratic Equations
  • Probability

lock

Is it likely? You be the judge!

  • The Basic Counting Principle
  • Combinations and Permutations

Sequences, Series and Partial Sums

A Sequence is a set of things (usually numbers) that are in order. We can also sum up a series, where Sigma Notation is very useful.

  • Sequences - Finding A Rule
  • Sigma Notation
  • Partial Sums
  • Arithmetic Sequences and Sums
  • Geometric Sequences and Sums

These last few subjects use what we have learned above.

  • Partial Fractions
  • Mathematical Induction
  • Pascal's Triangle
  • Binomial Theorem

And that is all!

But there are many other interesting algebra topics such as:

  • Euler's Formula for Complex Numbers
  • Taylor Series (needs a basic understanding of derivatives )

Illustration of a boy leaning against a blackboard and a girl writing Math 55 on it with chalk.

Demystifying Math 55

By anastasia yefremova.

Few undergraduate level classes have the distinction of nation-wide recognition that Harvard University’s Math 55 has. Officially comprised of Mathematics 55A “Studies in Algebra and Group Theory” and Mathematics 55B “Studies in Real and Complex Analysis,” it is technically an introductory level course. It is also a veritable legend among high schoolers and college students alike, renowned as — allegedly — the hardest undergraduate math class in the country. It has been mentioned in books and articles, has its own Wikipedia page, and has been the subject of countless social media posts and videos.

Most recently, Harvard junior Mahad Khan created a TikTok video dedicated to Math 55 that has received over 360,000 views to date. His is only one of many — his older brother created one, too — but it has the distinction of an insider’s perspective. “I thought it would be interesting if I cleared up the misconceptions about Math 55,” Khan said. While he hadn’t taken the course himself, he wanted to go beyond its reputation. “I wanted to get a real perspective by interviewing a former student and current course assistant.”

Over the years, perception of Math 55 has become based less on the reality of the course itself and more on a cumulative collection of lore and somewhat sensationalist rumors. It’s tempting to get swept up in the thrill of hearsay but while there might be kernels of truth to some of the stories, many of them are outdated and taken out of context. At the end of the day, however, Math 55 is a class like any other. Below, we take a stab at busting some of the more well known and persistent myths about the class. Or, at the very least, offering an extra layer of clarity. 

Myth #1: Math 55 is only for high school math geniuses

Most articles or mentions of Math 55 refer to it as filled with math competition champions and genius-level wunderkinds. The class is supposedly legendary among high school math prodigies, who hear terrifying stories about it in their computer camps and at the International Math Olympiad. There are even rumors of a special test students have to take before they are even allowed into Math 55. But while familiarity with proof-based mathematics is considered a plus for those interested in the course, there is no prerequisite for competition or research experience. 

In fact students whose only exposure to advanced math has been through olympiads and summer research programs can have a harder time adjusting. Their approach to the material tends to be understandably more solitary and that can be a disadvantage for the level of collaboration higher level mathematics require. “It has become a lot more open to people with different backgrounds,” said Professor Denis Auroux , who teaches Math 55,. “Our slogan is, if you’re reasonably good at math, you love it, and you have lots of time to devote to it, then Math 55 is completely fine for you.” 

Also, there is no extra test to get into the class.

Myth #2: Just take a graduate class, instead

Math 55 is hard. Whether you’re just 55-curious, or a past or present student in the class, this is something everyone agrees on. The course condenses four years of math into two semesters, after all. “For the first semester, you work on linear and abstract algebra with a bit of representation theory,” said sophomore math concentrator Dora Woodruff. “The second semester is real and complex analysis, and a little bit of algebraic topology. That’s almost the whole undergraduate curriculum.” Woodruff — incidentally, the student Khan interviewed — took Math 55 as a freshman and returned her second year as a course assistant. She is intimately familiar with the course’s difficulty level.

So why not just take an upper level undergraduate course to begin with or even one at a graduate level, if you’re really looking for a challenge? What justifies the existence of a class with the difficulty level of Math 55? One argument is that the course helps structure and systemize the knowledge with which many students come to Harvard. It gives them a firm background in preparation for the rest of their math education. Math 55 is difficult and it is purposefully structured that way as it’s meant to help students mature as mathematicians rather than as simple course takers.

But more importantly, “it’s just not true that Math 55 is at the level of a graduate class,” Auroux said. “It goes through several upper division undergraduate math classes with maybe a bit more advanced digressions into material here and there, but it sticks very close to what is taught in 100-level classes. The difference is we go through it at a faster pace, maybe with more challenging homework, and ideally as a community of people bringing our heads together.” 

A core goal of Math 55, according to Auroux, is to build a sense of community. Other schools might encourage advanced first-year students to take upper level undergraduate or even graduate classes, but Math 55 helps build a cohort of like-minded people who really like math, are good at it, and want to do a lot of it during their time at Harvard. That’s the experience Woodruff had, as well. “The community can be very strong,” she said. “You meet a lot of other people very interested in math and stay friends with them for the rest of college.”

Myth #3: Homework takes between 24 and 60 hours

Horror stories of endless homework are synonymous with the class. You’ll read or hear about “24 to 60 hours per week on homework” in almost every reference to Math 55. But one, there is a world of difference between 24 and 60 hours that is never explained, and two, this timeframe is quite misaligned with reality.

Auroux frequently sends out surveys to his students asking how long homework takes them and the average for most is closer to 15 hours a week. Those with more extensive prior math backgrounds can take as little as five to ten hours. The key factor is collaboration. “This class doesn’t lend itself to self-study,” Auroux stressed. Once they have thought about each problem set on their own, students are welcome and encouraged to talk to their friends and collaborate. “As soon as I see that something took over 30 hours I ask the student, do you know you’re supposed to be working with people and come ask me questions when you’re stuck?”

It is true that between reviewing lectures, digesting the material, and solving the problem sets, students usually end up devoting between 20 and 30 hours a week to the class. However, that includes the time dedicated to homework. So while students are discouraged from taking too many difficult classes and extracurriculars in the same semester as Math 55, they are also not expected to spend the time equivalent to a full-time job on their problem sets every week.

Myth #4: less than half of the class makes it to the second semester

Math 55 is just as infamous for its attrition rate as it is for its difficulty. Most sources like to cite the 1970 class, which began with 75 students and — between the advanced nature of the material and the time-constraints under which students had to work — ended with barely 20. Since then, the rumor has been that the Math 55 class shrinks by half its original size or more before the first semester is over. The reality is much less shocking and a bit more complicated.

Enrollment in this past fall semester’s Math 55A peaked at (ironically) 55 students. Well into the spring semester’s Math 55B, 47 students were still enrolled in the course. “On average, a drop of about 10-15 percent is much closer to what I would expect,” Auroux said. And those numbers become even more flexible if one takes into consideration the weeks math students have at the beginning of each semester to try out different classes and “shop” around before they have to commit to anything. This means students find their way in and out of Math 55 in a variety of ways over the course of the academic year.

According to Auroux, some students shop Math 55 in the fall and switch to the less intense Math 25 for the remainder of the semester. Others start out in Math 25 and, if not sufficiently challenged, switch to Math 55. Even people who end up in academia are not exempt from this. During his time as a student, our own Department of Mathematics’ Professor Emeritus Benedict Gross switched to the lower level Math 21 after two weeks in Math 55. In fact, those two weeks almost made him reconsider his desire to pursue mathematics. “By the beginning of sophomore year, I had decided to major in physics,” he recalled. “But during shopping period that fall, I walked past a math class taught by Andrew Gleason and stopped in to listen. It turned out to be Math 55.” He enrolled and by the end of the semester had found his vocation in mathematics.

All this means that Auroux sees student numbers vacillate up and down throughout the academic year. “There are about four or five students in this spring semester’s Math 55 that took Math 25 or even Math 22 in the fall, and they’re doing mostly fine,” he said. “It’s a lot of work, but I think they’re having a great time.”

Myth #5: 55-er culture is cult-y and exclusionary

Even though her experience with Math 55 was a positive one, Woodruff is very aware of the unhealthy culture the class has been rumored to cultivate. It’s easy for students to form exclusionary cliques that consist only of other Math 55 students, and some look down on anyone taking lower level math classes. But Woodruff also stressed that the instructors are very aware of this and actively take steps to curb that kind of toxic behavior. She said Auroux frequently brings up the importance of keeping the Math 55 community inclusive through Slack messages and lecture references.

Some students come to Harvard just for the opportunity to take Math 55. Some view enrolling in the class as proof of their mathematical gumption and competence. A Harvard Independent article called Math 55 the “premiere mathematical challenge for overachieving and…ridiculously mathy freshmen” and a piece in The Harvard Crimson referred to it as “a bit of a status thing as far as math majors here are concerned.” Over the years, the Harvard Department of Mathematics has taken steps to correct these assumptions. 

For one thing, neither the Math 55A nor the Math 55B official course descriptions boast the dubious honor of referring to it as “probably the most difficult undergraduate math class in the country” (don’t trust everything you read on Wikipedia). For another, “we’re trying to emphasize that there’s no magic to Math 55,” Auroux said. “It contains the same material as some of the other classes we have. People who take it are not intrinsically better or smarter than the ones who don’t.” 

Myth #6: You have to take Math 55 if you’re serious about going into academia

One reason math concentrators could feel pressured to enroll in Math 55 is because they view it as a prerequisite for a career in academia. It’s a sort of badge of honor and proof of their commitment to the field of mathematics. It is true that quite a few graduates of the course have gone on to pursue a career in mathematics. Woodruff herself believes that will be the most likely path for her, and several faculty members in our own Department of Mathematics took Math 55 during their days as Harvard freshmen.

“Several times in my research career when I understood something fundamental, I would realize that this was what Math 55 was trying to teach us,” Gross said. “It was an amazing introduction to the whole of mathematics and it was transformative for me.” In fact, Gross met Higgins Professor of Mathematics Joe Harris when they took the class together, forging a lifelong friendship. When they returned to Harvard as faculty, they took turns teaching Math 25 and Math 55. 

However, Auroux is quick to point out that while many graduates of the course do end up in academia, most professional mathematicians have likely never even heard of Math 55. “I would like to think that it’s a success story if people end up doing math, because the goal of Math 55 is to show students how beautiful math can be,” he said. “If they love it enough to go to grad school and become mathematicians, that’s wonderful. And if they want to take that math knowledge and do something else with their life, that’s just as wonderful.”

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From “The New York Times,” I’m Sabrina Tavernise, and this is “The Daily.”

A “Times” investigation has found that doctors are increasingly performing unnecessary medical procedures that generate huge profits while often harming patients.

Today, my colleague Katie Thomas — on the forces driving this emerging and troubling trend in American health care and the story of one family caught in the middle of it. It’s Monday, February 19.

So Katie, tell me about this investigation.

So I am a health care reporter who writes about the kind of intersection of health care and money. And I was working with two other colleagues, Sarah Kliff and Jessica Silver-Greenberg. And together, the three of us had long been interested in, are the medical procedures and the tests and other things that we get when we go to the doctor or into a hospital — are they always necessary?

But what we were really interested in exploring was not just are these procedures and are these tests, et cetera — are they necessary, but in some situations, could they actually be harmful to patients? And so that’s what we decided to try and take a look at. And so we had gotten started in our reporting when we got a tip. And it was from a mom in Boise, Idaho. And her name was Lauren Lavelle.

Nice to meet you.

Hi, how are you?

And my colleague Jessica Silver-Greenberg and I went to her house to meet with her.

And where does her story start?

I am a mom of two. I live in Boise. My daughter, June, is four, and I have a 17-month-old, Flora.

Her story starts when Lauren gets pregnant with her daughter, June.

So by the time we got pregnant with June, November of 2018, about eight months after we had the miscarriage, I think I was just more hesitant and nervous than anything.

Lauren and her husband had trouble conceiving, and so they were so happy when they learned that they were going to have June. And like most first-time parents, they were also a little bit nervous.

But being type-A and super prepared, I did all my homework. We hired a doula. I wanted an epidural. Having a natural childbirth absolutely was not for me.

And Lauren is very organized. She’s always on top of everything, and she makes all sorts of plans. And she gets a lot of different providers lined up ahead of time —

I didn’t know anything about breastfeeding, like zero things.

— including one that she has hired to help her with breastfeeding.

Where did you find out about her?

So I asked our doula for a list of recommendations, and she gave me a very short list. At the time, there were very few lactation consultants in the Valley. And Melanie was one of them.

She ended up deciding to work with Melanie Henstrom, who was a local lactation consultant in Boise.

She sold this package at the time. I don’t know if she still did, but it was like prenatal visit breastfeeding class. And then, she’ll come to the hospital and help you latch, and then she’ll come to the house a couple of times after. And I thought, well, this sounds perfect. Great. You know, I’m covered there.

So one week after her due date, she gives birth. And it was a difficult labor. It took 24 hours. Lauren was completely exhausted. But once June arrived, the family was very, very excited to have her.

And I remember June coming out and that surreal feeling have when you see your first baby for the first time, like oh, my God, there’s a baby in the room.

And June was a healthy baby, but she was having trouble breastfeeding.

She would not latch. Like, she wouldn’t even attempt. She would scream. It was the only time she ever cried — if you tried to make her to breastfeed.

And so as her pediatrician was making the rounds, they noticed that June was having trouble and said that June’s tongue is really tight.

We can clip it if you’d like.

And that they could clip it.

What does that mean exactly, Katie — clipping her tongue?

What it means is that there’s a small percentage of babies whose tongue is very tightly tethered to the bottom of their mouth. And for a very small percentage of babies, their tongue is almost tied so tightly down that they can’t nurse well.

So it makes breastfeeding very difficult if a baby has a tongue like this.

Exactly. If you bottle-feed your baby, the baby can basically adjust and make do. But if you want to breastfeed, some babies have trouble, basically, latching on to their mother when they don’t have that tongue motion. And so some version of clipping these tongue ties has been done for centuries. Midwives have been doing it. Pediatricians do it.

And traditionally, what it’s been is a very quick snip right underneath the tongue just to loosen up the tongue. And traditionally, that procedure is extremely straightforward. There’s little to no follow-up care. And basically, the baby naturally heals as it learns to breastfeed.

And so we said, OK. They explained that it was completely painless. They’d do it with scissors. She wouldn’t even feel it. And all of that was true. They clipped it. I don’t even think she woke up.

But in June’s case, it didn’t seem to help much, and she and Lauren were still having problems breastfeeding afterwards. So while she’s still in the hospital, she calls up the lactation consultant that she had hired — Melanie Henstrom — just to let her know what was going on. And from talking to her on the phone, Melanie said that the situation was actually much worse than Lauren had thought and that Lauren’s baby needed another tongue-tie procedure — a deeper cut under the tongue.

How did she make this diagnosis, Katie? Was it over the phone? How did she know this?

Yes, Lauren told us that it was from a phone conversation. And in addition to that, she also warned her that, basically, Lauren and her husband should really take this seriously and consider getting it done, because if she doesn’t get it fixed, it could lead to a whole host of problems beyond just problems breastfeeding.

She’ll have scoliosis, and she’ll suffer from migraines, and she’ll never eat, and she’ll have a speech impediment, and she won’t sleep — I mean, just, like, the long list of things over the phone.

And Lauren starts panicking.

I mean, first of all, I felt — I’ve never felt more terrible in my life than that first day or so after giving birth. Like, the comedown from the hormones, the drugs — all of it — the sleep deprivation. And then, here was this baby we’d wanted, we were told we probably would never have after one miscarriage. And she’s so perfect, like, the most beautiful baby I’d ever seen. And you think that she has some deformity that’s going to ruin her.

But Melanie says it’s OK. She has a solution. And she tells Lauren that there’s a dentist in town who can handle cases that are as severe as June’s.

A dentist? Why a dentist?

Well, there’s a procedure that’s done in a dentist’s office that’s a laser surgery. And dentists use this high-powered laser machine that can quickly cut the flesh that connects the lips and the cheeks to the gums. So according to Lauren, Melanie tells her that by chance, this dentist has an opening, because she said a family coming in from Oregon had just canceled their Saturday appointment.

So I thought, OK, wow, people are coming in from Oregon to see him. So we talked about it. We both felt unsure. But we said, well, let’s at least take the appointment, and then we can at least meet with the dentist, and also, someone can look at her mouth and assess.

And so Lauren agrees to go in and meet the dentist.

Like, I think some people, when they hear this story, think, like, why would you believe that? It just sounds so scammy. But to me, there is a lot of things that you hear in the hospital that sound insane. Like, it’s no different than someone saying, like, your baby’s orange because their bilirubin levels are too high, so we got to go put them under these lights. Like, that sounds insane. That sounds more insane than, your baby’s having a hard time eating because their tongue is too tight and it needs to be cut. Like, that seems rational, actually.

And all of this seemed really weird to Lauren at the time. But in the context of the hospital and having a baby, lots of things about health care are weird.

So one day after they got back home from the hospital, Lauren and her husband pack up the car and go to the office early in the morning.

You know, I was wearing my hospital diaper and an ice pack, took the elevator up to his office, and —

And what happens?

So Melanie greets them at the door. They sign some paperwork, and pretty soon, the dentist, Dr. Samuel Zink, arrives.

And then, he, like, very briefly — very briefly — looks in her mouth and is like, yeah, she’s got whatever — however he classified it — grade 4 or whatever he says — class 4 — and she has a lip tie, which — that had never been mentioned to us before, so it’s very much on the spot, this new piece of information.

You know, pretty quickly, the dentist diagnosed June as having a couple of ties. He confirmed that she had a tongue tie, and he said it was severe. He also said that she had tightness under her top lip, called a lip tie. And so the baby actually needed to get two cuts. And again, Lauren said that the dentist and the consultant told her how important it was for her to do this for her baby.

One of us says, like, what happens if we don’t do the procedure? , Like what are our alternatives? And it was basically like, there’s no alternative. Like, you have to do this. Otherwise, again, long —

So Lauren and her husband decided to do it. But before the procedure starts, Melanie actually stopped Lauren from coming into the room.

Melanie turned around and put a hand on my shoulder and said, oh, no. And I said, oh, am I not going with you? She goes, well, we can’t tell you no, but if you hear her cry, it’ll impact your milk supply, like, adversely.

What do I know? So I said, oh, OK. And she pulled out the white-noise machine and said, what do you want to listen to? And I had no idea what she was talking about. I had no idea what it was. And so then she just turned it on — white noise — and left.

What happens next is, Melanie turns on a white-noise machine in the room.

And that was the moment that I was like, get your baby and get out of here. And I didn’t listen to it. It was like all of my mom intuition firing, being like this isn’t right, you know. It’s like, I don’t know how to describe it, but your full body — you have to get your baby and get out of here. And I just ignored it.

She said her maternal instincts really kicked in, and she just had this instinctive fear about the procedure and whether June would be OK. But the procedure itself was very quick. Within just a couple of minutes, Melanie returns with June.

And she was screaming. Like, screaming, and so worked up. This was, like, hysterical, inconsolable. And she was also choking on something, like, gagging.

And June was so worked up. Lauren had only had her for a couple of days, but she said that this was on a different level than any other way she had ever seen June crying. And June just wouldn’t stop crying.

And she looked over to Melanie, and Lauren said that she remembered Melanie saying this was very typical. And so they pay the dentist. They pay $600 for the procedure, and then they go home.

Over the next several days, June did not get better as Melanie had assured them. You know, she was basically inconsolable, Lauren said — just crying hysterically. And Lauren and her husband — they don’t know how to comfort her. They’re new parents. They’ve only had a baby a couple of days. And they’re almost beside themselves.

There was nothing we could do. And I remember finally, I said, like, this is not normal. We’re going to an emergency room.

And they decided to go to the emergency room, where a doctor looks inside June’s mouth and finds a large sore in her mouth that he says is probably causing her so much pain.

And so he said, you know, it breaks my heart to see a sore that big in a baby this small. It was like the floodgates opened, and there was nothing but guilt and shame. Like, unmanageable guilt and shame.

Like, what have we done? Who are these people? What have I done to my baby? Will she ever be the same? Like, what did I do?

So at this point, Lauren is really understanding that her intuition about this surgery was probably right and that she and her husband may have really made a mistake with this. What does June’s recovery look like?

So June never did end up breastfeeding successfully, which was the main reason why Lauren and her husband had decided to do this procedure.

That was the whole point, right?

That was the whole point. Right. And over the next couple of years, June had a number of issues that there’s no official medical diagnosis for, but Lauren has attributed a lot of her behaviors to what had happened to her when she was just a few days old.

I mean, you couldn’t close a fridge door too loud, or else it would set her off. Or, we would attempt to take her for a stroller walk on the Greenbelt, which is the walking path, and she’d be asleep in her car seat, you know, stroller, and someone would try to pass us on their bike and ring their bell, and it would startle her, and it would just set her off. So she just was very, very, very fragile.

So Lauren just wanted to get answers, and she really wanted to hold Melanie and the dentist accountable. So she gathered all of the paperwork that she had — texts, emails, other correspondence — and she went to the Idaho Board of Dentistry, where she filed a complaint against the dentist. And then, she also went to a professional organization that certifies lactation consultants and filed a complaint with them as well.

And did she get anywhere with either of them?

At first, no. The Idaho dentistry board didn’t want to investigate, and Lauren appealed, and she lost her appeal. And she didn’t initially hear back at all from the lactation board.

No one wanted to take responsibility. That’s the thing. No one wanted to stick their neck out there. What’s the alternative? The story never gets told?

And that’s when she decided to reach out to us. And after our story came out, the lactation board finally told Lauren that they were investigating Melanie.

And Katie, you guys were reporting the story. I’m assuming you reached out to both the dentist and to Melanie. What did they say?

Beyond a very brief phone conversation that I had with Melanie in which she defended her work and she said that she had a number of very satisfied customers, she didn’t respond to detailed questions about Lauren’s story or the stories of her former clients. And Dr. Zink did not respond to our requests for comment, but he did tell the dentistry board that Lauren’s baby’s procedure was uneventful and that an extremely small percentage of patients do not respond well to the procedure.

And how big of an issue is this, Katie? I mean, how common is it for mothers to have an experience like Lauren’s?

So after we got the tip from Lauren and we dug deeper into her story, we found ourselves really surprised by how big this industry was for tongue-tie releases. And in part, it’s been driven by this movement for breastfeeding and the Breast is Best campaign and a growing number of parents who are choosing to breastfeed their children.

In turn, that has sparked this big boom in tongue-tie releases. One study that we found showed that these procedures have grown 800 percent in recent years.

Yeah. And also, as we started talking to other parents around the country, we learned that some of them had similar stories to what Lauren had told us. There’s plenty of instances where there’s no harm done to the baby at all when they get these procedures.

But we also found cases where babies were harmed, you know, where they developed oral aversions, which basically means that they don’t want to eat because they fear having anything put in their mouth, including a bottle. We found cases where babies became malnourished, had to be hospitalized. We found more than one instance in which babies had to be given a feeding tube just weeks after the procedure.

So these sounds so painful and awful for a newborn — these problems. But I guess there’s always a risk, Katie, in any medical procedure, right? I mean, how much of this is just the risk you sign up for when you agree that your baby should have a surgery?

Well, that’s true. I mean, there’s always a risk. But what you’re supposed to do is weigh the risks against what the potential benefits of a procedure are. And when we really started drilling down into what those benefits were and into the medical research, we found there just wasn’t a lot of potential benefit for these procedures, if at all, in many cases.

Really? So the procedures don’t have a medical reason to exist?

That’s right. We reviewed all of the best-quality medical research on this. And other than that old-fashioned snip under the tongue, which does show that in some cases, it can reduce pain for breastfeeding mothers, but otherwise, all of this growth and all of these other more invasive procedures — we found there just wasn’t good evidence that they helped babies. And the more we looked into tongue ties and started to connect it to the other reporting we were doing, we started to realize that it was driven by some really big forces in our health care system that really had the potential to harm patients.

We’ll be right back.

So Katie, we talked about this new surge in a procedure that surgically unties infants’ tongues from the bottom of their mouths, often needlessly, sometimes even harmfully. And you said your reporting found that this surgery was actually part of a broader trend. Tell me about this trend and what’s driving it.

So that’s what this investigation was really about — to find the procedures that are doing unnecessary harm to patients and to really understand why this is happening. You know, like, what’s driving the prevalence of these procedures? And there’s just a lot of unnecessary surgeries out there, but we decided to center our reporting on three particular surgeries that had the potential to harm patients, in addition to tongue ties. We focused on a particular hernia surgery, a bariatric surgery, which can be overdone and cause harm, and a vascular surgery done on patients’ legs to help us understand the forces that were at work that were driving all of this.

And what did you find when you dug deeper into those surgeries?

Well, it’s very complex, but we ultimately found three main drivers that were underlying all of these. First, there’s a financial incentive for the doctors to perform these surgeries. There’s also a real push from the medical device companies that make these surgeries possible. And last, there’s a huge information void for solid medical advice that a lot of these doctors and companies take advantage of in order to push the surgeries.

OK, so let’s start with the money, Katie. How exactly is that incentivizing doctors to perform a lot more of these procedures? Like, what are the mechanics of that?

So the reality of our health care industry today is that in many places, even in places like non-profit hospitals, the doctors who work there are not getting a salary, a straight salary that’s just kind of, you get paid for showing up to work that day. Instead, they’re actually getting paid based on the procedures that they’re doing, how complex those procedures are, or possibly how lucrative.

And it’s not every doctor. There are still doctors that get paid salaries. But it’s increasingly the case that doctors have — at least a part of their pay is tied to the procedures that they’re doing.

Interesting. So the procedure is growing in importance in terms of actual compensation for doctors.

Right. I mean, in part, it’s kind of baked into the health care system that we’ve always had. You can even think about it as the small-town doctor who operated his own independent practice or her own independent practice. It’s essentially a small business, and they would get paid based on the patients that they saw.

But increasingly, even in, for example, large hospital systems where you might think that a doctor is just getting paid a salary to work in a hospital, in fact, a chunk of their bonus, for example, can sometimes be tied to the procedures that they’re doing, and that is increasingly the case.

Interesting.

And so one particularly egregious example of this was at a hospital that’s in New York — Bellevue Hospital. And basically, what my colleagues found there was that they had basically turned their surgery department into an assembly line for bariatric surgery, which makes your stomach smaller and can lead to weight loss. But what we found was that they were greenlighting patients that, basically, didn’t meet the qualifications for the surgery, which is a serious surgery. And what they found was that there were several situations where people had very serious outcomes as a result of getting the bariatric surgery there.

OK, so this is an extreme case of a hospital turning to a particular surgery to drive profits. And it wasn’t uncommon in your reporting, it sounds like.

No, it wasn’t the only example, but it was the most striking. And when we reached out to Bellevue, they defended their work, and they said that their practices were helping patients who wouldn’t otherwise get care. But our reporting was pretty conclusive that the program was churning through a record number of surgeries.

So what else was driving this increase in harmful surgeries that you guys found?

So we found it wasn’t just the hospitals who were benefiting. The other major player that benefits are these companies that are making the tools and the products that doctors are using during the procedures. And in order for them to sell more of their products, a lot of time, what they end up doing is promoting the procedures themselves.

So like medical device makers, like the company that made the laser in June’s surgery.

Right. And they do this in a number of ways. They’re giving them loans to help them buy the equipment, and in some cases, they’re even lending them money to help set up those clinics where the procedures are performed.

So they’re really underwriting these doctors so that they can perform more surgeries and, ultimately, sell more machines.

Yes. And the other things that they do is — the laser companies, for example — they will host webinars where they will have dentists who frequently perform these procedures show other dentists how to do the procedures. We even discovered this conference that was created by one of the laser companies, and it had kind of a wild name. The name of the conference was Tongue Ties and Tequilas.

(CHUCKLING) Right. It brought in dentists to talk about how to make money off the procedures. You know, how to promote themselves on social media, how to actually perform the procedures, and of course, when they were all done, they got to celebrate with an open tequila bar.

OK, so a lot of this really amounts to these companies trying to popularize these procedures, basically, like, to get the word out, even if the procedures don’t really work or, in some cases, cause harm.

Right. But they also play a big role in the other factor that’s driving a lot of this, which is the information that they put out there about the surgeries. These companies often sponsor research, which doctors often rely on to guide their practices. And part of what we’ve found is that it can create this echo chamber where doctors feel more comfortable and justified in doing these procedures when they have this whole alternate universe that is telling them that it’s OK to do these procedures, and in fact, it’s beneficial to patients.

So tell me about this echo-chamber effect.

The best example of this we found was a doctor in Michigan named Dr. Jihad Mustapha. He calls himself “the Leg Saver.” And what we found was that he and several other doctors do these procedures called atherectomies, which is basically like inserting a tiny roto-rooter inside an artery to get the blood flowing.

And Dr. Mustapha in particular was not only a very prolific performer of these procedures, but he actually founded his own medical conference, and he even helped start a medical journal that was devoted to using these procedures. And you know, tongue ties — there’s really no good evidence that these are actually beneficial to patients. And in fact, despite his nickname as “the Leg Saver,” one insurance company told Michigan authorities that 45 people had lost their limbs after getting treated at Dr. Mustapha’s clinic over a four-year period.

45 people lost their limbs?

I mean, that is the ultimate version of harm, right?

Right. Now, he did speak to us, and he defended his work and said that he treats very sick people. And despite his best efforts, some of these patients are already so sick that they sometimes lose their limbs.

And how much did he receive for each procedure?

Doctors like him typically receive about $13,000 for each of these atherectomy procedures.

But we found that misinformation, or poor information, also applied when doctors were learning new types of surgeries.

Really? Like how?

So one of the areas we looked at was the area of hernia surgery that I mentioned. And there’s a particular type of surgery. It’s a very complex version of a hernia surgery, called component separation. And the expert surgeons that we spoke to said that it’s difficult to learn, and you have to practice it over and over and over again to get it right. But one recent survey of hernia surgeons said that one out of the four surgeons had taught themselves how to perform that operation.

Yeah, not by learning it from an experienced surgeon but by watching videos on Facebook and YouTube.

I mean, how unusual is that? I guess, to me, it strikes me as very unusual. I mean, I think of learning about how to take my kitchen faucet apart on YouTube, but I do not think of a doctor learning about how to perform a surgery on YouTube.

Right. And it has actually become increasingly popular in recent years, and there’s not good vetting of the quality of the instruction. We even found videos on a website run by a medical device company that was intended to be a how-to for how to do these surgeries, but the video contained serious mistakes.

Wow. And Katie, all of these videos — some of them with serious mistakes — I mean, is this something that would be subject to medical regulators? Like, is there any kind of rules of the road for this stuff?

You know, there’s less than you would expect. Sometimes hospitals have rules about what sort of education their doctors need before performing a surgery. But we were surprised that there was a lot less regulation than we thought there would be and much less vetting of these videos than we anticipated.

So essentially, what you found was this complex, oftentimes interconnected, group of forces — device companies pushing their products, hospitals bolstering their bottom line, and rampant misinformation that, as you said, all really trace back to the same motivating factor, which is money. But wouldn’t the fear of being sued for medical malpractice prevent a lot of this behavior?

You know, this kept popping up during the course of our reporting. I do think we have this idea that any time a doctor does anything wrong, they’re going to get sued. But it just wasn’t always the case in our reporting. There’s a lot of statutes of limitations, time limits on when somebody can file a lawsuit, and other ways that make it somewhat hard to really hold a doctor accountable.

One example is the regulatory organizations that oversee doctors. The one doctor that I mentioned earlier — Dr. Mustapha — state investigators had found that his overuse of procedures had led people to lose their legs. And yet, he ultimately settled with the state, and he was fined $25,000. That actually adds up to about two of these atherectomy procedures.

So it sounds like malpractice is not necessarily going to be the route to rectifying a lot of this. But I guess I’m wondering if the federal government could actually rein some of this in before the patients are harmed.

It’s possible. But this is just a very difficult issue. Some of the themes that we explored in this reporting are really just firmly embedded in our health care system in the way that it works. The fact is that we have a for-profit health care system, right? So everyone, from doctors to hospitals to the device companies, benefit when more procedures are done. All of the incentives are pointing in the same direction.

And so trying to find one or two simple solutions will probably not easily fix the issue, as much as we all hope that it could.

So is the lesson here, be much more discriminating and vigilant as a patient? I mean, to get a second opinion when you’re standing in front of a doctor — or a dentist — who’s telling you that you or your baby needs a procedure?

Yes. I think that is one of the takeaways. But look, we understood that even reporting on all of this was risky, because people could hear about these harmful surgeries and start wondering if everything that their doctors tells them is a scam. And of course, while some of these procedures are harmful, a lot of procedures are lifesaving. But ultimately, for now, patients are kind of left on their own to navigate what’s a pretty complex and opaque health care system. When you have somebody standing in front of you saying, you should do this, it can be very confusing.

And this is something that Lauren talked a lot about — just how confusing all of this was for her.

There’s a lot of information that you’re getting that is truly like someone is speaking a foreign language. And because they do it all day long, it’s not user-friendly. Like, it isn’t designed for the comfort or understanding of the person receiving the information.

There is so much blind trust and faith that you have in the system, in the providers who are giving you this information. You trust, like, this is what they do all day long. So there is no real reason to question. That is the system that we have in this country.

Katie, thank you.

Here’s what else you should know today. On Friday, the Russian authorities announced that opposition leader Alexei Navalny died in prison. He was 47.

Navalny, a charismatic anti-corruption activist, led the opposition to Vladimir Putin for more than a decade. His popularity was broad, extending far outside the realm of liberal Moscow. And that proved threatening to the Russian authorities, who attempted to poison him in 2020.

Navalny survived and later extracted a confession from his would-be assassin on tape. Navalny believed that Russia could be a free society, and he had the extraordinary ability, through sheer force of his personality, charisma, and confidence, to get others to believe it, too. Though he had been in prison since 2021, his death still came as a shock.

[SPEAKING RUSSIAN]

His wife, Yulia Navalnaya, made a surprise appearance at a security conference in Munich shortly after the Russian authorities announced her husband’s death.

She received an emotional standing ovation.

In Moscow, my colleague, Valerie Hopkins, spoke to Russians who were placing flowers in his honor —

— and expressing disbelief that he was gone.

Then I asked them if they believe in the beautiful Russia of the future that Navalny talked about. And they said, yes, but we don’t think we will survive to see it.

At least 400 people have been detained since his death, including a priest who had been scheduled to hold a memorial service in Saint Petersburg.

Today’s episode was produced by Asthaa Chaturvedi, Diana Nguyen, Will Reid, and Alex Stern, with help from Michael Simon Johnson. It was edited by Michael Benoist, with help from Brendan Klinkenberg, contains original music by Diane Wong and Dan Powell, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Sabrina Tavernise. See you tomorrow.

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Hosted by Sabrina Tavernise

Featuring Katie Thomas

Produced by Asthaa Chaturvedi ,  Diana Nguyen ,  Will Reid and Alex Stern

With Michael Simon Johnson

Edited by Michael Benoist and Brendan Klinkenberg

Original music by Diane Wong and Dan Powell

Engineered by Alyssa Moxley

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A Times investigation has found that dentists and lactation consultants around the country are pushing “tongue-tie releases” on new mothers struggling to breastfeed, generating huge profits while often harming patients.

Katie Thomas, an investigative health care reporter at The Times, discusses the forces driving this emerging trend in American health care and the story of one family in the middle of it.

On today’s episode

college math problems hard

Katie Thomas , an investigative health care reporter at The New York Times.

A woman holding a toddler sits on a bed. The bed has white sheets and pink pillows.

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Inside the booming business of cutting babies’ tongues .

What parents should know about tongue-tie releases .

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Super Bowl 2024: For Taylor Swift, biggest problem getting to Vegas to see Chiefs isn't the flight from Tokyo

This may be the biggest first-world problem that has ever existed.

swiftkelce.jpg

When it comes to the Super Bowl , it turns out that Taylor Swift's biggest challenge isn't going to be making it to the game on time from Tokyo. As we established earlier this week , Swift, who has a concert in Japan on Feb. 10, will have plenty of time to make it to Las Vegas for the big game thanks to the magic of the international dateline. 

Since getting to Vegas likely won't be an issue for Swift, that means her biggest problem might be trying to get her plane parked. In what might possibly be the biggest first-world problem that has ever existed, all the parking spaces for private planes in Las Vegas are currently booked, according to the Associated Press . 

There are four airports in the immediate Las Vegas area and those four airports have a combined 475 parking spaces, but every single of those spaces is spoken for as we get ready to head into Super Bowl week. 

Harry Reid International Airport is the main hub in Las Vegas: If you're flying commercial to the Super Bowl, that's almost certainly where you'll be landing. The city has several smaller executive airports -- with one in North Las Vegas and one in Henderson -- but those are booked for Super Bowl weekend. 

The problem for Swift is that the Super Bowl won't be the only sporting event in town during the week of the big game. LIV Golf is holding a tournament at Las Vegas Country Club and the Saudi-backed golf tour has likely booked several parking spaces for the event. The only upside for Swift is that the tournament ends on Saturday and there's a good chance that not everyone will want to stay for the Super Bowl, which means a few spaces could open up. 

Assuming she flies straight from Tokyo after her Feb. 10 concert, Swift is expected to arrive in Vegas late Saturday night after the conclusion of the LIV tournament, although that won't guarantee her a spot. 

Of course, this is Taylor Swift we're talking about, so there's a good chance that someone will make something happen. 

Michael Giordano, a partner at  Cirrus Aviation Service , a company that has a few spaces in Las Vegas, told the New York Post that Swift will definitely have a place to land. 

"Taylor Swift will definitely have a spot through the NFL to land, but not necessarily a place to park," Giordano said. 

That's what's known as a "drop-and-go" flight and if that happens, Swift and her crew will be dumped off in Vegas, but the plane will have to fly to an airport outside of Vegas to park. The NFL works with the FAA and local authorities for the week of the Super Bowl, so the league almost certainly has the power to find a landing spot for Swift, but an NFL spokesman wouldn't confirm any individual requests when asked by the AP. 

The most likely scenario here is that Swift's private jet will drop her off in Vegas and then head somewhere else to park. At that point, Swift will attend the game and then hang out with Travis Kelce after the Super Bowl, which mean she likely won't need her jet again until she heads to Australia for her next concert. Swift has a tour date in Melbourne on Feb. 16, so she'd likely head to the land down under on Feb. 13, so her jet could theoretically sit in Los Angeles or another airport until then. 

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