6 SAT Math Problem Solving and Data Analysis Concepts to Master

Problem Solving and Data Analysis is worth 29 percent of your SAT Math score.

SAT Math Concepts to Master

High Angle View Of Students Studying At Table

Key concepts to master include ratios, percentages and lines of best fit. (Getty Images)

The SAT Math Problem Solving and Data Analysis subsection is a significant area of focus on the college entrance exam. In fact, it accounts for 17 of 58 questions, or 29 percent of SAT Math. Given this subsection’s importance to an individual student’s score, it is critical to master its key concepts.

Below, we provide an overview of the test's six sections:

  • Percentages
  • Unit conversion and unit rate
  • Lines of best fit
  • Relationships between variables

A ratio is a numerical comparison that depicts the relationship between two or more values. When sitting and studying for the SAT, students should think of ratios in terms of their individual parts. This allows you to convert values into easy-to-manage fractions.

For instance, consider this recipe for instant soup: Each serving calls for two parts (or 2/3) water and one part (1/3) noodles, but you would like to make 15 servings of soup. How much water must you add? All you need to do is multiply 15 by 2/3, which yields a total of 10 cups. Recipes thus provide a simple way to practice with ratios.

To perform ratio calculations, students should be comfortable with multiplying and dividing fractions. Also note that ratios may be expressed in any of the following formats on the SAT: X to Y, X:Y, X/Y or X (when Y is equal to 1).

2. Percentages

First and foremost, students should remember that percentages are always relative to the number 100. This is true even for percentages that exceed 100 – 150 percent, for example, is 1.5 times 100.

To calculate any value related to percentages, students can memorize the following formula: IS/OF = %/100. “Is” represents a partial amount, such as 60 blue marbles, while “of” represents the total amount, such as 100 colored marbles. In this case, blue marbles represent 60 percent of all the marbles.

Students must also be comfortable with cross-multiplying to find missing values in the formula. Food nutrition labels are a practical outlet for familiarizing yourself with percentages.

3. Unit conversion and unit rate

Unit rate expresses one quantity as compared to another. Common examples include “miles per hour” or “dollars per year." Words like “per,” “each,” and “every” indicate unit rate. Unit rate problems often require you to convert from one unit, such as feet, to another unit, such as inches.

The SAT is known to draw on both the English system of feet and inches and the metric system of kilometers and meters. Therefore, students should be familiar with common units for both.

The metric system is conveniently based on the number 10, so performing calculations within this system is rather simple. Students must be more careful when calculating within the English system and between both systems. To get more comfortable with unit conversion, you can practice with everyday concepts: converting a person’s height from inches to centimeters, or a car’s speed from kilometers to miles.

Ace the SAT's Passport to Advanced Math

Sonya Ellis Nov. 19, 2018

LONDON, ENGLAND - DECEMBER 01:  A teacher writes an equation on a whiteboard during a maths lesson at a secondary school on December 1, 2014 in London, England. Education funding is expected to be an issue in the general election in 2015.  (Photo by Peter Macdiarmid/Getty Images)

4. Lines of best fit

A line of best fit represents the relationship between variables as a linear function. It is generally a straight line on a scatterplot that is expressed as y = a + bx.

It is crucial for SAT test-takers to know the best fit formula. Furthermore, students should understand the concept of best fit as a prediction of values. Practice specifically with scatterplot questions to gain familiarity with lines of best fit.

5. Relationships between variables

Relationships between variables are frequently expressed as equations or functions, which may be exponential, linear, or quadratic. You should also be aware that charts and tables are more simplistic representations of relationships between variables.

SAT test-takers should be extremely knowledgeable about basic equations and their components – namely those for lines (y = mx + b), quadratic equations (ax² + bx + c = 0), and exponential equations – as well as what they look like. A helpful way to practice this skill is by matching graphs with their equations.

6. Statistics

To start, students should be familiar with the more basic concepts in statistics: mean, or average; median, the number precisely in the middle when values are placed in order; mode, the value that appears most often; and range, the difference between the highest value and the lowest. Once these concepts are understood, students can make sense of the more complicated concepts in statistics. These include terms like population parameter, a characteristic of a population as described by a value, and standard deviation, or how far away points in the data set are from the mean.

Students can gain confidence in statistics by first analyzing relatively straightforward data sets, such as the distribution of test grades in a class. Once the student feels comfortable calculating statistical values for such a set, he or she can move on to looking at more complex data sets, such as those found in scientific studies.

For success on the SAT Math section, deeply familiarize yourself with the six concepts listed above. These are key areas in the Problem Solving and Data Analysis subsection, which amounts to nearly one-third of SAT Math.

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Unit 6: Lesson 2

The SAT Math Test: Problem Solving and Data Analysis

In this series of articles, we take a closer look at the SAT Math Test.

Sat math questions fall into different categories called "domains." one of these domains is problem solving and data analysis..


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problem solving and data analysis sat topics

Guide to SAT Math Problem Solving and Data Analysis + Practice Questions

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What’s Covered:

Overview of sat math problem solving and data analysis, how will the sat impact my college chances.

Final Tips and Strategies

Problem Solving and Data Analysis questions appear on the Calculator section of the SAT Math test and involve applying mathematical knowledge to real-world contexts. These problems can be tough, so if you want to improve your math score, here are some strategies and practice problems to help you out.

The SAT Math section contributes to half of the total SAT score. This section is scored out of 800 and includes three main categories, which each have a subscore out of 15.

Here is the breakdown of each category:

Additional Topics in Math covers the remaining 10% and consists of a variety of different mathematical topics.

The Problem Solving and Data Analysis section tests students’ ability to solve real-world problems using mathematical understanding and skills. This includes quantitative reasoning, interpreting and synthesizing data, and creating representations. These questions never appear on the SAT No Calculator section, so you’ll always be allowed a calculator for them.

Problem Solving and Data Analysis questions ask students to:

Many selective colleges use a metric called the Academic Index (AI) to assess an application’s strength. The AI is calculated based on GPA and SAT/ACT scores, so you should make sure your scores are competitive to increase your chances of admission. Some colleges even automatically reject applicants with AIs that are too low.

To see how your SAT score compares, use CollegeVine’s free Admissions Chances Calculator . This tool will let you know the impact of your SAT score on your chances and will even offer advice to improve other aspects of your profile.

problem solving and data analysis sat topics

Strategies to Solve Problem Solving and Data Analysis Problems

Problem Solving and Data Analysis problems often involve graphs or data tables, so it’s important to pay attention to titles and labels to make sure you don’t misinterpret the information. 

As you read the question, underline or circle any important numeric information. Also, pay close attention to what exactly the question is asking for.

Because Problem Solving and Data Analysis problems vary, there is no concrete algorithm to approach them. These questions are typically more conceptual than calculation-based, so though a calculator is allowed, you probably won’t need it aside from simple arithmetic. Therefore, the key to these problems is reading carefully and knowing concepts like proportions, median, mean, percent increase, etc.

10 Difficult Problem Solving and Data Analysis Questions

Here are some sample difficult Problem Solving and Data Analysis questions and explanations of how to solve them. Remember, these questions only appear on the Calculator section of the exam, so you will have access to a calculator for all of them.

1. Measures of Central Tendency (Mean/Median/Mode)

problem solving and data analysis sat topics

Correct Answer: B

This problem involves computing the median. If you have a graphing calculator, this could be done via lists, but since there are only 7 data values, it might be faster to just write this one out. The median is the measure of the middle of the data set, so start by ordering the values from smallest to largest. This results in the following list: 19.5%, 21.9%, 25.9%, 27.9%, 30.1%, 35.5%, 36.4%. From here, we can clearly see that the middle value is 27.9%, so that is our median. 

However, we’re not done here. We now have to compute the difference between the median we just calculated and the median for all 50 states, 26.95%. Subtracting these two values yields 0.95%, which corresponds to answer choice B. 

2. Percent Increase

problem solving and data analysis sat topics

Percent increase can be a tricky concept if you don’t remember this rule of thumb: “new minus old over old.” In this case, for the percent increase from 2012 to 2013, we take the “new” value, 5,880, and subtract the “old” value, 5,600. This is 280, which we then divide by the “old” to get .05, which is 5%. 

Since the percent increase from 2012 to 2013 was 5%, and this is double the predicted increase from 2013 to 2014, we know that the percent increase from 2013 to 2014 will be half of 5%, or 2.5%.

Then, to calculate the number of subscriptions sold in 2014, we multiply the value in 2013, 5,880, by 2.5%. This yields 147, which means that in 2014, 147 additional subscriptions were sold. So, the total amount of subscriptions sold in 2014 is 5,880 + 147 = 6,027.

3. Analyzing Graphical Data

problem solving and data analysis sat topics

Since we are presented with a graph, let’s make note of what this graph is showing us. On the y-axis we have speed, and on the x-axis we have time. So, this graph is showing us how Theresa’s speed varies with time. 

When the graph is flat, the speed is unchanging and is therefore constant. When the graph has a positive slope, the speed is increasing, and when the graph has a negative slope, the speed is decreasing. The rates at which it increases and decreases will be constant since the graph is composed of straight lines (and a line has a constant slope, which means it changes at a constant rate).

For questions asking which statement is not true, it’s crucial to take the time to read through each answer choice. First, choice A states that Theresa ran at a constant speed for five minutes. We can see that this is true, since from 5 to 10 minutes, the graph is flat. Next, choice C says that the speed decreased at a constant rate during the last five minutes. This is also true because from 25 to 30 minutes, the graph is a line with negative slope, which indicates decreasing speed. Finally, choice D claims the maximum speed occurs during the last 10 minutes. We can see that the maximum speed (the highest point on the graph) occurs at 25 minutes, which is within the last 10 minutes, so choice D is also true.

By process of elimination, choice B should be correct, but let’s verify. Choice B states that Theresa’s speed was increasing for a longer time than it was decreasing. Speed was increasing from 0 to 5 and 20 to 25 minutes, for a total of 10 minutes. Speed was decreasing from 10 to 20 and 25 to 30 minutes, for a total of 15 minutes. So, the speed decreased for a longer time than it increased, and choice B is false, making it the correct answer.

4. Inference

problem solving and data analysis sat topics

Correct Answer: D

For questions involving surveys, always remember that generalizations can only be made to the specific population studied. For example, if a study is given to a select group of 5th grade math students, when analyzing the results, you can only generalize to 5th grade math students, not all math students or all 5th graders. 

In this case, the group surveyed was people who liked the book. From these people, 95% disliked the movie. So, from this survey, most people who like the book will then dislike the movie, which corresponds to choice D.

Choices A and C are incorrect since they generalize to people who see movies and people who dislike the book, which doesn’t apply to the population studied. Choice B is incorrect since it falsely generalizes to all people who read books.

5. Proportions

problem solving and data analysis sat topics

Proportion problems are usually fairly quick, but easy to mess up on if not read carefully. For proportions, the denominator is the total number of things in the group we’re looking at, and the numerator is the specific characteristic we want. 

This question asks for the fraction of the dogs that are fed only dry food. So, the group we’re looking at is “dogs” and the characteristic we want is that they “are fed only dry food.” 

From the table, the total number of dogs is 25. This means the denominator will be 25. Next, we must find the number of dogs which are also only fed dry food, which is 2, according to the table. So, our numerator is 2, and the answer is 2/25.

problem solving and data analysis sat topics

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6. Scale Factors

problem solving and data analysis sat topics

For this question, let’s start with what we know: the column is 8 inches tall. We know that 50 coins creates a \(3\frac{7}{8}\) inch column, which is approximately 4 inches. Since this question asks for an approximation, we know that 8 is slightly more than double \(3\frac{7}{8}\), so we’ll need slightly more than \(50\:\cdot\: 2\) pennies to create an 8-inch column. Answer B is the closest number to our approximate value.

If you wanted to be precise, you could set up an equation:

\(8 in\:\cdot\:\frac{50 coins}{3\frac{7}{8} in}\)

Because 50 coins corresponds to a column which is \(3\frac{7}{8}\) inches tall, we set those two values up in a fraction. We decide which value goes in the numerator and which in the denominator based on the units: since we started with 8 inches, we need the \(3\frac{7}{8}\) inches to be in the denominator so that the inches cancel. Then, we are left with the unit in the numerator, which is coins. The question asks for the number of coins, so this is exactly what we want.

At this point, you would use your calculator to solve the expression, and get about 103 coins. This value is closest to answer choice B.

7. Line of Best Fit/Scatterplots

problem solving and data analysis sat topics

Correct Answer: A

Once again, since we have a graph, let’s take a moment to read the labels. The y-axis shows density and the x-axis shows distance from the sun.

We also see that the line of best fit is sloping downwards. As the distance increases, the density seems to decrease. So, choice A is correct in that larger distances correspond to lesser densities.

Though it wasn’t explicit in this question, an important thing to note about scatterplots is that these relationships show correlation, not causation. Choice C is incorrect because it falsely implies that changes in distance cause changes in density. Choice D is incorrect since though there is no cause and effect relationship, there is a correlation between these two variables.

8. Geometric Applications of Proportions

problem solving and data analysis sat topics

Correct Answer: 5/18, .277, .278

This problem could be confusing in that so little information is given. However, this problem requires that you recall that the proportion of degrees is equivalent to the proportion of area. So, for this problem, all you have to do is divide 100 by 360, which is the total number of degrees in a circle.

Then, the answer is 100/360, or 5/18. If you’re faced with a similar problem on the test, where there is little to no numeric information, try to work with the numbers you do have and find helpful relationships.

9. Unit Conversions

problem solving and data analysis sat topics

Correct Answer: 195

Unit conversions are fairly simple once you set up the expression correctly. Start with the information given.

For this problem, the price is $62,400, so we will start with this value. Next, we will multiply this value by fractions. Each fraction will consist of a numerator and denominator which are equivalent, so multiplying by these fractions is the same as multiplying by 1. Here is what the expression would look like:

\(\$62,400\:\cdot\:\frac{1 ounce}{\$20}\:\cdot\:\frac{1 pound}{16 ounces}\)

We decide which value to put in the denominator based on the units. In this case, the dollars and the ounces cancel, leaving us with pounds, which is what the question asked for. Solving this expression results in 195 pounds, the answer to the question. 

10. Probability

problem solving and data analysis sat topics

Correct Answer: 5/7, .714

Probability questions are similar to proportion questions in that the denominator should be the group we’re looking at and the numerator should be the characteristic we want.

In this case, it is given that we’re looking at contestants who received a score of 5 on one of the three days, and there is a total of 7 such contestants. The characteristic we want is that the contestant received a score of 5 on Day 2 or Day 3. The number of contestants who fit this description is 2 + 3 = 5, so the probability is 5/7.

For Problem Solving and Data Analysis problems, make sure that your answer addresses what the question asked for. Wrong answer choices on the SAT often reflect common student mistakes, so take the time to read Problem Solving and Data Analysis questions carefully.

When studying for the SAT Math section, try to do plenty of practice problems. The best way to get better at math is to do more math.

Here are some other articles that will help you prepare for the SAT Math section:

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problem solving and data analysis sat topics


Problem Solving and Data Analysis: SAT Math Tips & Practice

Zack Robinson

When was the last time you found yourself trying to run a proper parabola in a corn field? Or when did you last get really radical four times in a row, just to make the imaginary real?

Most of the SAT math section is about stuff you’re probably not going to use outside of a formal math environment.

For example, the Heart of Algebra section makes up 30% of the section, and it’s mostly linear algebra. It’s really important for math, but you’re probably not going to use y=mx+b to understand the world around you or make daily decisions. Passport to Advanced Math is also 30% of the section, and it mostly adds an exponent to all of that algebra, hardly changing its import to the regular world.

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But what if I told you there was a secret middle child, forgotten by its parents and ignored by its peers, only to be intuitive, surprisingly practical, worth just as many points as the other sections and possibly one of the best places to pick up points on the SAT math section?

Welcome to the dark side, people. Welcome to Problem Solving and Data Analysis.

problem solving and data analysis sat topics

Problem Solving and Data Analysis: What’s Included?

College Board created the Problem Solving and Data Analysis questions to test your ability to use your math understanding to solve real-world problems. Your SAT score report will list this domain as one of three scored domains (along with Heart of Algebra and Passport to Advanced Math), assigning you a score of 1-15.

Fortunately, College Board has a terrible poker face, and they’ve decided to tell you which cards they’re holding right up front. Here’s what you’ll see:

You will see a total of 17 questions in this section, which makes it worth about 30% of your SAT math score.

And since data and calculators go together like spam and mayo, you’ll only see these questions on section four (the calculator section). They could, however, appear in either the multiple choice or the open response portions of the section.

Think about that for a moment: 17 of the 38 questions in section 4 are Problem Solving and Data Analysis questions. That’s 45% of the section!

Data Analysis: Statistics and Data Collection

Let’s talk about politics for a second. Okay, sorry, I’m not trying to be your uncle Greg at Thanksgiving. Let me rephrase: I want to talk about political surveys for a second. We are all familiar with them, regardless of which side of the aisle you sit on, so they make a familiar example.

Presidential surveys are one of the great accomplishments in the history of statistics nerds. They can describe the views of hundreds of millions of people by asking about a thousand people.

And despite some recent bad press in the last few years, they typically do a surprisingly good job. Much like Taylor Swift on even her worst day, most of them are within a few percentage points of perfection.

How do they do this, though? A lot of people instinctively want to say that a survey must include everyone if it is going to be accurate. Fortunately, that’s not true. Instead, it just needs to follow a few basic principles.

Let’s start by defining a couple of key terms:

So, we ask questions of a sample in order to learn something about the population .

What’s the key to making this all work correctly? Picking your sample well. In fact, the single most important part of survey design is selecting a sample that accurately reflects the population.

A good sample has three features:

Those statistics nerds mentioned earlier spend a lot of their time trying to get samples that are reflective of the population. The success of a survey depends on it. But let’s say they do everything right. Their sample size is sufficiently large, they randomly selected their people and the sample is part of the population. What can they learn from the results?

There are a few caveats that should come with any survey results:

problem solving and data analysis sat topics

Statistics and Data Collection Practice

Let’s take a look at a practice problem:

problem solving and data analysis sat topics

Let’s think back through our three caveats regarding samples: they tell us what is likely to be true rather than what is necessarily true, two different samples can vary and correlation doesn’t equal causation.

The last sentence in the question is asking us what must be true rather than what is likely to be true. That creates an issue right off the bat, but let’s go through our answer choices to see what we have to work with:

Problem Solving and Data Analysis Basics: Mean, Median and Mode

Mean, median and mode on the SAT math section tends to only require two things:

Our comprehensive guide to mean, median and mode digs into these concepts more deeply, but I want to give the most important points here.

First, the definitions:

You should be able to calculate the mean, median and mode of any data set just by knowing the definitions. But you might also see questions about how they differ as the data set changes.

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Let’s take a look at that sort of question:

problem solving and data analysis sat topics

For these sorts of problems, it makes sense to actually go through the motions and create your new data set with the stated changes. Our new set would be 7, 9, 12, 15 and 16+.

Running through the answer chocies:

Visual Representations of Data

Numbers are boring, but pictures sell. This is why I’ve included pictures of a rooster, a little girl holding corn and Derek Zoolander in this post. I know your weaknesses. We humans may have opposable thumbs for smartphone use, but we have short attention spans. Just show us the pictures! And, surprisingly, College Board is sort of in on this, too. They love to figure out ways to express numbers with pretty pictures.

Okay, they aren’t pretty pictures. They are things like scatterplots, which are to great pictures like Kevin Hart is to great big people.

These questions often present a scatterplot followed by several questions. Here’s an example:

problem solving and data analysis sat topics

First, let’s look at the first question. The “line of best fit” is the line in the scatterplot that is shown, and it better reflects the data points than any other possible line. You can think of it as the most accurate straight line possible.

Based on this line, a price of $8.60 should result in about 110 books being sold. A price of $9.20 would result in about 70 books being sold, so the answer is 40.

On the second question, note that the text below the scatterplot tells us that we are working with twelve weeks. Data points above the scatterplot represent weeks where more books than predicted were sold, and points below the line represent weeks where fewer books than predicted were sold.

Six points are above the line, so more books were sold than predicted in 50% of weeks (6 divided by 12).

problem solving and data analysis sat topics

Models: The Scary Side of Problem Solving and Data Analysis

If you want to take some information that is otherwise-intuitive and easy to understand and make it such that a PhD is required to grasp it, figure out how to put it inside of a model.

Imagine you go to the fair, which costs $10 to get into, and you ride three rides at $5 each. That’s all pretty simple, right? You will spend $25 total, and if you want to ride an additional ride, that will be an extra $5. You could have understood this in elementary school.

But watch what happens when we put it into a model:

problem solving and data analysis sat topics

At this point, “given user” is probably giving up and going home. Sonic has corn dogs, too, you know.

But while this sounds complicated, remember that it is really the same scenario I laid out before: It’s ten bucks to get in, and five bucks per ride. If you have $30 to spend, how many rides can you ride?

It’s an easy question in a hard package. So what are we going to do? We’re going to take it out of the package.

We can do this by plugging in for the variable c and solving just as we would with any algebra problem:

problem solving and data analysis sat topics

The key to most model problems is to plug in values. Models can run two ways, though. They can either ask you to find a value when a model is given, or they can ask you to pick which model best describes data. Either way, plugging is your friend.

Problem Solving & Data Analysis: Final Thoughts

Problem Solving and Data Analysis is an opportunity to pick up points for many students because, unlike all of the algebra and geometry you are going to be tested on, these questions often involve some level of unfamiliarity.

Becoming familiar with an easy concept that you simply don’t know is about as easy as points gettin’ gets. And many of these concepts don’t involve much technical math; rather, they require you to understand some fundamental idea and then execute easy math.

Putting this together, you should see that this is a category where you should spend the time needed to excel. In my opinion, Problem Solving and Data Analysis is maybe the biggest opportunity for most SAT math students.

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Sat / act prep online guides and tips, problem solving and data analysis: key sat math concepts.

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SAT Math is divided intro three domains:

It's a good idea to get really familiar with what's going to be on the test, where it was derived, and what the SAT is really testing.

This post will focus on one domain— Problem Solving and Data Analysis . This is an opportunity to get cozy with these concepts, and with the overall tapes of information that test-makers are looking for. Problem Solving and Data Analysis problems are all about applying your math knowledge to practical situations and looking at actual statistics instead of abstract, theoretical scenarios.

Basic Information

There are 17 questions in this domain (out of 58 total math questions). They aren't labeled or otherwise indicated on the test— you're never told which type of question you're working on.

There are no Problem Solving and Data Analysis problems on the no-calculator section. You will always be permitted to use your approved calculator for questions from this domain —though you may not always need it.

You will receive a subscore on a scale of 1-15 on this domain.

There will be both multiple choice and grid-in questions.

You'll be dealing with both single-step and multistep problems; sometimes, it's just a matter of reading the data and parroting it back, while other problems require a bit more manipulation of the numbers.

General Concepts

The test-makers want to know that you understand math thoroughly enough to use your skills in real-world settings .

Quantitative reasoning is also crucial; you should be able to work with numbers and draw conclusions about what they imply.


You’ll be working with a lot of numbers.

Real-World Applications

You should be able to build a representation of a problem. If a scenario is described to you, you should be able to model it mathematically by describing it with expressions and equations.

You should know to consider the units involved. If there happens to be a shift of units (from feet to miles, or something like that), you should account for that as you calculate.

You should keep track of the practical meaning of quantities . You're going to be representing real values with variables: don't forget what those variables represent. Also, be sure you understand how a change in one of these variables or quantities affects what's happening in the equation. For example, in a line ($y=mx+b$) with a positive slope (or value of $m$), increasing $x$ will also result in an increase in $y$.

Data and Statistics

There are a lot of graphs, charts, and tables that could be covered on the test. You should be capable of analyzing one-variable data in bar graphs, histograms, line graphs, and box-and-whisker plots—as well as two-variable data in scatterplots and two-way tables. In other words, you should be fluent in reading these various representations of data.

You should be able to describe overall patterns. You'll have to identify positive and negative trends. You should be able to distinguish between linear and exponential growth.

Specific Skills

There are a number of skills that you'll want to be handy with on the day of the exam. In this section you'll find a discussion of these skills, including what they are and what they look like in action.


In fact, let’s hope these skills aren’t all that new!

Statistical Analysis

The measures of center are arithmetic mean (average) and median. If they can't be calculated from what's given, you may still need to draw some conclusions about them. Even if you can't find the actual number, there may be a question about what possible values are, or how the values compare to another set of data. Outliers typically affect the mean, but not the median.

The measure of spread to know is standard deviation. You've got to have the basic theory, but you won't need to calculate the exact value. You should be able to look at two sets of data and see which set is more spread out—that is, which has the greater standard deviation.

Insofar as the precision of estimates is concerned, everything depends on the variability of the data and the sample size; smaller variability combined with a larger sample size makes for estimates that are more precise in terms of the actual population.

Another tidbit of interest you should know is that randomization combats bias .

You should, in context, be able to work with margins of error, which are affected by sample size and standard deviation.

Confidence intervals should also be taken into account. The SAT always works with confidence intervals of 95%. This doesn’t mean that 95% of the population is necessarily described by the figure that’s been derived, just that we can be 95% certain that the descriptive figure that’s been reached is accurate.

When analyzing the relationship between two variables, remember, correlation is not causation. If subjects for a sample are selected randomly, we can generalize to the entire population reasonably well. If subjects are randomly assigned to test groups, we can reasonably speculate about cause and effect. Otherwise, though, we’re out of luck.

You should also be able to evaluate reports to make inferences, justify conclusions, and determine the appropriateness of data collection methods .

Take a look at this problem:


The correct answer here is (C). We know that removing one very high number from the set is not going to affect the median—the middle three values are all 12, so we know the median will still be 12, even if the middle of the data is shifted over one. The mean will shift somewhat if we don’t include the 24-inch measurement, but there are 20 other data points that anchor the mean at a relatively consistent value if any one value is removed. The range, however, will change from $24-8=16$ to $16-8=8$. The range gets cut in half if we remove the 24-inch measurement! That’s the measure that will change the most.

Mathematical Models

The domain of the SAT that we’re examining gives special attention to mathematical models. You must, therefore, be able to create and use a model.

When two variables are presented in a graph, table, or other chart , you should be handy with analyzing and drawing conclusions with regards to the relationship between these variables.

Relationships between variables can be modeled by functions, but remember the function is only a model! It may give scientifically accurate predictions, or it may just describe a general trend. You may be asked whether a model is good, acceptable, or entirely inappropriate.

Let’s look at this problem:


The geologist in the question provides a model regarding the country’s beach erosion. According to that model, beaches erode at a rate of 1.5 feet per year. 21 feet of erosion would therefore take 14 years, as $14(1.5)=21$.

The functions you’re asked to work with may be linear, quadratic, and/or exponential. Linear and exponential are discussed in more detail below.

An important example of linear growth is simple interest , where you earn interest on your principal, each period, but not on any interest that has been added since that first deposit. This is modeled by the function: $A = P(1+rt)$. $P$ is the principal, $r$ is the interest rate, and $t$ is the amount of time interest has been accruing.


An important example of exponential growth is compound interest , where you earn interest on the interest you’ve previously earned. This is modeled by the function: $A = P (1+r/n)^{nt}$, where $P$ is the principal, $r$ is the interest rate (typically annual), $n$ is the number of times the interest compounds per period (typically a year), and $t$ is the amount of time that has passed since the principal began accruing interest.

Be careful! The stated rate of change may not be the same as the rate of change over time. This is typical of compound interest:


Math and money are very closely linked.

Ratios, Proportions, Units, and Percentages

You’ll need to be familiar with direct proportionality/variation : $y = kx$, where $k$ is a unitless constant of proportion. This relationship may also be expressed as $x_1/y_1 = x_2/y_2$.

You’ll need to know how percent increase and percent decrease work. Be careful about how you approach these problems; they can get a little tricky. Remember, for instance, that if you have a 20%-off coupon for an item that’s on a 20%-off sale, you won’t save 40%. You’ll save 36%, as you’ll pay 80% of 80% of the original price: $x(.8)(.8) = (.64)x = x - (.36)x$.

Take a look at this pair of problems:


This is the sort of situation where you’re asked to deal with quantities in very practical terms. These aren’t just numbers floating in and out of an abstract function ; no, these numbers represent that annual budget, in thousands of dollars, for each of six different state programs in Kansas from 2007 to 2010.

The first problem is asking for the approximate average rate of change in the annual budget for agriculture/natural resources in Kansas from 2008 to 2010. From 2008 to 2009, the budget grew by 127,099 thousands of dollars, or $127,099,000. From 2009 to 2010, the budget grew by 2,299 thousands of dollars, or $2,299,000. Thus, we add those two figures together, divide by two, and find that the average growth was $64,699,000, or, when rounding, (B).

For the next problem, we are comparing the 2007-budget-to-2010-budget ratios across the various programs. We will first need to find those ratios:  

Agriculture/natural resources — $373,904/488,106=.766$

Education — $2,164,607/3,008,036=.7196$

Highways and transportation — $1,468,482/1,773,893=.8278$

Public safety — $263,463/464,233=.5675$

Out of these, the closest to human resources ($4,051,050/5,921,379=.6841$), is (B).

Two events are independent if one happening has nothing to do with another , like the sun shining and you eating a sandwich for lunch. The sun may shine, and you may eat a sandwich for lunch, but one does not cause or prevent the other.

Two events are mutually exclusive if they cannot both occur , like me wearing a hat and me not wearing a hat. I can’t do both.

For independent, non-mutually exclusive events: P(A and B) = P(A)*P(B), whereas P(A or B) = P(A) + P(B) - P(A and B).

For mutually exclusive events: P(A or B) = P(A) + P(B).

There are other formulas for more complicated scenarios, but these will get you pretty far — they’re all you should need on the SAT.

Let’s take a look at this problem:


Twenty-five people passed the bar exam; of these, seven did not take the review course. So, the probability that the interviewed person in question did not take the bar exam is 7/25, or (B).


Math and gambling are closely linked, too. It’s all about that probability.

This domain of the test is calculation-heavy, although there are some theoretical questions.

These questions compose almost a third of the test . They cover, roughly:

Your answers will be scored to yield one of three subscores for the Math section.

What’s Next?

That was a fair amount of information. Digest it a little; then, a great next stop would be perusing our overall guide to SAT Math , including directions to a number of other great posts .

Now, because Problem Solving and Data Analysis problems are so information-heavy , you may wish to practice with some word problems , to get used to that much verbal data being thrown your way.

As you’re trying these practice problems, you’ll want to know how to use them to your best advantage .

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SAT Math: Problem Solving and Data Analysis

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Covering 29% of the concepts on the SAT, the Problem Solving and Data Analysis section is the second most common on SAT Math , after Heart of Algebra. Keeping in line with real-world scenarios, these SAT problems will ask you to infer information based on a study with any number of participants or interpret data from a graph. This is actually beneficial for SAT students, given that you’ll be learning lots about cause and effect and inferential statistics in college.

Now, let’s talk about how to approach the various types of Problem Solving and Data Analysis questions on the SAT, plus some practice questions to get you started! Feel free to use the Table of Contents to navigate directly to the topics you want to learn.

Table of Contents

Quick facts about sat problem solving and data analysis, ratio, proportion, units, and percentage, subjects and treatments, sat statistics (mean, median, and mode), what’s important to know about averages on the sat, sat graphs: tips and tricks.

You can expect to see about 17 Problem Solving and Data Analysis questions on the SAT Math section, although they will not be outright labeled by question type. Some answers will be single-step questions, while others will be multistep problems. If you fear mental math, there’s no need to sweat it in this section—you’ll be allowed to use a calculator. Like the other SAT Math sections, you’ll receive a Problem Solving and Data Analysis subscore on a scale of 1 to 15.

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This Problem Solving and Data Analysis question type shouldn’t come as a surprise since it has probably been part of your math courses for the last five years (yep, you most likely went over this stuff, in some form, all the way back in early middle school). I’ll start with ratios.

Understanding Ratios

A good way to think of ratios is apples and oranges. Say I have two oranges and three apples, the ratio of oranges to apples is 2:3. Seems straightforward. What if I have four apples and 6 oranges? If you answered 4:6, that is not quite correct. You have to think of the ratio the way you would a fraction, in lowest terms. Both 4 and 6 can be divided by ‘2’, giving you 2:3. Notice how that is the same ratio as 2:3.

This highlights an important conceptual idea: ratio is not about total number . It is about the number of one thing, to the number of another thing, reduced, so that the ratio is expressed as two prime numbers.

One last thing about ratios. Let’s say you have a ratio of 1:2. This is not the same thing as ½. The bottom number in a fraction is always the total. The total of a ratio is always the parts of a ratio added together. In this case, 1:2 is 1 + 2 = 3. So if I have 1 apple to two oranges, 1/3 of the fruit are apples and 2/3 are oranges.

If you have more than two ratios, make sure to add up all of the ratios. For instance, if the ratio of blue marbles to red marbles to green marbles is 2 : 5: 7, red marbles account for 5/14 of the total (2 + 5 + 7 = 14).

Do you think you got that? Well, here are some practice questions to test your knowledge of Problem Solving and Data Analysis.

Understanding Percentages

Percentages can be surprisingly complicated on the SAT. Part of that is because we can’t always translate them into fractions, which are easier to work with algebraically. While it’s easy enough to think of \(50%\) as \(frac{1}{2}\), it’s rarely so easy to make the conversion on the SAT, especially when the percentages given are, say, 35% or 15%.

Finding a percent is pretty easy, as long as you have a calculator. Just divide the part by the whole and multiply the decimal that comes out by 100. So if you ate 10 out of a serving of 12 buffalo wings, then you ate (10/12)100=83.33%. Remembering that formula can save you some grief when you have to use it algebraically.

However, the SAT won’t just test you on the simple process of finding the percentage of a number (like calculating a tip). Instead, it’ll ask you to calculate in reverse (finding the whole from the part), find a combination of percentages, find a percent change, or give some other scenario-specific piece of information.

Being prepared for percent change questions, in particular, will take you far on Problem Solving and Data Analysis.

The equation for percent increase \( = frac{text{New Number – Original Number}}{text{Original Number}}*100 \).

The equation for percent decrease is \( = frac{text{Original Number – New Number}}{text{Original Number}}*100 \).

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Practice Questions

Easy questions.

A) 1 B) 2 C) 4 D) 5

Okay, this question is slightly evil, since shorts sounds like shirts and it is easy to get the two mixed up when you are reading fast. So always pay attention, even on easier questions!

Since we know that Kevin has 10 shirts and that 10, therefore, corresponds to the number ‘5’ in the ratio, that the actual number of shorts, shirts, etc., he owns is double the number in the ratio. Thus, he owns four shirts and six pairs of shoes. So he’ll have to give away two pairs of shoes so that he’ll have the same number of shoes as he does shorts. Answer: (B).

Medium Difficulty Question

A) 12,000 B) 120,000 C) 132,000 D) 176,000

# of voters who voted in 2008 election is equal to 200,000 x 60 = 120,000

In 2010, the number of overall eligible voters increased by 20%, so 20% of 200,000 is 40,000 giving us 240,000 total voters.

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Difficult Questions

We know that 7/3 of mile = one inch.

We also know that the area is 49 square miles, meaning that each side = 7: √49 = 7). To find how many inches correspond to 7 miles, we set up the following equation:

7 = 7/3x, x = 3

Another possible question type, and one that most are familiar with and probably dread, is the percent question.

To reduce something by a certain percentage, either turn that percent into a ratio over 100 or convert the percent into a decimal by moving the point back two spaces. For example, 40% equals both 40/100 and 0.40. So the answers are:

5% = .05, 5/100 or 1/20 (you don’t always have to reduce for quick calculations) 26% = .26, 26/100 or 13/50 37.5% = .375, 375/1000 or 3/8 125% = 1.25, 125/100, 5/4

A) 4% B) 6.25% C) 16% D) 36%

When you are not given a specific value for a percent problem, use 100 since it is easiest to increase or decrease in terms of %.

1 st discount: 20% off of 100 = 80.

2 nd discount: 20% off of 80 = 64.

Online, the coat sells for 40% off of the original department store price, which we assumed is 100.

Online discount: 40% of 100 = 60.

This is the tricky part. We are not comparing the price difference (which would be 4 dollars) but how much percent less 60 (online price) is than 64 (department store sale price).

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This is not an official title but the name I’m giving to questions that deal with studies trying to determine cause and effect .

In order to understand how to approach Subjects and Treatments questions, let’s talk about randomization. The idea of randomization is the essence, the beating heart, of determining cause and effect. It helps us more reliably answer the question of whether a certain form of treatment causes a predictable outcome in subjects.

Randomization can happen at two levels. First off, when researchers select from the population in general, they have to make sure that they are not unknowingly selecting a certain type of person. Say, for instance, that I want to know what percent of Americans use Instagram. If I walk on a college campus and ask students there, I’m not taking a randomized sample of Americans (think how different my response rate would be if I decide to poll the audience at a Rolling Stones concert).

On the other hand, if I went to a city phone directory and dropped a quarter on the page, choosing the name that the center of the quarter was closest to, I would be doing a much better job of randomizing (though, one would rightly argue, I’d still be skewing to an older age-group, assuming that most young people have only cell phones, which aren’t listed in city directories). For the sake of argument, let’s say our phone directory method is able to randomly choose for all ages.

After throwing the quarter a total of a hundred times on randomly selected pages (we wouldn’t want only people whose names begin with ‘C’, because they might share some common trait), our sample size consists of 100 subjects. If we were to ask them about their Instagram use, our findings would far more likely skew with the general population. Therefore, this method would allow us to make generalizations about the population at large.

Subjects and Treatments Practice Question

Which of the following is an appropriate conclusion?

A) The exercise bike regimen led to the reduction of the varsity runners’ time. B) The exercise bike regimen would have helped the junior varsity team become faster. C) No conclusion about cause and effect can be drawn because there might be fundamental differences between the way that varsity athletes respond to training in general and the way that junior varsity athletes respond. D) No conclusion about cause and effect can be drawn because junior varsity athletes might have decreased their speed on the 3-mile course by more than 30 seconds had they completed the biking regimen.

When dealing with cause and effect in a study, or what the SAT calls a treatment, researchers need to ensure that they randomly select amongst the participants. Imagine that we wanted to test the effects on the immune system of a new caffeinated beverage. If researchers were to break our 100 subjects into under-40 and over-40, the results would not be reliable.

First off, young people are known to generally have stronger immune systems. Therefore, once we have randomly selected a group for a study, we need to further ensure that, once in the study, researchers randomly break the subjects into two groups. In this case, those who drink the newfangled beverage and those who must make do with a placebo, or beverage that is not caffeinated.

At this point, we are likely to have a group that is both representative of the overall population and will allow us to draw reliable conclusions about cause and effect.

Another scenario and this will help us segue to the practice question above, are treatments/trials in which the subjects are not randomly chosen. For instance, in the question about the runners, clearly, they are not representative of the population as a whole (I’m sure many people would never dare peel themselves off their couches to something as daft as run three miles).

Nonetheless, we can still determine cause and effect from a non-representative population (in this case runners) as long as those runners are randomly broken into two groups, exercise bike vs. usual one hour run. The problem with the study is the runner coach did not randomly assign runners but gave the slower runners one treatment. Therefore, the observed results cannot be attributed to the bike regimen; they could likely result from the fact that the two groups are fundamentally different. Think about it: a varsity runner is already the faster runner, one who is likely to improve faster at running a three-mile course than his or her junior varsity teammate. Therefore, the answer is C).

Subjects and Treatments: A Summary

Here are the key points regarding subjects and treatments (aka cause and effect questions) on SAT Problem Solving and Data Analysis:

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The SAT Math test often asks you to do some statistics problems involving averages. Finding the mean is the most commonly used average and, as it so happens, the most commonly tested when it comes to SAT statistics. The formula is pretty simple:

{a+b+c+….}/n where n is the number of terms added in the numerator.  In the set of numbers {2,3,4,5}, 3.5 would be the mean, because 2+3+4+5=14, and \(14/4=3.5\)

If the numbers in a set are listed in order, the median is the middle number. In the set {1,5,130}, 5 is the median. In the set above, {2,3,4,5}, the median is 3.5, which is the mean of the middle two terms since there’s an odd number of them.

The mode is just the number that shows up the most often. It’s perfectly possible that there is no mode or that there are several modes. In the set {5,7,7,9,18,18}, both 7 and 18 are modes.

Averages come up in algebra or word problems. You’ll usually have to find some value using the formula for a mean, but it may not be as simple as finding the average of a few numbers.  Instead, you’ll have to plug some numbers into the formula and then use a bit of algebra or logic to get at what’s missing.

For example, you might see a question like this:

If the arithmetic mean of x , 2 x , and 6 x is 126, what is the value of x ?

To solve the question, you’ll need to plug it all into the formula and then do some variable manipulation.

\(frac {x+2x+6x}{3}=126\)   \({x+2x+6x}=378\)   \(9x = 378\)   \(x=42\)

Medians and modes, on the other hand, don’t show up all that often in Problem Solving and Data Analysis. Definitely be sure that you can remember which is which, but expect questions on means, most of the time. As for other types of statistical analysis, you may also be asked to solve some problems involving standard deviation .

If you’re careful to remember that the question is asking you for the sum of the sisters’ ages, you can solve this one pretty quickly. Keep in mind that we can’t find their individual ages, though. There’s not enough information for that. First we find the total combined age of the three, which must be 72, since \(24*3=72\). Careful not to fall for the trap that is (E), we take the last step and subtract 16 from that total age to find the leftover sum, which is 56, or (C).

What is a “weighted average”?

Basically, weighted means uneven, here; the numbers that you’re looking at don’t carry the same importance. For example, if I’m trying to find the average number of fleas that my pets have, and each cat has 150 while each dog has 200, then those two numbers have equal “weight” only if I have the same number of cats as dogs. Let’s say I have 1 of each.

That’s just a normal mean, so that’s no problem. Well, the fleas are a problem, I guess. And the fact that I’m counting fleas might have my family a little worried…anyway, the math is easy. But that’s a non- weighted average.

For a weighted average, I would have a different number of cats than dogs. Let’s say I had 3 cats and 2 dogs. (And they all have fleas…things are starting to get kinda gross. Sorry.)

In order to give them the appropriate weight, we’d have to multiply each piece appropriately and change the total (denominator) to reflect it.

But if you expand that, you’ll see that it’s the same as the standard mean formula.

Just make sure you divide by five (because I have five pets) not two (for two types of pets).

Finding Average Rates

Average rates are a type of weighted average. Your SAT Problem Solving and Data Analysis section will include a problem or two about these, and you need to be sure not to fall for the common trap.

Maria’s drive to the supermarket takes her 20 minutes, during which she averages a speed of 21 miles per hour. She takes the same route home, but it only takes 15 minutes to cover the equal distance. What was Maria’s average speed while driving?

This is a tricky, multi-step problem, and you can’t plug in the answer choices to solve it, sadly.

Let’s first find all of our information, because the question has only given you part of it. You need to know the formula r=d/t (rate = distance/time) , also expressed as d=rt (easily remembered as the “dirt” formula). We’re going to use it both ways.

Using that formula, let’s look at the first leg of her trip. She traveled for 1/3 of an hour at 21 mph, so she must have traveled 7 miles.

That’s \(21*0.333=7\)

Using that info, we can figure out the rate of her trip back home. Going 7 miles in 1/4 of an hour on the way home, she went an average of 28 mph.

That’s \(7/0.25=28\)

So now we need to find the total average. That’s not the average of the two numbers we have! Because each mile she traveled on the way there took more time than each mile on the way home, they have different weights!

✗ \(frac{21+28}{2}=24.5\)

Instead, you need to take the total of each piece—total time and total distance—to find the total, average rate.

✓ \(frac{14 text{ miles}}{.333 text{ hours} + .25 text{ hours}}=frac{14 text{ miles}}{.5833 text{ hours}}={24 text{ mph}}\)

Average Rate Practice Question

A) 55 miles per hour B) 65 miles per hour C) 70 miles per hour D) 75 miles per hour

To figure out the average speed of the entire trip, divide the total distance by the total number of hours. The handy equation D = rt, where D is total distance, r is rate, and t is time, will make this easier.

D = 910, r = ?, t = 9 + 4 = 13 hours.

Weighted Averages That You Won’t See on Your SAT

I’ve never seen an SAT Problem Solving and Data Analysis question that asks you to find an average based on percent weights (e.g. finding a final grade in a class where quizzes count for 70%, attendance for 20%, and participation for 10%). Finding that average is a little more complicated, so it’s nice that we don’t have to worry about it.

  Simply put… if you’re finding the average of two sets of information that already are averages in their own right, as the number of fleas per cat and the number fleas per dog, you can’t just take the mean of those averages. You have to find the totals and then plug them into the formula. You should be excited about these kinds of problems, if for nothing more than having the opportunity to bust out your handy-dandy, brand-spanking’ new SAT calculator . 😛

Chart with bell curve cut by dotted lines representing average - Magoosh

Among the math skills that SAT Problem Solving and Data Analysis tests, reading data from tables or graphs is one of the more straightforward tasks. But there are a number of simple mistakes that might make you miss out on points if you’re not careful. The best way to avoid those totally avoidable slip-ups is to train yourself to follow a pattern.

You’ll want to read the headings, the axes, and the units of measurement, then make note of any missing information or obvious patterns.

That’s all for the SAT Problem Solving and Data Analysis section! We hope this breakdown was helpful for you. To read up on the other two sections of SAT Math, check out our posts on Heart of Algebra and Passport to Advanced Math .

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Chris Lele is the Principal Curriculum Manager (and vocabulary wizard ) at Magoosh. Chris graduated from UCLA with a BA in Psychology and has 20 years of experience in the test prep industry. He's been quoted as a subject expert in many publications, including US News , GMAC , and Business Because . In his time at Magoosh, Chris has taught countless students how to tackle the GRE , GMAT, SAT, ACT, MCAT (CARS), and LSAT exams with confidence. Some of his students have even gone on to get near-perfect scores. You can find Chris on YouTube , LinkedIn , Twitter and Facebook !

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SAT Math : Data Analysis

Study concepts, example questions & explanations for sat math, all sat math resources, example questions, example question #1 : venn diagrams.

In a group of 83 gym members, 51 are taking kickboxing and 25 are taking yoga. Of the students taking kickboxing or yoga, 11 are taking both classes. How many members are not taking either course?

If 11 people are taking both courses, this means 51-11 or 40 are taking kickboxing only and 25-11 or 14 are taking yoga only. The number of people taking at least one course, therefore, is 40 + 14 + 11 = 65. The 83 members minus the 65 that are taking courses leaves 18 who are not taking any courses.

Example Question #2 : Venn Diagrams

Doug has a cow farm.  Some of Doug's cows are used for milk, some are used for reproduction and some are used for both. If he has a total of 40 cows and 10 are used only for milk and 3 are used for both milk and reproduction, then how many cows are used for reproduction?

Since we know that only 10 cows are for milk only we must subtract this number from the total amount of cows to get our answer: 40 – 10 = 30 cows. The cows that do both are still used for reproduction, so the correct answer is 30 cows.

Example Question #3 : Venn Diagrams

All students have to take at least one math class and one language class. Twenty students take calculus, and thirty students take statistics. Fifteen students take Spanish and twenty-five take French. If there are thirty-five students total, what is the maximum number of students taking both two math classes and two language classes.

Totalling the number in math there are 50 students on the rosters of all the math classes. With 35 total students this means that there are 15 students taking 2 math classes. For the language classes there are 40 students on the roster, showing that 5 students are taking 2 language classes. The maximum number of students taking two math classes and two language classes is only as great as the smallest number taking a double math or language class, which is 5 students (limited by the language doubles). 

Example Question #4 : Venn Diagrams

The class of 2034 at Make Believe High School graduated 50 students. 13 students studied only math. 35 students studied English.  30 students studied only 2 subjects. Only 4 students studied writing, it was the third subject for all of them. How many students did not study anything?

The answer is 2. Taking away the 4 writing students, the 13 math-only students, and the remaining 31 English students, we have 2 students remaining.

Example Question #5 : Venn Diagrams

In a school of 1250 students, 50% of the students take an art class and 50% of the students take a gym class. If 450 students take neither art nor gym class, then how many students take both art and gym?

problem solving and data analysis sat topics

You can construct a Venn diagran in which one circle represents art (A), the other represents gym (B), the region of overlap is designated (C), and the number of students not present in either circle is designated (S).

First, it is given that S=450.

50% of students take art and 50% take gym and the total number of students is 1250. 50% of 1250 is 625. Thus, A+C=625 and B+C=625.

Setting them equal to each other we get A+C=B+C.

Subtract C from both sides to get A=B; so the same number of students take art only and gym only.

Now 1250-450= A+B+C=800.

Since A=B we can use substitution to get 2A+C=800.

Finally you can solve the system of equations using the method of your choice (substitution or elimination) to solve the system with A+C=625 and 2A+C=800.

For substitution, we solve for C in the first equation to get C=625-A. Then we substitute this value into the second equation to get 2A+ (625-A)=800. Solve for A to get A=175, so B=175 also. Since A+B+C = 800, C=450.

Example Question #6 : Venn Diagrams

The universal set is positive counting numbers less than 11.  Set A = {1, 3, 5} and Set B = { 2, 4, 6}.

(A'\bigcap B)\bigcup(A\bigcap B')

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 3, 5}

A' = {2, 4, 6, 7, 8, 9, 10}

B = {2, 4, 6}

B' = {1, 3, 5, 7, 8, 9, 10}

(A'\bigcap B)

Sets P, Q, and R consist of the positive factors of 48, 90, and 56, respectively. If set T = P U (Q ∩ R), which of the following does NOT belong to T?

First, let's find the factors of 48, which will give us all of the elements in P. In order to find the factors of 48, list the pairs of numbers whose product is 48.

The pairs are as follows:

1 and 48; 2 and 24; 3 and 16; 4 and 12; 6 and 8

Therefore the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

Now we can write P = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.

Next, we need to find the factors of 90.

Again list the pairs:

1 and 90; 2 and 45; 3 and 30; 5 and 18; 6 and 15; 9 and 10

Then the factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

Thus, Q = {1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90}.

Next find the factors of 56:

1 and 56; 2 and 28; 4 and 14; 7 and 8

Set R = {1, 2, 4, 7, 8, 14, 28, 56}

Now, we need to find set T, which is P U (Q ∩ R).

We have to start inside the parantheses with Q ∩ R. The intersection of two sets consists of all of the elements that the two sets have in common. The only elements that Q and R have in common are 1 and 2. 

Q ∩ R = {1, 2} 

Lastly, we must find P U (Q ∩ R).

The union of two sets consists of any element that is in either of the two sets. Thus, the union of P and Q ∩ R will consist of the elements that are either in P or in Q ∩ R. The following elements are in either P or Q ∩ R:

{1, 2, 3, 4, 6, 8, 12, 16, 24, 48}

Therefore, T = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.

The problem asks us to determine which choice does NOT belong to T. The number 28 doesn't belong to T.

The answer is 28.

Example Question #215 : Data Analysis

problem solving and data analysis sat topics

Sixty high school seniors were polled to see if they were taking history and calculus. A total of 29 students said they were taking calculus, and a total of 50 students said they were taking history. What is the minimum number of students who take both history and calculus?

problem solving and data analysis sat topics

We can draw a Venn diagram to see these two sets of students.


We need to find the overlap between these two sets. To find that, add up the total number of students who are taking history and the total number of students who are taking calculus.

problem solving and data analysis sat topics

Notice that we have more students this way than the total number who were polled. That is because the students who are taking history AND calculus have been double counted. Subtract the total number of students polled to find out how many students were counted twice.

Example Question #8 : How To Interpret Venn Diagrams


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