Are you going to take the next SAT?
If yes, then you might want to increase your overall skills, knowledge, and confidence in academics. Or maybe you want to get into your dream university.
Whatever the case, Math has to be the hardest nut to crack in SATs. However, with a little practice, you can go a long way.
So, to begin your practice journey, the first thing you need to do is to stop focusing on the problems you know how to tackle. The main concern should be the “hard” questions that appear in the exam.
To improve your math practice, we have shared the top-most hard math questions that usually appear in SATs — along with their answers.
The SAT always includes two math sections, with the first one not allowing a calculator and the second one allowing it. Each section has 15 questions, with the difficulty increasing from the easiest (question 1) to the hardest (question 15).
However, for the second section, the difficulty resets for the grid-in questions. Generally, the hardest math problems are found towards the end of each section.
Here’s a breakdown of the SAT Math section:
Heart of Algebra: This section constitutes 33% of the test, with 19 questions focusing on linear equations, inequalities, graphs, and systems.
Problem Solving and Data Analysis: Making up 29% of the test, this section includes 17 questions related to ratios, proportions, percentages, units, analyzing graphical data, probabilities, and statistics.
Passport to Advanced Math: Comprising 28% of the test, this section contains 16 questions centered around identifying and creating equivalent expressions, quadratic and nonlinear equations/functions, and their graphs.
Additional Topics in Math: Accounting for 10% of the test, this section has 6 questions covering various topics like geometry, trigonometry, radians and the unit circle, and complex numbers.
Grid-in Questions: These questions, found at the end of each portion of the math test, both with and without a calculator, can test any of the above topics. They are generally considered more challenging than multiple-choice questions because you can't rely on answer choices. So, it’s really important that you improve your score in this section.
Practice Questions
From every category we have shared the most important and hardest questions you can find in your SAT Math exam.
So, grab your notebook and pencil and get ready to practice the following questions now!
Question 1:
𝐶=5/9 (𝐹−32)
The equation above shows how temperature 𝐹, measured in degrees Fahrenheit, relates to a temperature 𝐶, measured in degrees Celsius. Based on the equation, which of the following must be true?
I) A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of 5/9 degree Celsius.
II) A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.
III) A temperature increase of 5/9 degrees Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.
A) I only
B) II only
C) III only
D) I and II only
Answer: D
Explanation: Think of the equation as an equation for a line
𝑦=𝑚𝑥+𝑏
where in this case
𝐶=5/9(𝐹−32)
or
𝐶=59𝐹−59(32)
You can see the slope of the graph 59, which means that for an increase of 1 degree Fahrenheit, the increase is 59 of 1 degree Celsius.
𝐶=59(𝐹)
𝐶=59(1)=59
Therefore, statement I is true. This is the equivalent of saying that an increase of 1 degree Celsius is equal to an increase of 95 degrees Fahrenheit.
𝐶=59(𝐹)
1=59(𝐹)
(𝐹)=95
Since 95 = 1.8, statement II is true.
The only answer that has both statement I and statement II as true is D, but if you have time and want to be thorough, you can also check to see if statement III (an increase of 59 degree Fahrenheit is equal to a temperature increase of 1 degree Celsius) is true:
𝐶=59(𝐹)
𝐶=59(59)
𝐶=2581(which is≠1)
An increase of 59 degrees Fahrenheit leads to an increase of 2581, not 1 degree, Celsius, and so Statement III is not true.
Pro Tip #1: If you can't answer a question, don't spend too much time on it. Skip it and come back to it later. Remember to mark it in your booklet so you don't forget.
Question 2:
The equation24𝑥2+25𝑥−47𝑎𝑥−2=−8𝑥−3−53𝑎𝑥−2 is true for all values of 𝑥≠2/a, where 𝑎 is a constant.
What is the value of 𝑎?
A) -16
B) -3
C) 3
D) 16
Answer: B
Explanation: There are two ways to solve this question. The faster way is to multiply each side of the given equation by 𝑎𝑥−2 (so you can get rid of the fraction). When you multiply each side by 𝑎𝑥−2, you should have:
24x2+25𝑥−47=(−8𝑥−3)(𝑎𝑥−2)−53
You should then multiply (−8𝑥−3) and (𝑎𝑥−2) using FOIL.
24x2+25𝑥−47=−8𝑎x2−3𝑎𝑥+16𝑥+6−53
Then, reduce on the right side of the equation
24x2+25𝑥−47=−8𝑎x2−3𝑎𝑥+16𝑥−47
Since the coefficients of the x2-term have to be equal on both sides of the equation, −8𝑎=24, or 𝑎=−3.
The other option, which is longer and more tedious, is to attempt to plug in all of the answer choices for a and see which answer choice makes both sides of the equation equal. Again, this is the longer option, and I do not recommend it for the actual SAT as it will waste too much time.
Question 3:
Line ℓ is graphed in the xy-plane below.
If line ℓ is translated up 5 units and right 7 units, then what is the slope of the new line?
25
−32
−89
−1114
Answer: B
Explanation: The slope of a line can be determined by finding the difference in the y-coordinates divided by the difference in the x-coordinates for any two points on the line. Using the points indicated, the slope of line ℓ is −32. Translating line ℓ moves all the points on the line to the same distance in the same direction, and the image will be a line parallel to ℓ. Therefore, the slope of the image is also −32.
Pro Tip #2: Start with the easier questions first. This way, you'll have more time to tackle the tougher ones later on.
Question 4:
The average number of students per classroom, y, at Central High School can be estimated using the equation y = 0.8636x + 27.227, where x represents the number of years since 2004 and x ≤ 10. Which of the following statements is the best interpretation of the number 0.8636 in the context of this problem?
Answer: 4
Explanation: When an equation is written in the form y = mx + b, the coefficient of the x-term (in this case, 0.8636) is the slope of the graph of this equation in the xy-plane. The slope of the graph of this linear equation gives the amount that the average number of students per classroom (represented by y) changes per year (represented by x). The slope is positive, indicating an increase in the average number of students per classroom each year.
Question 5:
𝑄(𝑥) =P(2x)2
In the equation shown above, Q and P are functions. If (x0, y0) is a point on the graph of y = 𝑄(𝑥), which of the following is a point on the graph of y = P(x)?
A) (x0, y0)
B) (2x0, y02)
C) (x02, 2y0)
D) (2x0, 2y0)
Answer: D
Explanation: Don't be confused by the notation. Plug in 𝑥0 as if it were a number:
Q(x0) =P(2x0)2
Since
𝑄(x0)=𝑦0
y0 = 𝑃(2𝑥0)2
2y0=𝑃(2𝑥0)
Stop and take a look at what we have. What does this mean? When we plug in 2x0 into 𝑃(𝑥) we get 2y0. That means 2x0,2y0 is a point on 𝑦=𝑃(𝑥)
By the way, don't get confused by the notation 𝑦=𝑃(𝑥). It just refers to the function being a graph in the 𝑥𝑦-plane, much like 𝑦=𝑥2+1 is a function that can be graphed in the 𝑥𝑦-plane, with 𝑥 representing the input and 𝑦 representing the output. Here, 𝑥0 and 𝑦0 are the actual constants you actually want to be working with. It would have been pointless to do something like
𝑄(𝑥) =P(2x)2
y = 𝑃(2𝑥)2
where you're substituting 𝑄(𝑥) with 𝑦. While it's true that 𝑦 in this case would represent the output of 𝑄(𝑥), it doesn't lead anywhere. Think of 𝑦=𝑃(𝑥) and 𝑦=𝑄(𝑥) as more of a notational thing to designate that they are functions that have graphs in the 𝑥𝑦-plane.
Question 6:
The recommended daily calcium intake for a 20-year-old person is 1,000 milligrams (mg). One cup of milk contains 299 mg of calcium, and one cup of juice contains 261 mg of calcium. Which of the following inequalities represents the possible number of cups of milk, m, and cups of juice, j, a 20-year-old person could drink in a day to meet or exceed the recommended daily calcium intake from these drinks alone?
A. 299m + 261j ≥ 1,000
B. 299m + 261j > 1,000
C. 299m + 261j ≥ 1,000
D. 299m + 261j > 1,000
Answer: A
Explanation: Multiplying the number of cups of milk by the amount of calcium each cup contains and multiplying the number of cups of juice by the amount of calcium each cup contains gives the total amount of calcium from each source. You must then find the sum of these two numbers to find the total amount of calcium. Because the question asks for the calcium from these two sources to meet or exceed the recommended daily intake, the sum of these two products must be greater than or equal to 1,000.
Pro Tip #3: Since there's no negative marking in the SAT exam, make sure you answer every question! If a question seems hard, try to rule out some answer choices before making your choice.
Question 7:
A research assistant selected 75 undergraduate students at random from the list of all students enrolled in the psychology degree program at a large university. She asked each of the 75 students, “How many minutes per day do you typically spend reading?” The mean reading time in the sample was 89 minutes, and the margin of error for this estimate was 4.28 minutes. Another research assistant intends to replicate the survey and will attempt to get a smaller margin of error. Which of the following samples will most likely result in a smaller margin of error for the estimated mean time students in the psychology degree program read per day?
A. 40 undergraduate psychology degree program students selected at random
B. 40 undergraduate students selected at random from all degree programs at the university
C. 300 undergraduate psychology degree program students selected at random
D. 300 undergraduate students selected at random from all degree programs at the university
Answer: C
Explanation: Increasing the sample size while randomly selecting participants from the original population of interest will likely result in a decrease in the margin of error.
Read More: How to improve your math skills
Question 8:
A potato chip company produces the same number of snack-sized bags of potato chips every month. A manager at the company randomly selects a sample of snack-sized bags of potato chips to be weighed in a routine quality-assurance check each month. The January sample had a mean weight of 0.95 ounces and a margin of error of 0.06 ounces. The February sample had a mean weight of 0.98 ounces and a margin of error of 0.04 ounces. Based on these findings, which of the following is an appropriate conclusion?
A) Most of the snack-sized bags of potato chips produced by the company in January each had a weight of at least 0.95 ounces, while most of the snack-sized bags of potato chips produced by the company in February each had a weight of at least 0.98 ounces.
B) The mean weight of all the snack-sized bags of potato chips produced by the company in January must have been 0.03 ounces more than the mean weight of all the snack-sized bags of potato chips produced by the company in February.
C) The mean weight of all the snack-sized bags of potato chips produced by the company in January was 0.03 ounces less than the mean weight of all the snack-sized bags of potato chips produced by the company in February.
D) The number of snack-sized bags of potato chips in the January sample was less than the number of snack-sized bags of potato chips in the February sample.
Answer: D
Explanation: Answer (A) is wrong because the margin of error doesn't tell us anything about where most of the data are. We don't know whether most of the bags are above the mean or below the mean. Answer (B) is wrong because we can't infer from the samples that the mean for all the January bags was 0.03 ounces more than the mean for all the February bags.
In fact, the sample means suggest the opposite. In any case, what we got from the samples were estimated means. You can't know the exact mean for all bags unless you weigh all of them. Answer (C) is wrong for this same reason. Answer (D) is correct because the higher the sample size, the lower the margin of error.
Pro Tip #4: Don't spend too much time on one question. If you don’t know the answer, move on. Come back to the hard questions later on.
Question 9:
One evening, Maria walks, jogs, and runs for a total of 60 minutes. The graph above shows Maria's speed during the 60 minutes. Which segment of the graph represents the times when Maria's speed is the greatest?
A) The segment from (17, 6) to (19, 8)
B) The segment from (19, 8) to (34, 8)
C) The segment from (34, 8) to (35, 6)
D) The segment from (35, 6) to (54, 6)
Answer: B
Explanation: 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 increased 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 showed the longest segment when Maria was going at the same speed (6 mph). So, choice B is the only correct answer.
Question 10:
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
Answer: B
Explanation: 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.
Pro Tip #5: Understand how to work with the facts and formulas of basic statistical measures.
Question 11:
In triangle ABC, x < y < z. Side AC = 10 and side BC = 6. Which of the following could not be the perimeter of triangle ABC?
I. 22
II. 24
III. 26
I and II only
I and III only
II and III only
I, II, and III
Answer: B
Explanation: We use the rule that opposite the smallest angle is the smallest side of a triangle. Similarly, opposite the largest angle is the longest side of a triangle.
We know, since x < y < z, that angle x is the smallest angle. Thus, side BC, which is opposite angle x, is the shortest side of the triangle.
Similarly, we know that angle z is the largest angle, so side AC, which is opposite angle z, is the longest side of the triangle.
We know, therefore, that the length of side AB is between the lengths of BC and AC, so the length of side AB is between 6 and 10.
In a triangle, the longest side is opposite the greatest angle, and the shortest side is opposite the smallest angle.
We’ll use these facts to consider each possible perimeter.
I. Perimeter = 22
Perimeter = side AB + side AC + side BC
22 = AB + 10 + 6
AB = 6
We previously determined that AB must be any value greater than 6 and less than 10. Thus, side AB cannot be equal to 6. So, the perimeter of triangle ABC cannot be 22.
II. Perimeter = 24
Perimeter = side AB + side AC + side BC
24 = AB + 10 + 6
AB = 8
Since the length of side AB must be between 6 and 10, we see that a length of 8 for side AB is permissible. Thus, a perimeter of 24 is possible.
III. Perimeter = 26
Perimeter = side AB + side AC + side BC
26 = AB + 10 + 6
AB = 10
The length of side AB must be between 6 and 10. Thus, a length of 10 is not possible. The perimeter of triangle ABC cannot be 26
The perimeter of triangle ABC can be neither 22 nor 26.
Relevant Read: How MTS tutors can help you in improving your Mathematics?
Question 12:
The bar chart shows the amount (in millions of dollars) of income tax paid by six wealthy individuals in 2014.
If the 2014 income tax data for a seventh person, Ella Scott, were added to the group, the seven data values would have exactly one mode, and the mean, median, and mode would be equal. In 2014, how many dollars in income tax did Ella Scott pay?
A. 19 million
B. 20 million
C. 21 million
D. 22 million
Answer: D
Explanation: Let’s let x = the dollar amount (in millions) that Ella Scott paid in taxes in 2014. Thus, the values for taxes paid, in millions of dollars, are 6, 8, 10, 12, 12, 14, and x.
First, notice that since we’re told that there is exactly one mode, that mode must be 12. There are already two values of 12 in the set, and no other value occurs more than once. Now that we know the mode is 12, and since we know that the mean is equal to the mode, we can set the mean of these values equal to 12.
Mean = sum / quantity
12 = (6 + 8 + 10 + 12 + 12 + 14 + x) / 7
84 = 62 + x
22 = x
Thus, Ella Scott’s tax payment in 2014 was 22 million dollars.
The questions provided above give an overview of the SAT Math exam. It will help you understand what to expect on exam day.
Although practicing these questions will give you an edge over other students remember, that you still need to practice every question in the practice book to get the highest possible grade.
Note down your score after going through these questions and practicing yourself. After that, analyze your results. You will see which questions and areas you still need to work on.
If you need further help, contact an online math tutor to improve your grades quickly!