The SAT exam has around 5 to 7 SAT Problem-Solving and Data Analysis content domain questions out of 44 SAT Math questions. This makes up around 15% of the SAT Math. While the majority of the questions are easy to solve, some SAT Problem-Solving and Data Analysis questions are tricky. Knowing the important points about the content domain and practicing as many SAT Problem-Solving and Data Analysis practice questions as possible will double your chances of getting a high SAT Math score.

📌 Hint: Do not skip this article; you will find FREE SAT Math Prep resources throughout the article.

We’ve listed the most important and frequently occurring concepts in this SAT Problem-Solving and Data Analysis post. You will see SAT Problem-Solving and Data Analysis practice test questions and exercises, in total 15 SAT Problem-Solving and Data Analysis questions with rationales all for FREE. 

💡You might be interested in reading the Digital SAT Math Prep post.

SAT Problem-Solving and Data Analysis Content Domain

SAT Problem-Solving and Data Analysis content domain measures the ability to apply quantitative reasoning about ratios, rates, and proportional relationships; understand and apply unit rate; and analyze and interpret one- and two-variable data. This group of skills is about being quantitatively literate and demonstrating a command of math that resonates throughout college courses, career training programs, and everyday life.

💡You might be interested in reading the Digital SAT Math Ultimate Guide post. We have provided further details about the SAT Math structure, examples of easy, medium, and hard questions, answers, rationales, and frequently asked questions about the SAT Math.

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Anna B. is one of our thousands of successful SAT students. She scored 800 on SAT Math. You can watch her SAT story.

SAT Problem-Solving and Data Analysis Skills and Knowledge Testing Points

The SAT exam will have around 5 to 7 questions from the Problem-Solving and Data Analysis content domain. There are 7 skills and knowledge testing points in the SAT Problem-Solving and Data Analysis content domain:

1. 1. Ratios, rates, proportional relationships, and units
   2. Percentages
   3. One-variable data: distributions and measures of center and spread
   4. Two-variable data: models and scatter plots
   5. Probability and conditional probability
   6. Inference from sample statistics and margin of error
   7. Evaluating statistical claims: observational studies and experiments

Let’s review the important points you should know for each of the SAT Problem-Solving and Data Analysis skills and knowledge below.

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SAT Problem-Solving and Data Analysis Topic 1: Ratios, Rates, Proportional Relationships, and Units

A brief overview of the importance of understanding ratios, rates, proportional relationships, and units in everyday life and their significance in the SAT Math section.

1. Understanding Ratios

Definition: A ratio is a comparison of two quantities that shows the relative sizes of two or more values.

Examples:
The ratio of apples to oranges is 3:2.
The ratio of boys to girls in a class is 5:3.
Representation: Ratios can be represented as fractions , with a colon (a:b), or in words (“a to b”).

2. Exploring Rates

Definition: A rate is a special type of ratio where the two quantities being compared have different units.

Examples:
Speed as miles per hour (mph).
Price as dollars per pound.
Unit Rate: The rate per one unit (e.g., cost per item).

3. Proportional Relationships

Definition: Two quantities are proportional if they maintain a constant ratio or rate.
Mathematical Form: If , then a.d = b.c (cross multiplication).
Solving Proportions: How to solve for an unknown in a proportional relationship using cross multiplication.

4. Units and Unit Conversions

Importance of Units: Units provide context to numerical values and are crucial in understanding and solving real-world problems.
Dimensional Analysis: A technique to convert one unit to another using conversion factors.
Examples of Unit Conversions:
Converting inches to centimeters.
Converting gallons to liters.
Converting miles per hour to feet per second.

5. Applications of Ratios, Rates, and Proportional Relationships

Mixture Problems: Determining concentrations in solutions.
Scale Models and Drawings: Understanding maps and architectural drawings.
Speed, Distance, and Time Problems: Using the formula Distance=Speed x Time.

6. Common Mistakes and How to Avoid Them When Solving SAT Problem-Solving and Data Analysis Questions

• Misinterpretation of Ratios and Rates: Common errors in understanding or applying ratios and rates.
  Unit Inconsistencies: Importance of consistent units when performing calculations.
  Errors in Proportional Reasoning: Avoiding mistakes in setting up and solving proportions.

7. Practice Questions

• Ratios: If a recipe requires a 3:2 ratio of flour to sugar and you have 9 cups of flour, how much sugar do you need?
• Rates: A car travels 150 miles in 3 hours. What is the car’s speed in miles per hour? How far will it travel in 5 hours at this speed?
• Proportional Relationships: On a map, 1 inch represents 5 miles. If two cities are 7 inches apart on the map, what is the actual distance between them?
• Unit Conversions: A runner completes a race in 45 minutes at an average speed of 8 kilometers per hour. How long is the race in kilometers?

💡We’ve prepared a 7-Step Digital SAT Math Study Guide helping students to prepare their unique SAT Math Study Guide.

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SAT Problem-Solving and Data Analysis Topic 2: Percentages

Percentages are a way to express numbers as parts of a whole, where the whole is 100. For example, 50% means 50 out of 100, or half of a whole.

Conversion Tips:

To convert a percentage to a decimal, divide by 100. (e.g., 25% = 0.25)
To convert a decimal to a percentage, multiply by 100. (e.g., 0.75 = 75%)
To convert a fraction to a percentage, divide the numerator by the denominator and multiply by 100. (e.g., 1/4 = 0.25 = 25%)

Basic Percentage Calculations

Finding a Percentage of a Number: To find x% of a number, multiply the number by (x100).
Example: 20% of
Percentage Increase or Decrease:
Example: If a shirt price increases from $40 to $50, the percentage increase is

Advanced Percentage Problems

Problems with Unknowns: These often involve setting up an equation. For example, “What number is 15% of x?” translates to 0.15x = Given Number
Compound Percentages: When calculating multiple percentage changes, it’s essential to apply them sequentially, not add them.
Example: A price increases by 10% and then decreases by 10% is not a net 0% change but rather: 1.10 . 0.90=0.99, a 1% decrease.

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SAT Problem-Solving and Data Analysis Topic 3: One-variable Data: Distributions and Measures of Center and Spread

One-variable data (or univariate data) consists of observations on a single characteristic or attribute. Examples include test scores, heights, weights, or any data that can be listed in a single column.

Data Distributions

A distribution tells us how data values are spread out. It is vital to understand the shape of the data distribution as it informs which measures of center and spread are most appropriate.

Types of Distributions:

• Uniform Distribution: All values occur with roughly the same frequency.
• Symmetric Distribution: The left and right sides of the distribution are approximately mirror images.
• Skewed Distribution: Data is not symmetric. Skewed left means the tail is longer on the left side, and skewed right means the tail is longer on the right side.

Visual Representations:

• Histograms: Useful for showing frequency distributions for continuous data.
• Dot Plots: Show each individual data point, useful for small data sets.
• Box Plots: Display the five-number summary (minimum, first quartile, median, third quartile, maximum) and are useful for showing spread and identifying outliers.

Measures of Center

Mean: The average of all data points.

Formula: Mean= ⅀Data Points / Number of Data Points

Median: The middle value when the data points are arranged in order. If there is an even number of data points, the median is the average of the two middle numbers.

Mode: The most frequently occurring value(s) in the data set.

🖋️ Mean, Median, and Mode is a frequently occurring concept in SAT Problem-Solving and Data Analysis questions. 

Measures of Spread

Range: The difference between the maximum and minimum values in a data set.
Formula: Range=Maximum−Minimum
Interquartile Range (IQR): The range of the middle 50% of the data.
Formula: IQR = Q3 − Q1 (where Q1 is the first quartile and Q3 is the third quartile)

Standard Deviation: A measure of the average distance of each data point from the mean. A higher standard deviation indicates more spread-out data.

Variance: The square of the standard deviation, represents the spread of the data points.

Analyzing Data Using Measures of Center and Spread
When analyzing data:

• Use the mean for symmetric distributions without outliers.
• Use the median for skewed distributions or distributions with outliers.
• Use the mode when the most common item or category is needed.
• The range gives a basic measure of spread, but the IQR is better for skewed data or data with outliers.
• Standard deviation is useful for understanding the spread of data relative to the mean.

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SAT Problem-Solving and Data Analysis Topic 4: Two-variable Data: Models and Scatterplots

Two-variable data (bivariate data) involves pairs of linked numerical observations. For example, analyzing the relationship between hours studied and test scores, or the relationship between height and weight.

Independent Variable (Explanatory): The variable that is presumed to cause or influence the other variable.
Dependent Variable (Response): The variable that is affected or influenced by the independent variable.

Understanding Scatterplots
A scatterplot is a graph that shows the relationship between two sets of data. Each point on the graph represents a pair of values.

Constructing a Scatterplot:

1. Identify the variables (independent and dependent).
2. Label the x-axis (independent variable) and y-axis (dependent variable).
3. Plot each data pair as a point on the graph.

Interpreting Scatterplots:

1. Look for patterns or trends.
2. Identify clusters of data points.
3. Note any outliers that don’t fit the general pattern.

Types of Relationships in Scatterplots

1. Positive Correlation: As one variable increases, the other variable also increases.
2. Negative Correlation: As one variable increases, the other variable decreases.
3. No Correlation: There is no apparent relationship between the two variables.

Linear Relationship: Points roughly follow a straight line.
Non-Linear Relationship: Points form a curve or some other shape.

Line of Best Fit (Trend Line)

The line of best fit (trend line) is a straight line that best represents the data on a scatterplot. It is used to predict values and understand the relationship between variables.

Drawing a Line of Best Fit:

1. Ensure the line has about as many points above it as below.
2. The line should follow the general direction of the data points.

🖋️ Line of Best Fit is a frequently occurring concept in SAT Problem-Solving and Data Analysis questions. 

Using the Line of Best Fit:

• Make predictions based on existing data.
• Understand trends and potential relationships.

Calculating the Equation of a Line of Best Fit
Slope-Intercept Form: The equation of a line can be written as y=mx+b, where:

• m is the slope (rise over run).
• b is the y-intercept (where the line crosses the y-axis).

Calculating the Slope (m):

Finding the Intercept (b):
Use a point on the line (x,y) and the slope m to solve for b.

Residuals: The difference between observed values and the values predicted by the line of best fit. A smaller sum of squared residuals indicates a better fit.

Correlation Coefficient and Strength of Relationship

The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables.
Interpreting r:

• r = 1 : Perfect positive linear correlation.
• r = −1 : Perfect negative linear correlation.
• r = 0 : No linear correlation.
• 0.7 < |r| ≤ 1: Strong correlation.
• 0.3 < |r| ≤ 0.7: Moderate correlation.
• 0 ≤ |r| ≤ 0.3: Weak correlation.

Note that this topic is strongly related to linear equations in two variables. You may revisit that section in our SAT Algebra post to memorize it.

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SAT Problem-Solving and Data Analysis Topic 5: Probability and Conditional Probability

Probability measures the likelihood that a specific event will occur. It is expressed as a number between 0 (impossible event) and 1 (certain event).

Key Terms:

• Experiment: A procedure that can be infinitely repeated and has a well-defined set of possible outcomes (e.g., flipping a coin).
• Outcome: A possible result of an experiment (e.g., getting heads in a coin flip).
• Event: A set of one or more outcomes (e.g., getting heads in a coin flip is an event).
• Sample Space: The set of all possible outcomes (e.g., for a coin flip, the sample space is {Heads, Tails}).

1. Basic Probability Principles

Simple Probability: The probability of an event A occurring is calculated as:

Example: The probability of rolling a 3 on a standard 6-sided die:
P(3) = 16

Complementary Events: The probability of an event not occurring is:
P(A′) = 1− P(A)

2. Compound Events

Compound events involve the combination of two or more events.

Independent Events: Two events are independent if the occurrence of one does not affect the occurrence of the other.
P(A∩B) = P(A) × P(B)

Dependent Events: Two events are dependent if the occurrence of one event affects the occurrence of the other.
P(A∩B) = P(A) x P(B | A)

Addition Rule:

• Mutually Exclusive Events: Events that cannot happen at the same time.
  P(A∪B) = P(A) + P(B)
• Non-Mutually Exclusive Events: Events that can occur at the same time.
  P(A ∪ B) = P(A) + P(B) − P(A∩B)

3. Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred.
Formula:
P(A∣B) = P(A∩B) / P(B)
Example: The probability of drawing a red card from a deck given that the card is a face card.

4. Using Venn Diagrams and Tables
Venn Diagrams: Useful for visualizing relationships and overlaps between events.
Two-Way Tables: Organize data to calculate joint, marginal, and conditional probabilities.

5. The Multiplication Rule for Independent and Dependent Events
Independent Events:
P(A∩B) = P(A) × P(B)
Dependent Events:
P(A∩B) = P(A) × P(B∣A)

6. Bayes’ Theorem (Introduction)
Bayes’ Theorem helps calculate conditional probabilities in reverse.

Formula:
P(A∣B) = [P(B∣A)×P(A)] /P(B)
Application: Used in various problems to update the probability of an event based on new evidence.

Tips for SAT Problem-Solving and Data Analysis Probability Problems

• Be careful with “at least” and “at most” phrases.
• Check if events are mutually exclusive or independent.
• Practice using the formulas and understanding when to apply each.

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SAT Problem-Solving and Data Analysis Topic 6: Inference from Sample Statistics and Margin of Error

Statistical inference involves drawing conclusions about a population based on a sample taken from that population. Since it’s often impractical to collect data from an entire population, statisticians use samples to make educated guesses (inferences) about population parameters.

Population: The entire group that you want to draw conclusions about.
Sample: A subset of the population that is used to make inferences about the population.

1. Sample Statistics

Sample Statistics are numerical values calculated from a sample, which are used to estimate population parameters (e.g., population mean or population proportion).

Key Sample Statistics:

• Sample Mean (x̄): The average of all sample observations.
• Sample Proportion (p̂): The proportion of sample observations that meet a specific criterion.
• Sample Standard Deviation (s): A measure of the variability or spread of sample observations.
• Law of Large Numbers: As the size of the sample increases, the sample mean will get closer to the population mean.

2. Sampling Distribution

Sampling Distribution: The probability distribution of a sample statistic (like the mean) based on many samples from the same population.
Central Limit Theorem: Regardless of the population’s distribution, the sampling distribution of the sample mean will approach a normal distribution as the sample size increases.

Standard Error (SE): The standard deviation of the sampling distribution of a sample statistic.

where s is the sample standard deviation and n is the sample size.

3. Confidence Intervals
Confidence Interval (CI): A range of values, derived from a sample statistic, that is likely to contain the population parameter.
Confidence Interval for a Population Mean:

CI=x̄ ±
where z is the z-score corresponding to the desired confidence level.

Interpretation: A 95% confidence interval means that if we were to take many samples and compute a confidence interval for each sample, approximately 95% of those intervals would contain the true population mean.

4. Margin of Error
Margin of Error (MOE): The maximum amount by which the sample statistic is expected to differ from the true population parameter.

Calculating the Margin of Error:
MOE=

Factors Affecting the Margin of Error:

• Sample Size: Larger samples produce smaller margins of error.
• Variability: Greater variability in the data increases the margin of error.
• Confidence Level: Higher confidence levels produce larger margins of error.

5. Inference from Sample Statistics
Inference: Drawing conclusions about the population based on sample data. Confidence intervals and margins of error help quantify the uncertainty in these inferences.
Example: Estimating the average height of students in a school by measuring a sample of 50 students. Using the sample mean and margin of error, you can estimate the population mean height with a specified level of confidence.

SAT Problem-Solving and Data Analysis Topic 7: Evaluating Statistical Claims: Observational Studies and Experiments

Statistical claims are assertions about data or a population based on a study or research. Evaluating these claims critically is crucial because data can be misrepresented or misunderstood.

Key Questions to Ask:

• What is the source of the data?
• How was the data collected?
• Is the claim based on an observational study or an experiment?
• Are there any biases or confounding factors that could affect the results?

1. Types of Studies in Statistics

There are two main types of studies used to collect data and make statistical claims: observational studies and experiments.

Observational Studies: The researcher observes and records data without manipulating variables.
Experiments: The researcher manipulates one or more variables to observe their effect on other variables.

Key Differences:

• Experiments can establish causality; observational studies cannot.
• Observational studies are often easier and less expensive to conduct.

2. Observational Studies

Characteristics:

• No manipulation of variables by the researcher.
• Data is collected through observation or surveys.
• Can identify associations but not causality.

Types of Observational Studies:

• Cross-Sectional Studies: Data collected at a single point in time.
• Longitudinal Studies: Data collected over a period to observe changes over time.
• Case-Control Studies: Comparing groups with and without a certain outcome to identify factors associated with the outcome.

Strengths:

• Can study variables that are unethical or impractical to manipulate.
• Useful for studying multiple outcomes.

Limitations:
Prone to bias and confounding variables.
Cannot establish causation.

3. Experiments
Definition and Features:
Researchers manipulate one or more independent variables and measure their effect on dependent variables.
Often involve control groups and random assignment to reduce bias.

Randomized Controlled Trials (RCTs):
Considered the “gold standard” for experiments.
Participants are randomly assigned to different groups to ensure results are not biased by confounding factors.

Blinding and Placebos:
Blinding: Participants (and sometimes researchers) do not know which group participants are in to prevent bias.
Placebos: Inactive substances given to control groups to mimic the experimental conditions.

Strengths:
Can establish causal relationships.
A high level of control over variables reduces bias.

Limitations:
Often expensive and time-consuming.
Ethical concerns may limit the scope of experiments.

4. Evaluating the Validity of Statistical Claims

Biases to Consider:

• Selection Bias: When the sample is not representative of the population.
• Response Bias: When participants give inaccurate responses.
• Confounding Variables: Other variables that may affect the outcome.

Sample Size and Randomization:
Larger, randomized samples tend to provide more reliable results.
Replicability: The ability of a study’s findings to be replicated by others; a crucial component of scientific validity.

5. Common Misinterpretations and Misuses of Statistics

Correlation vs. Causation: Just because two variables are correlated does not mean one causes the other.
Misleading Graphs: Graphs can be manipulated to make data appear more significant than it is. Look for scales that are manipulated or data that is cherry-picked.
Selective Reporting: Only reporting data that supports a claim while ignoring data that does not.

Tips for SAT Math Problems on Evaluating Statistical Claims

• Distinguish between observational studies and experiments.
• Be skeptical of conclusions drawn from small sample sizes.
• Look for confounding variables that may affect the validity of claims.

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SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

SAT Problem-Solving and Data Analysis Exercises

Exercise I. Alisa purchased a box of 100 tea bags. She uses one tea bag for each cup of tea. If Alisa drinks 3 cups of tea every day, in how many days will the number of tea bags in the box drop below 20?

Exercise II. A store offers a 20% discount on a certain bag. During the Black Friday promotion, an additional 10% discount is applied on all products in the store. If the final price of the bag is x % of the initial price, what is the value of x?

Exercise III. 1, 3, 7, 7, 8, 5, 2, 11

What is the sum of the median and mean of the data set shown?

Exercise IV. 

The scatterplot shows the relationship between two variables, x and y. A line of best fit for the data is also shown. What is the difference between the y-coordinate of the data point with x = 4 and the y-value predicted by the line of best fit at x = 4?

Exercise V. The following table shows the number of students in each grade in a High School.

25% of the Grade 12 students attend French club. If a student is picked randomly, what is the probability of selecting a Grade 12 student not attending the French club?

Exercise VI. A random sample of 60 people from a town with a population of 18,756 were asked for their opinion on a recent government policy. If 34 people in the sample support the government policy, what is the expected number difference between the supporters and non-supporters in the town?

Note: You can use a calculator in this SAT Math Exercise

Exercise VII. A study is conducted in the state of Utah. A sample of people over 50 years old are asked how many times they visit a doctor each year. What is the largest population to which the result of the survey can be generalized?

💡You can see our Free SAT Math Exercises which has 50 exercises on all SAT Math domains. 

SAT Problem-Solving and Data Analysis Math Practice Test

We’ve listed 3 hard SAT Problem-Solving and Data Analysis practice test questions below. Note that this test does not resemble the typical question difficulty distribution in an SAT Problem-Solving and Data Analysis domain. Instead, we wanted to show you the hardest SAT Problem-Solving and Data Analysis questions you may see on the SAT.

Besides, since these are the hardest questions for the SAT Problem-Solving and Data Analysis, it is very normal that you will spend longer than usual time to solve each question. It is also super normal that you may score lower than your previous SAT Problem-Solving and Data Analysis Practice tests in this one. Because a typical Digital SAT Math Practice Test covers easy, medium, and hard questions. However, this one contains only the hardest questions.

Question 1

Anita created a batch of green paint by mixing 2 ounces of blue paint with 3 ounces of yellow paint. She must mix a second batch using the same ratio of blue and yellow paint as the first batch. If she uses 5 ounces of blue paint for the second batch, how much yellow paint should Anita use?

A. Exactly 5 ounces

B. 3 ounces more than the amount of yellow paint used in the first batch

C. 1.5 times the amount of yellow paint used in the first batch

D. 1.5 times the amount of blue paint used in the second batch

Skill and Knowledge Testing Point: Ratios, rates, proportional relationships, and units

Question 2

37% of the items in a box are green. Of those, 37% are also rectangular. Of the green rectangular items, 42% are also metal. Which of the following is closest to the percentage of the items in the box that are not rectangular green metal items?

A. 1.16%

B. 57.50%

C. 94.25%

D. 98.84%

Skill and Knowledge Testing Point: Percentages

Question 3

The mean amount of time that the 20 employees of a construction company have worked for the company is 6.7 years. After one of the employees leaves the company, the mean amount of time that the remaining employees have worked for the company is reduced to 6.25 years. How many years did the employee who left the company work for the company?

A. 0.45

B. 2.30

C. 9.00

D. 15.25

Content Domain: Problem-Solving and Data Analysis

Skill and Knowledge Testing Point: One-variable data: distributions and measures of center and spread

Question 4

The scatterplot above shows the size x and the sale price y of 25 houses for sale in Town H. Which of the following could be an equation for a line of best fit for the data?

A. y = 200x + 100

B. y = 100x + 100

C. y = 50x + 100

D. y = 100x

Skill and Knowledge Testing Point: Two-variable data: models and scatterplots

Question 5

The table summarizes the distribution of age and assigned group for 90 participants in a study.

One of these participants will be selected at random. What is the probability of selecting a participant from group A, given that the participant is at least 10 years of age? (Express your answer as a decimal or fraction, not as a percent.)

Skill and Knowledge Testing Point: Probability and conditional probability

Question 6

The results of two random samples of votes for a proposition are shown above. The samples were selected from the same population, and the margins of error were calculated using the same method. Which of the following is the most appropriate reason that the margin of error for sample A is greater than the margin of error for sample B?

A. Sample A had a smaller number of votes that could not be recorded.

B. Sample A had a higher percentage of favorable responses.

C. Sample A had a larger sample size.

D. Sample A had a smaller sample size.

Skill and Knowledge Testing Point: Inference from sample statistics and margin of error

Question 7

To determine the mean number of children per household in a community, Tabitha surveyed 20 families at a playground. For the 20 families surveyed, the mean number of children per household was 2.4. Which of the following statements must be true?

A. The mean number of children per household in the community is 2.4.

B. A determination about the mean number of children per household in the community should not be made because the sample size is too small.

C. The sampling method is flawed and may produce a biased estimate of the mean number of children per household in the community.

D. The sampling method is not flawed and is likely to produce an unbiased estimate of the mean number of children per household in the community.

Skill and Knowledge Testing Point: Evaluating statistical claims: observational studies and experiments

SAT Problem-Solving and Data Analysis Practice Test Answers and Rationales

We’ve created a comprehensive answers and rationales PDF file for these SAT Problem-Solving and Data Analysis questions. If you can fill in your name and email below, we can send it to your email in minutes. Note that, the PDF you will receive will have 19 questions from all SAT Math domains. Questions 9, 10, 11, 12, 13, 14, and 15  (Questions 9-15) are answers and rationales for this SAT Problem-Solving and Data Analysis Practice Test.

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There are 44 SAT Math questions in the SAT Exam. 13 to 15 of these 44 questions come from the SAT Advanced Math content domain. Approximately 30-35% of the SAT Math questions come from the SAT Advanced Math. Advanced Math is a little bit harder than SAT Algebra. However, having a solid background and solving as many SAT Advanced Math practices as possible will improve your SAT Advanced Math scores.

📌 Hint: Do not skip this article, you will find FREE Digital SAT Math Prep resources throughout the article.

We’ve listed the most important and frequently occurring concepts in this SAT Advanced Math post. You will see SAT Advanced Math practice test questions and exercises, totaling 27 SAT Advanced Math questions with rationales for FREE. 

💡You might be interested in reading the Digital SAT Math Prep post.

SAT Advanced Math Content Domain

Advanced Math focuses on the math you’ll need to pursue further study in disciplines such as science or economics and for career opportunities in the STEM fields of science, technology, engineering, and math. SAT Advanced Math area measures skills and knowledge central for progression to more advanced math courses, including demonstrating an understanding of absolute value, quadratic, exponential, polynomial, rational, radical, and other nonlinear equations.

💡You might be interested in reading the Digital SAT Math Ultimate Guide post. We have provided further details about the SAT Math structure, examples of easy, medium, and hard questions, answers, rationales, and frequently asked questions about the SAT Math.

👉 Take our full-length FREE SAT Practice Test, see where you stand!

Anna B. Scored 800 on SAT Math!

Anna B. is one of our thousands of successful SAT students. She scored 800 on SAT Math. You can watch her SAT story.

SAT Advanced Math Skills and Knowledge Testing Points

The SAT exam will have around 13 to 15 questions from the Advanced Math content domain. There are 3 skills and knowledge testing points in the SAT Advanced Math content domain:

1. Equivalent expressions
2. Nonlinear equations in one variable and systems of equations in two variables
3. Nonlinear functions

Let’s review each skill and knowledge point and see some SAT Advanced Math Exercises for each.

🗎 Download the 15-page Digital SAT Math Formula Sheet.

1. Equivalent Expressions

Equivalent expressions are algebraic expressions that, despite having different forms, produce the same result for any value of the variable(s).

Importance: Understanding equivalent expressions is crucial for simplifying algebraic SAT Advanced Math problems, solving equations, and performing algebraic manipulations.

Basic Principles of Equivalent Expressions

Commutative Property:
Addition: a+b = b+a
Multiplication: a×b = b×a

Associative Property:
Addition: (a+b)+c=a+(b+c)
Multiplication: (a×b)×c=a×(b×c)

Distributive Property:
a(b+c) = ab + ac

Recognizing Equivalent Expressions

Simplification: Combine like terms and use properties to simplify expressions.

Example 1: Simplify 2x + 3x −4 + x.

Solution: Combine like terms: 2x+3x+x−4=6x−4.
Equivalent Expression: 6x−4

Techniques for Finding Equivalent Expressions

Factoring and Expanding: Use the distributive property to factor and expand expressions.

Example 2: Find an equivalent expression for 3(x+2)+4x.

Solution: Expand 3(x+2)=3x+6, then add 4x:
3x+6+4x = 7x+6
Equivalent Expression: 7x+6

Equivalent Expressions Practice Problems

Example 3: Identify if the expressions 4(x+1)−2 and 4x+2 are equivalent.

Solution: Expand 4(x+1)−2=4x+4−2 = 4x+2
Equivalent: Yes, both expressions simplify to 4x+2.

Example 4: Determine the equivalent expression for 2(a+b)−3(b−a).

Solution:
Expand: 2a+2b−3b+3a = 5a−b
Equivalent Expression: 5a−b.

Application in SAT Advanced Math Problems

Strategy: Recognize and use equivalent expressions to simplify complex problems and solve equations efficiently.

Example 5: Solve for x if 3x+4=2x+8.
Solution: Rearrange to form an equivalent expression: 3x−2x = 8−4.
Solve: x = 4.

Common Mistakes to Avoid

Ignoring Parentheses: Remember to apply the distributive property correctly when parentheses are involved.
Forgetting to Combine Like Terms: Always combine like terms to simplify the expression fully.

Summary and Key Takeaways

• Equivalent expressions represent the same quantity and are essential for algebraic manipulation.
• Mastery of properties (commutative, associative, distributive) is crucial in recognizing and forming equivalent expressions.
• Practice identifying and creating equivalent expressions to build confidence for the SAT Advanced Math section.

💡We’ve prepared a 7-Step Digital SAT Math Study Guide helping students to prepare their unique SAT Math Study Guide.

SAT Advanced Math Exercises for Equivalent Expressions

To improve your math skills, we do not recommend using a calculator when solving these SAT Advanced Math Exercises.

Exercise I.

The given equation relates the positive numbers p, x, and y. Write the p-value in terms of x and y.

Exercise II.

What is the value of ?

Exercise III.

What is the value of x?

Exercise IV. 

Simplify the given expression.

💡You can see our Free SAT Math Exercises which has 50 exercises on all SAT Math domains. 

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Free Digital SAT Prep Course

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

2. Nonlinear Equations in One Variable and Systems of Equations in Two Variables

Nonlinear Equations in One Variable

A nonlinear equation in one variable is an equation in which the variable is raised to a power other than one, appears in the denominator, or is part of a function like a square root or absolute value. Common examples include quadratic equations, cubic equations, and equations involving square roots or absolute values.

We’ve listed types of nonlinear equations and how to solve these SAT Advanced Math questions below.

Types of Nonlinear Equations

Quadratic Equations:
Cubic Equations:
Equations Involving Square Roots:
Equations with Absolute Values: |x-3| = 5

Solving Nonlinear Equations

Quadratic Equations:
Factoring: Find two numbers that multiply to ac (coefficient of x2 times constant term) and add to b (coefficient of x).
Quadratic Formula:
Completing the Square: Rewriting the equation in the form

💡Tip: Quadratic Equations and Formula is a frequently occurring SAT Advanced Math concept in SAT exam.

Example 1: Solve

Solution:
Therefore, x = 2 or x = 3.

Equations with Square Roots:
Isolate the square root on one side of the equation and then square both sides.

Example 2: Solve
Solution: Square both sides: x + 2 = 9
x = 7

Equations with Absolute Values:
Split into two cases, one where the expression inside the absolute value is positive and one where it is negative.

Example 3: Solve ∣x−3∣ = 5.

Solution: x – 3 = 5 or x – 3 = – 5
x = 8 or x = -2

Systems of Equations in Two Variables

A system of equations consists of two or more equations with the same set of variables. Solving systems of equations means finding the set of values for the variables that satisfy all equations in the system.

Types of Systems

Linear-Linear Systems: Both equations are linear.
Linear-Nonlinear Systems: One equation is linear, and the other is nonlinear (like a quadratic).

Solving Systems of Equations in SAT Advanced Math

Substitution Method:
Solve one equation for one variable and substitute that expression into the other equation.

Example 4: Solve the system:
y=2x+3

Solution: Substitute y = 2x + 3 into the second equation:

Expand and simplify:


Solve the quadratic equation using the quadratic formula or factoring.

Elimination Method:
Add or subtract the equations to eliminate one of the variables.

Example 5: Solve the system:
x+y=7
x−y=1

Solution: Add the two equations to eliminate y:
2x = 8 ⇒ x = 4

Substitute x=4 back into x+y=7
4 + y = 7 ⇒ y = 3

The solution is x = 4, y = 3

💡You can see our Free SAT Math Exercises which has 50 exercises on all SAT Math domains. 

Watch SAT Math Prep Online Course – Sample Lecture on YouTube

We have a sample 8-minute video lecture from our SAT Math Prep Online Course on YouTube. You can watch below.

SAT Advanced Math Exercises for Nonlinear Equations in One Variable and Systems of Equations in Two Variables

To improve your math skills, we do not recommend using a calculator when solving these SAT Advanced Math Exercises.

Exercise I. Two variables, x, and y are related such that for each increase of 1 in the value of x, the value of y increases by a factor of 5. When x=0, y=10. Write the y in terms of x.

Exercise II. .

One solution to the given equation can be written as , where k is a constant. What is the value of k?

Exercise III. .

In the given equation, p is a constant. The equation has exactly one solution. What is the value of p?

Exercise IV.
.

A solution to the given system of equations is (x, y), where x>0. What is the value of x?

Digital SAT Prep Online Course Program

San Francisco Business School offers a comprehensive Digital SAT Prep Online Course taught by 99th-percentile SAT Instructors and exam experts. The program cracks down each content domain, skills, and knowledge testing point through 1,000+ realistic Digital SAT Exam questions. You will see all the different types of questions that may appear in Digital SAT.

3. Nonlinear Functions

Nonlinear functions are those whose graphs are not straight lines. These functions have at least one variable raised to a power other than one, multiplied by itself, or in some other configuration that creates a curve rather than a line. In SAT Advanced Math, understanding nonlinear functions involves recognizing their different forms and how they behave on a graph.

Key Characteristics of Nonlinear Functions:

• Nonlinear Graphs: The graphs of nonlinear functions can be parabolas, circles, ellipses, hyperbolas, or any other shape that is not a straight line.
• Variable Powers: At least one variable is raised to a power other than one (e.g., x², x³).
• Multiple Solutions: Nonlinear functions can have multiple x-intercepts, y-intercepts, or roots.
• Changes in Direction: Nonlinear graphs can change direction, unlike linear graphs that are consistently increasing or decreasing.

Common Types of Nonlinear Functions in SAT Advanced Math

1. Quadratic Functions

Form: f(x) = ax² + bx + c
Graph: Parabola (U-shaped curve)
Vertex: The highest or lowest point of the parabola.
Examples:
f(x) = x² – 4x + 3
Graph this function: The parabola opens upwards because the coefficient of x² is positive. The roots are where the function crosses the x-axis.

2. Cubic Functions

Form: f(x) = ax³ + bx² + cx + d
Graph: S-shaped curve with one or more turns.
Examples:
f(x) = x³ − 3x² + 2x
Graph this function: The curve starts from the lower left, turns upward, turns again, and moves downward or upward depending on the coefficients.

3. Exponential Functions

Form: f(x) =   where b > 0 and b≠1
Graph: Exponential growth or decay curve.
Examples:
f(x)= shows exponential growth.
f(x)= shows exponential decay.
These functions rapidly increase or decrease and never touch the x-axis.

4. Absolute Value Functions

Form: f(x) = ∣ax+b∣
Graph: V-shaped graph.
Examples:
f(x) = ∣x−2∣
Graph this function: The graph has a vertex at x=2 and opens upwards.

5. Rational Functions

Form: f(x) = where p(x) and q(x) are polynomials, and q(x) ≠ 0
Graph: Can have asymptotes and undefined points.
Examples:

Graph this function: The function has a vertical asymptote at x=1 and a horizontal asymptote at y=0.

Examples and Practice Problems

Example 1: Quadratic Function
Problem: Graph the function
Solution:
Identify the coefficients: a=1, b=−4, c=3.
Find the roots using the quadratic formula: x=1 and x=3.
The vertex is at (2,−1).
The parabola opens upwards.

Example 2: Exponential Function
Problem: Solve for x in the equation
Solution: Rewrite 16 as , so , thus x = 4.

Example 3: Rational Function
Problem: Determine the domain of the function
Solution: The function is undefined when the denominator is zero, so x ≠ 3. The domain is all real numbers except x = 3.

Graphing Nonlinear Functions
Graphing nonlinear functions involves plotting points and understanding the shape of the function. Here are the steps to graph a nonlinear function:

1. Identify the function type (quadratic, cubic, etc.)
2. Determine key features: roots, intercepts, asymptotes, vertex, etc.
3. Plot critical points and sketch the graph based on these points.

SAT Advanced Math Hack Points & Exercises YouTube Video

You can view our SAT Advanced Math YouTube video. We’ve gone through each of the 5 SAT Advanced Math topics, provided the important points to know, and exercises for each as well.

SAT Advanced Math Exercises for Nonlinear Functions

To improve your math skills, we do not recommend using a calculator when solving these SAT Advanced Math Exercises.

Exercise I.

The given equation defines the function f. What is the minimum value of f(x)?

Exercise II. The function f is defined by . What is the value of f(6)?

Exercise III. The function gives a spring’s length, in feet, when an object of w kilograms is hung, where . What is the best interpretation of 15 in this context?

Free Digital SAT Prep Course

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

SAT Advanced Math Practice Test

We’ve listed 3 hard SAT Advanced Math practice test questions below. Note that this test does not resemble the typical question difficulty distribution on an SAT Advanced Math domain. Instead, we wanted to show you the hardest SAT Advanced Math questions you may see on the SAT.

Besides, since these are the hardest questions for the SAT Advanced Math, it is very normal that you will spend longer than usual time to solve each question. It is also super normal that you may score lower than your previous SAT Advanced Math Practice tests in this one. Because a typical Digital SAT Math Practice Test covers easy, medium, and hard questions. However, this one contains only the hardest questions.

Question 1

In the expression above, b and c are positive integers. If the expression is equivalent to x+b and x ≠ b, which of the following could be the value of c ?

A. 4

B. 6

C. 8

D. 10

Skill and Knowledge Testing Point: Equivalent expressions

Question 2

If and are the two solutions to the system of equations above, what is the value of ?

A. -3

B. -2

C. -1

D. 1

Skill and Knowledge Testing Point: Nonlinear equations in one variable and systems of equations in two variables

Question 3

x
1
2
3

For the exponential function , the table above shows several values of and their corresponding values of , where is a constant greater than 1. If is a constant and , what is the value of ?

Skill and Knowledge Testing Point: Nonlinear functions

SAT Advanced Math Practice Test Answers and Rationales

We’ve created a comprehensive answers and rationales PDF file for these SAT Advanced Math questions. If you can fill in your name and email below, we can send it to your email in minutes. Note that, the PDF you will receive will have 19 questions from all SAT Math domains. Questions 6, 7, and 8 (Questions 6-8) are answers and rationales for this SAT Advanced Math Practice Test.

Note that, the email may hit your junk or spam folders, please check your junk and spam folders and if you did not receive it, please email us at support@sanfranciscobs.com.

💡Do not forget to visit SAT Math Practice Test – Hardest Questions. Assess your SAT Math skills with the hardest questions you may see on SAT Math.

The post SAT Advanced Math – FREE SAT Advanced Math Practice appeared first on San Francisco Business School.

There are 44 SAT Math questions in the SAT Exam. 13 to 15 of these 44 questions come from the SAT Algebra content domain. This makes the Algebra content domain 30-35% of the SAT Math. Therefore, having a solid SAT Algebra background and solving as many SAT Algebra Practice Tests as possible is crucial to having a high SAT Math score.

📌 Hint: Do not skip this article, you will find FREE Digital SAT Math Prep resources throughout the article.

We’ve listed the most important and frequently occurring concepts in this SAT Algebra post. You will see SAT Algebra practice test questions and exercises, in total 31 SAT Algebra questions with rationales all for FREE. 

💡You might be interested in reading the Digital SAT Math Prep post.

SAT Algebra Content Domain

Algebra is a branch of mathematics in which arithmetical operations and formal manipulations are applied to abstract symbols rather than specific numbers. Algebra is the first content domain in the SAT Math. In the SAT Algebra domain, the College Board assesses the abilities of students in solving and creating linear equations and inequalities as well as analyzing and fluently solving equations and systems of equations using multiple techniques.

💡You might be interested in reading the Digital SAT Math Ultimate Guide post. We have provided further details about the SAT Math structure, examples of easy, medium, and hard questions, answers, rationales, and frequently asked questions about the SAT Math.

👉 Take our full-length FREE SAT Practice Test, see where you stand!

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Anna B. is one of our thousands of successful SAT students. She scored 800 on SAT Math. You can watch her SAT story.

SAT Algebra Skills and Knowledge Testing Points

The SAT exam will have around 13 to 15 questions from the Algebra content domain. There are 5 skills and knowledge testing points in the Algebra content domain:

1. Linear equations in one variable
2. Linear equations in two variables
3. Linear functions
4. Systems of two linear equations in two variables
5. Linear inequalities in one or two variables

Let’s review each skill and knowledge point and see some SAT Algebra Exercises for each.

🗎 Download the 15-page Digital SAT Math Formula Sheet.

1. Linear Equations in One Variable – Important Points

A linear equation in one variable is an equation that can be expressed in the form:

Where:

• and are constants and ≠ Error: WPMathPub plugin shortcode cannot be empty.
• is the variable.

The general solution to a linear equation in one variable will yield a single value for .

How to Solve SAT Algebra Linear Equations

There are three simple steps to solve linear equations:

1. Isolate the variable: Use addition, subtraction, multiplication, and division to get the variable by itself on one side of the equation.
2. Simplify: Combine like terms and simplify both sides if necessary.
3. Check your solution: Substitute the found value into the original equation to verify correctness.

💡We’ve prepared a 7-Step Digital SAT Math Study Guide helping students to prepare their unique SAT Math Study Guide.

SAT Math Algebra Exercises for Linear Equations in One Variable

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Algebra Exercises.

Exercise I. 

Exercise II. 

Exercise III. 

Exercise IV. 

Exercise V. 

Exercise VI.  Ethan has $15. He gets dollars from his grandfather. If Ethan has $35 now, what is the value of ?

💡You can see our Free SAT Math Exercises which has 50 exercises on all SAT Math domains. 

2. Linear Equations in Two Variables – Important Points

A linear equation in two variables can be expressed in the form:

Where:

• , and are constants.
• and are variables.

Graphing SAT Algebra Linear Equations

• The graph of a linear equation in two variables is a straight line.
• The coefficients and determine the slope of the line, while affects its position on the graph.

Slope-Intercept Form:

• A common way to express a linear equation is the slope-intercept form: where is the slope and is the y-intercept (the point where the line crosses the y-axis).

🖋️ Slope-intercept form is a frequently occurring concept in SAT Math. 

• Parallel Lines: Slopes of parallel lines are the same.
• Perpendicular Lines: The product of the perpendicular lines is (-1).

Free Digital SAT Prep Course

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

SAT Algebra Exercises for Linear Equations in Two Variables

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I.  . What is the y-intercept of this graph?

Exercise II.  . What is the x-intercept of this graph?

Exercise III.  . and are two possible solutions to the equation. What is the value of ?

Exercise IV.  . Grapf of line m is given. A line k is parallel to line m. What is the slope of line k?

Exercise V.  . If the line h is perpendicular to the given equation’s graph, what is the slope of line h?

Exercise VI. A line passes through (0, 4) and the slope of the line is 2. What is the equation of this line?

SAT Algebra Hack Points & Exercises YouTube Video

You can view our SAT Algebra YouTube video. We’ve gone through each of the 5 SAT Algebra topics, provided the important points to know, and exercises for each as well.

💡You might be interested in seeing our Digital SAT Math Practice Test. It covers the hardest questions you may see on the SAT Math exam for each skill and knowledge testing point!

3. Linear Functions – Important Points

A linear function is a function that creates a straight line when graphed on the xy-plane.

• The general form of a linear function is: where is the slope and is the y-intercept ((0, b), the value of f(x) when x = 0).
• The slope (m) of a linear function indicates the steepness of the line. It can be calculated as:

The slope can be positive, negative, zero, or undefined:

• • Positive slope: The line rises as it moves from left to right.
  • Negative slope: The line falls as it moves from left to right.
  • Zero slope: The line is horizontal.
  • Undefined slope: The line is vertical.

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San Francisco Business School offers a comprehensive Digital SAT Prep Online Course taught by 99th-percentile SAT Instructors and exam experts. The program cracks down each content domain, skills, and knowledge testing point through 1,000+ realistic Digital SAT Exam questions. You will see all the different types of questions that may appear in Digital SAT.

SAT Algebra Exercises for Linear Functions

To improve your math skills, we do not recommend using a calculator when solving these SAT Algebra Exercises.

Exercise I.   The graph of y = f(x) passes through the points (1, 4) and (3, 10). What is the function f ?

Exercise II.  The function f is defined by the equation . What is the value of f(x) when ?

Exercise III. An electrician charges a $40 fixed fee plus $25 per hour. If the function h models the total fee for the electrician for t hours of work, write the function h.

Exercise IV. The function f is defined by . What is the y-intercept of the graph of in the xy-plane?

Exercise V. The graph of y = f(x) and y=h(x) are perpendicular to each other in the xy-plane. Function f is defined by . If , write the y=h(x) function.

Exercise VI. The function models the altitude of an airplane m minutes after takeoff. According to the model, what is the altitude of the airport where the airplane took off?

📚 San Francisco Business School offers a vast amount of FREE Digital SAT Prep Online materials. See it on the Free Digital SAT Prep Online Library.

4. Systems of Two Linear Equations in Two Variables – Important Points

A system of two linear equations in two variables consists of two equations that can be represented in the form:

where x and y are variables, and , , , , , and are constants.

Graphical Interpretation: Each equation represents a straight line in the coordinate plane. The solution to the system is the point where the two lines intersect.

How to Solve SAT Algebra Systems of Equations

There are three main methods to solve a system of linear equations:

1. Graphing:
   • Rewrite both equations in slope-intercept form (y = mx + b).
   • Graph each line on the same coordinate plane.
   • Identify the intersection point, which is the solution.
     
2. Substitution
   • Solve one equation for one variable.
   • Substitute that expression into the other equation.
   • Solve for the remaining variable.
   • Substitute back to find the other variable.
3. Elimination
   • Align the equations.
   • Multiply one or both equations to make the coefficients of one variable opposite.
   • Add or subtract the equations to eliminate one variable.
   • Solve for the remaining variable.
   • Substitute back to find the other variable.

Watch SAT Math Prep Online Course – Sample Lecture on YouTube

We have a sample 8-minute video lecture from our SAT Math Prep Online Course on YouTube. You can watch below.

Free Digital SAT Prep Course

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

SAT Algebra Exercises for Systems of Two Linear Equations in Two Variables

To improve your math skills, we do not recommend using a calculator when solving these SAT Algebra Exercises.

Exercise I. 

The solution to the given system of equations is (x, y). What is the value of y)

Exercise II. 

For the given system of equations, what is the value of y ?

Exercise III. 

The solution to the given system of equations is (x, y). What is the value of x + y?

Exercise IV.   

In the given system of equations, k is a constant. If the system has no solution, what is the value of k?

💡You might be interested in seeing our Digital SAT Math Practice Test. It covers the hardest questions you may see on the SAT Math exam for each skill and knowledge testing point!

5. Linear Inequalities in One or Two Variables – Important Points

A linear inequality is similar to a linear equation but uses inequality signs (>, <, ≥, ≤) instead of the equals sign (=).

Example: Linear inequality: ( 2x + 3 < 7 )

Solving SAT Algebra Linear Inequalities in One Variable

You can apply the following basic steps to solve SAT Algebra Linear Inequalities in One Variable:

• Isolate the variable on one side of the inequality.
• Perform the same operations on both sides (addition, subtraction, multiplication, division).
• Important: If you multiply or divide by a negative number, reverse the inequality sign.

Example: Solve ( 3x – 5 ≥ 4 ).

• Step 1: Add 5 to both sides: ( 3x ≥ 9 )
• Step 2: Divide by 3: ( x ≥ 3 )
• Graphing the Solution: On a number line, you would represent ( x ≥ 3 ) with a closed circle at 3 shaded to the right.

SAT Algebra Compound Inequalities

These involve two inequalities connected by “and” or “or”. Types:

• Conjunction (And): True if both inequalities are true.
• Disjunction (Or): True if at least one inequality is true.

Example (Conjunction): Solve ( 1 < 2x + 1 < 7 ).

• Break it into two inequalities:
  • ( 1 < 2x + 1 )
    ( 2x + 1 < 7 )
• Solve both:
  • First: ( 0 < 2x  ->  x > 0 )
  • Second: ( 2x < 6  ->  x < 3 )
• Combined solution: ( 0 < x < 3 )
• Graphing: Represent this with an open interval on a number line from 0 to 3.

Linear Inequalities in Two Variables

Linear inequality uses two variables, represented in the form ( Ax + By < C ), ( Ax + By > C ), etc.

Example: Linear inequality: ( x + 2y ≤ 4 )

Systems of Linear Inequalities

A set of two or more inequalities that can be graphed on the same coordinate plane.

Example:
( y > x + 1 )
( y < -x + 3 )

SFBS Digital SAT Prep Online Course Program

San Francisco Business School offers a comprehensive Digital SAT Prep Online Course taught by 99th-percentile SAT Instructors and exam experts. The program cracks down each content domain, skills, and knowledge testing point through 1,000+ realistic Digital SAT Exam questions. You will see all the different types of questions that may appear in Digital SAT.

SAT Algebra Exercises for Linear Inequalities in One or Two Variables

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I. Ryan has $100 and wants to purchase balls. A store sells basketballs and baseballs. Basketballs are $15 each and baseballs are $20 each.  If x represents the number of basketballs and y represents the number of baseballs Ryan can purchase, what is the inequality representing this situation?

Exercise II. The maximum value of is 13 greater than another number k. What is the inequality for x in terms of k?

Exercise III. A High School has students whose height is between 160 cm to 186cm. If h represents the height of a student in this High School, what is the inequality representing the height of a student?

Exercise IV.

(2, p) is a solution to the given system of inequalities. What is the maximum integer value for p?

💡You can see our Free SAT Math Exercises which has 50 exercises on all SAT Math domains. 

SAT Algebra Practice Test

We’ve listed 5 hard SAT Algebra practice test questions below. Note that this test does not resemble the typical question difficulty distribution on a SAT Algebra domain. Instead, we wanted to show you the hardest SAT Math Algebra questions you may see on the SAT.

Besides, since these are the hardest questions for the SAT Algebra, it is very normal that you will spend longer than usual time to solve each question. It is also super normal that you may score lower than your previous SAT Algebra Practice tests in this one. Because a typical Digital SAT Math Practice Test covers easy, medium, and hard questions. However, this one contains only the hardest questions.

Question 1

The equation 9x + 5 =a(x+b), where a and b are constants, has no solutions. Which of the following must be true?

I. a = 9
II. b = 5
III. b ≠

A. None

B. I only

C. I and II only

D. I and III only

Skill and Knowledge Testing Point: Linear equations in one variable

Question 2

To earn money for college, Avery works two part-time jobs: A and B. She earns $10 per hour working at job A and $20 per hour working at job B. In one week, Avery earned a total of s dollars for working at job B. In one week, Avery earned a total of s dollars for working at the two part-time jobs. The graph above represents all possible combinations of the number of hours Avery could have worked at the two jobs to earn s dollars. What is the value of s ?

A. 128

B. 160

C. 200

D. 320

Skill and Knowledge Testing Point: Linear equations in two variables

Question 3

An object hangs from a spring. The formula relates the length , in centimeters, of the spring to the weight , in newtons, of the object. Which of the following describes the meaning of the 2 in this context?

A. The length, in centimeters, of the spring with no weight attached

B. The weight, in newtons, of an object that will stretch the spring 30 centimeters

C. The increase in the weight, in newtons, of the object for each one-centimeter increase in the length of the spring

D. The increase in the length, in centimeters, of the spring for each one-newton increase in the weight of the object

Skill and Knowledge Testing Point: Linear functions

SAT Algebra Practice Test – Question 4

Store A sells raspberries for $5.50 per pint and blackberries for $3.00 per pint. Store B sells raspberries for $6.50 per pint and blackberries for $8.00 per pint. A certain purchase of raspberries and blackberries would cost $37.00 at Store A or $66.00 at Store B. How many pints of blackberries are in this purchase?

A. 4

B. 5

C. 8

D. 12

Skill and Knowledge Testing Point: Systems of two linear equations in two variables

Question 5

Ken is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Ken saves 90% of his earnings from each week. What is the least number of hours he must work the rest of the week to save at least $270 for the week?

A. 38

B. 33

C. 22

D. 16

Skill and Knowledge Testing Point: Linear inequalities in one or two variables

SAT Algebra Practice Test Answers and Rationales

We’ve created a comprehensive answers and rationales PDF file for these SAT Algebra Practice questions. If you can fill in your name and email below, we can send it to your email in minutes. Note that, the PDF you will receive will have 19 questions from SAT Math domains. The first 5 questions are answers and rationales for this SAT Algebra Practice Test.

Note that, the email may hit your junk or spam folders, please check your junk and spam folders and if you did not receive it, please email us at support@sanfranciscobs.com.

💡You might be interested in seeing our Digital SAT Math Practice Test. It covers the hardest questions you may see on the SAT Math exam for each skill and knowledge testing point!

The post SAT Algebra – Hack Points and FREE SAT Algebra Practice appeared first on San Francisco Business School.

After teaching thousands of SAT exam students, we’ve revealed that a high SAT Math score strongly depends on students’ practice on SAT Math exercises. We’ve seen many students increase their SAT Math scores by around 200 points. This is a huge increase, and it is only possible by doing as many SAT Math exercises and practices as possible.

📌 Hint: Do not skip this article, you will find FREE Digital SAT Math Prep resources throughout the article.

Ethan is one of our thousands of successful SAT students. He perfectly scored 1600 on the SAT, the highest score a student can get! We were with thousands of students, like Ethan, in their SAT Math Prep journey. We’ve witnessed how they approach SAT Math Exercises and practices and that is why created this SAT Math Exercices post to help many others!

💡You might be interested in reading the Digital SAT Math Prep post.

Digital SAT Math Structure

Before diving into SAT Math Exercises, you must understand the SAT Math structure, question types, and how to approach different types of questions.

The SAT exam consists of two modules, Module I and Module II. Each module consists of 22 questions, and there will be a total of 44 questions. You will have 35 minutes for each module. Going through comprehensive SAT Math Exercises will help you to get a higher score on SAT Math. We prepared the following table to summarize the structure of the  SAT Math sections.

👉 Take our full-length FREE SAT Practice Test, see where you stand!

The most critical aspect of the Digital SAT is being adaptive. In SAT Math Module I, you will be asked a broad mix of easy, medium, and hard questions. Then, the difficulty of the SAT Math Module II will depend on your score in the SAT Math Module I. This means that the test “adapts” to present questions that are more appropriate to a student’s performance level. Going through several SAT Math Exercises and Practices will improve your scores. The following figure depicts the Digital SAT adaptive testing model.

💡You might be interested in reading the Digital SAT Math Ultimate Guide post. We have provided further details about the SAT Math structure, examples of easy, medium, and hard questions, answers, rationales, and frequently asked questions about the SAT Math.

Anna B. Scored 800 on SAT Math!

Anna B. is one of our thousands of successful SAT students. She scored 800 on SAT Math. You can watch her SAT story.

Now, let’s go through the SAT Math Exercises!

🗎 Download the 15-page Digital SAT Math Formula Sheet.

Digital SAT Math Exercises

We’ve created these SAT Math Exercises to provide exercises for the commonly tested concepts on the SAT exam. You can consider enrolling in our Online Digital SAT Math Prep Course for a comprehensive SAT Math Prep.

We’ve listed the SAT Math Exercises for each content domain and skills and knowledge testing points respectively. There are a total of 50 SAT Math Exercises in 10 sets. We do not recommend using a calculator for many of the questions. In case a question needs complex calculations, we mentioned as a note that you can use a calculator for that particular SAT Math Exercise.

💡We’ve prepared a 7-Step Digital SAT Math Study Guide helping students to prepare their unique SAT Math Study Guide.

SAT Math Exercises for Algebra

Algebra is the first content domain in the SAT Math. The SAT exam will have around 13 to 15 questions from the Algebra content domain. There are 5 skills and knowledge testing points in the Algebra content domain:

1. Linear equations in one variable
2. Linear equations in two variables
3. Linear functions
4. Systems of two linear equations in two variables
5. Linear inequalities in one or two variables

💡You might be interested in seeing our SAT Algebra post. It covers the hack points you should know and 31 SAT Algebra Practice Questions!

Let’s review each skill and knowledge point and see some SAT Math Exercises for each.

SAT Math Exercise Set 1: Linear Equations in One Variable

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I. 

Exercise II. 

Exercise III. 

Exercise IV. 

Exercise V. 

Exercise VI.  Ethan has $15. He gets dollars from his grandfather. If Ethan has $35 now, what is the value of ?

💡You might be interested in seeing our Digital SAT Math Practice Test. It covers the hardest questions you may see on the SAT Math exam for each skill and knowledge testing point!

📚 San Francisco Business School offers a vast amount of FREE Digital SAT Prep Online materials. See it on the Free Digital SAT Prep Online Library.

Free Digital SAT Prep Course

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

SAT Math Exercise Set 2: Linear Equations in Two Variables

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I.  . What is the y-intercept of this graph?

Exercise II.  . What is the x-intercept of this graph?

Exercise III.  . and are two possible solutions to the equation. What is the value of ?

Exercise IV.  . Grapf of line m is given. A line k is parallel to line m. What is the slope of line k?

Exercise V.  . If the line h is perpendicular to the given equation’s graph, what is the slope of line h?

Exercise VI. A line passes through (0, 4) and the slope of the line is 2. What is the equation of this line?

Watch SAT Math Prep Online Course – Sample Lecture on YouTube

We have a sample 8-minute video lecture from our SAT Math Prep Online Course on YouTube. You can watch below.

SAT Math Exercise Set 3: Linear Functions

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I.   The graph of y = f(x) passes through the points (1, 4) and (3, 10). What is the function f ?

Exercise II.  The function f is defined by the equation . What is the value of f(x) when ?

Exercise III. An electrician charges a $40 fixed fee plus $25 per hour. If the function h models the total fee for the electrician for t hours of work, write the function h.

Exercise IV. The function f is defined by . What is the y-intercept of the graph of in the xy-plane?

Exercise V. The graph of y = f(x) and y=h(x) are perpendicular to each other in the xy-plane. Function f is defined by . If , write the y=h(x) function.

Exercise VI. The function models the altitude of an airplane m minutes after takeoff. According to the model, what is the altitude of the airport where the airplane took off?

SAT Math Exercise Set 4: Systems of Two Linear Equations in Two Variables

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I. 

The solution to the given system of equations is (x, y). What is the value of y)

Exercise II. 

For the given system of equations, what is the value of y ?

Exercise III. 

The solution to the given system of equations is (x, y). What is the value of x + y?

Exercise IV.   

In the given system of equations, k is a constant. If the system has no solution, what is the value of k?

SAT Math Exercise Set 5: Linear Inequalities in One or Two Variables

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I. Ryan has $100 and wants to purchase balls. A store sells basketballs and baseballs. Basketballs are $15 each and baseballs are $20 each.  If x represents the number of basketballs and y represents the number of baseballs Ryan can purchase, what is the inequality representing this situation?

Exercise II. The maximum value of is 13 greater than another number k. What is the inequality for x in terms of k?

Exercise III. A High School has students whose height is between 160 cm to 186cm. If h represents the height of a student in this High School, what is the inequality representing the height of a student?

Exercise IV.

(2, p) is a solution to the given system of inequalities. What is the maximum integer value for p?

SAT Math Exercises for Advanced Math

Advanced Math is the second content domain in SAT Math. In the SAT exam, there will be around 13 to 15 questions from the Advanced Math content domain. There are 3 skills and knowledge testing points in the Advanced Math content domain:

1. 1. Equivalent expressions
   2. Nonlinear equations in one variable and systems of equations in two variables
   3. Nonlinear functions

💡You might be interested in seeing our SAT Advanced Math post. It covers the hack points you should know and 27 SAT Advanced Math Practice Questions!

Let’s go through each skill and knowledge point and see some SAT Math Exercises for each one.

SAT Math Exercise Set 6: Equivalent Expressions

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I.

The given equation relates the positive numbers p, x, and y. Write the p-value in terms of x and y.

Exercise II.

What is the value of ?

Exercise III.

What is the value of x?

Exercise IV. 

Simplify the given expression.

Free Digital SAT Prep Course

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

SAT Math Exercise Set 7: Nonlinear Equations in One Variable and Systems of Equations in Two Variables

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I. Two variables, x, and y are related such that for each increase of 1 in the value of x, the value of y increases by a factor of 5. When x=0, y=10. Write the y in terms of x.

Exercise II. .

One solution to the given equation can be written as , where k is a constant. What is the value of k?

Exercise III. .

In the given equation, p is a constant. The equation has exactly one solution. What is the value of p?

Exercise IV.
.

A solution to the given system of equations is (x, y), where x>0. What is the value of x?

SAT Math Exercise Set 8: Nonlinear Functions

To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

Exercise I.

The given equation defines the function f. What is the minimum value of f(x)?

Exercise II. The function f is defined by . What is the value of f(6)?

Exercise III. The function gives a spring’s length, in feet, when an object of w kilograms is hung, where . What is the best interpretation of 15 in this context?

Digital SAT Prep Online Course Program

San Francisco Business School offers a comprehensive Digital SAT Prep Online Course taught by 99th-percentile SAT Instructors and exam experts. The program cracks down each content domain, skills, and knowledge testing point through 1,000+ realistic Digital SAT Exam questions. You will see all the different types of questions that may appear in Digital SAT.

SAT Math Exercises for Problem-Solving and Data Analysis

Problem-solving and Data Analysis is the third content domain in SAT Math. In the SAT exam, there will be around 5 to 7 questions from the Problem-solving and Data Analysis content domain. There are 7 skills and knowledge testing points in the Problem-solving and Data Analysis content domain:

1. 1. Ratios, rates, proportional relationships, and units
   2. Percentages
   3. One-variable data: distributions and measures of center and spread
   4. Two-variable data: models and scatterplots
   5. Probability and conditional probability
   6. Inference from sample statistics and margin of error
   7. Evaluating statistical claims: observational studies and experiments

💡You might be interested in seeing our SAT Problem-Solving and Data Analysis post. It covers the hack points you should know and 15 SAT Problem-Solving and Data Analysis Practice Questions!

We’ve listed one SAT Math Exercise for each skill and knowledge testing below. To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

SAT Math Exercise Set 9: Problem-Solving and Data Analysis

Exercise I. Alisa purchased a box of 100 tea bags. She uses one tea bag for each cup of tea. If Alisa drinks 3 cups of tea every day, in how many days will the number of tea bags in the box drop below 20?

Exercise II. A store offers a 20% discount on a certain bag. During the Black Friday promotion, an additional 10% discount is applied on all products in the store. If the final price of the bag is x % of the initial price, what is the value of x?

Exercise III. 1, 3, 7, 7, 8, 5, 2, 11

What is the sum of the median and mean of the data set shown?

Exercise IV. 

The scatterplot shows the relationship between two variables, x and y. A line of best fit for the data is also shown. What is the difference between the y-coordinate of the data point with x = 4 and the y-value predicted by the line of best fit at x = 4?

Exercise V. The following table shows the number of students in each grade in a High School.

25% of the Grade 12 students attend French club. If a student is picked randomly, what is the probability of selecting a Grade 12 student not attending the French club?

Exercise VI. A random sample of 60 people from a town with a population of 18,756 were asked for their opinion on a recent government policy. If 34 people in the sample support the government policy, what is the expected number difference between the supporters and non-supporters in the town?

Note: You can use a calculator in this SAT Math Exercise

Exercise VII. A study is conducted in the state of Utah. A sample of people over 50 years old are asked how many times they visit a doctor each year. What is the largest population to which the result of the survey can be generalized?

Free Digital SAT Prep Course

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

SAT Math Exercises for Geometry and Trigonometry

Geometry and Trigonometry is the fourth and last content domain in SAT Math. The SAT exam will have around 5 to 7 questions from the Geometry and Trigonometry content domain. There are 4 skills and knowledge testing points in the Geometry and Trigonometry content domain:

1. 1. Area and Volume
   2. Lines, angles, and triangles
   3. Right triangles and trigonometry
   4. Circles

💡You might be interested in seeing our SAT Geometry and Trigonometry post. It covers the hack points you should know and SAT Geometry Practice Questions!

We’ve listed one SAT Math Exercise for each skill and knowledge testing below. To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.

SAT Math Exercise Set 10: Geometry and Trigonometry

Exercise I. One side of a rectangle and square are common. The area of the rectangle is two times the area of the square. If the perimeter of the rectangle is 10 units greater than the perimeter of the square, what is the length of the rectangle in units?

Exercise II.

In the figure, line m is parallel to line n, and line k intersects both lines. What is the value of x + y ?

Exercise III. One leg of an isosceles right triangle A is common with the shortest leg of another right triangle B. The length of the longest side of the triangle B is 17, and longer leg length is 15. What is the length of the longest side of triangle A?

Exercise IV. The graph of in the xy-plane is a circle. What is the area of the circle?

Note: You can use a calculator in this SAT Math Exercise

💡You might be interested in seeing our Digital SAT Math Practice Test. It covers the hardest questions you may see on the SAT Math exam for each skill and knowledge testing point!

Digital SAT Prep Online Course Program

San Francisco Business School offers a comprehensive Digital SAT Prep Online Course taught by 99th-percentile SAT Instructors and exam experts. The program cracks down each content domain, skills, and knowledge testing point through 1,000+ realistic Digital SAT Exam questions. You will see all the different types of questions that may appear in Digital SAT.

The post SAT Math Exercises – FREE 50 Math Exercises – All Domains appeared first on San Francisco Business School.

Nearly two million students take the SAT every year. A good Digital SAT Math Prep is crucial to high scores on the SAT Math section. The majority of the students wobble on the SAT Math Prep. We have listed the 7 Digital SAT Math Prep ways leading to a higher SAT Math score.

📌 Hint: Do not skip this article, you will find FREE Digital SAT Math Prep resources throughout the article.

Ethan is one of our thousands of successful SAT students. He perfectly scored 1600 on the SAT, the highest SAT score a student can get! We were with thousands of students, like Ethan, in their Digital SAT Math Prep journey and that is why created this post to help many others!

💡You might be interested in reading the Digital SAT Math Ultimate Guide post.

Digital SAT Math Section

Before diving into Digital SAT Math Prep ways, you must understand the SAT Math structure, question types, and how to approach different types of questions.

The SAT exam consists of two modules, Module I and Module II. Each module consists of 22 questions, and there will be a total of 44 questions. You will have 35 minutes for each module. There are two types of questions:

1. Four-option multiple-choice questions: Around 75% of the questions will be in this format. There is only one option correct. You should find the correct option and mark it as the answer.
2. Student-Produced Response questions: Around 25% of the questions will be in this format. You should find the answer and type the answer.

👉 Take our full-length FREE SAT Practice Test, see where you stand!

A solid Digital SAT Math Prep will help you to get a higher score on SAT Math. We prepared the following table to summarize the structure of the  SAT Math sections.

The most critical aspect of the Digital SAT is being adaptive. In SAT Math Module I, you will be asked a broad mix of easy, medium, and hard questions. Then, the difficulty of the SAT Math Module II will depend on your score in the SAT Math Module I. This means that the test “adapts” to present questions that are more appropriate to a student’s performance level. The following figure depicts the Digital SAT adaptive testing model.

💡You might be interested in reading the Digital SAT Math Ultimate Guide post. We have provided further details about the SAT Math structure, examples of easy, medium, and hard questions, answers, rationales, and frequently asked questions about the SAT Math.

Anna B. Scored 800 on SAT Math!

Anna B. is one of our thousands of successful SAT students. She scored 800 on SAT Math. You can watch her SAT story.

Now, let’s go through the 7 steps for a perfect Digital SAT Math Prep.

Digital SAT Math Prep – 7 Ways to Get 800 on SAT Math

After helping thousands of SAT exam students, we are confident to say that it is not impossible to get a high SAT Math score. We’ve noticed that high-scoring students went through similar SAT Math Prep paths and we’ve exposed their paths with you in this post. Are you ready for a higher SAT Math score?

Step 1. A Solid SAT Math Study Guide

The first and most important point to SAT Math success is having a solid study plan. A solid SAT Math Study Guide covers:

• Initial SAT scores from your SAT Math Practice Tests. You should score your initial scores to monitor your progress during your Digital SAT Math Prep journey.
• Target SAT Score for your dream college. Each college has a different SAT score percentile and acceptance rate. If you are aiming for a popular college and a popular department, you should score higher respectively. You can look for Colleges’ SAT Score Percentiles and Acceptance rates in our post.
• Plan Off-Weeks and Finalize Your Study Guide. On average, successful students spend around 150 hours for the SAT exam prep. You should plan your weeks and days to make sure you are ready for the exam day.

💡We’ve prepared a 7-Step Digital SAT Math Study Guide helping students to prepare their unique SAT Math Study Guide.

Free Digital SAT Prep Course

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

Step 2. Determine Your Weak Areas

College Board assesses the students’ attainment of critical college and career readiness knowledge and skills in math in math sections of the Digital SAT.

SAT Mathematics covers four content domains.

1. Algebra. In this domain, SAT measures the ability to analyze, fluently solve, and create linear equations and inequalities and analyze and fluently solve equations and systems of equations using multiple techniques. You might be interested in seeing our SAT Algebra post. It covers the hack points you should know and 31 SAT Algebra Practice Questions!
2. Advanced Math. This domain measures skills and knowledge central for progression to more advanced math courses, including demonstrating an understanding of absolute value, quadratic, exponential, polynomial, rational, radical, and other nonlinear equations. You might be interested in seeing our SAT Advanced Math post. It covers the hack points you should know and 27 SAT Advanced Math Practice Questions!
3. Problem-solving and Data Analysis. In this domain, SAT measures the ability to apply quantitative reasoning about ratios, rates, and proportional relationships; understand and apply unit rates; and analyze and interpret one- and two-variable data. You might be interested in seeing our SAT Problem-Solving and Data Analysis post. It covers the hack points you should know and 15 SAT Problem-Solving and Data Analysis Practice Questions!
4. Geometry and Trigonometry. This fourth domain measures the ability to solve problems focusing on area and volume; angles, triangles, trigonometry; and circles. You might be interested in seeing our SAT Geometry and Trigonometry post. It covers the hack points you should know and SAT Geometry Practice Questions!

💡You can use our SAT Math Exercises which has 50 exercises on all SAT Math domains. You can identify your weak areas with the help of SAT Math Exercises.

Watch SAT Math Prep Online Course – Sample Lecture on YouTube

We have a sample 8-minute video lecture from our SAT Math Prep Online Course on YouTube. You can watch below.

SAT Math Content Domains, Skills, and Knowledge Testing Points

Under each content domain, there are several skills and knowledge testing domains with a total of 19 skill and knowledge testing points. The following table summarizes the Digital SAT Math contain domains, each skill and knowledge testing point under each content domain and approximately how many questions appear from each content domain.

Content Domains	Algebra	Advanced Math	Problem-solving and Data Analysis	Geometry and Trigonometry
Skill and Knowledge Testing Points	Linear equations in one variable	Equivalent expressions	Ratios, rates, proportional relationships, and units	Area and volume
	Linear equations in two variables	Nonlinear equations in one variable and systems of equations in two variables	Percentages	Lines, angles, and triangles
	Linear functions	Nonlinear functions	One-variable data: distributions and measures of center and spread	Right triangles and trigonometry
	Systems of two linear equations in two variables		Two-variable data: models and scatterplots	Circles
	Linear inequalities in one or two variables		Probability and conditional probability
			Inference from sample statistics and margin of error
			Evaluating statistical claims: observational studies and experiments
Number of Questions	13 to 15 questions.	13-15 Questions	5-7 Questions	5-7 Questions

Once you grasp the details of how to approach each skill and knowledge testing point question, you will double your chances of reaching an 800 score on the SAT Math test.

Make a Full-Length SAT Math Practice Test

To determine your weak areas, make a full-length SAT Math Prep Practice Test. Note down your scores in each content domain, skill, and knowledge testing point. The lowest score percentages are your weakest areas. For instance, consider the results for an SAT Math Practice test as follows.

	Algebra	Advanced Math	Problem-Solving and Data Analysis	Geometry and Trigonometry	Total
Number of Questions	15	15	7	7	44
Number of Correct Answers	13	12	5	2	32
% of Correct Answers	86.67%	80.00%	71.43%	28.57%

The weakest area of this student is Geometry and Trigonometry. Therefore, he or she should focus on studying Geometry and Trigonometry. Improving scores on this content domain will make the biggest difference to this student’s SAT scores.

💡You might be interested in seeing our Digital SAT Math Practice Test. It covers the hardest questions you may see on the SAT Math exam for each skill and knowledge testing point!

Step 3. Go Through Your Weak Areas

If your scores are typically high in the SAT Math Practice Test in the Algebra content domain, and lower in the SAT Geometry and Trigonometry content domain, studying and making more practice tests in the Algebra content domain will not improve your SAT Math scores. Getting better in your weak areas will make a difference in your SAT score.

Journeys of Two SAT Exam Prep Students

Consider two friends, Jen and Noah who are good at Algebra. They each score around 12 out of ~14 Algebra questions on a full-length SAT Math Practice test of 44 questions. Both of their Advanced Math content domain are weak and they score around 6 out of ~14 questions. During their Digital SAT Math Prep journey, they choose different paths.

Like the majority of the SAT exam prep students do, Jen focuses on the content domains she knows better, and feels better as she scores higher on these content domains. So, she studies and practices more Algebra.

On the other hand, Noah goes through his weak areas, studies more Advanced Math topics, and practices more on this topic.

On their actual SAT exam, Jen improved her Algebra scores and answered all Algebra questions correctly. So, she improved her scores by 2 more questions correctly. Noah improved his SAT Advanced Math scores while his level for Algebra remained the same. He improved his Advanced Math scores by 5 more questions correctly.

In total number of correct questions for Algebra and Advanced Math, Both Jen and Noah started with 18. Jen improved to 20 and Noah improved to 23. Most probably, Noah’s SAT Math Score will be higher than Jen’s.

List your weak content domains and skill and knowledge testing points. Do as much practice as possible to improve your results in your weak areas. Improvement in your weaker areas will bring you the highest score improvements. You can use our Digital SAT Score Calculator to calculate your scores in SAT practice tests and quizzes.

Step 4. Attend a Digital SAT Math Prep Course

There are several SAT Math Prep books, SAT Math classes, SAT tutors, and Online SAT Math Prep courses. Each student’s study habits and tactics are different. While some students prefer studying morning or during the weekend, some students prefer the evening. Many students find it hard to sit for predefined SAT Math class hours. Because, if they are not in the mood, tired, or have other priorities, they cannot postpone a class. This demotivates the majority of the SAT students.

After teaching thousands of SAT students for the exam, we can tell that most of them liked the self-paced delivery so they could study when they wanted to, not when they were forced to sit for a class. This is very important. Students must study when they feel the best time to do so. Otherwise, they will lose their concentration easily during their study.

How to Choose an Online SAT Math Prep Course

Attending an online SAT Math Prep course is the most convenient and budget-friendly option for SAT exam prep students. On average, self-paced online SAT Math prep courses start from ~$100 and go up. When choosing an online SAT Math prep course, you can consider the following points:

• Student Reviews: Check if there are any student testimonials for the SAT Math prep course. Typically, successful programs have positive student reviews.
• Curriculum: Many SAT math prep classes and courses claim they cover all content domains and skill and knowledge testing points, however, they don’t. Go through their curriculum and make sure it covers all SAT Math topics.
• Broad Mix of Questions: SAT tests the students’ Math knowledge and skills on 4 content domains and 19 skills and knowledge testing points. While testing these, the SAT categorizes each question into three difficulty levels: easy, medium, and hard. Make sure the program you are planning to attend covers all types of questions. We see many SAT Math Prep Course providers only show easy questions and that leaves students alone with the hard questions in their actual SAT exam.
• Step-by-Step Walkthrough: There are different tactics and strategies to solve each question on SAT Math. The SAT Math Prep Course must walk through the students on each question step-by-step so students will be well-prepared for the exam day.
• Free Demo: Only the bold SAT exam prep course providers offer a Free SAT Math Prep Course sample version of their course. Because they are confident with their content and they are happy to give a free demo of their comprehensive content. You can check the free demo of the course provider and make sure their style fits you.
• Money-back Guarantee: Some providers do not offer a money-back guarantee. Make sure the provider offers a refund if you are not satisfied with their content.

Digital SAT Prep Online Course Program

San Francisco Business School offers a comprehensive Digital SAT Prep Online Course taught by 99th-percentile SAT Instructors and exam experts. The program cracks down each content domain, skills, and knowledge testing point through 1,000+ realistic Digital SAT Exam questions. You will see all the different types of questions that may appear in Digital SAT.

What is the Best SAT Math Prep Course For Me?

This is a hard question to answer. While the majority of our students love our self-paced delivery method, some students prefer studying through a book, online classes, or tutor-guided SAT math prep. The Best SAT Math Prep Course for you all depends on your preferences and budget.

SAT Math Prep Books cost around $50 on average. However, they are not interactive. Many do not provide step-by-step rationale for each question and even if they provide, in case you will have questions, there is no place to ask your questions. You can see our Digital SAT Math Prep Book.

SAT Math prep classes cost around $1,200 on average and typically for around 16 hours. However, you have to force yourself to go for an in-person class and typically these courses are grueling as you need to attend after school or during the weekend. Many students find it ineffective.

Online SAT Math Prep classes cost around $700 on average. While you can attend these classes from your home, you will be forced to sit for a class for a determined time. In many cases, if you miss a class, you will not have an option to make up.

While tutor-guided SAT Math prep is exclusive and in many cases, one-on-one, the average cost for SAT tutoring is $70 per hour. However, depending on the region, it may be as much as $150 per hour or even higher. This leaves out as an option for the majority of the students as it will cost a significant amount of money.

Self-paced Online SAT Math Prep Courses cost around $100. You can study whenever you would like to study and you can revisit any lecture as many times as you wish. Therefore, self-paced online SAT Math Prep courses provide great flexibility to students. Besides, they are at a fraction of other course options.

Our Recommendation for the Best SAT Math Prep Course

Considering all these, here is our method for the best SAT Math Prep:

1. Find a Self-Paced Online SAT Math Prep Course. This will be budget-friendly, easy to attend, and flexible. Complete the course.
2. Practice with a good SAT Math Prep Book. Practicing as many questions as possible is key to the SAT exam success.
3. In case you still feel weak in particular topics, find a tutor for those particular topics only. This will limit the amount of money you will spend on SAT Math tutors.

📚 San Francisco Business School offers a vast amount of FREE Digital SAT Prep Online materials. See it on the Free Digital SAT Prep Online Library.

Step 5. Note Down and Revisit Your Wrong Answers

Going through your wrong answers will improve your skills in your weak areas the most. Because, you will remember the mistakes you make, and in case you see a similar question next time, you will decrease your chances of making the same mistake again.

When you are practicing SAT Math Prep questions, mark your wrong answers. Once you finish the book, quiz, or SAT practice test go back and review your wrong answers one more time. Take notes on why you make those mistakes and go through the notes frequently.

Step 6. Take Notes During Your SAT Math Prep

There are several SAT Math Prep cheat sheets, formulas, or guides. However, none of them will be as good as you will prepare for yourself.

When studying for the SAT Math, take notes for important points. Go through your notes every week or at least a month. Going through your notes will keep your memory fresh around those topics.

Some students take notes in their notebooks, and some write on sticky notes and paste them on the walls in their rooms. Choose your own way, and you will know the best one for you.

🗎 Download the 15-page Digital SAT Math Formula Sheet.

Step 7. Go and Crack the SAT Exam

You spent hours on your SAT exam and practiced thousands of SAT Math Prep questions. You are ready for the big day!

Try not to study hard on your exam week. You can go through your notes, practice a few questions, or maybe go over your wrong answers. Make sure you will have a good sleep before the exam day and relax. If you practice enough, the questions will be similar to the ones you practiced several times before.

We recommend using your own laptop which you use regularly for the SAT Exam. While the College Board allows school-owned devices or they can provide a device for the exam day, using a new device may cost you additional time as you will need time to get used to a new keyboard, screen, etc. Therefore, we recommend using your own laptop for the exam. Make sure you bring your device fully charged on the exam as there may not be enough power outlets in your testing room.

The post Digital SAT Math Prep – 7 Steps to Get 800 on SAT Math appeared first on San Francisco Business School.

In this Digital SAT Math Ultimate Guide, we’ve listed what’s tested on SAT Math, how to prepare for a higher Digital SAT Math Score, different types of questions you may see on SAT exams, common mistakes students make, and all frequently asked questions by SAT exam students.

📌 Hint: Do not skip this article, you will find FREE Digital SAT Math Prep resources throughout the article.

SAT stands for Scholastic Aptitude Test. Every year, nearly two million students sit for the SAT exam. A high SAT score is vital to getting you to the college you dream of. After teaching thousands of students, we’ve experienced that most students falter in the SAT Math Prep. Therefore, we’ve prepared this Digital SAT Math Ultimate Guide. This guide will help you to reach your target scores.

What is the Digital SAT?

Recently, the SAT exam structure changed, and the new exam is called Digital SAT. The new format of the SAT is more student-friendly, so no need to be afraid of the recent changes. The Digital SAT measures the knowledge and skills that students are learning in school and that matter most for college and career readiness. College Board, the administrator of the SAT exam, states that the Digital SAT exam will be easier to take, easier to give, more secure, and more relevant.

💡You might be interested in reading the 7-Step Digital SAT Math Study Guide post!

The biggest change in the Digital SAT is adaptive testing. In the previous paper format, all SAT exam questions were fixed when you started the test. However, in the digital SAT, the adaptive testing will show you medium-difficulty questions first, if you answer them correctly, the next questions will be getting harder. Similarly, as you answer wrong, the next questions will get easier.

👉 Take our full-length FREE SAT Practice Test, see where you stand!

Digital SAT Adaptive Testing Model

SAT is divided into two equal-length and separately timed stages, each composed of a module of questions. As illustrated in the figure below, students begin each test section by answering the set of questions in the first module. This module contains a broad mix of easy, medium, and hard questions that allow students to demonstrate their achievement before moving on to the second module. The questions in this second module are broadly targeted to the test taker’s achievement level based on how they performed in the first module; questions are either (on average) higher difficulty or lower difficulty than questions in the first module. This means that the test “adapts” to present questions that are more appropriate to a student’s performance level.

The following figure depicts the Digital SAT adaptive testing model.

Anna B. Scored 800 on SAT Math!

Anna B. is one of our thousands of successful SAT students. She scored 800 on SAT Math. You can watch her SAT story.

Digital SAT Math Modules

There are two Math modules in the Digital SAT exam. Digital SAT Math is a two-stage adaptive test and one Math section administered via two separately timed modules. Module 1 and Module 2. Each module consists of 22 questions. 20 of these questions are operational and scored. 2 of the questions are for pretest and for testing purposes of College Board, and not scored. However, students will not know which questions are operational and which ones are for pretest. Therefore, answer each question carefully.

The total time allotted for each Math Module is 35 minutes. In total, you will be allowed 70 minutes to answer 44 SAT Math questions. The calculator is allowed in both Math modules and there will be two types of questions:

1. Discrete; four-option multiple-choice: Approximately 75% of the questions will be this type. There will be four options and only one of the options will be true. Students should select the correct answer.
2. Student-produced response (SPR): Approximately 25% of the questions will be this type. The student will have to do the solution and type the answer.

Note that, there might be questions including science, social science, or real-world stimulus topics as well. For instance, the following is an example SAT Math question.

SAT Math Example Question Including a Stimulus Topic

A sample of oak has a density of 807 kilograms per cubic meter. The sample is in the shape of a cube, where each edge has a length of 0.90 meters. To the nearest whole number, what is the mass, in kilograms, of this sample?
A) 588
B) 726
C) 897
D) 1,107

For example, this question refers to density. To solve this question, the student must know that . This is an example of a question including a science stimulus topic.

Note: The answer and rationale for this question are at the end of the post in Appendix 1.

🗎 Download the 15-page Digital SAT Math Formula Sheet.

Free Digital SAT Prep Course

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

Summary of Format and Delivery of the Digital SAT Math

We prepared the following table to summarize the format and delivery of the SAT Math sections.

Digital SAT Math Content Domains, Skills, and Knowledge Testing Points

College Board assesses the students’ attainment of critical college and career readiness knowledge and skills in math in math sections of the Digital SAT.

SAT Mathematics covers four content domains.

1. Algebra. In this domain, SAT measures the ability to analyze, fluently solve, and create linear equations and inequalities and analyze and fluently solve equations and systems of equations using multiple techniques. You might be interested in seeing our SAT Algebra post. It covers the hack points you should know and 31 SAT Algebra Practice Questions!
2. Advanced Math. This domain measures skills and knowledge central for progression to more advanced math courses, including demonstrating an understanding of absolute value, quadratic, exponential, polynomial, rational, radical, and other nonlinear equations. You might be interested in seeing our SAT Advanced Math post. It covers the hack points you should know and 27 SAT Advanced Math Practice Questions!
3. Problem-solving and Data Analysis. In this domain, SAT measures the ability to apply quantitative reasoning about ratios, rates, and proportional relationships; understand and apply unit rates; and analyze and interpret one- and two-variable data. You might be interested in seeing our SAT Problem-Solving and Data Analysis post. It covers the hack points you should know and 15 SAT Problem-Solving and Data Analysis Practice Questions!
4. Geometry and Trigonometry. This fourth domain measures the ability to solve problems focusing on area and volume; angles, triangles, trigonometry; and circles. 💡You might be interested in seeing our SAT Geometry and Trigonometry post. It covers the hack points you should know and SAT Geometry Practice Questions!

💡You can see our Free SAT Math Exercises which has 50 exercises on all SAT Math domains. 

Watch SAT Math Prep Online Course – Sample Lecture on YouTube

We have a sample 8-minute video lecture from our SAT Math Prep Online Course on YouTube. You can watch below.

SAT Mathematics Content Domains, Skills, and Knowledge Testing Points

Under each content domain, there are several skills and knowledge testing domains with a total of 19 skill and knowledge testing points. The following table summarizes the Digital SAT Math contain domains, each skill and knowledge testing point under each content domain and approximately how many questions appear from each content domain.

Once you grasp the details of how to approach each skill and knowledge testing point question, you will double your chances of reaching an 800 score on the SAT Math test.

💡You might be interested in seeing our Digital SAT Math Practice Test. It covers the hardest questions you may see on the SAT Math exam for each skill and knowledge testing point!

Digital SAT Prep Online Course Program

San Francisco Business School offers a comprehensive Digital SAT Prep Online Course taught by 99th-percentile SAT Instructors and exam experts. The program cracks down each content domain, skills, and knowledge testing point through 1,000+ realistic Digital SAT Exam questions. You will see all the different types of questions that may appear in Digital SAT.

How Hard Are the Actual Digital SAT Math Questions?

In your actual Digital SAT test, there will be two Math sections and in each, there will be 22 questions. You will be allowed to use a calculator in each module. The Digital SAT is an adaptive exam. It starts with easy questions, and as you answer each question correctly, the difficulty of the questions will get harder. Since it is a computer-adaptive test, it is crucial to respond to the first questions in the exam correctly.

College Board categorizes the Digital SAT Math questions into three difficulty levels: easy, medium, and hard.  Typically, the Digital SAT exam starts with medium-difficulty questions, and depending on the correct or wrong answers of the test-taker, the rest of the questions appear.

Below, we shared three easy, medium, and hard questions respectively. To provide consistency, we used all questions from the same content domain, Algebra, and the same skill and knowledge testing point, linear equations in one variable.

SAT Math EASY Question Example 🟢

What is the solution to the equation ?

A. 1
B. 3
C. 5
D. 15

Note: The answer and rationale for this question are at the end of the post in Appendix 2.

SAT Math MEDIUM Difficulty Question Example 🟡

How many solutions does the given equation have?

A. Exactly one
B. Exactly two
C. Infinitely many
D. Zero

Note: The answer and rationale for this question are at the end of the post in Appendix 3.

SAT Math HARD Question Example 🔴

The equation 9x + 5 =a(x+b), where a and b are constants, has no solutions. Which of the following must be true?

I. a = 9
II. b = 5
III. b ≠

A. None
B. I only
C. I and II only
D. I and III only

Note: The answer and rationale for this question are at the end of the post in Appendix 4.

💡You might be interested in seeing our Digital SAT Math Practice Test. It covers the hardest questions you may see on the SAT Math exam for each skill and knowledge testing point!

Frequently Asked Questions About the Digital SAT

We’ve listed some common frequently asked questions of SAT students below.

What changed from the Paper version of the SAT to the Digital SAT?

The topics and content of the SAT exam did not change. However, the questions and their structure have changed. Most sections now ask fewer questions and take less time. In the paper SAT, there were 500+ words long passages asking ~10 questions for each passage. In the Digital SAT, passages are around 100 words and they ask only one question per passage.

The Digital SAT is “module adaptive,” meaning that how you perform on the first module determines the difficulty of the 2nd module. For example, if you do well in Digital SAT Math Module 1, you’ll see harder questions in Math Module 2.

In the paper SAT, Reading and Writing used to be separate sections. In the Digital SAT, they are now combined.

The paper SAT had a “no calculator” math section. In the Digital SAT, a calculator can be used on any math question.

📚 San Francisco Business School offers a vast amount of FREE Digital SAT Prep Online materials. See it on the Free Digital SAT Prep Online Library.

Is the Digital SAT easier than the old version paper SAT?

Yes, the digital SAT is easier than the paper-and-pencil version according to students who have taken the exam and experts at College Board. As mentioned by Priscilla Rodriguez, College Readiness Assessments at College Board: “The digital SAT will be easier to take, easier to give, and more relevant”.

How much should I score on the Digital SAT?

The maximum score a student can get on the SAT is 1600, and the minimum score is 400. While the average SAT score is around 1060, the SAT score you need depends on the school you dream of. If you are dreaming of an Ivy League university or college, you need to score over 1400.

We’ve listed some of the colleges’ SAT score percentiles below. You can look for other Colleges’ SAT Score Percentiles and Acceptance rates in our post.

For instance, if you are planning to apply to Boston College, you should be scoring around 1330 and 1500. Note that, that is the average of all college applications. If you are planning to apply for a popular department of a college, you need to score more than these averages. For instance,  Economics, Finance, and Computer Science are popular departments at Boston College. Therefore, the score percentiles of the accepted students in these departments are expected to be above the 1330 – 1500 range.

Depending on your initial result, if there is a huge gap from the target score, you should be studying harder. You can use our Digital SAT Score Calculator to calculate your scores in SAT practice tests and quizzes.

Where can I take the Digital SAT?

You can take the digital SAT at a school or test center. A proctor will be present during the test. You cannot take the digital SAT at home.

How long is the Digital SAT?

Digital SAT lasts 2 hours and 15 minutes without the essay. The test has four sections:

• Reading & Writing Module I
• Reading and Writing Module II
• Math Module I
• Math Module II.

With the digital SAT, a calculator is allowed in both of the Math modules.

Can I use scratch paper when I am taking the digital SAT?

Yes, scratch paper will be provided to you on test day.

What Kind of Devices Can I Use to Take the Digital SAT?

Students can take the digital SAT Suite tests on a wide range of devices, including their own laptops (Windows or MacOS), iPads, school-owned desktops and laptops, and school-managed Chromebooks. Students will take the digital SAT Suite using a custom-built digital testing application that they’ll download in advance of test day.

Make sure you bring your device fully charged on the exam as there may not be enough power outlets in your testing room. The SAT exam application works even if internet connectivity drops. Therefore, your exam progress won’t be affected by internet outages.

If you do not have your own device, students taking the SAT on a weekend who do not have access to a device can request to borrow one from the College Board. College Board will provide one for use on test day. You must request a device when you are registering for the SAT.

What Are the Tools Provided in the SAT Exam?

The digital testing application will include many test tools for students. Examples include:

• Mark for review: Students can flag and return to any question within a given test module they want to come back to later.
• Testing timer: A clock counts down the time remaining in each module. Students can hide the timer, and they get an alert when 5 minutes remain in the module.
• Calculator: A built-in graphing calculator is available in the entire Math section. (Students can also bring their own approved calculator.)
• Reference sheet: In the Math section, students have access to a list of common formulas.
• Annotation: Students can highlight any part of a question and leave themselves a note.

How many times can I take the Digital SAT?

We recommend taking the PSAT or a practice test your sophomore year. You can plan to take the SAT in the spring or winter of your junior year. This will leave you some time to get ready and take it again if you are unhappy with your first results.

What is the validity of the Digital SAT?

Once you get SAT scores, your scores are good for 5 years. If you took the SAT more than 5 years ago, you need to retake the SAT.

Appendix 1: Answer & Rationale for SAT Math Example Question Including a Stimulus Topic

Answer: A

Rationale: It’s given that the shape of the oak is a cube and each edge has a length of 0.90 meters. The volume of a cube formula is as follows:

. If we substitute the edge size of the cube in this formula;

cubic meters.

. Therefore, if we multiply the volume of the cube-shaped oak by the density of the oak, we can find the mass.

kilograms. Rounding this mass to the nearest whole number gives 588 kilograms.

Appendix 2: Answer & Rationale for SAT Math 🟢 EASY Question Example

Answer C.

If we subtract 5 from both sides of the given equation;

yields;

. If we divide both sides by 3;

. This yields;

Appendix 3: Answer & Rationale for SAT Math 🟡 MEDIUM Difficulty Question Example

Answer C.

If two sides of a linear equation are equivalent, then the equation is true for any value. If an equation is true for any value, it has infinitely many solutions.

Since the two sides of the given linear equation are equivalent, the given equation has infinitely many solutions.

Appendix 4: Answer & Rationale for SAT Math 🔴 HARD Question Example

Answer D.
Rationale: If we expand the right side of the equation;

9x + 5 = ax + ab.

For a linear equation in the form of ax + b = cx + d, if the coefficients of x are the same and if the remaining terms are not equal, there are no solutions. Therefore;

If a = 9; and ab ≠ 5, there are no solutions. → I is true.

If a is 9, ab = 9b ≠ 5  → b ≠   → III is true.

💡You can see our SAT Math Exercises which has 50 exercises on all SAT Math domains. It’s for FREE!

Disclaimer: SAT® is a registered trademark of the College Board, which is not affiliated with San Francisco Business School and was not involved in the production of, and does not endorse this product or website.

The post Digital SAT Math Ultimate Guide – Crack the SAT Math appeared first on San Francisco Business School.

Digital SAT Math covers two modules each containing 22 questions. In total, SAT test takers have to answer 44 Math questions. Solving as many Digital SAT Math practice tests as possible is key to SAT success. We’ve created a Digital SAT Math Practice Test of the 19 hardest questions in this test. You can practice this Digital SAT Math Practice test and assess how sharp your SAT Math skills are.

💡You might be interested in reading the Digital SAT Math Ultimate Guide post!

First, we explain the structure of Digital SAT Math and how we picked the hardest questions for the Digital SAT Math Practice Test, and then you will see the questions.

📌 Hint: Do not skip this article, you will find FREE Digital SAT Math Prep resources throughout the article.

Before You Start The Digital SAT Math Practice Test

Before you start the Digital SAT Math Practice Test, we would like to explain how we created the test and how you should interpret your results. College Board assesses the students’ attainment of critical college and career readiness knowledge and skills in math in math sections of the Digital SAT.

👉 Take our full-length FREE SAT Practice Test, see where you stand!

Digital SAT Math covers four content domains.

1. Algebra. In this domain, SAT measures the ability to analyze, fluently solve, and create linear equations and inequalities and analyze and fluently solve equations and systems of equations using multiple techniques. You might be interested in seeing our SAT Algebra post. It covers the hack points you should know and 31 SAT Algebra Practice Questions!
2. Advanced Math. This domain measures skills and knowledge central for progression to more advanced math courses, including demonstrating an understanding of absolute value, quadratic, exponential, polynomial, rational, radical, and other nonlinear equations. You might be interested in seeing our SAT Advanced Math post. It covers the hack points you should know and 27 SAT Advanced Math Practice Questions!
3. Problem-solving and Data Analysis. In this domain, SAT measures the ability to apply quantitative reasoning about ratios, rates, and proportional relationships; understand and apply unit rates; and analyze and interpret one- and two-variable data. You might be interested in seeing our SAT Problem-Solving and Data Analysis post. It covers the hack points you should know and 15 SAT Problem-Solving and Data Analysis Practice Questions!
4. Geometry and Trigonometry. This fourth domain measures the ability to solve problems focusing on area and volume; angles, triangles, trigonometry; and circles. You might be interested in seeing our SAT Geometry and Trigonometry post. It covers the hack points you should know and SAT Geometry Practice Questions!

💡You might be interested in reading the 7-Step Digital SAT Math Study Guide post!

Anna B. Scored 800 on SAT Math!

Anna B. is one of our thousands of successful SAT students. She scored 800 on SAT Math. You can watch her SAT story.

🗎 Download the 15-page Digital SAT Math Formula Sheet.

SAT Math Content Domains, Skills, and Knowledge Testing Points

Under each content domain, there are several skills and knowledge testing domains with a total of 19 skill and knowledge testing points. The following table summarizes the Digital SAT Math contain domains, each skill and knowledge testing point under each content domain and approximately how many questions appear from each content domain.

Once you grasp the details of how to approach each skill and knowledge testing point question, you will double your chances of reaching an 800 score on the SAT Math test.

💡You can see our Free SAT Math Exercises which has 50 exercises on all SAT Math domains. 

Digital SAT Prep Online Course Program

San Francisco Business School offers a comprehensive Digital SAT Prep Online Course taught by 99th-percentile SAT Instructors and exam experts. The program cracks down each content domain, skills, and knowledge testing point through 1,000+ realistic Digital SAT Exam questions. You will see all the different types of questions that may appear in Digital SAT.

How Hard Are the Actual Digital SAT Math Questions?

In your actual Digital SAT test, there will be two Math sections and in each, there will be 22 questions. You will be allowed to use a calculator in each module. The Digital SAT is an adaptive exam. It starts with easy questions, and as you answer each question correctly, the difficulty of the questions will get harder. Since it is a computer-adaptive test, it is crucial to respond to the first questions in the exam correctly.

College Board categorizes the Digital SAT Math questions into three difficulty levels: easy, medium, and hard.  Typically, the Digital SAT exam starts with medium-difficulty questions, and depending on the correct or wrong answers of the test-taker, the rest of the questions appear.

💡You might be interested in reading the Digital SAT Math Prep: 7 Steps to Get 800 on SAT Math post!

Watch SAT Math Prep Online Course – Sample Lecture on YouTube

We have a sample 8-minute video lecture from our SAT Math Prep Online Course on YouTube. You can watch below.

Important Points About This Test

In this Digital SAT Math Practice Test, we picked 19 hard questions for each skill and knowledge testing point. So, this test does not resemble the typical skill and knowledge testing point distribution on a Digital SAT Math practice test. Instead, we wanted to show you the hardest questions you may see in each skill and knowledge testing point. Therefore, there are fewer questions from the SAT Algebra and Advanced Math content domains and more questions from problem-solving and data analysis and SAT Geometry and Trigonometry content domains.

📚 San Francisco Business School offers a vast amount of FREE Digital SAT Prep Online materials. See it on the Free Digital SAT Prep Online Library.

Besides, since these are the hardest questions for each skill and knowledge testing point, it is very normal that you will spend longer than usual time to solve each question. It is also super normal that you may score lower than your previous Digital SAT Math Practice tests in this one. Because a typical Digital SAT Math Practice Test covers easy, medium, and hard questions. However, this one contains only the hardest questions.

Are you ready to take a challenging Digital SAT Math Practice Test Now?

SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.

Digital SAT Math Practice Test – Hardest Questions

Question 1

The equation 9x + 5 =a(x+b), where a and b are constants, has no solutions. Which of the following must be true?

I. a = 9
II. b = 5
III. b ≠

A. None

B. I only

C. I and II only

D. I and III only

Content Domain: Algebra

Skill and Knowledge Testing Point: Linear equations in one variable

Question 2

To earn money for college, Avery works two part-time jobs: A and B. She earns $10 per hour working at job A and $20 per hour working at job B. In one week, Avery earned a total of s dollars for working at job B. In one week, Avery earned a total of s dollars for working at the two part-time jobs. The graph above represents all possible combinations of the number of hours Avery could have worked at the two jobs to earn s dollars. What is the value of s ?

A. 128

B. 160

C. 200

D. 320

Content Domain: Algebra

Skill and Knowledge Testing Point: Linear equations in two variables

Question 3

An object hangs from a spring. The formula relates the length , in centimeters, of the spring to the weight , in newtons, of the object. Which of the following describes the meaning of the 2 in this context?

A. The length, in centimeters, of the spring with no weight attached

B. The weight, in newtons, of an object that will stretch the spring 30 centimeters

C. The increase in the weight, in newtons, of the object for each one-centimeter increase in the length of the spring

D. The increase in the length, in centimeters, of the spring for each one-newton increase in the weight of the object

Content Domain: Algebra

Skill and Knowledge Testing Point: Linear functions

Digital SAT Math Practice Test – Question 4

Store A sells raspberries for $5.50 per pint and blackberries for $3.00 per pint. Store B sells raspberries for $6.50 per pint and blackberries for $8.00 per pint. A certain purchase of raspberries and blackberries would cost $37.00 at Store A or $66.00 at Store B. How many pints of blackberries are in this purchase?

A. 4

B. 5

C. 8

D. 12

Content Domain: Algebra

Skill and Knowledge Testing Point: Systems of two linear equations in two variables

Question 5

Ken is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Ken saves 90% of his earnings from each week. What is the least number of hours he must work the rest of the week to save at least $270 for the week?

A. 38

B. 33

C. 22

D. 16

Content Domain: Algebra

Skill and Knowledge Testing Point: Linear inequalities in one or two variables

Question 6

In the expression above, b and c are positive integers. If the expression is equivalent to x+b and x ≠ b, which of the following could be the value of c ?

A. 4

B. 6

C. 8

D. 10

Content Domain: Advanced Math

Skill and Knowledge Testing Point: Equivalent expressions

Question 7

If and are the two solutions to the system of equations above, what is the value of ?

A. -3

B. -2

C. -1

D. 1

Content Domain: Advanced Math

Skill and Knowledge Testing Point: Nonlinear equations in one variable and systems of equations in two variables

Question 8

x
1
2
3

For the exponential function , the table above shows several values of and their corresponding values of , where is a constant greater than 1. If is a constant and , what is the value of ?

Content Domain: Advanced Math

Skill and Knowledge Testing Point: Nonlinear functions

Question 9

Anita created a batch of green paint by mixing 2 ounces of blue paint with 3 ounces of yellow paint. She must mix a second batch using the same ratio of blue and yellow paint as the first batch. If she uses 5 ounces of blue paint for the second batch, how much yellow paint should Anita use?

A. Exactly 5 ounces

B. 3 ounces more than the amount of yellow paint used in the first batch

C. 1.5 times the amount of yellow paint used in the first batch

D. 1.5 times the amount of blue paint used in the second batch

Content Domain: Problem-Solving and Data Analysis

Skill and Knowledge Testing Point: Ratios, rates, proportional relationships, and units

Digital SAT Math Practice Test – Question 10

37% of the items in a box are green. Of those, 37% are also rectangular. Of the green rectangular items, 42% are also metal. Which of the following is closest to the percentage of the items in the box that are not rectangular green metal items?

A. 1.16%

B. 57.50%

C. 94.25%

D. 98.84%

Content Domain: Problem-Solving and Data Analysis

Skill and Knowledge Testing Point: Percentages

Question 11

The mean amount of time that the 20 employees of a construction company have worked for the company is 6.7 years. After one of the employees leaves the company, the mean amount of time that the remaining employees have worked for the company is reduced to 6.25 years. How many years did the employee who left the company work for the company?

A. 0.45

B. 2.30

C. 9.00

D. 15.25

Content Domain: Problem-Solving and Data Analysis

Skill and Knowledge Testing Point: One-variable data: distributions and measures of center and spread

Question 12

The scatterplot above shows the size x and the sale price y of 25 houses for sale in Town H. Which of the following could be an equation for a line of best fit for the data?

A. y = 200x + 100

B. y = 100x + 100

C. y = 50x + 100

D. y = 100x

Content Domain: Problem-Solving and Data Analysis

Skill and Knowledge Testing Point: Two-variable data: models and scatterplots

Digital SAT Math Practice Test – Question 13

The table summarizes the distribution of age and assigned group for 90 participants in a study.

One of these participants will be selected at random. What is the probability of selecting a participant from group A, given that the participant is at least 10 years of age? (Express your answer as a decimal or fraction, not as a percent.)

Content Domain: Problem-Solving and Data Analysis

Skill and Knowledge Testing Point: Probability and conditional probability

Question 14

The results of two random samples of votes for a proposition are shown above. The samples were selected from the same population, and the margins of error were calculated using the same method. Which of the following is the most appropriate reason that the margin of error for sample A is greater than the margin of error for sample B?

A. Sample A had a smaller number of votes that could not be recorded.

B. Sample A had a higher percentage of favorable responses.

C. Sample A had a larger sample size.

D. Sample A had a smaller sample size.

Content Domain: Problem-Solving and Data Analysis

Skill and Knowledge Testing Point: Inference from sample statistics and margin of error

Question 15

To determine the mean number of children per household in a community, Tabitha surveyed 20 families at a playground. For the 20 families surveyed, the mean number of children per household was 2.4. Which of the following statements must be true?

A. The mean number of children per household in the community is 2.4.

B. A determination about the mean number of children per household in the community should not be made because the sample size is too small.

C. The sampling method is flawed and may produce a biased estimate of the mean number of children per household in the community.

D. The sampling method is not flawed and is likely to produce an unbiased estimate of the mean number of children per household in the community.

Content Domain: Problem-Solving and Data Analysis

Skill and Knowledge Testing Point: Evaluating statistical claims: observational studies and experiments

Question 16

What is the area, in square units, of the triangle formed by connecting the three points shown?

Content Domain: Geometry and Trigonometry

Skill and Knowledge Testing Point: Area and volume

Digital SAT Math Practice Test – Question 17

In triangle RST, angle T is a right angle, point L lies on , point K lies on , and is parallel to . If the length of is 72 units, the length of is 24 units, and the area of triangle RST is 792 square units, what is the length of , in units?

Content Domain: Geometry and Trigonometry

Skill and Knowledge Testing Point: Lines, angles, and triangles

Question 18

Triangle ABC is similar to triangle DEF, where A corresponds to D and C corresponds to F. Angles C and F are right angles. If and DF = 125, what is the length of ?

A.

B.

C.

D.

Content Domain: Geometry and Trigonometry

Skill and Knowledge Testing Point: Right triangles and trigonometry

Question 19

A circle in the xy-plane, the graph of is a circle. What is the radius of the circle?

A. 5

B. 6.5

C.

D.

Content Domain: Geometry and Trigonometry

Skill and Knowledge Testing Point: Circles

👨‍💻You can use our Free Digital SAT Score Calculator to calculate your scores in practice tests and quizzes.

Digital SAT Math Practice – 19 Hardest Questions YouTube Video

View our 19 Hardest Digital SAT Math Practice Questions YouTube Video.

Answers and Rationales for the Digital SAT Math Practice Test

We’ve created a comprehensive answers and rationales PDF file for these Digital SAT Math Practice questions. If you can fill in your name and email below, we can send it to your email in minutes.

Note that, the email may hit your junk or spam folders, please check your junk and spam folders and if you did not receive it, please email us at support@sanfranciscobs.com.

💡Do not forget to visit SAT Math Exercises which has 50 exercises on all SAT Math domains. It’s for free!

The post Digital SAT Math Practice Test – Hardest Questions appeared first on San Francisco Business School.