problem solving and data analysis drill 3 answers

Problem Solving and Data Analysis

We have lots of free resources and videos to help you prepare for the SAT. These materials are for the redesigned SAT which is for you if you are taking the SAT in March 2016 and beyond.

Related Pages More Lessons for SAT Math More Resources for SAT

Problem Solving and Data Analysis includes questions that test your ability to

• create a representation of the problem.
• consider the units involved.
• pay attention to the meaning of quantities.
• know and use different properties of mathematical properties and representations.
• apply key principles of statistics.
• estimate the probability of a simple or compound event.

There are many ways that you can be tested and practicing different types of questions will help you to be prepared for the SAT.

The following video lessons will show you how to solve a variety of problem solving and data analysis questions in different situations.

Ratio, Proportion, Units and Percentages

There will be questions on ratios. A ratio represents the proportional relationship between quantities. Fractions can be used to represent ratios.

There will also be questions involving percentages. Percent is a type proportion that means “per 100”.

You will need to convert units when required by the question. One way to perform unit conversion is to write it out as a series of multiplication steps.

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8

Charts, Graphs and Tables

The questions in Problem Solving and Data Analysis focus on linear, quadratic and exponential relationships which may be represented by charts, graphs or tables. A model is linear if the difference in quantity is constant. A model is exponential if the ratio in the quantity is constant.

A line of best fit is a straight line that best represents the data on a scatterplot. It is written in y = mx + c.

You need to know the formulas for simple and compound interest. Simple Interest: A = P(1 + rt) Compound Interest: A = P(1 + r/n) nt where A is the total amount, P is the principal, r is the interest rate, t is the time period and n is the number of times the interest is compounded per year.

Probability measures how likely an event is. The formula to calculate the probability of an event is: Probability = (number of favorable outcomes)/(total number of possible outcomes)

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question 14 Question 15

Data and Statistics

You need to know that mean, median, and mode are measures of center for a data set, while range and standard deviation are measures of spread. You will not be asked to calculate the standard deviation of a set of data, but you do need to understand that a larger standard deviation means that the values are more spread out from the mean. You may be asked to compare the standard deviation of two data sets by approximating the spread from the mean.

You do not need to calculate the margins of error or confidence level, but you do need to know what these concepts are and how to interpret them in context. Take note that the questions in the SAT will always use 95% confidence levels. Some questions may give you the confidence level and ask you to find the value for which the interval applies. When the confidence level is kept the same, the size of the margin of error is affected by the standard deviation and the sample size. The larger the standard deviation, the larger the margin of error. The larger the sample size, the smaller the margin of error. The margin of error and confidence interval are estimates for the entire population and do not apply to values of individual objects in the populations.

The results of a sample can be generalized to the entire population only if the subjects in the sample are selected randomly. Conclusions about cause and effect can appropriately be drawn only if the subjects are randomly assigned to treatment.

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9

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Guide to SAT Math Heart of Algebra + Practice Questions

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What’s Covered:

Overview of sat math heart of algebra, how will the sat impact my college chances, strategies to solve heart of algebra problems, 10 difficult heart of algebra questions, final tips and strategies.

The Heart of Algebra questions make up a third of the SAT Math section. If you’re looking to improve your score or even aiming for a perfect 800, here are some strategies and tough questions to test your skills.

The SAT Math section is scored out of 800 and constitutes half of the total SAT score. There are three main categories: Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math. When you take the SAT, you’ll receive a subscore out of 15 for each of these categories. 

Here is the breakdown of each section:

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

The remaining 10% falls under Additional Topics in Math, which contains a wide array of topics. 

The Heart of Algebra section focuses on linear equations, inequalities, graphs, and systems. These questions range from simple calculations to more complex word problems. 

Heart of Algebra questions require that students:

• Create, solve, or interpret a linear expression or equation in one variable. 
• Create, solve, or interpret linear inequalities in one variable.
• Build a linear function that models a linear relationship between two quantities. 
• Create, solve, and interpret systems of linear inequalities in two variables. 
• Create, solve, and interpret systems of two linear equations in two variables. 
• Algebraically solve linear equations (or inequalities) in one variable. 
• Algebraically solve systems of two linear equations in two variables. 
• Interpret the variables and constants in expressions for linear functions within the context presented. 
• Understand connections between algebraic and graphical representations. 

In this post, we will explore general strategies to solving Heart of Algebra Problems and cover some of the more difficult problems that you may encounter on the SAT.

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Though Heart of Algebra problems appear in many different shapes and forms, the basic strategy to solve them is relatively the same from question to question. In general, when taking the SAT math section, remember to skip problems that seem too difficult or that may take too long to answer. Since all questions are weighted the same, answering many easy questions is more beneficial than answering one difficult question.

For Heart of Algebra problems specifically, since they center on linear equations, start by underlining or circling relevant information (i.e. any quantities, variables, or relationships that are defined). Then, make sure you understand what the question is asking. Begin by defining variables, then if an equation is not already given to you, construct as many equations as you can, based on the information given. 

Once your equations are established, use them to solve for what the question initially asked. After solving each problem, make sure to check back and ensure you actually answered the question , or even double check your calculations if you have the time.

Here are some sample Heart of Algebra questions which are on the more difficult side, along with a detailed description of how to solve them.

1. System of Inequalities (No Calculator, Multiple Choice)

Correct Answer: B

For multiple choice problems, looking at the answer choices can be helpful in determining how to solve the problem. For this question, the answer choices show us that the question is looking for a system of inequalities. So, when reading the question, let’s pay close attention to any relationships between quantities. The last sentence of the problem defines variables for us: \(x\) is for junior directors and \(y\) is for senior directors. 

The first sentence says “at least 10 staff members” should be hired. The term “at least” mathematically translates to “greater than or equal to.” So, we have that the total number of directors (which is the number of junior directors and the number of senior directors) must be greater than or equal to 10: \(x+y\:\geq\:10\) . This eliminates choices A and D.

To choose between B and C, we can save time by noticing that one difference is whether \(x\:\geq\:3\) or \(x\:\leq\:3\) . The problem says, “She must hire at least 3 junior directors.” Like before, this indicates \(x\:\geq\:3\), so the correct answer is B. This question is an example of how you don’t always need to solve the entire problem to determine the correct answer.

2. Constructing Linear Equations (No Calculator, Multiple Choice)

Correct Answer: C

A lot of information is given in this equation, so our first instinct should be to try to relate that information. We know that the initial value of the equipment is $32,400 and that every year it depreciates some (unknown) amount. We are also given that at 12 years, the value is $0.

If we were to relate value with time, we’d have that \(V=32,400-kt\), where \(V\) represents value in dollars, \(t\) represents time after the purchase in years, and \(k\) represents the unknown amount it depreciates each year. We know this equation should be linear since the question states that the equipment depreciates at a “constant rate.” The reason why we subtract \(kt\) is because the value decreases each year. We also have 32,400 in the equation since, when \(t=0\), we should have that \(V=32,400\) (since at the time of purchase, the value was $32,400).

Now, we can use more information to determine \(k\) and complete our equation. We have that when \(t=12, V=0\). Substituting this into our equation, we get: \(0=32,400-k(12)\). When we solve for \(k\), we have that \(k=2.700\). So our equation is \(V=32,400-2,700t\).

We must make sure that we actually answer the question, which asks for the value 4 years after it was purchased. If we plug \(t=4\) into our equation, we get \(V=32,400-2,700(4)\). So, \(V=21,600\).

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3. Geometric Application of Linear Inequalities (No Calculator, Multiple Choice)

Correct Answer: A

Again, looking at the answer choices indicates we will be dealing with equalities. Let’s see what information we can extract from the problem. Note that we are dealing with rectangular prisms, so you should have any relevant formulas on hand. The first constraint is that “the sum of the perimeter of the base of the box and the height of the box cannot exceed 130 inches.” This translates to \(P+h\:\leq\:130\), where \(P\) is perimeter and \(h\) is height. This sum must be less than or equal to 130, since 130 is the maximum possible sum allowed.

Next, since the “perimeter of the base is determined using the width and length of the box,” and we know the base is a rectangle since we’re dealing with a rectangular prism, we have that \(P=2l+2w\), where \(l\) is the length and \(w\) is the width. This formula is given to you on the SAT, but it would be helpful to have it memorized.

Substituting this new equation into our first one, we get: \(2l+2w+h\:\leq\:130\). The next constraints are that the height is 60 inches and the length is 2.5 times the width. This means \(h=60\) and \(l=2.5w\). If we plug this into our inequality, we have: \(2(2.5w)+2w+60\:\leq\:130\). When we simplify, \(w\:\leq\:10\). Because a measurement can never be negative, we also have that \(w\:\geq\:0\). So, \(0\:\leq\:w\:\leq\:10\), which corresponds to A.

4. Word Problems Involving Linear Equations (No Calculator, Multiple Choice)

This problem also contains a lot of numeric information, so we should handle it one line at a time. First, we know that Ken earned $8 per hour for 10 hours of work. This means that Ken has already earned $80 dollars this week. Since Ken will now earn $10 for each additional hour of work, his total earnings for the week will be \(80+10h\), where \(h\) is the number of hours worked during the rest of the week.

Since Ken saves 90% of these earnings, we have that his savings are: \(.9(80+10h)\). If we want to save “at least $270,” we must have that \(.9(80+10h)\:\geq\:270\). We can do algebra to solve for \(h\):

\(80+10h\:\geq\:\frac{270}{0.9}\)

\(80+10h\:\geq\:300\)

\(10h\:\geq\:220\)

\(h\:\geq\:22\)

5. Properties of Linear Graphs (No Calculator, Free Response)

Correct Answer: 8

This problem requires knowledge of linear graphs, specifically the formula of a line and how to find a linear slope. 

From any two points \((x_1,y_1)\) and \((x_2,y_2)\), the slope of a line, \(m\), can be determined using \(m=\frac{y_2-y_1}{x_2-x_1}\). So, for our two points, \(m=\frac{0-4}{2-1}=-4\). This part of the problem is easy to mess up on, so make sure to double check your work here.

So far, we have the equation \(y=-4x+b\) for our line, where \(b\) is the coordinate of the y-intercept.

Now, we can pick one of the existing points on the line and plug in the values to find \(b\). Let’s use \((2,0)\). We have: \(0=-4(2)+b\). So, we get that \(b=8\).

6. Solving for Quantities within a Linear Equation (Calculator, Multiple Choice)

This problem can seem complicated, because having multiple variables might tempt students to solve for one at a time. However, the key to solving this problem is treating \(ax+b\) as just one variable.

We can do this by factoring out 9: \(9(ax+b)-6=21\). Then, we can isolate \(ax+b\) on one side:

\(9(ax+b)=21+6=27\)

\(ax+b=27/9=3\)

We can choose A as the answer.

7. Real-World Applications of Linear Equations (Calculator, Multiple Choice)

This problem may seem daunting given the length of the paragraph, but if we carefully read, we can see that the answer is more simple. The problem asks for what the y-intercept means. From our knowledge that the y-intercept occurs when \(x=0\), we know that the y-intercept occurs when the truck is driven 0 miles (since \(x\) represents miles driven). Because the total cost “consists of a flat fee plus a charge per mile driven,” if the number of miles driven is 0, the total cost in this situation is only equal to the flat fee. Thus, the y-intercept represents the flat fee, which corresponds to answer choice A.

8. Analyzing Linear Graphs (Calculator, Multiple Choice)

For this problem, I’d start by doing something familiar: finding the equation of the line. Since the line passes through \((0,0)\) and \((3,6)\) we can start by finding the slope: \(m=\frac{6-0}{3-0}=2\).

So far, we have \(y=2x+b\) for the equation of our line.

Next, we can use this slope and the point \((0,0)\) to construct an equation. Plug in \((0,0)\) into our equation, and we get \(0=2(0)+b\). So, \(b=0\) and \(y=2x\).

So, for a point \((s,t)\) we have that \(t=2s\). This means that whatever \(s\) is equal to, \(t\) will be twice as much. It’s important to note that the question asks for the ratio of \(t\) to \(s\). Because \(t\) is twice as large as \(s\), the ratio should be 2:1.

9. Constructing Multivariable Linear Equations (Calculator, Multiple Choice)

This question is a bit more complicated since it involves multiple variables. Again, the problem defines the variables for us. For a given food, the amount of calories is \(4p+9f+4c\), since we have 4 food calories for every 1 gram of protein, 9 food calories for every 1 gram of fat, and 4 food calories for every 1 gram of carbohydrate, according to the table.

Since we were told the granola bar had 180 food calories, we can set our expression equal to this value: \(4p+9f+4c=180\). Since we want the equation in terms of \(f\), we now need to do some algebra to isolate for \(f\):

\(4p+9f+4c=180\)

\(9f=180-4p-4c\)

\(9f=180-4(p+c)\)

\(f=20-\frac{4}{9}(p+c)\)

10. Creating and Solving a Linear System (Calculator, Free Response)

Correct Answer: 30

This situation may not seem like a linear system, since so little numeric information is given, but upon taking a closer look, we can extract additional information. First, let’s define \(C\) to be the number of correct answers and \(I\) to be the number of incorrect answers. Then, since any given question produces either a correct or incorrect answer, the total number of questions is \(C+I\).

Next, the score is determined by “subtracting the number of incorrect answers from twice the number of correct answers.” So, the score is \(2C-I\).

Since we are given that 40 questions were answered and the score is 50, we now have two equations:

\(2C-I=50\)

We can solve this system by adding the two equations together:

\(C+I+2C-I=40+50\)

We should make sure that the question was indeed asking for the number of correct answers (not the number of incorrect ones), and since it was, we are done.

When approaching Heart of Algebra problems, remember to check to make sure you answered the question. The SAT will often anticipate common student errors and misconceptions and will construct the wrong answer choices based on them. So, it’s crucial that you make sure to answer the questions carefully, double checking your work if you have the time.

The best way to study for the SAT Math section is to complete as many practice problems as you can. If you find yourself struggling on Heart of Algebra problems more than the other sections, try focusing your study solely on Heart of Algebra content.

Check out some of these other posts to help you as you study for the SAT Math section:

• 15 Hardest SAT Math Questions
• 30 SAT Math Formulas You Need to Know
• 5 Common SAT Math Mistakes to Avoid
• 5 Tips to Boost Your Math SAT Score
• Which Formulas Are Given on the SAT Math Section?
• What Does the Math Section of the SAT Test?
• Ultimate Guide to the New SAT Math Test

Related CollegeVine Blog Posts

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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Last week's blog article provided an extensive overview and look into Heart of Algebra, a major section of the SAT Math test ( it constitutes for 33% of the examination's questions! ).

If you have not yet taken a read of that piece yet, go ahead and do so before moving on with today's blog. We will be moving forward to the next section that is covered on the SAT Math, Problem Solving and Data Analysis.

The Problem Solving and Data Analysis section of SAT Math accounts for the second-largest area of the college entrance examination, roughly 29% to be exact.

As an introductory refresher, these sub-categories of SAT Math is used by the College Board in order to help break down your math score into useful subscores on your comprehensive results.

What Is Tested On The SAT Math Section?

Within the SAT Math section, you will find the format is structured by multiple-choice questions and "grid-ins." While the majority of SAT math is multiple-choice, 22% of the math examination portion will constitute grid-ins.

Grid-ins are student-produced, meaning that they require you to first solve the mathematical problem and then enter your answer into the grid provided on the answer sheet.

In total there are 58 questions, with 45 being multiple-choice and 13 as grid-ins.

The SAT Math exam is separated into four main component areas:

Heart of Algebra (19 questions)

Problem Solving and Data Analysis (17 questions)

Passport to Advanced Math (16 questions)

Additional Topics in Math (6 questions)

Need help building a study plan? Look no further, click here to read our previous blog!

Last week, we went over the first domain, Heart of Algebra. Now we will proceed towards the second-largest component of the SAT Math portion.

What Is Problem Solving And Data Analysis?

Students can anticipate working on a total of 17 Problem Solving and Data Analysis questions on the SAT Math exam. You will never be told by the College Board what type of question you are working on from the four major component areas, but you can identify them if you understand what to look for.

Problem Solving and Data Analysis problems revolve around applying your mathematical understanding beside realistic scenarios and analyzing the data rather than working through conceptual situations.

TIP: An important thing to remember for this category of SAT Math is that there will be no Problem Solving and Data Analysis questions on the no-calculator portion. This means that for every Problem Solving and Data Analysis problem you encounter, you will always be allowed to utilize an approved calculator in order to solve the question. You may not always require the aid of a calculator, but it will be permitted if you choose to use it for any Problem Solving and Data Analysis question.

Straight from the College Board , the Problem Solving and Data Analysis section test students' abilities to:

"Create a representation of a problem, consider the units involved, attend to the meaning of quantities, and know and use different properties of operations and objects. Problems in this category will require significant quantitative reasoning about ratios, rates, and proportional relationships and will place a premium on understanding and applying unit rate."

In broad terms, this category is used to illustrate a student's math comprehension and capability to solve problems that they could potentially encounter in real-life.

Struggling with reading? Get some insights on what the ACT 4 core reading passages are with our recent blog article !

You should expect that the majority of these problems will be set in either academic or career contexts and are more than likely to reference the sciences and social science.

What Will Problem Solving And Data Analysis Questions Look Like?

According to the official College Board, you can expect this sub-category to inquire about 10 major types of questions:

Use of ratios, rates, proportional relationships, and scale drawings to solve single- and multistep problems.

Solve single- and multistep problems involving percentages

Solve single- and multistep 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 the appropriateness of data collection methods.

To condense the areas above, we will provide an overview of Problem Solving and Data Analysis three major concepts:

Ratios, units, and percentages

Relationships in Scatterplots, Graphs, Tables, and Equations

Data and Statistics Relationships

We will reference the Official College Board SAT Study Guide, Chapter 17, Problem Solving and Data Analysis .

Ratios, Units, and Percentages

A ratio is a numerical comparison that illustrates some sort of relationship between two or more values. In order to complete ratio calculations, you should increase your comfort level when it comes to multiplying and dividing fractions.

A ratio can be expressed in any of the following formats on SAT Math:

X (when Y is equal to 1)

You may see something like this:

This means that there were 5 adults for every 1 child. Of every 6 people who attended the show, 5 were adults and 1 was a child. Turning this problem into fractions, 5/6 of the 240 people that attended were adults and 1/6 were kids. 1/6 * 240 would equal 40 children attended the show, leaving you with the answer choice A.

You will encounter units in terms of oƒ unit conversion and unit rates. Unit rates express one quantity compared to another ("miles per hour", "dollars per year", etc.) Look for terms like "per" or "each." More often than not, they will require that you convert from one unit to another.

You are given that 1 mile = 1.6 kilometers. Using that information, you solve for the distance in miles. Take 1,060 kilometers and multiply it by (1 mile/1.6 kilometers) and you are left with 664.5 miles, roughly around 663 miles. The correct answer choice is C.

TIP: The College Board usually draws on both the English system of feet and inches as well as the metric system of kilometers and meters. As such, you should be well-versed and familiar with the common units for both. Practice with everyday concepts such as changing your height from inches to centimeters or concerting a car's speed from kilometers to miles.

As for percentages, students should always remember that these are relative to the number 100 and that they are a form of proportions. Given that, these questions involve usually the concepts of percentage increase and percentage decrease.

The sale price of the table is $299. Consider x as the cost from the wholesaler. Your sale price of the table is equal to the cost from the wholesaler plus 15%. Therefore, you will have $299 = 1.15x. Solve for this and the cost from the wholesaler will be $299/1.15 which equals to $260. You are told that the usual price is the cost from the wholesale (which we now know is $260) plus a percentage increase of 75%. The usual price the store charges for the table will be 1.75 * $260, which equals $455. Your answer choice is B.

TIP : In order to do well on Problem Solving and Data Analysis percentage problems, you can brush up on your practice with cross-multiplication when solving for missing values in a formula. Also, remember to work through the problems step-by-step. In the previous example, we had to solve for the cost from the wholesaler first before finally solving for the usual store price with its percentage increase.

Here is a handy list of all the important formulas given on the SAT!

Scatterplots, Graphs, Tables, and Equations

You can anticipate that questions in the Problem Solving and Data Analysis will concentrate on linear, quadratic, and exponential relationships. These are usually depicted by scatterplots, graphs, tables, or tables.

TIP: You can differentiate between linear and exponential as such,

A model is linear if the difference in quantity is constant.

A model is exponential if the ratio in the quantity is constant

The first thing you should think of when you see either a scatterplot, graph, table, or chart is how you can analyze and draw a relational conclusion between its variables.

Be knowledgable about the foundational equations and its components such as for lines ( y = mx + b ), quadratic equations ( ax² + bx + c = 0 ).

TIP : If you are unsure whether it is linear or exponential, try to follow this type of thinking.

If a quantity is increasing linearly over time, then the difference in the quantity between each time period should be the same.

If a quantity is increasing exponentially over time, then the ratio in the quantity between each time period should be the same.

Using this logic for the problem above, you see that after each hour the number of bacteria in the culture is 4 times as great as it was the prior hour. We see a constant ratio between each time period that increases the number of bacteria over time so answer choice D is the correct answer.

For the Problem Solving and Data Analysis category of SAT Math, you will most likely run into data that involves the measures of the center (the mean, average, median).

If the problem you are given does not provide enough information to calculate the for the measures of center, you may still be asked to pull a conclusion about them.

For a beginner refresher:

Mean - the average

Median - the number exactly in the middle when values are placed in order

Mode - the value that appears the most often

Range - the difference between the highest value and the lowest

If you have a strong grasp on these concepts, then it will be much easier for you to make sense and move forward with more challenging statistical concepts on the SAT Math.

The more complicated ideas revolve around terms like population parameter, standard deviation, or how far apart points in a data set are from the mean.

Let's take a look at the concept of standard deviation. It is the measure of spread and generally, the College Board will ask that you compare two sets of data to define which has a more spread out distribution (a greater standard deviation). In these particular cases, do not worry about calculating the exact value. You will need to know the foundational theory and comprehension in order to solve this type of problem.

You see that Class A has a mean score between 3 and 4. The bulk of the scores are 3 and 4, with a very few landing in 0, 1, 2, and 5. As for Class B, the mean score is 2.5 with all scores being equally distributed across all possible scores. Not many of these quiz scores are close to the mean score. Since Class A has quiz scores that are grouped closely together around the mean, the spread (or standard deviation) of the scores in Class A is smaller. The correct answer choice is A.

Covering approximately 29% of the concepts found on the SAT Math test, Problem Solving and Data Analysis assesses a student's math ability alongside real-world scenarios. You will be expected to pull information from the concepts and apply reasoning that is more practical for your future professional careers.

As for any portion of SAT Math, we recommend our students to take each step at a time to ensure that they do not make minor mistakes. One little, but crucial, error when working through a mathematical problem can result in the wrong answer choice.

Be sure to brush up on foundational concepts and equations that you should be aware of for this domain of SAT Math and start practice problems or exercises that can help you prepare.

7EDU Impact Academy offers many college preparation services including one-on-ones, counseling, and specific courses.

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SAT Math Problems with Solutions

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

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

Here are the topics we’ll cover in this article:

An overview of the sat math section, the ttp and sat breakdown of sat math topics, sat math subtopics, using your calculator on the sat, the two major question types on the sat, multiple-choice question 1, multiple-choice question 2, multiple-choice question 3, multiple-choice question 4, multiple-choice question 5, multiple-choice question 6, multiple-choice question 7.

• Grid-In Question 1

Grid-In Question 2

Grid-in question 3, grid-in question 4, grid-in question 5, grid-in question 6, grid-in question 7.

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

Preparing for SAT Math

Topical learning with ttp, what math problems are on the sat, is the math on the sat hard, what is the hardest math subject in sat tests, what’s next.

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

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

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

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

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

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

SAT Math Topics

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

Many SAT math topics should be familiar to you.

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

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

• Heart of Algebra
• Problem Solving and Data Analysis
• Passport to Advanced Math
• Additional Topics in Math

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

Heart of Algebra:

• Solving Linear Equations
• Inequalities and Absolute Value
• Coordinate Geometry
• Linear Functions
• Systems of Linear Equations

Problem Solving and Data Analysis:

• General Word Problems
• Unit Conversions
• Percentages

Passport to Advanced Math :

• Quadratic Equations
• Graph Interpretation

Additional Topics in Math :

• Trigonometry
• Complex Numbers

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

There are 22 major SAT math topics.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiple-Choice Questions

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

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

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

SAT Major Topic: Passport to Advanced Math

TTP Major Topic: Quadratic Equations

TTP Subtopic: The Discriminant

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

• More than 2

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

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

x 2 – 4x + 2 = 0

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

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

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

Problem Solving Question 2

SAT Major Topic: Heart of Algebra

TTP Major Topic: Arithmetic Word Problems

TTP Subtopic: Inequality Word Problems

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

50 + 3y ≤ 90

• 50 – 3y ≤ 90

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

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

Next, we can fill in the given numbers accordingly:

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

SAT Major Topic: Problem Solving and Data Analysis

TTP Major Topic: Statistics

TTP Subtopic: Solving for an Unknown in an Average Equation

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

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

average = sum/number

⇒ 32 = (30 + 32 + x)/3

⇒ 96 = 62 + x

TTP Major Topic: Functions

TTP Subtopic: Domain of a Function

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

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

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

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

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

TTP Major Topic: General Word Problems

TTP Subtopic: Age problems

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

First, let’s define two variables:

H = Harold’s age today

P = Paloma’s age today

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

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

⇒ H = P + 30

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

⇒ H + 10 = 3(P + 10)

⇒ H + 10 = 3P + 30

⇒ H = 3P + 20

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

⇒ P + 30 = 3P + 20

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

SAT Major Topic: Additional Topics

TTP Major Topic: Complex Numbers

TTP Subtopic: Combining Complex Numbers

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

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

⇒ 3 + 5 = 8

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

⇒ 2i + (-7i) = -5i

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

TTP Major Topic: Trigonometry

TTP Subtopic: Radian Measure

What is the equivalent of 72° in radians?

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

𝞹 radians / 180 degrees = x / 72 degrees

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

⇒ 72𝞹 = 180x

⇒ 72𝞹 / 180 = x

⇒ 2𝞹 / 5 = x

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

Grid-In Questions

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

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

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

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

An example of a blank grid is shown below:

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

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

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

TTP Major Topic: Exponents

TTP Subtopic: Multiplication of Exponents with the Same Base

If a and b are positive integers and 3 12 = 3 a 3 b , what is one possible value of a x b?

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

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

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

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

Let’s try another question.

TTP Major Topic: Arithmetic

TTP Subtopic: Fractions

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

The problem gives us these two pieces of information:

Number of recovered marbles = 24

Total number of marbles = 120

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

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

• Reduce the fraction:

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

⇒ 24/120 = 1/5

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

⇒ 24/120 = 0.2

Answer: 1/5 or 0.2

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

TTP Major Topic: Linear Equations

TTP Subtopic: Solving For Expressions

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

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

⇒ 15x + 24y = 5

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

⇒ 5x + 8y = 5/3

Answer: 5/3

TTP Subtopic: Compound Inequalities

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

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

⇒ 5 < n < 9

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

Answer: 6, 7, 8

TTP Major Topic: Rate Questions

TTP Subtopic: Rate Questions Not Involving Distance

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

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

⇒ rate x time = amount

⇒ 240 x 6 = 1,440 gallons

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

TTP Subtopic: Symbolism Questions

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

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

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

⇒ 5 2 + 2(3) = 25 + 6 = 31

SAT Major Topic: Heart of Algebra:

TTP Major Topic: Coordinate Geometry

TTP Subtopic: The Y-Intercept

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

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

⇒ 5(0) + (2/3)y = 10

⇒ 0 + 2y/3 = 10

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

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

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

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

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

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

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

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

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

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

Learn each SAT math topic one at a time.

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

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

For example, mission three is the Quadratic Equations chapter. So, the first task in that mission is to learn all about quadratic equations. Those topics include FOILing quadratics, factoring quadratics, quadratic identities, the quadratic formula, completing the square, etc.

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

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

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

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

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

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

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

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

Frequently Asked Questions (FAQ)

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

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

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

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

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

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

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About The Author

Jeffrey Miller is the head SAT instructor for Target Test Prep. Jeff has more than 14 years of experience in the business of helping students with low SAT scores hurdle the seemingly impossible and achieve the scores they need to get into the top 20 colleges in the world. Jeff has cultivated many successful college graduates through his SAT instruction, and will be a pivotal resource for many more to follow.

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PWN Test Prep

Problem solving and data analysis.

Your SAT will have only 5-7 Problem Solving and Data Analysis questions. This is way fewer questions than Algebra or Advanced Math. In this unit you’ll practice a bunch of different kinds of things—some stuff like percents and ratios will probably be pretty familiar, and other things like interpreting scatterplots might be a bit more…

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Digital SAT Math

Course: digital sat math   >   unit 7.

• Quiz 1 Medium: Problem solving and data analysis