# SAT Math Problems with Solutions

If you are in the middle of your SAT studying, then I’m sure you understand the importance of practicing SAT math problems as you learn the various SAT math topics. In this article, we will provide some math background and present various SAT math practice problems and answers.

The problems presented here are just a start, but you can get additional practice by checking out the Target Test Prep online SAT course.

Before we jump into SAT math practice questions, let’s discuss some basics of SAT math.

## An Overview of the SAT Math Section

There are four sections on the SAT, and SAT math is tested in section 3 and section 4. Section 3 presents 20 questions, which you have 25 minutes to answer. Section 4 contains 38 questions, which you have 55 minutes to answer.

Furthermore, section 3 does not permit the use of a calculator, while section 4 does.

Additionally, each section contains both multiple-choice and grid-in questions, both of which we’ll discuss shortly.

KEY FACT:

Sections 3 and 4 of the SAT are math and are 25 minutes and 55 minutes long, respectively.

Next, let’s discuss the SAT math topics that could show up on your exam.

## SAT Math Topics

So, here is some good news! Many math topics that can show up on your SAT are topics that you have already studied or will study soon in your high school math curriculum. So, as you jump into your SAT math prep, some topics should not be new to you, at least on a foundational level. And don’t worry, if some are new, you can learn them!

TTP PRO TIP:

Many SAT math topics should be familiar to you.

Now, let’s discuss the breakdown of the math concepts tested on the SAT.

### The TTP and SAT Breakdown of SAT Math Topics

It’s important to note that although the College Board has its way of categorizing SAT math topics, we at TTP break them down a bit more granularly. For now, we can start with the College Board’s categorization, which includes four major SAT math categories:

1. Heart of Algebra
2. Problem Solving and Data Analysis

Let’s subdivide these categories, the “TTP way,” so you can get a better overview of what is tested.

Heart of Algebra:

1. Solving Linear Equations
2. Inequalities and Absolute Value
3. Coordinate Geometry
4. Linear Functions
5. Systems of Linear Equations

Problem Solving and Data Analysis:

1. General Word Problems
2. Rates
3. Unit Conversions
4. Ratios
5. Statistics
6. Percentages

1. Exponents
2. Roots
4. Functions
5. Coordinate Geometry
6. Graph Interpretation
7. Table Data

1. Geometry
2. Trigonometry
3. Complex Numbers

So, among the College Board’s four major categories, there are 22 math topics tested. However, keep in mind that these are the main topics. Each main topic branches into hundreds of subtopics, so let’s discuss that now.

TTP PRO TIP:

There are 22 major SAT math topics.

## SAT Math Subtopics

You may think that studying just 22 math topics is a piece of cake. However, you can’t forget that there are hundreds of subtopics! For example, consider the major math topic of Linear Equations. Branching off from that major topic are smaller subtopics such as:

• Solving an Equation for One Variable
• Solving a System of Equations for Two Variables
• The Combination Method
• Solving for One Variable in Terms of Other Variables
• Equations with Fractions
• When a System of Linear Equations Has No Solution

In TTP, the chapter on Linear Equations actually has 23 subtopics! So, when you’re prepping for the SAT math test, just make sure to dig deep into the major topics and come out on the other side having also mastered the subtopics. This approach to your prep will make you an absolute force on test day!

TTP PRO TIP:

When studying each major SAT math topic, learn the subtopics too.

Before jumping into our discussion and practice of the two major question types in SAT math, with answer explanations, let’s discuss calculator use on the SAT.

## Using Your Calculator on the SAT

We have already discussed that you can use your calculator only in section 4 of the SAT. For all other sections, it must be put away and out of sight.

It’s also important to understand that your calculator can be a tool that can either work to your advantage or hurt you, depending on how and when you are using it.

For instance, if you find that you start using your calculator to do simple math calculations, then you likely are wasting time. For example, you should not need your calculator to calculate the sum of 11 and 13 or the product of 10 and 12, right? Conversely, you do want to use your calculator when determining the product of 3.56 and 11.2 or when dividing 0.348 by 0.12.

Certainly, you have to use your judgment, but getting a feel for when to use the calculator and when not to will be huge for test day. Regarding which calculator to use, check out the list from College Board to see the exact calculators allowed for the SAT. Our recommendation is to use either a graphing calculator or a scientific calculator, as those have more functionality than a basic 4-function calculator.

No matter which calculator you choose, make sure you practice using it well before test day, so you do not have any unnecessary issues during your SAT.

TTP PRO TIP:

Ensure that you are strategically using your calculator on the SAT.

Now, let’s discuss the two major SAT question types.

## The Two Major Question Types on the SAT

SAT math questions are of two types: multiple-choice and grid-in. Of the 58 math questions on the SAT, you will encounter 45 multiple-choice and 13 grid-in questions.

If we break down the number by each section, section 3 (no calculator) contains 15 multiple-choice questions and 5 grid-in questions, while section 4 (calculator) contains 30 multiple-choice and 8 grid-in questions.

KEY FACT:

There are 45 multiple-choice and 13 grid-in questions on the SAT.

Next, let’s discuss multiple-choice questions in further detail and practice some questions!

## Multiple-Choice Questions

Multiple-choice questions on the SAT are pretty much what you would expect, with one small difference. In a traditional multiple-choice question, you have five answer choices (A, B, C, D, and E). However, in SAT multiple-choice questions, you have just four answer choices (A, B, C, and D).

KEY FACT:

There are four answer choices in multiple-choice questions on the SAT.

Now, let’s practice some SAT multiple-choice questions.

### Multiple-Choice Question 1

SAT Major Topic: Passport to Advanced Math

TTP Subtopic: The Discriminant

x2 + 2 = 4x has how many real solutions?

• 0
• 1
• 2
• More than 2

#### Solution:

It isn’t necessary to solve this equation for x. Note that we need only determine the number of real solutions. Thus, we can use discriminant analysis to efficiently answer the question.

First, let’s express the given equation in general form:

x2 – 4x + 2 = 0

So a = 1, b = -4, and c = 2. Thus, for this equation, the discriminant is:

b2 – 4ac = (-4)2 – 4(1)(2) = 16 – 8 = 8

Because the discriminant’s value is positive, we know there are two real roots to the quadratic equation.

### Multiple-Choice Question 2

SAT Major Topic: Heart of Algebra

TTP Major Topic: Arithmetic Word Problems

TTP Subtopic: Inequality Word Problems

A particular department store has 50 displays, and the total number of displays in the store cannot exceed 90. If no old displays are taken down and new ones are added at a rate of 3 per day, and if y represents time in days, which of the following inequalities represents the situation?

• 3y ≤ 90
• 50 + 3y ≤ 90
• 50 – 3y ≤ 90
• 50y ≤ 90

#### Solution:

First, we should note that the situation described has all of the components of a linear inequality: There is an initial value (50 displays), a total capacity (90 displays), and a constant rate of increase (3 displays per day).

Both sides of the inequality will represent the number of displays:

[number of displays on any given day] is at or below [display capacity].

Therefore:

[number of displays on any given day] ≤ [display capacity].

Next, we can fill in the given numbers accordingly:

[number of displays on any given day] = 50 original displays + 3 new displays per day = 50 + 3y

[display capacity] = 90

Finally, when we assemble these into the final inequality, we get:

50 + 3y ≤ 90

### Multiple-Choice Question 3

SAT Major Topic: Problem Solving and Data Analysis

TTP Major Topic: Statistics

TTP Subtopic: Solving for an Unknown in an Average Equation

Tabitha swims the 200-meter freestyle three times. Her completion times for the first two swims are 30 seconds and 36 seconds. What is the completion time for her third swim?

• 33
• 34
• 35
• 36

#### Solution:

We know that Tabitha’s average is 32 seconds. Her first two times are 30 seconds and 36 seconds. Let’s let x = the time of her third swim. So, using the formula for the average, we have:

average = sum/number

⇒ 32 = (30 + 32 + x)/3

⇒ 96 = 62 + x

⇒ 34 = x

### Multiple-Choice Question 4

SAT Major Topic: Passport to Advanced Math

TTP Major Topic: Functions

TTP Subtopic: Domain of a Function

Which of the following describes the domain of the function v(x) = x^5 + x^3?

• x > 5
• x < 3
• 3 < x < 5.
• The domain is all real numbers.

#### Solution:

First, recall that the domain of a function specifies all allowable x values. Generally, for SAT purposes, we have domain restrictions when we have (1) a denominator that is equal to 0 or (2) a function for which we would take the square root of a negative number.

For example, if we had f(x) = 2/(x – 4), we would have a domain restriction of x = 4 because that value would make the denominator of the fraction equal to 0. A second example would be sqrt(x – 5). If x were any number less than 5, then we would be taking the square root of a negative number.

Clearly, the domain of the function v(x) has no restrictions, since we can take any real number to the fifth power or the third power, and we can add these two quantities for any real value of x. Thus, the domain of v(x) is all real numbers.

### Multiple-Choice Question 5

SAT Major Topic: Problem Solving and Data Analysis

TTP Major Topic: General Word Problems

TTP Subtopic: Age problems

Harold is 30 years older than Paloma. If in 10 years, Harold will be 3 times as old as Paloma will be then, how old will Harold be in 3 years?

• 38
• 33
• 28
• 24

#### Solution:

First, let’s define two variables:

H = Harold’s age today

P = Paloma’s age today

Next, we can create two equations from the information presented in the problem stem.

Since Harold is 30 years older than Paloma, we have:

⇒ H = P + 30

In 10 years, Harold will be (H + 10) years old, and Paloma will be (P + 10) years old. Thus, at that time, Harold will be 3 times as old as Paloma, and we have:

⇒ H + 10 = 3(P + 10)

⇒ H + 10 = 3P + 30

⇒ H = 3P + 20

Next, we can substitute P + 30 for H in the second equation:

⇒ P + 30 = 3P + 20

⇒ 10 = 2P

⇒ 5 = P

Therefore, Harold is currently 5 + 30 = 35 years old, so in 3 years, he will be 38 years old.

### Multiple-Choice Question 6

TTP Major Topic: Complex Numbers

TTP Subtopic: Combining Complex Numbers

What is the sum of the complex numbers 3 + 2i and 5 – 7i, where i = ?

• 8 + 9i
• 8 – 5
• 2 + 5i
• 15 – 14i

#### Solution:

First, we can add the real parts of 3 + 2i and 5 – 7i (3 and 5, respectively):

⇒ 3 + 5 = 8

Now, we can add the imaginary parts of 3 + 2i and 5 – 7i. Note that because 5 – 7i has subtraction between its two parts, the imaginary part of this complex number is -7. (We can perhaps see this more clearly if we rewrite 5 – 7i as the equivalent expression 5 + (-7i).) The imaginary part of 3 + 2i is 2:

⇒ 2i + (-7i) = -5i

Thus, the sum of 3 + 2i and 5 – 7i is 8 + (-5i), or, equivalently, 8 – 5i.

### Multiple-Choice Question 7

TTP Major Topic: Trigonometry

What is the equivalent of 72° in radians?

• 𝞹/3
• 𝞹/5
• 2𝞹/5
• 3𝞹/5

#### Solution:

To convert 72˚ to radians, we can create the following proportion:

𝞹 radians / 180 degrees = x / 72 degrees

Next, we can remove the units and cross-multiply to solve for x:

⇒ 72𝞹 = 180x

⇒ 72𝞹 / 180 = x

⇒ 2𝞹 / 5 = x

Now, let’s discuss SAT grid-in questions.

## Grid-In Questions

Technically, the make-up of grid-in questions differs from that of multiple-choice questions. A grid-in question will always have a numerical answer with no variables in the answer. It’s important to note, too, that some grid-in questions have multiple possible answers. You need to grid in only one of these answers.

Despite those differences, the skills needed to answer a grid-in question do not differ from those required to answer a multiple-choice question. The primary difference between multiple-choice questions and grid-in questions is that there are no answer choices to select from in a grid-in question.

There are 8 grid-in calculator questions and 5 grid-in non-calculator questions. Also, grid-in questions are always presented at the end of a section.

KEY FACT:

There are 8 grid-in calculator questions and 5 grid-in non-calculator questions.

An example of a blank grid is shown below:

There are a few rules to learn about filling in the grid:

2. Only positive numbers can be bubbled. Thus, if your answer is a negative number, you have made an error in your calculations.
3. If your answer is a decimal number, round it to 3 decimal places, if necessary, and make sure you bubble one position for your decimal point. A leading zero is not required for decimal numbers between 0 and 1.
4. Fractions need not be reduced. They can be entered in traditional fraction fashion, with the “slash” mark bubbled to separate the numerator and the denominator. Alternatively, you can express your fractional answer as a decimal number.

Now, let’s practice with a couple of grid-in examples.

SAT Major Topic: Passport to Advanced Math

TTP Major Topic: Exponents

TTP Subtopic: Multiplication of Exponents with the Same Base

If a and b are positive integers and 312 = 3a3b, what is one possible value of a x b?

#### Solution:

Since 3a3b = 3a+b, we can rewrite the given equation as 312 = 3a+b. Since our base is 3 on both sides of the equation, a + b must be equal to 12. We need to determine the two values whose sum is 12. Because a and b must be positive integers, there are only a handful of options for their values: {1, 11}, {2, 10}, {3, 9}, {4, 8}, {5, 7}, and {6, 6}. The products of these pairs are: 11, 20, 27, 32, 35, 36.

Therefore, any of the values 11, 20, 27, 32, 35, or 36 are correct.

Answer: 11, 20, 27, 32, 35, or 36

Remember that sometimes there are multiple correct answers to a grid-in question. You need to bubble only one of them into the grid to get the question correct.

Let’s try another question.

### Grid-In Question 2

SAT Major Topic: Problem Solving and Data Analysis

TTP Major Topic: Arithmetic

TTP Subtopic: Fractions

Alvin accidentally spilled his marble collection on the floor. If he was able to recover 24 of the original 120 marbles, what fraction of his marble collection was he able to recover?

#### Solution:

The problem gives us these two pieces of information:

Number of recovered marbles = 24

Total number of marbles = 120

Thus, we know that the fraction of recovered marbles is 24/120.

Because this is a grid-in question, we cannot directly grid in the value 24/128 because there are not enough positions in the grid to accommodate this fraction. Thus, we have two options: (1) reduce the fraction or (2) convert the fraction to a decimal number.

1. Reduce the fraction:

Since both 24 and 120 are divisible by 24, we can divide the numerator and denominator by 24:

⇒ 24/120 = 1/5

1. We can convert the fraction 24/120 to a decimal value:

⇒ 24/120 = 0.2

We could bubble either 1/5 or 0.2 into the grid.

### Grid-In Question 3

SAT Major Topic: Heart of Algebra

TTP Major Topic: Linear Equations

TTP Subtopic: Solving For Expressions

If 15x + 24y + 10 = 15, then what is the value of 5x + 8y?

#### Solution:

To start, we can move 10 from the left side of the equation to the right side:

⇒ 15x + 24y = 5

Next, we see that each term in the equation on the left is a multiple of 3. Thus, if we divide the entire equation by 3, we have:

⇒ 5x + 8y = 5/3

### Grid-In Question 4

SAT Major Topic: Heart of Algebra

TTP Major Topic: Arithmetic Word Problems

TTP Subtopic: Compound Inequalities

If n is an integer and 10 < 2n < 18, then what could be a value of n?

#### Solution:

First, we need to isolate n. To do so, we can divide the entire inequality by 2, and we have:

⇒ 5 < n < 9

Since n is an integer, we see that n can be 6, 7, or 8. Any of these answers is correct.

### Grid-In Question 5

SAT Major Topic: Problem Solving and Data Analysis

TTP Major Topic: Rate Questions

TTP Subtopic: Rate Questions Not Involving Distance

A tank is filled with 12,000 gallons of water. If a drain is opened that allows water to drain from the pool at a constant rate of 240 gallons per hour, how many gallons are drained from the pool after 6 hours?

#### Solution:

We are given that a pool is filled with 12,000 gallons of water and that, when a drain is opened, water drains at a rate of 240 gallons per hour. So, to determine how much water has drained after 6 hours, we can use the following form of the general rate formula:

⇒ rate x time = amount

⇒ 240 x 6 = 1,440 gallons

Answer: 1,440 (Note: you will grid in 1440 with no comma)

### Grid-In Question 6

SAT Major Topic: Passport to Advanced Math

TTP Major Topic: Functions

TTP Subtopic: Symbolism Questions

If x@y = x2 + 2y, what is 5@3?

#### Solution:

The function statement x@y = x2 + 2y sets up the instruction for evaluating the function. The second function statement tells us that we are to evaluate the function when x = 5 and y = 3.

When x = 5 and y = 3, we substitute these values into the function instruction, which is x2 + 2y:

⇒ 52 + 2(3) = 25 + 6 = 31

### Grid-In Question 7

SAT Major Topic: Heart of Algebra:

TTP Major Topic: Coordinate Geometry

TTP Subtopic: The Y-Intercept

In the xy-plane, the equation of line k is given as 5x + (2/3)y = 10. What is the y-intercept of line k?

#### Solution:

First, we can find the y-intercept by recalling that the y-intercept can be found by letting x = 0 and solving for y. So, we have:

⇒ 5(0) + (2/3)y = 10

⇒ 0 + 2y/3 = 10

⇒ 2y = 30

⇒ y = 15

## Don’t Try to Predict Which SAT Math Topics You’ll See

If you have been studying for the SAT for some time, my bet is that you have a solid sense of the kinds of questions you might see on the SAT. However, the exact makeup of your SAT is difficult to predict. The fact is, what you see is going to be somewhat random.

Despite College Board’s releasing numerous official SATs, you can’t assume that what you see on those exams will be exactly what you see on test day. For example, if you see a unit circle question on a practice exam, there is no guarantee that you will see one on your exam. There are 8 official practice exams that the College Board has released, for a total of 58 x 8 = 464 math questions on those exams. Your SAT will contain just 58 questions. So, there are many questions that you will have seen on practice tests that you will not see on your SAT. Likewise, those 464 practice questions do not encompass every question type that you could encounter on the SAT.

Don’t get me wrong; it’s very useful to carefully review those 8 practice exams. But if you bank your SAT math score just on those 464 questions, you will most likely have an unpleasant surprise when your SAT score is posted.

So, if you’re wondering, “What math should I practice for SAT test day?,” unfortunately, there is no specific answer to that question. What does all this mean? Well, let’s not play roulette with what you learn for the SAT. To set yourself up for success, learn all SAT math topics.

TTP PRO TIP:

Don’t try to predict the exact makeup of your SAT.

With that point in mind, let’s discuss a great way to learn SAT math.

## Preparing for SAT Math

Thus far, we’ve discussed how learning a wide range of concepts is necessary to succeed in SAT math. A great way to learn all of these topics is through topical learning. Topical learning entails learning one topic at a time and focusing solely on that topic until you have mastered it. By studying in this way, you can ensure that you truly learn each topic before moving to the next one.

Ask yourself, would it be effective to jump from Linear Equation questions to Functions to Geometry questions? I think you know the answer …

Remember, SAT math prep takes time, care, and attention. So, jumping around from topic to topic will hinder your ability to learn.

TTP PRO TIP:

Learn each SAT math topic one at a time.

To get a better idea of how topical learning works, let’s take a look at the Target Test Prep (TTP) study plan.

## Topical Learning With TTP

The cornerstone of the TTP study plan is topical learning and practice. The study plan is broken up into missions, each of which contains one major math topic. Students learn that topic, and then answer practice questions about that topic until they master it.

After finishing a particular section, you answer a few example questions to practice what you have just learned. Then, at the end of the chapter, you take chapter tests rated by level of difficulty, to drill every concept that was presented in that chapter.

Now that you see topical learning in action in the TTP prep course, you should have a good idea of how to structure your math studying.

## Summary

In this article, we provided an overview of the SAT and presented you with 14 SAT math practice questions.

Of the four sections on the SAT, two test you on math. Section 3 is the no-calculator section, for which you must answer 20 questions in 25 minutes. Section 4 is the calculator section, and you are presented 38 questions, which you must complete in 55 minutes.

The College Board has created 4 major math conceptual topical areas: Heart of Algebra, Problem Solving and Data Analysis, Passport to Advanced Math, and Additional Topics in Math. These encompass 22 main topics, including lines, systems of equations, arithmetic, statistics, exponents, functions, and trigonometry.

To test the 22 main topics, the SAT employs two question types:

1. Multiple-choice questions present 4 answer choices. Of the 58 math questions on the SAT, 45 of them are multiple-choice, including 15 in the no-calculator section and 30 in the calculator section.
2. Grid-In questions require you to bubble in a 4-column answer to a question. Of the 58 math questions on the SAT, 13 of them are grid-in, including 5 in the no-calculator section and 8 in the calculator section.

There is no way to accurately predict which topics will be tested on your particular test, so you need to give attention to all of the topics. A particularly effective technique is topical studying, in which you study one topic at a time. After a topic is mastered, you move to the next one. This technique ensures that nothing slips through the cracks. You’ll be fully prepared to do your best on exam day!

### What Math Problems Are on the SAT?

Earlier in this article, we outlined the 22 main topics that are tested on the SAT. Remember, there are many subtopics as well; SAT math encompasses a huge number of concepts that might be tested.

### Is the Math on the SAT Hard?

You will encounter easy, medium, and hard questions on the SAT. A good number of questions should be relatively easy to answer, as you saw in the practice questions in this article. But to obtain a great SAT score, you will have to study each topic thoroughly, so that you are able to answer even the most challenging problems.

### What Is the Hardest Math Subject in SAT Tests?

Many students taking the SAT have not yet studied trigonometry, so this topic is often identified as difficult. However, the level of difficulty of the trigonometry questions is generally not high, so a quick overview of basic trig may suffice, even if you have never taken a trigonometry course.

Other areas identified as difficult by students include complex numbers, functions, and factoring of quadratics. These subtopics are usually encountered in the second year of algebra, and thus may give some students more trouble than easier topics, such as linear equations and percentages.

## What’s Next?

Now that you’ve had lots of practice with the two math question types on the SAT, you might want to read some tips for improving your SAT math score.

Additionally, from the College Board, you can access free SAT math practice tests to hone your newly acquired skills.

Good luck!