## 6 SAT Math Problem Solving and Data Analysis Concepts to Master

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

SAT Math Concepts to Master

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

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

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

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

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

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

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

2. Percentages

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

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

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

3. Unit conversion and unit rate

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

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

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

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## Unit 6: Lesson 2

## The SAT Math Test: Problem Solving and Data Analysis

- The SAT Math Test: Passport to Advanced Math
- The SAT Math Test: Additional Topics in Math
- Controlling careless errors on the SAT Math Test
- SAT Math Test Strategies Share Space
- SAT Math Test inside scoop: Meet the Maker

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

- Converting units (for example changing km/hr to meters/second)
- Choosing appropriate graphical representations for data sets
- Interpreting the slope and intercepts of a line
- Computing and interpreting probability
- Evaluating statistical claims or the results of a study
- Using percentages in a variety of contexts, including discounts, interest rates, taxes, and tips
- (Choice A) $ 359 \$359 $ 3 5 9 dollar sign, 359 A $ 359 \$359 $ 3 5 9 dollar sign, 359
- (Choice B) $ 455 \$455 $ 4 5 5 dollar sign, 455 B $ 455 \$455 $ 4 5 5 dollar sign, 455
- (Choice C) $ 479 \$479 $ 4 7 9 dollar sign, 479 C $ 479 \$479 $ 4 7 9 dollar sign, 479
- (Choice D) $ 524 \$524 $ 5 2 4 dollar sign, 524 D $ 524 \$524 $ 5 2 4 dollar sign, 524
- Comparing distributions
- (Choice A) The standard deviation of the scores in Class A \text{A} A start text, A, end text is smaller. A The standard deviation of the scores in Class A \text{A} A start text, A, end text is smaller.
- (Choice B) The standard deviation of the scores in Class B \text{B} B start text, B, end text is smaller. B The standard deviation of the scores in Class B \text{B} B start text, B, end text is smaller.
- (Choice C) The standard deviation of the scores in Class A \text{A} A start text, A, end text and Class B \text{B} B start text, B, end text is the same. C The standard deviation of the scores in Class A \text{A} A start text, A, end text and Class B \text{B} B start text, B, end text is the same.
- (Choice D) The relationship cannot be determined from the information given. D The relationship cannot be determined from the information given.

## Attributions

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

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Show me what areas I need to improve

## 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

Here is the breakdown of each category:

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.

## Strategies to Solve Problem Solving and Data Analysis Problems

## 10 Difficult Problem Solving and Data Analysis Questions

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

## 2. Percent Increase

## 3. Analyzing Graphical Data

## 4. Inference

## 5. Proportions

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## 6. Scale Factors

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

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

## 7. Line of Best Fit/Scatterplots

## 8. Geometric Applications of Proportions

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

## 9. Unit Conversions

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

## 10. Probability

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

## Problem Solving and Data Analysis: SAT Math Tips & Practice

## Kickstart Your SAT Prep with Test Geek’s Free SAT Study Guide.

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

## Problem Solving and Data Analysis: What’s Included?

- 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

## Data Analysis: Statistics and Data Collection

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 .

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.

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:

- 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

- 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.

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

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

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

## Models: The Scary Side of Problem Solving and Data Analysis

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

## Problem Solving & Data Analysis: Final Thoughts

## SAT Math Complex Numbers & Imaginary Numbers

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## Choose Your Test

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

SAT Math is divided intro three domains:

## Basic Information

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

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

## General Concepts

You’ll be working with a lot of numbers.

## Real-World Applications

## Data and Statistics

## Specific Skills

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

## Statistical Analysis

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

## Mathematical Models

- You might take a loan at 9%, but if it compounds monthly, you’re really taking a loan at $(1+.09/12)^12 – 1 = 9.38%$ at the end of the year.
- On the other hand, you might make a deposit that accrues interest at a rate of 5%, but it compounds quarterly, so you’re really getting $(1+.05/4)^4 – 1 = 5.095%$ at the end of the year.

Math and money are very closely linked.

## Ratios, Proportions, Units, and Percentages

Take a look at this pair of problems:

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

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

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

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

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

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

Let’s take a look at this problem:

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

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

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

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

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## SAT Math: Problem Solving and Data Analysis

## Table of Contents

## Understanding Ratios

## Understanding Percentages

## Practice Questions

## Medium Difficulty Question

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

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

## Difficult Questions

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

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.

## Subjects and Treatments Practice Question

Which of the following is an appropriate conclusion?

## Subjects and Treatments: A Summary

- 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.

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 ?

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

## What is a “weighted average”?

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

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

## Finding Average Rates

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

## Average Rate Practice Question

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

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

## Weighted Averages That You Won’t See on Your SAT

- 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.

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## SAT Math : Data Analysis

## Example Question #2 : Venn Diagrams

## Example Question #3 : Venn Diagrams

## Example Question #4 : Venn Diagrams

## Example Question #5 : Venn Diagrams

First, it is given that S=450.

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

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

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

## Example Question #6 : Venn Diagrams

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

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

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

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

Next, we need to find the factors of 90.

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

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

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

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

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

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

Lastly, we must find P U (Q ∩ R).

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

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

## Example Question #215 : Data Analysis

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

## Example Question #8 : How To Interpret Venn Diagrams

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