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If you are currently in the thick of studying for the SAT, you already know how important it is to do a lot of math problems. And because you have chosen to read this article, you’re probably a good student looking to get even the hardest SAT problems right. To help with your practice, we’re covering a representative selection of the hardest SAT Math questions and their solutions. Of course, practicing the questions here is just a start. If you need more practice after completing what we offer in this article, please check out our Target Test Prep SAT course.
Here is what we’ll cover in this article:
- SAT Math Topics: An Overview
- The Two Types of SAT Math Questions
- SAT Math Question 1: Successive Percent Discounts
- SAT Math Question 2: The Path of a Projectile
- SAT Math Question 3: Exponential Versus Linear Growth/Decay
- SAT Math Question 4: Factoring by Grouping
- SAT Math Question 5: Radian Measure
- SAT Math Question 6: Angles and Sides of a Triangle
- SAT Math Question 7: Data Interpretation
- Summary
- What’s Next?
Let’s first take a look at the math topics that are tested on the digital SAT.
SAT Math Topics: An Overview
If you’re motivated to get the hardest SAT questions right, you need to master every topic that you might encounter. Knowing what’s on the test allows you to focus your time and attention on studying the right things. The College Board has identified 4 main math categories that are tested. We have listed those and added subtopics to that list:
- Algebra: linear functions, linear equations, systems of linear equations, linear inequalities, word problems (linear)
- Advanced Math (advanced algebra): functions, function notation and interpretation, factoring, the FOIL technique, quadratic functions and their graphs, higher-order polynomials and their graphs, rational functions, exponents and exponential equations, square root equations, absolute value, exponential and nonlinear functions, exponential growth and linear growth, systems of equations (one linear and one nonlinear).
- Problem-Solving and Data Analysis: ratios and proportions, percentages, rates, unit conversion, Statistics: measures of center, data charts and graphs, rules of probability, scatterplots, making inferences, observational studies versus experiments.
- Geometry/Trigonometry: two- and three-dimensional geometry, including area and volume, Pythagorean theorem, lines, angles, triangles, circles, right triangles, basic trigonometric functions.
This is not an exhaustive list of questions on SAT Math, but it gives you a good idea of the topics that are tested on the digital SAT. Do notice that the digital SAT does not test you on complex or imaginary numbers, a math topic that the previous version of the SAT included.
TTP PRO TIP:
Knowing the topics on the SAT Math section will give you a roadmap for what to study.
The Two Types of SAT Math Questions
The digital SAT Math section will present you with a total of 44 questions in two modules. Roughly 75% of the questions will be traditional multiple choice with 4 answer choices. The remaining 25% of the questions, formerly called grid-in questions, are now referred to as “student-produced response” questions. For this question type, you will calculate an answer and type it into a box.
KEY FACT:
About 75% of the math questions are multiple-choice questions with 4 answer choices. The remaining 25% of questions are student-produced responses.
Special Rules for Entering Student-Produced Responses
Ensure that you know these rules for entering a student-produced response prior to test day.
- If you find more than 1 answer, enter only one. For example, if you determine that 1 < x < 3, you may enter any number greater than 1 and less than 3. You might answer 2.45 or just 2. Any answer in the interval will count as a correct response.
- A change from the previous SAT is that a student-produced response can be negative. If that is the case, make sure to use the negative sign before your answer. Positive answers can be as long as 6 digits, and negative numbers can be as long as 5 digits (excluding the negative sign).
- If a fraction has many digits, you might need to reduce it or enter it as a decimal number. For example, 137/8845 exceeds the character capacity of the answer box, so you should enter its decimal equivalent.
- Round or truncate decimal numbers to the fourth digit.
- You may not enter mixed numbers into the answer box. Enter a mixed number such as 4 2/3 as 14/3.
- Don’t enter commas, dollar signs, or percent symbols.
TTP PRO TIP:
Know the rules for entering student-produced responses!
Now that we have a good understanding of what’s tested on the Math section of the digital SAT, let’s practice some of the hardest SAT Math questions.
SAT Math Question 1: Successive Percent Discounts
A jacket’s original selling price was x dollars. It was put on sale for 35% off, and it didn’t sell. The store owner reduced the discounted price by an additional 20%, and the jacket finally sold for $78.00. What is the value of x, the original price of the jacket?
- $130
- $140
- $150
- $160
Solution:
The most efficient way to solve this problem is to recall that if an item is reduced by 35%, then its discounted price is 100% – 35% = 65% of its original price. Decimally, this is expressed as 1 – 0.35 = 0.65. Using this fact and applying it twice to the two discounts allows us to set up a straightforward equation to solve for x.
After the first discount, the sale price will be (x)(1 – 0.35) = (x)(0.65). Now, for the second reduction, that reduced price of (x)(0.65) is further reduced by 20%, which means that the final (twice-reduced) price will be (x)(0.65)(1 – 0.20) = (x)(0.65)(0.80).
We are given that the final selling price after the two discounts were applied, is $78.00.Thus, we have:
78 = (x)(0.65)(0.80)
78 = 0.52x
150 = x
Answer: C
KEY FACT:
If an item is reduced by x%, then the sale price is (100 – x)%.
SAT Math Question 2: The Path of a Projectile
An object is launched directly upward at 64 feet per second from a platform 80 feet high. The equation for its height, s(t), after t seconds is given by the equation
s(t) = -16t^2 + 64t + 80
After how many seconds will the object reach the ground?
- -1
- 3
- 5
- 10
Solution:
The key to solving projectile problems is to keep in mind that s(t) is the height of the projectile at time t during its trajectory. Before we solve this problem, let’s make sure we understand what the equation tells us. For example, at time t = 0, we plug in 0 for t. We see that the height is
s(0) = -16(0)^2 + 64(0) + 80
s(0) = 80
Notice that this value of 80 agrees with the information provided in the question, that the height of the platform from which the projectile was launched was 80 feet.
KEY FACT:
For a projectile equation of the form s(t) = at^2 + bt + c , the height of the projectile at time t is given as s(t).
Now, to answer the question that was asked, we have to note that when the projectile hits the ground, its height is 0, so we know that s(t) = 0. And our goal is to determine the value of t, the time in seconds when this happens. When we plug in 0 for the height, we have
0 = -16t^2 + 64t + 80
In order to find the value of t, we must factor the quadratic. To make the factoring easy, first factor out -16 from each term and then factor the trinomial inside the parentheses.
0 = -16(t^2 – 4t – 5)
0 = (-16)(t – 5)(t + 1)
t – 5 = 0 or t + 1 = 0
t = 5 or t = -1
Since time cannot be negative, t cannot equal -1, so we see that the projectile will hit the ground after 5 seconds.
Answer: C
SAT Math Question 3: Exponential Versus Linear Growth/Decay
A wildlife biologist has counted the number of a particular species of fish in a lake for the past 5 years. On average, the number of fish he has counted each year has been 92% of the number he counted the previous year. Which of the following functions best describes how the number of fish that he has counted has changed over time?
- Decreasing linear
- Decreasing exponential
- Increasing linear
- Increasing exponential
Solution:
To get a sense of what is happening in this problem, let’s assume that the biologist counts 100 fish the first year. Since each year he counts 92% of the number from the previous year, we can make a chart of the number of fish over time.
Year 1: 100
Year 2: (100) (0.92) = 92
Year 3: (100) (0.92)(0.92) = (100) (0.92)^2 = 85
Year 4: (100) (0.92)(0.92)(0.92) = (100) (0.92)^3 = 78
Year 5: (100) (0.92)(0.92)(0.92)(0.92) = (100) (0.92)^4 = 72
First, we see that the number of fish counted decreases over time, so we can rule out answer choices C and D. Choices A and B remain.
Next, we have to determine whether the function is linear or exponential. To determine this, we can look at the decrease in the fish count each year.
Year 1: Count = 100
Year 2: Count = 92, which is a decrease of 8 from the previous year
Year 3: Count = 85, which is a decrease of 7 from the previous year
Year 4: Count = 78, which is a decrease of 7 from the previous year
Year 5: Count = 72, which is a decrease of 6 from the previous year
For the function to be linear, the decrease from one year to the next would have to be constant. We see that the decrease is not constant. (Note that in years 2, 3, 4, and 5, the decreases from the previous year were 8, 7, 7, and 6, so the decreases were not the same each year.) So we rule out choice A (decreasing linear), leaving choice B (decreasing exponential) as the correct answer.
Answer: B
Alternate Solution:
An alternative approach to verify that the function is exponential is to recall that the format of an exponential function is
f(x) = ab^x
In this exponential function, a is the initial value and b is the multiplier.
In our fish counting example, we see that a = 100 and b = 0.92, so we have
f(x) = (100) (0.92)^x
Because the data fit the exponential function format, we are assured that the function is indeed exponential. We also know that an exponential function is decreasing when the value of the multiplier is between 0 and 1. In this case, the multiplier is b = 0.92, so we verify that the exponential function is decreasing.
Answer: B
KEY FACT:
The standard form of an exponential function is f(x) = ab^x, where a = the initial value and b = the multiplier.
SAT Math Question 4: Factoring by Grouping
What is the product of the solutions of the following equation?
0 = 2x^3 + x^2 – 6x – 3
- 1/2
- 3/2
- 5/2
- 9/2
Solution:
To obtain the solutions of the equation, we have to factor the cubic equation, and the most efficient way is to use “factoring by grouping.”
Step 1: First, we group the first two terms together and the last two terms together:
0 = (2x^3 + x^2) – (6x + 3)
(Note that when we enclosed (-6x – 3) inside parentheses, we actually factored the negative from both terms.)
Step 2: Next, we find the common factor in each binomial.
In the binomial (2x^3 + x^2), we see that the common factor is x^2. We factor out the x^2 to obtain:
(2x^3 + x^2) = x^2(2x + 1)
In the binomial (6x + 3), the common factor is 3. After factoring out the 3, we have:
(6x + 3) = 3(2x + 1)
Step 3: We re-express the original equation with the factored expressions from Step 2.
0 = (2x^3 + x^2) – (6x + 3)
0 = x^2(2x + 1) – 3(2x + 1)
Step 4: We see that (2x +1) is common to each term, so we factor it out.
0 = (2x + 1)(x^2 – 3)
Step 5: We finalize the factoring and find the solutions.
We factor the quadratic x^2 – 3 as a difference of squares:
0 = (2x + 1)(x – √3)( x + √3)
We can now find the zeros (solutions) of the equation.
2x + 1 = 0 x – √3 = 0 x + √3 = 0
x = -1/2 x = √3 x = -√3
The product of these three solutions is:
(-1/2)(√3)(- √3) = 3/2
Answer: B
TTP PRO TIP:
Use the 5-step procedure for factoring by grouping to solve a cubic equation.
SAT Math Question 5: Radian Measure
Which of the following is closest to the measure, in degrees, of an angle measuring π/7 radians?
- 26
- 39
- 45
- 54
Solution:
Radian measure is an alternate way of expressing the measure of an angle. If you’ve taken a trigonometry course, radian measure is most likely second nature to you. If you haven’t, it might sound like a foreign language!
Memorize this fact: There are π radians in 180 degrees. Thus, we can express the conversion factor for radians to degrees as π radians / 180 degrees.
KEY FACT:
To convert from radians to degrees, use the conversion factor that π radians equals 180 degrees.
To convert π/7 radians to x degrees, we create the proportion:
π radians / 180 degrees = (π/7 radians) / x
Now, solve for the value of x by cross-multiplication:
(π)(x) = (180)(π/7)
x = 180/7 = 25.71 degrees
Answer: A
SAT Math Question 6: Angles and Sides of a Triangle
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
Solution:
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.
KEY FACT:
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.
Answer: B
SAT Math Question 7: Data Interpretation
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?
- 19 million
- 20 million
- 21 million
- 22 million
Solution:
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.
Answer: D
TTP PRO TIP:
Know how to manipulate the facts and formulas of basic statistical measures.
Summary
If you want to get the highest score on the SAT, you will need to master all of the math topics on the exam. You also need to understand the structure and format of the exam and the question types.
In this article, we have presented you with details of the Math section of the digital SAT. In addition, we have covered 7 of the hardest SAT Math questions that you may encounter:
successive percent discounts, projectile path, exponential/linear growth or decay, factoring by grouping, radian measure, triangle sides, and data interpretation.
These are certainly not exhaustive of the tough questions you may see, but they give you an idea of the level of mastery that you will need if you are motivated to score in the highest SAT percentiles.
What’s Next?
You already know that SAT Math is not a walk in the park! In order to get a high score, you need to have a great working knowledge of all the math topics. But, just as important, you need to be proficient with all aspects of the SAT Reading and Writing section.
The digital SAT will challenge you because it tests many concepts you have learned throughout your years in school. The more you prepare, the better you’ll perform. Good luck!
