problem solving and data analysis pdf

SAT Math: Problem Solving and Data Analysis

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.

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 \( = frac{text{Original Number – New Number}}{text{Original Number}}*100 \).

Practice Questions

Easy questions.

• The ratio of shirts to shorts to pairs of shoes in Kevin’s closet is 5 : 2 : 3. If Kevin owns 10 shirts, how many pairs of shoes does he have to give away so that he ends up having the same number of shorts as he does pairs of shoes?

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

• There are 200,000 eligible voters in district X, 60% of whom voted in the 2008 state election. In 2010 state election, the number of eligible voters in district X increased by 20% but if only 55% voted in this election, how many total votes were cast in the 2010 state election, assuming that no voter can cast more than one vote?

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.

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

• In a popular department store, a designer coat is discounted 20% off of the original price. After not selling for three months, the coat is further marked down another 20%. If the same coat sells online for 40% lower than the original department store price, what percent less would somebody pay if they were to buy the coat directly online than if they were to buy the coat after it has been discounted twice at the department store?

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).

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:

• Results from a study can only be generalized to the population at large if the group of subjects was randomly selected.
• Once subjects have been selected, whether or not they were randomly selected, cause and effect can only be determined if the subjects were randomly assigned to the groups within the experiment/study/treatment.
• There are three basic types of averages on the SAT that you should be pretty comfortable with at this point, and all of them start with the letter “m.” Those are the mean, the median, and the mode. In case those aren’t second nature, let’s define them, quickly.

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.

\(frac {150+200}{2}=175\)

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.

\(frac {3(150)+2(200)}{5}=170\)

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

\(frac {150+150+150+200+200}{5}=170\)

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 . 😛

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.

• Add in Any Information From the Question As is often true for other types of SAT math problems, the written question might have some info in it that the figure doesn’t include. Just like you would write in angle measurements , fill in any extra info; there’s no reason to try to keep it in your head.
• Write the Math Out If you’re asked about relationships between two things, look carefully at the relationships between 4-6 pieces of information (two x s and two y s), and write out the pattern. If you’re looking for some variable, write out the equation. If it’s not clear how to go about that, maybe you should try plugging answer choices in to see if they work.

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 .

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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Problem Solving and Data Analysis: SAT Math Tips & Practice

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: 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:

• Ratios, rates and proportions
• Percentages
• Measurement quantities and units
• Scatterplots
• The relationship between variables and graphs in a data context
• Linear vs. exponential growth
• Data tables
• Sample, population, inferences and other statistical survey topics
• Mean, median, mode and other related concepts

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:

• Sample : The group of people being surveyed. In a presidential election survey, the sample is the survey respondents.
• Population : The group of people being described. In a United States presidential election survey, the population is all American voters.

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:

• Randomly selected: Any population has outliers in it. There are probably a thousand people in America who particularly enjoy wearing matching outfits with their dog and only talking in bark. But those people are kind of weird, and if we only asked them who they were voting for, our sample probably wouldn’t be reflective of all of America.
• Large enough to represent the population: You don’t need to get too into the details here, but you can’t accurately describe a big group by asking six people. You can, however, describe a class of 400 students by asking 80 kids, and you can describe a country of 300 million by asking a thousand. Normally, sample size issues will be glaring (if they exist).
• Part of the population: If you want to learn something about how America is going to vote, don’t poll Canadians and definitely don’t poll the French.

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:

• Samples tell us what is likely, not what is necessary. I’m not going to get too into things like confidence intervals because those concepts are beyond the scope of the SAT. But surveys are ultimately just telling us what is very likely to be true, not what is guaranteed to be true. For example, every presidential election survey comes with a “margin of error” disclaimer. That is telling that the results are not a concrete number but are instead likely to fall within a certain range.
• Different samples might yield different results, even if they are both done correctly. This follows directly from the first point. If surveys are totally precise, we shouldn’t expect them to give us exactly the same results every time.
• Correlation doesn’t equal causation. For example, a recent study showed that a rooster crows every single morning. Also, get this, the sun rises every single morning, typically right after the roosters crow. Better correlation doesn’t exist. But is all of this crowing actually causing the sun to rise? Of course not. They are perfectly correlated, but that’s not the same thing as causation.

Statistics and Data Collection Practice

Let’s take a look at a practice problem:

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:

• A: Can we say that the entire studenty body must have the same preference as the sample? Of course not.
• B: Must a second sample yield exactly the same results as the first? Of course not.
• C: This is just an incorrect interpretation of the survey itself. The university didn’t ask students what they use, it asked what they prefer. Those are two different things, so this can’t be the right answer.
• D : This is all we have left, so that’s our answer.

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:

• Basic concept understanding (calculate the mean or median given a data set)
• An understanding of how mean, median and mode vary when things change

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:

• Mean is the average of a set of numbers. Add them up to get the sum, and then divide the sum by the number of numbers you have.
• Median is the middle number when the numbers are sorted from least to greatest. In the set 1, 2, 5, the median is 2.
• Mode is the number that occurs most frequently. In the set 1, 2, 2, 5, 6, the mode is 2.

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:

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:

• A: Can we calculate the mean if we don’t know exactly what the last number is? No, so this can’t be our answer.
• B: Can we calculate the median if we don’t know exactly what the last number is? Yes, we can. The median doesn’t depend on the outside numbers. In this case, our median is 12, so this is the right answer.
• C: We can’t calculate the mean, so this can’t be right.
• D: B is correct, so this is incorrect.

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:

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).

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:

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:

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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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.

• Strategies for Problem Solving and Data Analysis Questions
• Practice Questions for Problem Solving and Data Analysis

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:

• Heart of Algebra: 33%
• Problem Solving and Data Analysis: 29%
• Passport to Advanced Math: 28%

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:

• Use ratios, rates, proportional relationships, and scale drawings to solve single- and multistep problems.
• Solve single- and multi-step problems involving percentages.
• Solve single- and multi-step problems involving measurement quantities, units, and unit conversion. 
• Given a scatterplot, use linear, quadratic, or exponential models to describe how the variables are related.
• Use the relationship between two variables to investigate key features of the graph.
• Compare linear growth with exponential growth. 
• Use two-way tables to summarize categorical data and relative frequencies, and calculate conditional probability. 
• Make inferences about population parameters based on sample data. 
• Use statistics to investigate measures of center of data and analyze shape, center, and spread. 
• Evaluate reports to make inferences, justify conclusions, and determine appropriateness of data collection methods.

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.

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)

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

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

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

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

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.

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6. Scale Factors

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

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

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

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

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:

• 15 Hardest SAT Math Questions
• 30 SAT Math Formulas You Need to Know
• Guide to SAT Math Heart of Algebra + Practice Questions
• 5 Common SAT Math Mistakes to Avoid
• 5 Tips to Boost Your Math SAT Score

Related CollegeVine Blog Posts

• Test Preparation
• College & High School

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Problem Solving and Data Analysis: Sat Math Practice Test

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This practice book includes 400 questions (350 multiple-choice, 50 free-response) about the Problem Solving and Data Analysis in the SAT Math section.

It has been carefully developed to improve your SAT Math score and help you focus on your weak areas.

The book focuses only on carefully selected practice questions, similar to the commonly asked SAT Math questions. You will use your time wisely by avoiding questions you won't find on the actual test.

Simplified SOLUTIONS to the questions will make you love this workbook. You will discover some shortcuts in our solutions that will help you approach questions differently.

Five main topics: RATIOS, RATES & PROPORTIONS, PERCENTS & UNITS, TABLE DATA & PROBABILITY, SCATTERPLOTS, and STATISTICS & DATA INFERENCES are covered in this prep book.

• ISBN-10 171779369X
• ISBN-13 978-1717793690
• Publication date July 16, 2018
• Language English
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• Print length 235 pages
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