15 Hardest SAT Math Questions in 2023-2024
July 1, 2023
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
|Time Allotted||55 minutes|
SAT Math: Calculator Inactive
|Time Allotted||25 minutes|
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?
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?
A) 3 and 5
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?
- A) 12
- B) 15
- C) 20
- D) 22
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
|5 out of 5||4 out of 5||3 out of 5||2 out of 5||1 out of 5||0 out of 5||Total|
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.
|Region||Number of Earthworms||Region||Number of Earthworms|
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?
- A) 150
- B) 1,500
- C) 15,000
- D) 150,000
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.
Let s1 and r1 be the standard deviation and range, respectively, of the data before exercise, and let s2 and r2 be the standard deviation and range, respectively, of the data after exercise. Which of the following is true?
- s1 = s2 and r1 = r2
- s1 < s2 and r1 < r2
- s1 > s2 and r1 > r2
- s1 ≠ s2 and r1 = r2
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.
Hardest SAT Math Problems (Continued)
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
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: