The Hardest SAT Math Questions & How to Solve Them

You must have heard that if you can handle the most complicated part of an exam, you will tackle the rest of it with no problem. It’s hard to argue with that: the toughest questions often slow you down and lower your confidence. 

When it comes to the SAT, the challenge often lies in conquering the most intricate math questions. Yet, there is nothing to worry about. The hardest SAT questions in this scenario may seem to you just a myth! If you prepare well, know what to expect on the test day, and keep practicing, it will go easy. Keep reading to learn more about SAT math questions and check the examples of hard math problems with answers.  

The SAT Math Section 

On the upcoming digital SAT (starting in spring 2024), you will face two 35-minute math modules with 22 questions each. The difficulty of the second module tasks depends on how well you did in the first one. It is the feature of the new, adaptive SAT. If you struggle in the first set, the second set will be easier. Of course, you should take your time to learn more about the other sections as well. 

The SAT math section includes:

• Algebra (?35%). Here, you will focus on linear equations or systems of equations, inequalities, graphs, and systems.
• Advanced Math (?35%). It involves identifying and creating equivalent expressions, quadratic and nonlinear equations/functions, and their graphs.
• Problem-solving and Data Analysis (?15%). In this section, be ready to solve questions on ratios, proportions, percentages, units, graphical data, probabilities, and statistics.
• Geometry and Trigonometry (?15%). This part offers you to solve problems that focus on perimeter, area, and volume; angles, triangles, and circles.

Hard Math Questions: Everything You Need to Know to Solve Them

Understanding the types of questions from each section is an important part of getting ready to solve them. If you face even the medium-difficulty questions for the first time on the exam, it could take you longer to solve them. Let’s consider one question from each section.

Algebra

Task example:

In 2014, county X had 783 miles of paved roads. Since 2015, the county has been building 8 miles of new paved roads each year. At this rate, if n is the number of years after 2014, which of the following functions f indicates the number of miles of paved road that will be in county X? (Assume that no paved roads go out of service).

A) f (n) = 8 + 783n

B) f (n) = 2,014 + 783n

C) f (n) = 783 + 8n

D) f (n) = 2,014 + 8n

The hardest algebra question may look like this one. To solve it, you’ll need to take the following steps:

1. Figure out what the question is about and give names to the things involved using letters (like x or y).
2. Write down math sentences or symbols that show how these things are related in the question.
3. Solve these math sentences to find the letters’ values.
4. Think about what these values mean in the context of the question and use your conclusions to answer the question.

Advanced Math

Here is an example of the question from this section: 

For a certain reservoir, the function f gives the water level f (n) to the nearest whole percent of capacity, on the nth day of 2016. Which of the following is the best interpretation of f (37) = 70?

A) The water level of the reservoir was at 37% capacity for 70 days in 2016.

B) The water level of the reservoir was at 70% capacity for 37 days in 2016.

C) On the 37th day of 2016, the water level of the reservoir was at 70% capacity.

D) On the 70th day of 2016, the water level of the reservoir was at 37% capacity.

The right answer is going to be C. Let’s figure out why that is. So, the function f gives the water level on the nth day of 2016. The statement “f(37) = 70” implies that on the 37th day of 2016, the water level of the reservoir was at 70% capacity. In other words, if you input the value 37 into the function f, it outputs the value 70, representing the water level percentage on that specific day.

Problem-solving and Data Analysis

Take a look at the task below:

Source: SATsuite

The correct answer is B. Why? Let’s try to understand. The graph shows the changes in Maria’s speed. The horizontal segment of the graph, which shows Maria’s movement from 19,8 to 34,8 minutes, is the period when Maria was moving at 8 miles per hour (mph), which is the highest speed she had. 

Choice A shows that Maria’s speed was increasing from 6 to 8 mph between 17,6 and 19,8 minutes of her route. Choice C indicates that the speed was decreasing from 8 to 6 mph during the correspondent period. While the answer D shows the longest segment when Maria was going at the same speed (6 mph). So, choice B is the only correct answer.

Geometry and Trigonometry 

For questions in the Geometry and Trigonometry content area, students will deal with problems related to:

• Finding the area and volume of shapes.
• Working with angles, triangles, and trigonometry.
• Solving problems involving circles.

Here’s an example of the test: 

Two nearby trees are perpendicular to the ground, which is flat. One of these trees is 10 feet tall and has a shadow that is 5 feet long. At the same time, the shadow of the other tree is 2 feet long. How tall, in feet, is the other tree?

A) 3

B) 4

C) 8

D) 27

Choice B is the right answer. We can think of each tree and its shadow as making a right triangle. The tree’s height and the length of its shadow are like the sides of this triangle. When we look at two nearby trees, the triangles they make with their shadows are similar. 

So, if one tree is 10 feet tall and its shadow is 5 feet long, and we want to find out the height of the other tree (let’s call it “x“), we can set up a proportion like this: 10/5 = x/2. This means 2 = x/2. As a result, we get 4 = x. So, the other tree is 4 feet tall.

Remember – the concept of proportion means that two ratios are equal.

How to Answer Grid-Ins?

Successfully preparing for the SAT is possible, even if you have a limited amount of time for that. However, you must know well what to expect at each stage of your exam. For example, most questions on the math test have multiple-choice answers. Yet, around 22% of them are different. These are called grid-in questions. Instead of picking an answer from a list, you have to solve the problem and write your answer in a grid on the answer sheet.

Here are some important things to remember about this type of question:

• Mark only one bubble in each column;
• Only answers with filled-in bubbles will count. Don’t write anything in the boxes above the bubbles;
• Start entering your answers in any column, as long as it’s within the grid area, to get credit;
• The grid can only hold up to four decimal places; 
• You can use only positive numbers or zero;
• Unless a problem says otherwise, you can write your answer as a decimal or a fraction on the grid.
• You don’t have to simplify fractions like 3 over 24; just write them as they are. 

To lessen the number of potential problems on the SAT, you should know well how to answer problems from a technical point of view. 

Conclusion

Preparing for the SAT requires a lot of practice. Remember, in order to do better, you can always consider tutoring. The hard SAT questions should not scare you but motivate you to prepare better. Make sure you know well where you may fail and work on those areas. Besides, if you know what to expect from the hardest SAT math problems, you will be more confident, which is sometimes as important as knowing how to handle hard tasks. All the luck!