Graduate Management Admission Test (GMAT)

Preparing to take the Graduate Management Admission Test?


You’ve found the right page. We will answer every question you have and tell you exactly what you need to study to pass the Graduate Management Admission Test.

Quick Facts

Get the “need to know” information at a quick glance.


The Graduate Management Admission Test (GMAT) exam was designed by business schools to test candidates’ abilities to successfully earn an MBA. Many other postgraduate programs outside of the business industry evaluate GMAT scores to determine whether students are good candidates for admission, as well.


Check out the chart below for details about the time you will have to take each section of the test, the number of questions you’ll answer, and the question types you’ll see.


The current cost to take the GMAT is $250.


Check out the screenshot below for information about GMAT scores, including the score range:

Approximately two-thirds of GMAT test-takers score between 400 and 600 on the exam. The mean score is 561.

You will have the opportunity to view your score before it is sent to your graduate program(s) of choice. You may cancel your score on the test day if you choose not to send it to the program(s) to which you are applying. Test-takers who cancel their scores on their exam day may reinstate their scores within 60 days of taking the test.

Study Time

It is recommended that test-takers register for the GMAT two to three months before taking the test in order to sufficiently prepare. The amount of time you will need to spend preparing for the GMAT depends upon your existing content knowledge.

Although it’s best to study a little bit at a time instead of cramming for the test, you should also take a complete, timed practice test at least once before the actual exam. Doing this will give you a better idea of how to pace yourself.

What test takers wish they would’ve known:

  • Place equal importance on verbal questions and other questions. Many test-takers do not realize that being able to communicate properly is crucial to success in both the business world and in other fields. Remember, the more you can improve your responses to the verbal questions, the more you can improve your overall score.
  • Most of the math on the GMAT addresses concepts which you learned in high school. Therefore, it’s not a bad idea to take the GMAT during your first couple of years as an undergraduate.
  • It’s important to do your best when answering every question. The GMAT is a computer adaptive test. That means that each question you answer correctly or incorrectly will determine subsequent questions you receive.
  • If you are unsure of the answer to a question, you should try to eliminate as many answer choices as possible before selecting one. This increases your odds of answering the question correctly.
  • It is best to relax and focus on the main ideas of the passages which you are presented with. Remember, you do not need to memorize the details of the content.
  • After reading the question, do not immediately select an answer choice. Instead, “pre-think” the answer before reading the answer choices. This will actually work to save you time.
  • If you find yourself staring at a question for 30 seconds and it is still not making sense to you, just choose the best answer and move on. While you want to consider questions carefully, if an answer is not coming to you, it is better to keep moving than to waste time.

Information and screenshots obtained from the GMAC GMAT website.

Analytical Writing Assessment

The Analytical Writing Assessment includes one question. You will be given 30 minutes, about one-sixth of your test-taking time, to answer the question.

 This part has two main objectives:

  • Argument Analysis
  • Writing Style and Mechanics

So, let’s start with Argument Analysis.

Sub-area I: Argument Analysis

This objective tests your ability to spot conclusions in an argumentative essay and to identify the evidence and assumptions the author relies upon to support those conclusions.

Take a look at some concepts that may appear on the test.

Argument Identification

Argumentative passages on the test have three components:

  1. Evidence: Premises stated in the passage
  2. Assumptions: Unstated premises used for justification
  3. Conclusion: The claim drawn from evidence and assumptions

Let’s take a look at an example of an argumentative passage.

“According to national survey data, more people choose to go to frozen yogurt shops instead of ice cream parlors than ever before. In order to boost profits at Here’s The Scoop, we should begin selling frozen yogurt instead of ice cream. Frosty’s, our top competitor, has begun selling frozen yogurt instead of ice cream and their business is thriving. It should also be said that Polar Bear Frozen Yogurt is one of the most popular small businesses in town. By switching from ice cream to frozen yogurt, our business will be sure to attract more patrons.”

Analysis of Claims

Now, let’s determine the claims that the author is making:

         a) “In order to boost profits… we should begin selling frozen yogurt instead of ice cream.”

         b) “By switching from ice cream to frozen yogurt, our business will be sure to attract more patrons.”

Claim “b” is the intermediate conclusion. Claim “a” is the final conclusion because it rests in part upon claim “b.” Not every argumentative passage has an intermediate conclusion and an intermediate conclusion may be presented after the final conclusion.

Use your scratch paper to jot down the final conclusion, as well as the intermediate conclusion (if it exists).

Supporting Reasons and Evidence

Remember that claims or conclusions are supported by evidence. There are three examples of evidence in the passage:

    1. “According to national survey data, more people choose to go to frozen yogurt shops instead of ice cream parlors than ever before.”
    2. “…our top competitor has begun selling frozen yogurt instead of ice cream and their business is thriving.”
    3. “…Polar Bear Frozen Yogurt is one of the most popular small businesses in town.”

Identification of Assumptions

Clearly, the writer is making several assumptions. We’ll look at three common types of assumptions and how they appear in the passage.

  1. False Cause 

This is the assumption that an incident which occurs after another incident was caused by the prior incident.

We don’t know that Frosty’s decision to sell frozen yogurt has had a desirable impact upon business. Maybe they just finally got around to fixing the bathrooms.

  1.   False Analogy

This is the assumption that because two ideas are similar in one way, they are similar in a different way, as well.

Polar Bear Frozen Yogurt sells frozen yogurt and they are a successful business.

Here’s the Scoop can sell frozen yogurt and yet not be a successful business.

  1.   Misuse of Statistics

Remember that one sample of a population does not fully represent the whole population.

Even if frozen yogurt is becoming popular across the United States, it might not be becoming popular in the small town in question.

When identifying assumptions, note them briefly on your scratch paper.


While determining the conclusion(s), evidence, and assumptions in the article, you will have taken notes to refer to as you write. Take a moment to also write down any other problems which you noticed while reading.

For example, did the author of the frozen yogurt argument consider that the town might not be able to support three different frozen yogurt shops? Did she consider the costs associated with changing the merchandise and advertising materials?

Next, number the points you are going to address in a logical sequence. It’s time to start your critique!


Your critique should start with a concise introductory paragraph. Do not waste time giving details about what is stated in the essay. Instead, your introduction should state the conclusion(s), briefly refer to the evidence you found, and allude to the assumptions you identified.

Next, compose the body of your critique by addressing the points that you numbered. About three sentences on each of the points is sufficient. Each point in your critique should have its own paragraph, unless it is closely tied to another point. Be sure to provide reasoning and support for each of the points that you make. Each of your body paragraphs should immediately state the point in order to keep the reader oriented.

Finally, it is important to include a concluding paragraph – otherwise, a reader might think that you ran out of time. Your conclusion should not introduce any new topics. Instead, the conclusion should reiterate why the argument was weak. If you have time, you might also suggest how the argument could have been improved.

Sub-area II: Writing Style and Mechanics

This objective tests your ability to write with task-appropriate tone and diction while avoiding wordiness, redundancy, and idioms. While the main purpose of the Argument Analysis is to identify conclusions and flaws, you will not be able to show what you know without writing properly.

Take a look at some concepts that may appear on the test.


As you may remember from your English classes, tone is the speaker’s attitude towards a subject. The tone of your essay should be professional and persuasive, not emotional and inflammatory. Although you will be criticizing the reasoning in a passage, do not use harsh language.

It is also important that your statements are very direct and do not have double meanings. Ironic, sarcastic, or humorous tones are not appropriate for the GMAT. If you use these tones, the reader might miss what you’re saying and even think that you mean the opposite of what you write.


Wordiness refers to using too many words to get your point across. Remember, you want your writing to be concise so that the reader can get right to the point without confusion. Writing long, wordy sentences is a waste of your precious time, as well!

Here are a few examples of wordy sentences:

Too wordy: It is an establishment that is new and it is well-liked.

More appropriate: It is a new and well-liked establishment.

Too wordy: James Evans, who was my former co-worker, won the lawsuit.

More appropriate: James Evans, my former co-worker, won the lawsuit.

Too wordy: Dr. Banks is reading the newspaper while Mr. Gonzalez is performing research on the case.

More appropriate: Dr. Banks reads the newspaper while Mr. Gonzalez researches the case.


Redundancy is very similar to wordiness. When a writer is redundant, she is using words or phrases that are synonymous with each other. There is no need to include synonymous words and phrases. When you do so, your writing is less concise. Also, redundant writing will use up your precious testing time.

Let’s look at a few examples of redundancy.

Redundant: Each and every tourist agreed that they would like to visit again.

Rationale: “Each” and “every” are synonyms.

Redundant: The two teams will join together for a discussion.

Rationale: “Together” can be omitted. Things that join must do so together.

Redundant: Proceeding forward, we will plan for an event like this in advance.

Rationale: “Proceeding” and “forward” are synonymous. All plans are made in advance.

Redundant: The reason that stock prices have changed is caused by rising unemployment rates.

Rationale: A reason (in this context) is the same as a cause.


Diction refers to a writer’s word choice. During the assessment, try to use precise language in order to best convey your ideas. Do not choose lengthy or difficult words only to impress the reader. Instead, use your vocabulary effectively and meaningfully. If you are at all unsure of the meaning of a word, do not include it in your essay.

Even great writers can struggle with some commonly confused words. Let’s take a look at some of those words now:


An idiom is a phrase or statement which has an intended meaning different from its figurative meaning. Some common English idioms are “raining cats and dogs,” “put your foot in your mouth,” “beat around the bush,” and “kill two birds with one stone.”

While we use idioms in our daily lives, they should not be included in formal, standard English writing. When someone uses idioms in formal writing, the reader is given the impression that the author’s vocabulary is lacking.

You should also never assume that a reader knows the same idioms that you do. If the reader is left wondering why someone “let the cat out of the bag” and why the cat was confined to start with, you’ve missed the mark. Remember, just be clear and say exactly what you mean!

Making Connections

In order to receive a great score on your essay, your ideas must be clearly connected in order to provide the reader with smooth transitions between points. Sometimes, you will use transitional words and phrases in order to show that one idea is building upon another. Other times, you’ll use transitional phrases to make comparisons or draw conclusions.

Here are a few examples of transitional words and phrases:


Be sure to reserve five minutes after you finish writing to check over your essay. Decide if any ideas need to be introduced in a different order and watch for glaring technical and spelling mistakes. Delete any idioms that you find in your work and replace any terminology which you are uncertain of.

Do not waste your editing time looking for tiny errors such as extra spaces between words, and certainly do not try to rework your entire essay!

And that’s some basic info about the Analytical Writing Assessment.

Section II: Integrated Reasoning Assessment

The Integrated Reasoning Assessment includes 12 questions. You will be given 30 minutes, about one-sixth of your test-taking time, to answer the questions on this section.

This part has three main objectives:

  • Graphics Interpretation and Table Analysis
  • Multi-Source Reasoning
  • Two-Part Analysis


So, let’s start with Graphics Interpretation.

Sub-area I: Graphics Interpretation and Table Analysis

This objective tests your ability to interpret graphs and tables and to use those visuals in order to answer specific questions, such as questions about measures of center and correlational relationships.

On this section of the test, it is very important to analyze not only the graphs and tables but also to carefully consider what the accompanying text is asking you.

Take a look at some concepts that may appear on the test.

Measures of Center

Sometimes, you will be presented with a graph and asked questions about measures of center. Measures of the center include the mean, median, and mode of a set of values. Let’s use an example set of values and take a look at the mean, median, and mode:

Example values: 4, 35, 37, 38, 39, 39, 45

Mean – The mean is the average. You can determine the mean by adding all of the values and dividing by the number of values. In this case, the mean is 38.89.

Keep in mind that outliers, values which are very different from the other values, can greatly impact the mean.

Median – When values are listed from least to greatest, the median is the number in the center of the list. The median for this list is 38. (When there are two values in the middle, the median is the average of those two values.)

Mode – The mode is the value that occurs most frequently. Since 39 is the only value which appears more than once, 39 is the mode. It is important to note that a data set may have more than one mode.


Scatterplots can be used to determine the relationship between two variables. A scatterplot is a graph with an x-axis and a y-axis. Values appear on the scatterplot in the form of dots.

A scatterplot may show a positive correlation between variables, a negative correlation between variables, or no correlation between variables.

Let’s look at some examples of each.

Positive correlation
As the y-variable tends to increase, the x-variable tends to increase.
Note: a correlational relationship does not necessarily indicate a causational relationship.

Negative correlation
As the y-variable tends to increase, the x-variable tends to decrease.

Note: a correlational relationship does not necessarily indicate a causational relationship.

No correlation
The x-variable and the y-variable do not appear to have a relationship.

Stacked Bar Graphs

A stacked bar graph uses bars to show comparisons between categories of data. Unlike a traditional bar graph, a stacked bar graph gives the viewer the ability to break down parts of a whole. Every bar represents the whole and the segments of the bars represent categories of that whole.

Here is an example of a stacked bar graph:

If asked to describe the popularity of products between 2016 and 2018, we can say that the sale of Product A has decreased, the sale of Product C has increased, and the sale of Product B has remained roughly the same.


Some of the questions on the exam will ask you to refer to tables. You will have the ability to sort the tables, but you should only sort tables in a way which makes the most sense in order to determine an answer. Let’s look at a simple example of a table:

Sometimes, it is not necessary to sort tables at all. For example, if you were asked which breed of fish earned the least amount of money per sale in proportion to its wholesale cost (Gizmo 4 in this case), sorting the table would not help you.

In that case, you are considering the ratio of the wholesale cost to the ratio of the retail price. Since it’s the ratio and not the monetary value that counts, it’s not a really good use of time to sort by any of the given categories.

If, however, you were asked which type of gizmo attracted the greatest amount of customer attention in August, it would make sense to sort by “Number of Fish Sold.”

As you work with tables, read each question and decide how to sort the table, or whether to leave it unsorted. Then, find the best answer to the question.

Also, keep in mind that tables may include more than one unit. For example, a table might contain a column of pounds, a column of millions of dollars, and a column of tons. Be sure to pay attention to the units used!

Sub-area II: Multi-Source Reasoning

This objective tests your ability to work with multiple sources of data to reach conclusions. The questions on this section are accompanied by two to three tabbed pages with a different source of data on each. Prompts may include data in some combination of text, charts, or tables.

The questions on this section may be multiple-choice, true/false, or inferable/not inferable. On this section of the test, it is important that you focus on the data given to you that is relevant to the question. This means that you should carefully read the question first, then locate the information which will help you to answer the question and ignore any unrelated information.

Sometimes, you will have the opportunity to work backward on multiple-choice questions, which can be a time-saving shortcut. However, even though you will have access to a calculator, do not try to set up complicated equations. Often, estimation is a powerful enough strategy to help you determine the correct answer.

Remember that the multi-source reasoning section is included for you to have the opportunity to show that you can act decisively and reasonably when given incomplete or irrelevant information.

Let’s take a look at an example question which is similar to the questions that will appear on the test.

For each statement, choose Inferable if the statement may be reasonably inferred from the information provided. Otherwise, choose Not Inferable.

Now, let’s consider each of the three statements individually.

Statement 1: Not Inferable

Analyst A does not mention profits in Europe. He only predicts a boost in revenues. Therefore, we should not infer that he expects Best Gizmo to make a profit in Europe in the coming year.

Statement 2: Not Inferable

Analyst B mentions Max Gizmo as an up-and-coming entity that should give Best Gizmo increased competition in the future. However, he makes no mention of Max Gizmo’s stock.

Sub-area III: Two-Part Analysis

This objective may test your quantitative, critical thinking, or reading skills. Often, you will be asked questions that require knowledge of percentages or probability. On this section of the test, you should carefully read the question to determine what tasks you will need to complete. This section is called a two-part analysis because the solution to each question requires two components.

Answer choices will appear in the format of a table. Each component will be shown as a column in the table. You will select the correct answer for each component by marking the rows which contain the correct information.

The columns which contain your answer choices may be independent or dependent upon one another. Decide whether the information in each of the columns is independent or dependent before answering questions on this section of the test.

Let’s take a look at an example question that is similar to the questions which will appear on the test.

You are taking a business course and your grade is determined by the following formula:

Weighted Grade = 0.6T + 0.4A

T is your final test score, and A is your weekly assignment average.

Select the scores which will give you a weighted grade between 80% and 85%.

In this case, 60% of your grade would be based on a value in the T column and 40% of your grade would be based on a value listed in the A column. Let’s start by eliminating some extremes.

If you were to score 48% on your final test, your weighted grade would not be above 80%, even if your weekly assignment average was 100%. This means that you can eliminate all possible pairings in which T is equal to 48%.

Let’s move up to T = 65%.

So, let’s use T = 65% and A = 100% to find the weighted average:

0.6 x 65% + 40%

= 39% + 40% = 79%

Since this is still too low, let’s try out a higher value for both T and A. When we plug in 93% for T and 65% in for A, we will reach 81.8%.

In the T column, we should mark 93% and in the A column, we should mark 65%. In order to answer this question correctly, it is important that both of the two columns are marked with the correct answer.

Section III: Quantitative Reasoning Assessment

The Quantitative Reasoning Assessment includes 31 questions. You will be given 62 minutes, about one-third of your total testing time, to answer the questions on this section.

This part has five main objectives:

  • Problem Solving
  • Data Sufficiency and Analysis
  • Number Forms, Relationships, and Sets
  • Number Theory and Algebra
  • Geometry


So, let’s start with Problem Solving.

Sub-area I: Problem Solving

This objective tests your knowledge of the fundamental rules of arithmetic and algebra. In order to perform well in this section of the test, you should able to work quickly and decisively in order to answer questions as efficiently as possible. However, you should always check your answer by plugging it into the original problem whenever possible!

Take a look at some concepts that may appear on the test.

Fractions and Decimals

Some questions on the test will challenge your ability to work with fractions.

Remember that when multiplying fractions, you should multiply the numerators by each other and the denominators by each other. In order to divide two fractions, flip one of the fractions and multiply the new numerators by each other and the new denominators by each other. When adding or subtracting fractions, don’t forget to start by making the denominators equivalent.

Remember that you can always turn a fraction into a decimal or mixed number by making the denominator 100. For example, 1/5 = 20/100 or .20.

Let’s take a look at an example problem.

Let’s take a closer look at what the question is really asking: if you can buy an eraser for a dollar, how many erasers can you buy for a cent?

Now, the answer choices are presented as fractions, but we need to remember that 1 dollar is 100 cents. In other words, 1 cent is equal to .01 dollars or 1c is 1/100 dollars.

Therefore, we know we are looking for a fraction with a number larger than 100 in the denominator. B can only be the correct answer choice.


You already know that a percent is just a converted fraction. For example, .30 = 30%. Some of the questions on the test will require you to use that knowledge to find out what percent of one quantity is to another.

You may also need to find out the percent change, which could be a percent increase or a percent decrease. Here’s a sample problem which requires you to calculate the percent decrease:

If Joe drinks 10% of the coffee from a 16-ounce cup before breakfast and 20% of the remaining amount with breakfast, approximately how many ounces of coffee are left after breakfast?

A. 5.3 ounces
B. 6.2 ounces
C. 11.2 ounces
D. 11.5 ounces
E. 13 ounces

In order to solve this problem, we will need to perform two separate steps. It would be a mistake to add 10% and 20% and calculate 30% of 16. Remember, there is less coffee in the cup the second time that Joe drinks from it.

First, we need to find 10% of 16 and subtract that amount from 16 to get our first solution. Then, we need to find 20% of our first solution and subtract that amount from the first solution in order to reach our final solution:

16 × 0.1 = 1.6
16 – 1.6 = 14.4 (first solution)
14.4 × 0.2 = 2.88
14.4 – 2.88 = 11.52 (final solution)

Therefore, the correct answer is B. Sometimes, as in this example, it is necessary to round off to a value. Choice B is the closest in value to our final solution.

Order of Operations

If you have not taken a math course recently, it may be helpful for you to remember “PEMDAS” while working on problem-solving questions.

This acronym stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. The mnemonic phrase, “Please Excuse My Dear Aunt Sally” can help you to remember PEMDAS.

Let’s practice with a sample equation and solve it step by step:

8 + (2 x 5) x 3⁴ ÷ 9

  1. Parentheses: 8 + (10) x 3⁴ ÷ 9
  2. Exponents: 8 + (10) x 81 ÷ 9
  3. Multiplication/Division: 8 + 810 ÷ 9
           = 8 + 90
  4. Addition/Subtraction: 98

While working with the order of operations, solve problems left to right. Notice that in the multiplication/division step, we applied multiplication (left) before the division (right).

If you are a little out of practice with order of operations, it’s very important to hone your skills – you will need to use the order of operations on other subsections of the test, as well!

Defined Operations

Defined operations are very different from the order of operations. In a defined operations question, the test makers will tell you how to plug in numbers in order to solve an equation. You must follow their instructions to get the correct answer.

Defined operations questions can look a little funny. Here’s an example of one:

In defining the figure as “ac – bd,” the test makers are giving you instructions for how to apply numbers and variables whenever you see them in the diamond arrangement.

First, we should determine the value of x. Based on the last figure, a = 5, b = 4, c = 2, and d = 1. Let’s plug those numbers into the equation we have been given:

x = (5 × 2) – (4 × 1)
x = 10 – 4
x = 6

Now, we must solve the same equation using x = 6. Based on our answer and the second figure, which we must solve to find the final answer, a = 6, b = 10, c = 2, and d = 1.

(6 × 2) – (10 × 1)
= 12 – 10
= 2

Therefore, our final answer is 2. Defined operations questions may look tricky, but you are being given the instructions to solve them. All you need to do is to make sure that you pay attention to what the instructions are.

Sub-area II: Data Sufficiency and Analysis

This area tests your ability to determine whether or not an answer can be found based upon given information. Data sufficiency and analysis questions have prompts which contain two pieces of information.

The answer options will appear in the following format:

A. (1) ALONE is sufficient to answer the question. (2) ALONE is NOT sufficient to answer the question.
B. (2) ALONE is sufficient to answer the question. (1) ALONE is NOT sufficient to answer the question.
C. BOTH (1) and (2) are needed to answer the question.
D. BOTH (1) and (2) are sufficient to answer the question independently.
E. The question cannot be answered based on the information given.

Take a look at some concepts that may appear on the test.

Three-Step Elimination

In order to answer data sufficiency and analysis questions, it is best to have a plan to tackle them. Use the following plan to eliminate answer choices:

Step 1: Size up the prompt.

Step 2: Consider information (1) thoroughly.

Step 3: Consider information (2) thoroughly.

Step 1: Size up the prompt.

Determine what is being asked. Gather a general idea of the information that (1) and (2) are giving.

If (1) and (2) are essentially giving the same information, you can eliminate choices A – C. If the information is basically the same, (1) and (2) do not need to work together; one would not add any more information to the other. Furthermore, if one of them is not sufficient, the other will not be either.

If the information is the same, choose either (1) or (2) and decide whether it is sufficient or insufficient to answer the question. If it is sufficient, choose D. If not, choose E.

Step 2: Consider information (1) thoroughly.

If (1) and (2) appear to be giving different information, you will need to move on to the second step. Consider the information in (1) a little more deeply.

If (1) is sufficient to answer the question, either A or D is the correct answer.

If (1) is NOT sufficient to answer the question, you know that either:

  • You will need (2), with or without (1), to answer the question.
  • The question cannot be answered.

At this point, based on how much time you have spent on the question and the test in general, make a judgment call on whether or not to invest further time on the question.

To save time but possibly lose a point, choose A or D. Otherwise, move on to the final step.

Step 3: Consider information (2) thoroughly.

Answer the question does (2) prove to be sufficient alone?

Yes, (2) is sufficient alone:

  • If (1) was ALSO sufficient alone, the answer is D.
  • If (1) was NOT sufficient alone, the answer is B.

No, (2) is NOT sufficient alone.

  • Eliminate D and B.
  • If (1) was sufficient to answer the question, choose A.
  • If (1) was NOT sufficient to answer the question alone, can you figure out the answer by using both (1) and (2)? If so, the answer is C. If you still cannot figure out the answer, E is correct.

Now that you’ve developed a strategy, let’s try it out on a sample question:

A. (1) ALONE is sufficient to answer the question. (2) ALONE is NOT sufficient to answer the question.
B. (2) ALONE is sufficient to answer the question. (1) ALONE is NOT sufficient to answer the question.
C. BOTH (1) and (2) are needed to answer the question.
D. BOTH (1) and (2) are sufficient to answer the question independently.
E. The question cannot be answered based on the information given.

Now, let’s use our plan to come to a solution.

Step 1: Size up the prompt.

This is a percent increase question; Ted paid more than the dealer, but we don’t know by how much. The information given is similar, but not the same.

Step 2: Consider information (1) thoroughly.

We can determine how much Ted paid by calculating the profit (50%) the dealer made on his $10,000 purchase.

$10,000 + 0.5($10,000) = Amount paid by Ted

Since (1) is sufficient to answer the question, we can eliminate all choices EXCEPT A and D.

Step 3: Consider information (2) thoroughly.

Based on (2), we can determine how much Ted paid for the wardrobe with the following equation:

$10,000 x 3/2 = Amount paid by Ted

Without performing any actual calculations, we can determine that both (1) or (2) can answer the question independently. Choice D is correct.

Sub-area III: Number Forms, Relationships, and Sets

This objective tests your knowledge of ratios and standard deviation, as well as your ability to determine rates, such as production rates and speed.

Take a look at some concepts that may appear on the test.


A ratio is a part compared to a whole. For example, if 5:6 represents the ratio of oranges in a bowl to all of the fruit in the bowl, 5 oranges + 1 additional fruit = 11 pieces of fruit. 5/11 pieces of fruit in the bowl are oranges.

Here’s an example problem that is similar to what you may see on the test:

A basket contains only pens and pencils. If 18 of the items are pens, and if the ratio of the number of pencils to the number of pens in the basket is 5:3, how many items are in the basket?

A. 34
B. 36
C. 44
D. 48
E. 52

First, let’s look at the ratio.

5 pencils + 3 pens = 8 items
(Remember, this is the ratio, not the answer to the question.)

If 18 of the items are pens, then 18 = 3 parts of the whole. If we divide by 3, we find that each part is equal to 6 items.

6 (1 part) x 8 = 48 items

Choice D is the correct answer.

Rate of Production

The rate describes the relationship between a quantity and an amount of time. We can determine the rate of production by using the following formula:

Rate = Quantity ÷ Time

Here’s an example scenario:

If an editor can edit 5 pages per hour, how many pages can she edit in 150 minutes?


It is likely that you will see several problems on the test which will require you to perform computations that include different monetary values. Let’s look an example problem now.

Mrs. Jones has $2.05 in dimes and quarters. If she has four more quarters than dimes, how much money does she have in dimes?

A. 30 cents
B. 70 cents
C. 20 cents
D. 80 cents
E. 60 cents

If we use Z to represent the total number of dimes, Z + 4 must be the total number of quarters. Since the value of a dime is 10 cents, we will use 10Z to represent the total value of the dimes and 25(Z + 4) to represent the total value of the quarters:

10Z + 25(Z + 4) = 205
10Z + 25Z + 100 = 205
35Z + 100 = 205
35Z = 105
Z = 3

3 x 10 = 30 cents

The correct answer to this question is A.

Time, Speed, and Distance

Calculating the rate of speed is very similar to calculating the rate of production, which we recently reviewed. The following formula may be used to calculate the rate of speed:

Speed = Distance ÷ Time

Using the formula, we can determine that if a car travels 110 miles in 2 hours, the car is going 55 miles per hour (given that no one needs to make a rest stop!):

Speed = 110 ÷ 2
55 = 110 ÷ 2

We can adjust the same formula to determine the two other variables:

Distance = Speed x Time
Time = Speed x Distance

Sub-area IV: Number Theory and Algebra

This objective tests your algebra skills and your ability to work with linear and nonlinear equations, inequalities, and distance and midpoint formulas.

Take a look at some concepts that may appear on the test.


Some of the questions on the test may require you to calculate the root of a number. A root is simply the inverse of an exponent. Remember “root” means “reverse”!

You are most likely to encounter square roots and cube roots on the test.

A square root indicates a number multiplied by itself. For example, the square root of 9 is 3.

√9 = 3
9 = 3 x 3

A cube root indicates a number multiplied by itself 3 times. For example, the cube root of 27 is 3.

³√27 = 3
27 = 3 x 3 x 3

Linear Equations with Two Variables

We’ve already practiced solving some equations to find out the value of a variable (such as when we reviewed time, speed, and distance).

Sometimes, you will need to solve equations which contain more than one variable, or unknown value. In these equations, the value of one variable depends upon the value of another. Consider the following equation:

3x + 4y = – 8 and x – 2y = ½
Determine the value of each x and y.

We can work to solve one variable at a time. First, let’s find the value of x by eliminating y. We need to multiply the terms in the second equation by 2 to get 2x – 4y = 1.

Now, we can add our equations:

3x + 4y = – 8
2x – 4y = 1
5x + 0y = -7
x = -7/5

We have found the value of x. Next, we can find the value of y by multiplying the second equation by 3 (to get 3x – 6y = 3/2) and subtracting it from the first equation.

3x + 4y = – 8 
3x – 6y = 3/2
10y = -9½
y = – 19/20

Quadratic Equations

A quadratic equation contains variables that are squared. These equations can be identified in the following format:

ax² + bx + c = 0

These equations create a parabola (which looks like a smile or a frown) when graphed. Sometimes, quadratic equations are disguised. Take a look at the following example, which is not in the traditional form:

2a² = 16

There appears to be no b value in that equation, but we can rearrange it to get:

2a² – 16 = 0

Let’s practice solving a quadratic equation. In order to do so, we will need to use the quadratic formula:

+ 4x – 14 = 0
Solve for x.

In order to find the solution or the value of x, we need to plug the given values into the formula:

Algebraic Inequalities

On the test, you will likely need to solve inequalities, as well as equations. In order to do so, you will need to isolate a variable, just as you do when you solve equations.

It is important to keep in mind that when you divide or multiply by a negative number, you must reverse the greater than (>) or less than (﹤) symbol. Let’s take a look at an example inequality which requires us to reverse the sign:

12 – 4x < 8 (subtract 12 from each side)
–4x < –4 (divide each side by -4)
x 1 (flip the less than sign)

The same rule for flipping less than and greater than signs holds true for less than or equal to ( ≤ ) and greater than or equal to ( ≥ ) signs; we must reverse them when we multiply or divide by a negative number.

Let’s try something a little more challenging and solve two inequalities at the same time:

-2  ≤ 6 – 2x   ≤ 4

Multiply each part by 3 to get:

-6  ≤ 6 – 2x   ≤ 12

Next, subtract the 6:

-12  ≤ -2x   ≤ 6

Finally, isolate the x:

6    x  -3


-3  ≤ x   ≤ 6

Overlapping Sets

Sometimes, you may be presented with sets of data that overlap, or contain the same numbers. Overlapping sets may not be immediately recognizable because they may be described in word problems.

Let’s look at an example question containing an overlapping set:

A closet contains 24 shirts. Each shirt has either a collar, long sleeves, or both a collar and long sleeves. If 10 of the shirts have collars and 19 of the shirts have long sleeves, how many of the shirts have long sleeves but no collars?

The question includes three mutually exclusive sets: (1) collared shirts without long sleeves (2) long-sleeved shirts without collars (3) shirts with both collars and long sleeves. The total number of people in these three sets is 24.

If we use s to represent the number of shirts with both long sleeves and collars, we can create the following equation:

(10 – s) + n(19 – s) = 24
29 – s = 24
s = 5

Now that we have found the number of shirts with both long sleeves and collars, we can subtract that number from the total number of shirts with long sleeves:

19 – 5 = 14

Sub-area V: Geometry

This objective tests your knowledge of concepts such as lines, angles, shapes, and coordinate planes. If you haven’t had a geometry class in a while, don’t worry – we will review some basic concepts right now.

Lines and Angles

The concept of lines and their intersections is a basic building block for many geometry problems. Remember that if lines are perpendicular, they create right angles. If lines are parallel, they never intersect.

A transversal is a line which crosses two parallel lines. The following diagram is an example of two parallel lines crossed by a transversal:

Using the diagram, review the following types of angles:

  1. Exterior angles: ∠A∠F∠G∠D
  2. Interior angles: ∠B∠E∠H∠C
  3. Consecutive interior angle pairs: [∠B and∠E] [∠H and∠C]
  4. Alternate exterior angle pairs: [∠A and∠G] [∠Fand∠D]
  5. Alternate interior angle pairs: [∠E and∠C] [∠H and∠B]
  6. Corresponding angle pairs: [∠A and∠E] [∠C and∠G] [∠D and∠H] [∠F and∠B]

Since we know that lines equal 180°, we can use that information to find the unknown values of angles.

For example, if ∠A in the previous figure is equal to 141°, you know that ∠E is also equal to 141° and that ∠B and ∠F are equal to 39°.

When it comes to intersecting lines, often you only need to know the value of one angle to solve the rest of the “puzzle!”

The Pythagorean Theorem

Some questions on the test will evaluate your ability to work with triangles, which are three-sided geometric shapes with three angles. The three angles of any triangle add up to 180°.

The most important type of triangle to know is the right triangle, a triangle with an angle which measures 90°. A right angle of a right triangle may be identified by a small box in the corner of the triangle:

When working with right triangles, you will often need to apply the Pythagorean Theorem by using the Pythagorean equation a² + b² = c².

According to this theorem, side c is the side of the triangle opposite the right angle; side a and side b are the other two sides of the right triangle. Here’s a simple example of how the Pythagorean equation can be used.

If side a of a triangle is 4 inches and side b is 3 inches and side c is the hypotenuse, what is the length of side c?

To solve this problem, just plug the numbers into the Pythagorean equation:

4² + 3² = c²

Apply the exponents:

16 + 9 = c²

Add the values on the left:

25 = c²

And find the square root of c:

c = 5

Pythagorean Triplets

A Pythagorean triplet is a ratio which describes each of the three sides of a right triangle. The first two numbers refer to the shorter legs of the triangle and the final number refers to the hypotenuse. Here are some common Pythagorean triplets:

Since you are working with ratios, you know that a triangle measuring 3:4:5 has the same proportions as a triangle measuring 6:8:10 or 15:20:25.

The first two Pythagorean triplets (1:1:√2 and 1:√3:2) also have Pythagorean angle triplets. The angles of these two types of right angles respectively measure 45°/45°/90° and 30°/60°/90°.

Let’s look at an example problem:

The following diagram contains two right triangles. How long is line BD?

Before we worry about BD, we need to find the value of AD. Since triangle ADC is a 30°/60°/90° triangle, its ratio must be 1:√3:2.

Using the ratio 1:√3:2, and considering that 10 is the value of the hypotenuse, side AD should be 5 inches because the ratio 5:10 is the ratio 1:2.

If side AD is 5 inches, then side BD is also 5 inches because triangle ADB is a 45°/45°/90° with a ratio of 1:1:√2. In other words, the two shorter legs of triangle ADB are exactly the same length!


On the test, you’ll work with circles and calculate their circumferences and surface area. To calculate the circumference of a circle, use the formula C = 2πr. The “r” represents the radius, a segment that connects the center of the circle to the edge of the circle.

For example, imagine that you have a circle with a radius of 4 feet.

In order to determine the circumference, just plug 4 into the C = 2πr formula. When you do this, you will find that the circumference of the circle is 25.23.

Remember that the radius is half of the diameter. If we had been given the diameter of the circle instead of the radius, we could easily reach the same conclusion by dividing the diameter in half and plugging that answer into the same formula.

C = 2πr is the same as C = πd

In order to determine the surface area of a circle, we would use the formula A = πr² . The area of the same circle (with a radius of 4 feet) would be 50.27 square feet.


Any geometric figure which contains only straight lines is an example of a polygon. Triangles, rectangles, octagons, and parallelograms are all examples of polygons.

Using the following formula, in which n represents the number of sides, we can find the sum of the interior angles of a polygon:

(n – 2)(180°) = sum of interior angles

Here are a few examples of how this formula may be applied:

Triangle (3 sides): (3 – 2)(180°) = 180° ÷ 3 = 60°
Quadrilateral (4 sides): (4 – 2)(180°) = 360° ÷ 4 = 90°
Pentagon (5 sides): (5 – 2)(180°) = 540° ÷ 5 = 108°
Hexagon (6 sides): (6 – 2)(180°) = 720° ÷ 6 = 120°
Heptagon (7 sides): (7 – 2)(180°) = 900° ÷ 7 = 129°
Octagon (8 sides): (8 – 2)(180°) = 1080° ÷ 8 = 135°

Notice the underlined text above. It reflects the measure of each angle in a regular polygon. A regular polygon is a polygon which contains only angles which are equal in measure.

Now, let’s use the formula for interior angles to solve an actual problem.

A hexagon has six internal angles with the following measurements: 124°, 122°, 54°, x°, x°, ⅘x°. What is the least possible sum of any two of the hexagon’s internal angles?

First, let’s apply our formula:

(6 – 2)(180°) = 720°

Subtract the known angles from 720°.

720°- (124° + 122° + 54°) = 420°

The three remaining angles must add up to 420°. Let’s set up an equation to determine their values:

x + x + ⅘x = 420°
14/5x = 420°
x = (420)5/14 = (30)(5) = 150

Therefore, two of the angles are equal to 150° and one of the angles is equal to 120° (⅘ of 150°).

Now that we know the measures of all of the angles, we know that the smallest angles measure 120° and 54°.

Therefore, the smallest possible sum of any two of the angles is 174°.

Coordinate Planes

A coordinate plane is a two-dimensional number line. On a coordinate plane, the vertical line is called the y-axis and the horizontal line is called the x-axis. The lines on a coordinate plane are perpendicular and intersect at the zero point. This point is called the origin.

Let’s look at an example of how to graph a point on a coordinate plane. To graph the point (2, 4), you need to know that the first number is always x and the second number is always y (x, y).

Start at the origin (0,0). Now, count over to the “2” on the x-axis and up to the “4” on the y-axis. You’ll end up with this answer:

You can use y = mx + b to describe any line on a coordinate plane. In this equation, the x and y values of any point on the line may be used. We use m to represent the slope and b to represent the y-intercept. The y-intercept is where the line crosses the y-axis.

In order to determine the slope of a line, you simply need to find two points on the line. We subtract the y value of the first point from the y value of the second point. Underneath, we subtract the x value of the first point from the x value of the second point.

You can also do this in reverse and subtract the first point from the first if you wish. Just remember to keep the y values on top!

As an example, let’s calculate the slope of two points, A and B, on the same line.

Let’s look at an example question which requires us to work with two points on a line:

On a coordinate plane, at what point along the y-axis does a line passing through point E (5, –2) and point J (3, 4) intersect that axis?

This question is asking us to find the y-intercept (b) of a line. We already know the values of two points, so let’s start with finding the slope (m) using the method you just learned.

Now that we have the slope, let’s go back to the formula y = mx + b. We can use either point E (5, –2) or point J (3, 4) to find the answer. Let’s plug in the values from point J:

y = mx + b
4 = (-3)3 + b
4 = -9 + b
13 = b

The y-intercept of our line is 13!

Section IV: Verbal Reasoning Assessment

The Verbal Reasoning Assessment includes 36 questions. You will be given 65 minutes, about one-third of your total testing time, to answer the questions on this section.

This part has three main objectives:

  • Critical Reasoning
  • Sentence Correction
  • Reading Comprehension


So, let’s start with Critical Reasoning.

Sub-area I: Critical Reasoning

This objective tests your ability to make logical inferences about arguments and conclusions without being distracted by irrelevant statements or assumptions. You should also be able to locate conclusions within prompts and to understand how arguments are strengthened or weakened.

Take a look at some concepts that may appear on the test.

Making Inferences

An inference is a reasonable conclusion made by examining incomplete information. Inference questions usually include words such as “conclude,” “conclusion,” and “infer.” When given an inference question, avoid assumptions and making great leaps; the answer will be a conclusion that does not go far beyond the given information in the prompt.

Let’s look at an example question:

The New Hanover County government promised its citizens that the new state highway will not pass through existing sites in which Species B, a protected species of waterfowl breeds, lives, including Zone Z. However, the new highway will pass within a mile of Site Z, which is the center of several important breeding sites for members of Species B, though this has not been discovered yet. Which of the following inferences is best supported by the statements made above?

A. The government has broken its promise to the citizens of New Hanover County.
B. The citizens of New Hanover County are worried that the noise the new highway will create might disturb the Species B individuals which breed in Zone Z.
C. Because the government is committed to preserving the breeding sites of Species B, there is no threat to Zone Z, nor any undiscovered nearby breeding sites.
D. It does not matter whether breeding sites around Zone Z are damaged by the highway since New Hanover County needs the new highway in order to bring in more tourists, and this factor is more important than the thrival of Species B.
E. Unless the state highway plan changes, undiscovered breeding sites will likely be threatened during the construction of the road.

Let’s consider each of the answers together:

Choice A contradicts the premise that the county government promised that the state highway will not pass through the existing breeding sites of Species B. Although the state highway will pass through undiscovered breeding sites, it is incorrect to determine that the government has broken its promise; the government does not plan to pass through Zone Z, only near that site.

Choice B presents a new challenge regarding another concern of the citizens. There is no information in the existing premises to support the statement about noise, so this cannot be the correct answer.

Choice C addresses the premise that the government has promised to preserve the breeding sites of Species B. However, consider the premise that the highway will pass near Zone Z, which is the center of several breeding sites. This fact reveals that other breeding sites could be threatened. Therefore, choice C is incorrect.

Choice D may be ruled out almost automatically because it is based on assumptions and is only loosely linked to the information given. Remember, avoid making great leaps; use what you know to be true to make an inference!

Choice E is the most logical choice because it reflects a probable conclusion to the given premises. You know that the area around Zone Z includes several undiscovered breeding sites. You know that the highway will be built near Zone Z – in other words, it will be built in the area around Zone Z. Therefore, you can conclude that undiscovered breeding sites may be threatened unless construction plans change.


When answering assumption questions, you will be given a prompt which includes premises, a conclusion, and a list of assumptions.

The correct answer contains an assumption that would logically make the conclusion in the prompt true. The incorrect answers contain assumptions that do not make the conclusion in the prompt correction.

Most assumption questions will actually contain the word “assumption.” This is good news for you because this will make this type of task easier to identify.

Let’s look at an example question:

For many consecutive years, the price of avocados at each of four statewide grocery store chains has been about 50% higher than the national average price. Furthermore, the per-pound difference in avocado prices among the four stores were never more than five cents. Among grocery stores in other states, the price of avocados reflected as much as a dollar difference per pound during the same time period. The four grocery store chains must have made arrangements in order to not compete among themselves and to agree to fixed levels at which to set their avocado prices.

The claim that the four grocery store chains fixed avocado prices rests on which of the following assumptions?

A. No other grocery store sold avocados at a higher price than the four chains.
B. The average price of avocados in all grocery stores in the state where the four chains operated greatly exceeded the national average.
C. Consumers in the same state as the four chains also purchased avocados from many other grocery stores in the state.
D. Grocery shoppers in the state where the four chains conduct business generally prefer avocados over other product options, even if avocados are more expensive than the other product options.
E. The wholesale price that grocery stores paid for avocados varied significantly between states during the time period of interest.

Now, we can consider each of the five assumptions and decide which assumption makes the most sense with respect to the prompt.

Choice A might give some credibility to the conclusion that the four chains agreed to fix the price of avocados. However, the assumption is vague. We do not know whether to consider stores across the nation or only stores which compete in the same state as the four chains. Also, just one store charging consumers a higher or lower price would not create a statistically significant impact.

Choice B does not support the argument that the four stores held a monopoly and fixed prices. Rather, it would support the idea that the four stores only charged high prices for avocados out of necessity.

Choice C weakens the argument that the four chains held a monopoly on avocados and agreed upon fixed prices because the presence of more competitors would make a monopoly less likely.

Choice D is not relevant to the argument, which is concerned with the cost of avocados only, not the cost of avocados in comparison with other produce options.

Choice E is the best answer because it eliminates a factor (wholesale pricing) that would logically impact the prices which consumers paid for avocados. With this factor eliminated, we can make a better argument for the conclusion that the four chains agreed to fix prices.

Weakening Evidence

Just like the assumption questions, weakening evidence questions will present you with an argument in which premises and assumptions are used to reach a conclusion. However, you will identify which of the answer choices weakens the argument the most.

Weakening evidence questions will include phrases such as “weaken the argument,” “damage the argument,” and “flawed reasoning.”

Let’s look at an example question together and find the correct answer:

An undocumented painting, supposedly Italian and from the 15th century, has been offered to an art museum. The painting may be genuine because it comes from a private collection. However, yellow hues in 15th-century paintings tend to look dull, whereas this painting has comparatively very bright yellow tones. Therefore, the painting has more than likely been forged.

Which of the following options, if correct, most seriously weakens the argument?

A. The art museum has a policy in place to accept only international pieces which are accompanied by completed and valid export forms.
B. The landscape in the background of the painting is very similar to other landscapes depicted in the background of paintings which were completed in Italy during the 15th century.
C. Recently, a similar painting with bright yellow hues was revealed to be forged by a modern artist and fraudulently sold to a museum as a historical piece.
D. Expert art appraisers believe that foragers cannot convincingly reproduce the wooden panels which were typically used as canvases by painters in the 15th century.
E. Some solutions which art restorers and collectors use to clean very old paintings can make the hues of the paintings appear brighter.

Now, before we begin looking at the answer choices, remember: we want to find the choice which weakens the argument that the painting is probably a forgery.

Choice A does not address forgery, and therefore it does not weaken the argument.

Choice B somewhat weakens the argument that the painting is a forgery. If the background of the painting in question looks similar to the backgrounds of legitimate paintings, it may seem less likely that the painting was forged. However, a convincing forgerist would have knowledge of the appearance of legitimate paintings from the same area and era, and he would probably try to emulate the same style.

Choice C strengthens rather than weakens the argument that the painting is a forgery, so it is not the correct answer.

Choice D addresses the forgery of paintings, but it brings up a new topic, wooden panels, which were never addressed in the prompt.

Choice E is the correct answer because it gives a possible legitimate reason why a 15th century painting might display bright yellow hues.

Supporting Evidence

Think of a supporting evidence question as the opposite of a weakening evidence question. Once again, you will be given a prompt which uses premises to support a conclusion. However, you will identify which of the answer choices strengthens the argument the most.

Look for phrases such as “best supports,” “most strongly supports,” and “effectively strengthens” in order to identify a supporting evidence question.

Let’s look at an example of a supporting evidence question:

In a blind study of individuals suffering from dust mite allergies, each subject was administered AllerCalm or a placebo. Six months later, fewer than 25% of the individuals who were administered AllerCalm continued to experience allergy symptoms, in comparison with about 75% of the subjects who were given the placebo. This confirms that AllerCalm is effective in curing allergies to dust mites.

Which of the following statements, if correct, most strongly supports the argument?

A. None of the participants who received AllerCalm reported any negative side effects to the medication.
B. Proven research indicates that if a study is blind to the researchers, but not the experimenters, the researchers may treat the control group differently than the group receiving the actual treatment.
C. During the six months of the experiment, some of the subjects took other allergy medications to help control their symptoms.
D. Six months after the experiment, the total number of subjects who reported allergy symptoms were fewer than at the beginning of the experiment, when all subjects reported allergy symptoms.
E. During the six months of the experiment, researchers frequently monitored the subjects’ homes and reported no significant change with regard to evidence of dust mites.

Let’s think about which of the choices provides evidence to support the idea that AllerCalm cures dust mite allergies:

Choice A is irrelevant because it shows that AllerCalm is probably safe, but it does not support the argument that AllerCalm is effective.

Choice B is incorrect because it actually weakens the argument. If researchers knew which subjects were receiving AllerCalm, they may have been more likely to suggest to these individuals that their symptoms were improving, thus altering the results reported by the subjects.

Choice C could also be used to weaken the argument. If all or even some of the subjects who received AllerCalm took other allergy medications, the other medications could account for why these subjects were less likely to report allergy symptoms.

Choice D is incorrect because it neither provides new information, nor distinguishes between the control group and the group receiving AllerCalm.

Choice E is the correct answer because it supports the idea that there was no significant change to the subjects’ environment, thus lending credibility to the study.

Sub-area II: Sentence Correction

This objective tests your knowledge of English grammar and sentence structure by presenting you with sentences in need of correction. You will evaluate a sentence which contains errors, as well as a list of different versions of the original sentence.

After reading the revised sentences in the list, you will select the best version of the sentence. When selecting your answer, choose the version of the sentence which is concise and best conveys the meaning of the sentence.

Always read the original sentence before selecting your answer. As you do so, ask yourself if something about the sentence seems awkward.

Take a look at some concepts that may appear on the test.

Adjectives and Adverbs

Some sentences will show an error in the choice between an adjective and an adverb. Remember, adjectives describe nouns and adverbs describe verbs and adjectives. Generally, adverbs end in -ly.

Take a look at the chart below to review the examples of these two types of words:

Now, let’s look at an example sentence and choose the appropriate correction:

The data suggests that over the next decade, demand for the product will increase at an unexpectedly fast rate, and greatly surpasses the previous predictions of most economists.

A. increase at an unexpected fast rate, and greatly surpasses
B. be at an unexpected increased rate and will greatly surpass
C. increase at an unexpectedly fast rate, and even surpasses
D. increase at an unexpected fast rate, greatly surpassing
E. increase at an unexpectedly fast rate, greatly surpassing

In this example, E is the correct answer. This is because the adverb “unexpectedly,” instead of the adjective “unexpected,” should be used to describe the word “fast.” We can rule out A, B, and D because they both use the verb “unexpected.”

We can rule out C because of the use of “surpasses;” the sentence clearly refers to a future event, not a current trend. The use of “surpasses” indicates that the demand is currently increasing at an unexpected rate.

Personal Pronouns

Sometimes, you will be presented with a sentence which incorrectly uses a personal pronoun. Personal pronouns, such as “itself,” “his,” and “they” refer to specific nouns.

Now, let’s look at a GMAT-type sentence which contains an error with a personal pronoun:

Those of the lawmakers opposing the woodland protection bill have only theirselves to blame for the outrage in the local community.

A. Those of the lawmakers opposing the woodland protection bill have only theirselves to blame
B. Those lawmakers, who opposed the woodland protection bill, have only themselves to blame
C. Those of the lawmakers who are opposing the woodland protection bill have only theirselves to blame
D. Those of the lawmakers who are opposing the woodland protection bill have only themself to blame
E. Those lawmakers who opposed the woodland protection bill have only themselves to blame

In this example, answer choice E is correct. The nonword “theirselves” should be replaced with the pronoun “themselves.” Furthermore, choice D uses a more concise phrase to replace the refusing phrase “those of the lawmakers opposing.”

Choices A and C may automatically be ruled out because they contain the nonword “theirselves.” Choice B contains superfluous commas which indicate that the reader should pause at awkward moments. Choice D is incorrect because it uses the singular pronoun “themself” to refer to the plural noun “lawmakers.”

Pronoun-Antecedent Agreement

An antecedent is simply the noun to which a pronoun refers to. As we saw in incorrect answer choice D in the previous section, pronouns should agree with their antecedents. For example, if an antecedent is singular, the pronoun should be singular.

Let’s take a look at a question similar to what you will encounter on the test:

Many successful entrepreneurs throughout history, such as Steve Jobs, are still recognized for his own achievements.

A. Many successful entrepreneurs throughout history, such as Steve Jobs, are still recognized for his own achievements.
B. Many a successful entrepreneur throughout history, such as Steve Jobs, are still recognized for their own achievements.
C. Many successful entrepreneurs throughout history, such as Steve Jobs, is still recognized for his own achievements.
D. Many successful entrepreneurs throughout history, such as Steve Jobs, is still recognized for their own achievements.
E. Many successful entrepreneurs throughout history, such as Steve Jobs, are still recognized for their own achievements.

In this example, choice E is correct because the pronoun “their” appropriately addresses the plural antecedent “entrepreneurs” and well as the singular antecedent “Steve Jobs.”

Choices A and C are incorrect because the pronoun “his” only addresses Steve Jobs.

Choice B is incorrect because “entrepreneurs” becomes singular. Therefore, the plural pronoun “their” is not the correct match.

Choice D would be correct, except that “is” is singular instead of plural and does not agree with the plural antecedents. The error involving “is” also appears in choice C.

Subject-Verb Agreement

Just as pronouns and antecedents should agree, verbs and subjects should agree, too. When a subject is plural, the verb describing its action should also be plural. If a subject is singular, the corresponding verb should also be singular.

Let’s look at some examples in which the subjects and verbs are underlined:

Incorrect: The Senior Vice President, along with a few other executives, were in a meeting on Tuesday.
Correct: The Senior Vice President, along with a few other executives, was in a meeting on Tuesday.

Incorrect: Professionalism, particularly with respect to timeliness and communication, are necessary in order to complete the task.
Correct: Professionalism, particularly with respect to timeliness and communication, is necessary in order to complete the task.

When completing this portion of the test, remember to carefully look for the subject of the sentence. Do not be confused by other nouns in the sentence which may not agree with the actual subject. Identifying the subject first is a great strategy for success!

Misplaced and Dangling Modifiers

A modifier is a word, phrase, or clause that describes another word. Usually, modifiers are adjectives, like “conscientious” or “personable.” Sometimes, a modifier is included in the wrong part of a sentence. Consider this example:

“The live bowl of fish looks great in her office.”

Since the fish are alive and the bowl is not, “live” is a misplaced modifier. You could revise the sentence by moving the modifier:

“The bowl of live fish looks great in her office.”

A dangling modifier is a modifier which does not clearly refer to another word in a sentence. The word that a dangling modifier is meant to describe is missing. Consider this example:

“Set by an arsonist, the firefighters were able to save the burning store.”

We can assume that a fire is what the arsonist set, but let’s try clarifying the sentence:

“The firefighters were able to save the burning store from the fire set by an arsonist.”

Sub-area III: Reading Comprehension

This objective tests your ability to determine the main idea of a passage, to find supporting ideas and details, and to recognize when you are asked for information which is “out of context” and does not resonate with the passage. You should also be able to recognize and describe an author’s tone and viewpoint.

Take a look at some concepts that may appear on the test.


One excellent way to read passages strategically is to make notes as you move through them. Creating this “map” will help you to guide yourself through the passage.

While mapping, note the purpose of each paragraph. Remember to look for major themes. You should record the overall function of the paragraph, not minor details. If you need to refer to details later, you may use your map to find them.

Here is an example of what a passage map might look like. Notice that it is divided into paragraphs and gives only the overarching ideas in each.

¶1 Introduction – zoning conflict
¶2 Argument for new zoning
¶3 Argument against new zoning
¶4 Solutions/compromises
Purpose – Offers solutions related to zoning conflict, compromises suggest favoritism towards individuals against zoning

Note that the map also includes a final section, “Purpose,” which should include the author’s overall message. If you are asked a question about the main idea of the passage, you would refer to your notes for this part of the map.

If you were asked a question about why individuals were against the new zoning, you could refer to your map and immediately know to go to the third paragraph for more details.

Main Idea

Finding the main idea of a text is a crucial skill when reading. You can expect nearly every passage on the Reading Comprehension section of the exam to be accompanied by a question which requires you to identify the main idea.

Finding a central idea or theme in a text is all about getting to the point. As you read, think about the “big picture” created by the details in the text in order to determine the overall message. Is the author trying to make a general statement about life, to explain a major concept, or to argue a point? Before reading the answer choices, summarize the text in your own words. This is also an excellent time to refer back to the passage map you created!

Let’s take a look at an example main idea question. The following passage is an excerpt from Crime: Its Cause and Treatment by Clarence Darrow:

The question of capital punishment has been the subject of endless discussion and will probably never be settled so long as men believe in punishment. Some states have abolished and then reinstated it; some have enjoyed capital punishment for long periods of time and finally prohibited the use of it. The reasons why it cannot be settled are plain. There is first of all no agreement as to the objects of punishment. Next there is no way to determine the results of punishment. If the object is assumed it is a matter of conjecture as to what will be most likely to bring the result. If it could be shown that any form of punishment would bring the immediate result, it would be impossible to show its indirect result although indirect results are as certain as direct ones. Even if all of this could be clearly proven, the world would be no nearer the solution. Questions of this sort, or perhaps of any sort, are not settled by reason; they are settled by prejudices and sentiments or by emotion. When they are settled they do not stay settled, for the emotions change as new stimuli are applied to the machine.

A state may provide for life imprisonment in place of death. Some especially atrocious murder may occur and be fully exploited in the press. Public feeling will be fanned to a flame. Bitter hatred will be aroused against the murderer. It is perfectly obvious to the multitude that if other men had been hanged for murder, this victim would not have been killed. A legislature meets before the hatred has had time to cool and the law is changed. Again, a community may have capital punishment and nothing notable happens. Now and then hangings occur. Juries acquit because of the severity of the penalty. A feeling of shame or some bungling execution may arouse a community against it. A deep-seated doubt may arise as to the guilt of a man who has been put to death. The sentimental people triumph. The law is changed. Nothing has been found out; no question has been settled; science has made no contribution; the public has changed its mind, or, speaking more correctly, has had another emotion and passed another law.

Which of the following options best summarizes the content of the passage?

A. Decisions regarding the punishment of criminals are often based on emotional reactions and punishments often generate further emotional reactions.
B. The criminals responsible for atrocious crimes often elicit intense emotional responses from communities.
C. Members of the public may feel a sense of shame and guilt after severe punishments are inflicted upon criminals.
D. Although laws regarding capital punishment vary by government, people generally feel the same emotional response when presented with the topic.
E. With respect to financial consequences, the execution of a criminal is a lesser burden on a community than the confinement of the individual to lifelong imprisonment.

In this case, options B and C are ideas which may be gathered by the passage. However, they both support a larger idea – the idea which is presented in option A, the correct answer.

Option D is incorrect because it actually states the opposite of the ideas presented in the first four sentences.

Option E is incorrect because it is completely off topic; financial consequences are never addressed in the passage.

When searching for the correct answer choice for a main idea question, seek to eliminate any choices which fall within the following four categories:

  1. Too broad – expands beyond the scope of the passage
  2. Too narrow – focuses on a particular detail or point
  3. Distorted – shows a misunderstanding of the main idea
  4. Irrelevant – indicates ideas which are not addressed in the passage

Supporting Ideas and Details

Supporting ideas and details questions are related to the main idea question type. The main idea drives the whole passage. Therefore, the details mentioned in the passage support its main idea in some form.

Correctly answering a detail question requires you to build upon the skills you need to answer a main idea question; first, you must identify the main idea. Afterwards, you may need to reread part of the passage to identify how the details relate to the main idea. Once again, remember to use your passage map for help!

You can identify detail questions by phrases such as “the role of the third paragraph is…” or “the life cycle of grasshoppers is mentioned because…” Basically, keep an eye out for when you are asked about the purpose of a specific idea or paragraph.

When searching for the correct answer choice for a main idea question, seek to eliminate any choices which fall within the following three categories:

  1. Inference – not directly stated in the passage
  2. Distorted – shows a misunderstanding of the detail or its purpose
  3. Extreme Language – simplified statement about a detail which includes words like “never,” “none,” or “always”


As you reviewed when reading about the Analytical Writing portion of the exam, tone is the author’s attitude towards a subject. The passages on the Reading Comprehension section will not contain, obviously show, or openly state how an author feels.

However, you will still need to identify the authors’ tones on this section of the exam even though the authors’ attitudes will be very subtly expressed.

To discover an author’s tone, you will need to pay careful attention to word choice. This means being aware of the connotation of words.

The denotation of a word is its dictionary definition. Connotation refers to the overtones or associations of a word. To understand connotation, think about how words are often connected with certain emotions.

Let’s look at some examples of specific words which have different connotations though they may share the same denotation.

Evaluating the author’s word choice will help you to answer tone questions such as the following example questions:

The author’s attitude toward capitalism would best be described as which of the following?

Which of the following options best describes the tone of the passage?

Based on the statements in the second paragraph, which of the following could be inferred about the author’s attitude toward the use of fossil fuels?


You can expect to encounter two types of out-of-context questions on the test. Both types of questions will require you to select the answer which best resonates with the author’s main idea.

Let’s take a look at each of the two types of out-of-context questions:

  1. Author’s perspective

These questions will present an idea which is not included in the passage and ask what the author would think about the idea. When answering this question type, you will need to use your skills of determining an author’s perspective.

How would the author of the passage most likely respond to (new theory or assertion)?

Based on the passage, which of the following statements describes how the author would react to (new idea or assertion)?

Which of the following options best describes how the author would most likely respond if presented with (new situation or concept)?

  1. Unrelated situation

Some out-of-context questions will ask you to compare an idea in the passage to a completely different idea or situation. This type of question will be similar to one of the following examples:

Which of the following situations is most similar to the situation described in the passage?

Identify the hypothetical situation that is most comparable to a situation presented in the passage.

Which example is most similar to the example provided in the passage?

Out-of-context questions test your ability to recognize an argument or an idea, and then recognize a similar idea in a different context. The correct answer will be the answer which most clearly resonates with the author’s ideas. Basically, you will make the most logical inference possible in order to come to a conclusion.

Practice Questions & Answers

Question 1

The table below displays the percentage of permits issued by building size and geographic location in Los Angeles in 2017 and 2018. The Totals columns show the total number of permits issued for each neighborhood for the corresponding years.

For each of the following statements, select Yes if the statement is true based on the information in the table. Otherwise, select No.

Find the number of permits for each of the specified neighborhoods by multiplying the total number of buildings for those neighborhoods in 2018 by the percentages indicated in the columns which correspond to the building sizes addressed in the statement:

1.1 A

Explanation: First, add the total number of buildings for 2018 (the final column) to get 611. Next, find the total number of 5+ Family buildings for 2018 using the percentages in the appropriate column:

92.9%(27) = 25.08

25.3%(175) = 44.28

44.1%(67) = 29.55

0.3%(342) = 1.03

Add these answers to get 99.94. Next, you will need to find the percentage of 611 that 99.94 is:

99.94 = x%(611)

x = 16.36

The statement is correct because 16.36% is between 15% and 17%.

1.2 B

Explanation: Compare the number of buildings using the following formulas:

(Northern Areas 5+ Family buildings) .003 x 342 = 1.026

(Downtown LA 3-4 Family buildings) .036 x 27 = .972

1.026 > 972

1.3 B

Explanation: Find the decreases mentioned using the following formulas:

(Downtown LA 3-4 Family buildings) .091(11) – .036(27) = 1 – .97

(Central LA 5+ Family buildings) .371(140) – .253(175) = 51.94 – 44.28

You can save time by skipping the subtraction since the decrease for 5+ Family buildings in Central LA is obviously greater.

Question 2

For the past few years, the number of transactions on a site which sells shoes have been declining steadily. However, after a new marketing team took over this year, there have been promising indicators of a recovery. This year, customers responded to a survey asking them to rate, on a scale of 1 to 10 (10 being highest), their experience shopping on the site. Among these survey participants, 85% of them rated their experience 8 or higher. This is an improvement over the survey results last year, when only 30% of customers rated their satisfaction 8 or higher. The site’s marketing team is attributing this improved customer experience to the new advertising campaign. Therefore, the site is in position to reverse the recent trend and increase its number of transactions this year.

In the table below, make only two selections, one in each column.

Select Support for the statement that, if true, would provide the strongest support for the argument. Select Weaken for the statement that, if true, would most seriously weaken the argument.

2.1 D

Explanation: The satisfaction of customers who shop at the site has improved, and the marketing team believes that their new approach is responsible for this improvement. Choice D provides evidence that this is true and suggests that they may result in increased transactions. If people who do not shop at the site think better of the site now than they used to, there is a possibility that they will be converted to customers of the site.

2.2 E

Explanation: The conclusion that the number of transactions will increase rests partly on the improved customer satisfaction results. Choice B says that only a score of 10 in customer satisfaction correlates strongly with an increase in transactions. Therefore, if most of the 85% of respondents who rated their satisfaction an 8 or higher rated it 8 or 9, then the survey results, in terms of predicting increased transactions, becomes much weaker.

Question 3

The following emails are exchanged between an experienced pet fish store owner and his nephew, who has just taken over his business.

For each of the following statements, select Inferable if the statement is reasonably inferable from the information provided. Otherwise, select Not Inferable.

3.1 A

Explanation: Longhorn cowfish and cubicus boxfish are similar species, according to the brief descriptions. However, cubicus boxfish tend to be cheaper than longhorn cowfish. Because the new owner has expressed doubts that his customers will buy more expensive fish, it is reasonable to infer that he would choose cubicus boxfish over longhorn cowfish.

3.2 B

Explanation: You cannot infer this. The descriptions of the species are almost identical. There is no reason to believe the new owner would choose one over the other.

3.3 A

Explanation: You can infer that the new owner will choose coral sand over other substrates for two reasons: a) the experienced owner recommends this substrate over all others and b) the new owner mentions an interest in invertebrates, which need the calcium found in the coral sand. 

Question 4

If the new owner decides to keep offering gobies to customers, which of the following strategies would be most likely to increase the number of customers who choose this species?

  1. Increasing the price of the gobies to make them seem more exotic
  2. Adding the information that gobies are non-fussy eaters to the sign on their tank
  3. Placing the gobies in the same tank as the clownfish and prominently labeling the two species as excellent tank mates
  4. Ensuring that the part-time employees know which pH level is best for gobies
  5. Placing the gobies in a tank closer to the front of the store


Explanation: If the clownfish are popular, people will be likely to go to the clownfish tank and therefore they will see the gobies. Furthermore, people looking for fish compatible with clownfish will immediately see the gobies as an option.

Question 5

The graph shows the total payroll and number of wins for 15 popular National Football teams during the 2018 season.

Based on the graph, fill in the blank in the following statements.

5.1. The likelihood that a team with a payroll greater than $125 million won more than 10 games was _____ the likelihood that a team with a payroll less than $100 million won more than 8 games.

  1. greater than
  2. equal to
  3. less than

5.2. If a team is selected at random, the probability that it will be one of the top three teams in terms of both payroll and number of wins is _____.

  1. 2 out of 15
  2. 1 out of 5
  3. 3 out of 15
  4. 4 out of 15

5.1 A

Explanation: 3 out of the 5 teams with payrolls over $125 million won more than 10 games.
⅗ = .6

4 out of the 7 teams with payrolls less than $100 million won more than 8 games.
4/7 = .57

.6 > .57

5.2 A

Explanation: The top three teams in terms of wins have the following numbers of wins: 11, 11, 14.

However, one of the teams with 11 wins has a payroll less than both of the other teams, as well as less than two teams with less wins. Therefore, one of the teams with 11 wins is in 5th place in terms of payroll and must be eliminated. Therefore, there are 2 out of 15 teams which fit the description.

Question 6

Most of the attendees at Johnston Public Library are either children or senior citizens. The library will be offering an advanced watercolor painting class to the community. The director of the library is considering two options: offer the class on a Monday morning or offer the class on a Monday evening after public school hours, so that children will be able to attend.

However, senior citizens who visit the library are statistically more likely to indicate a preference for morning programs, so the library will be better off hosting the class in the morning.

Select only one statement for the Assumption column and only one statement for the Fact column. The Assumption column should indicate the assumption required by the argument. The Fact column should indicate a statement that, if true, would provide significant logical support for the required assumption.

6.1 D
Explanation: The argument is that the library will be better off hosting the class in the morning. Since seniors prefer morning classes, the underlying assumption that makes the most sense is that most of the people who will be going to the class are seniors.

6.2 B
Explanation: If it is a fact that most young children are not capable of using advanced watercolor techniques, the class would not be appropriate for them. Therefore, there would be no reason to host the class after school hours. This supports the argument that the library is better off hosting the class in the morning.

Question 7

The table shows food critics’ ratings of various cheeses. Outstanding cheeses are those which received a grade of 96 or higher; Excellent, the cheeses that received a grade between 91 and 95, inclusive; Very Good, the cheeses that received a grade between 86 and 90, inclusive; Good, the cheeses that received a grade between 81 and 85, inclusive; Average, the cheeses that received a grade between 71 and 80, inclusive; and Below Average, the cheeses that received a grade of 70 or lower.

N/A refers to an absence of a rating by a critic.

Select Yes if the statement is true based on the table. Otherwise, select No.

7.1 A

Explanation: A vintage is considered Average if it receives a grade between 71 and 80, inclusive. Critic 4 gave 1 Average rating and Critic 5 gave 0. Keep in mind that you would be able to sort the columns on the test.

7.2 A

Explanation: Critic 1 rated 15 cheeses, and 2 received an Outstanding rating (96 or higher). Critic 2 and Critic 5 each gave 2 cheeses an Outstanding rating, but they each rated more cheeses (20 and 23, respectively). Therefore, the probability of an Outstanding rating by Critic 1 was more likely than an Outstanding rating by any other critic.

7.3 B

Explanation: Critic 5 rated 1 French cheese a 90; the statement is incorrect.

Question 8

An animal shelter is soliciting donations from the community for its program to trap, spay/neuter, and release feral animals in order to control the populations of these animals.

The total number of the community members solicited is C, and a donation of $0.10 per month for one year from each community member solicited will fund exactly F feral animals treated for the following year.

In terms of C and F, select the expression that represents how many feral animals will be funded if half the community members solicited donate $0.15 per month each for one year, and the other half of the community members do not donate. (Column 8.1)

Next, select the expression that represents how many of the community members who were solicited must donate $0.15 per month each for one year in order to fund collectively 1.5F feral animals treated for the following year. (Column 8.2)

8.1 C

Explanation: If each of the C community members solicited donates $0.10 per month, that is, $1.20 for one year, then all the community members solicited will collectively donate $1.20C.

If half the community members solicited donate $.15 per month each for oneyear, and the other half do not give donations, that means that the total of donations for the year will be 9C.

$1.20C ÷ F feral animals = $.90C ÷ ? feral animals
? = $.90CF ÷ $1.20C
? = ¾(F)

8.2 D

Explanation: Next, let x be the number of community members who must donate $0.15 per month each for one year ($1.80 each, in total) in order to fund collectively exactly 1.5F feral animals. Set up a new proportion in order to find x:

$1.20C ÷ F feral animals = $1.80x ÷ 1.5F feral animals
2C ÷ 1 = 3x ÷ 1.5
x = C

Question 9

An entertainment analyst has predicted that the family drama film Home Again will have weak box office revenue during its initial release.

Home Again will open the same Friday as two big-budget films: the highly anticipated romance Wait for Me in Paris and Wild Goose Chase 2, the sequel to an extremely popular comedy. Furthermore, Home Again is scheduled to be shown in only half as many theaters as Wild Goose Chase 2 during the week that the films are released, so its box office potential is limited.

Select Strengthen for the statement that, if true, would most strengthen the analyst’s argument, and select Weaken for the statement that, if true, would most weaken the analyst’s argument.

9.1 A

Explanation: The analyst concludes that Home Again will not do well at the box office because of its competition and the limited number of screens on which it will be initially shown. Choice A says that family dramas, such as Home Again, do not fare well at the box office when they go up against big-budget romances, such as Wait for Me in Paris. This statement reinforces the analyst’s conclusion.

9.2 D

Explanation: Choice D weakens the argument: If Home Again will be shown on more screens after the first week, then its box office potential will increase later in its theatrical run.

Question 10

Wildlife biologist: Pumas, also known as mountain lions, face many threats. Much of their territory has been taken over by humans and they compete with other large prey animals, such as wolves and bears, for prey. Many of the deer species which pumas prey on most have experienced declines in populations which overlap with puma populations in terms of territory. This population decline has been caused by hunting.

The greatest danger for adult pumas comes from farmers, who view the cats as a threat to their cattle and other livestock; farmers have been killing pumas in great numbers over several decades. As a result, puma populations have been decreasing precipitously, and are wiped out altogether from entire regions in which they used to roam, such as most of the Appalachian mountains. Pumas will be nearly extinct in the wild within a few years.

Indicate two different statements as follows: one statement identifies an assumption required by the wildlife biologist’s argument, and the other identifies a possible fact that, if true, would provide significant logical support for the required assumption.

10.1 E

Explanation: The wildlife biologist gives many reasons why pumas are vulnerable, states that puma populations have been declining, and concludes that the species will soon be nearly extinct in the wild. The argument assumes that nothing will change in the near future in order to help puma populations thrive. Choice E identifies the assumption that conservation efforts will not alter the current trend.

10.2 C

Explanation: A fact suggesting that conservation efforts will not alter the current trend is supplied by choice C Farmers, who are the biggest threat to pumas, have resisted conservation efforts.

Question 11

What is ¼% of 54?

  1. .046
  2. .14
  3. 13.5
  4. .03


(¼%) 54 = (¼ x 54)  ÷ 100 = 13.5 ÷ 100 = .135

Round to .14.

Question 12

The total area of a warehouse is 150,000 square feet. The chart below shows warehouse units Q, R, S, and T as proportions of total warehouse area.

By approximately how many square feet does the size of Unit Q exceed that of Unit S?

  1. 12,000 square feet
  2. 13,000 square feet
  3. 13,500 square feet
  4. 14,000 square feet


Explanation: To determine the size of Unit S, first determine the size of Unit T as a percentage of the total warehouse size. Unit T occupies 14,500 square feet, or approximately 10%, of the total 150,000 square feet in the warehouse. Thus, Unit Q occupies 20% of that total (100% – 28% – 42% – 10% = 20%).

The question asks for the difference in size between Unit Q (28%) and Unit S (20%). Unit Q is 8% larger than Unit S.

150,000 total square feet x .08 = 12,000 square feet.

Question 13

Customers at Carolyn’s Cafe can select two of three appetizers—soup, chips with salsa, and garden salad— along with two of three sides—pasta salad, fruit, and bread. What is the statistical probability that any customer will select soup, garden salad, fruit, and bread?

  1. 1 in 3
  2. 1 in 6
  3. 1 in 9
  4. 1 in 12

Explanation: In each set are three member pairs. Thus the probability of selecting any pair is ⅓.

⅓ x ⅓ = 1/9

The answer would be the same for regardless of which specific menu items were ordered.

Question 14

On the xy-plane below, if the equation of w is defined as y = ½x and if point B is defined by the xy-coordinate pair (10, 0), what is the area of triangle CAB?

  1. 4 square units
  2. 10 square units
  3. 16 square units
  4. 20 square units


Explanation: The slope of w is ½ which means that every 2 units from left to right corresponds to 1 unit upward. Since angle A is a right angle, the slope of line AB must be a drop of -2 for every 1 unit from left to right. Drawing a plumb line down from point A reveals that the height of triangle CAB must be 2.

Given a base (CB) of 10 and an height of 2, the area of triangle CAB is equal to 10.

Every triangle is defined as ½ (base x height).

Question 15

If e > f, and if g > h, then

  1. e – f > g – h
  2. e – g > f – h
  3. g + h < e – f
  4. e – g < f + h
  5. f + h < e + g


Explanation: If unequal quantities (g and h) are added to unequal quantities of the same order (e and f), the result is an inequality of the same order.

Question 16

If the value of Widget Company stock drops from $50 per share to $45 per share, what is the percent of decrease?

  1. 5%
  2. 10%
  3. 12%
  4. 15%


Explanation: The amount of the decrease is $5. The percent of the decrease is 5 ÷ 50 or 10 ÷ 100, which is 10%.

Question 17

Greenfield Middle School has 1,200 students. The number of students in each of the grade levels is equivalent. The following chart reflects evaluations of the students after being assessed for meeting grade-level math expectations at the beginning of the school year.

17.1. If a Greenfield Middle School student is selected at random, there is a ______ probability that he or she exceeds grade-level expectations for math.

  1. .28
  2. .29
  3. .30
  4. .33

17.2. Based on the previous chart, _____ 8th graders need improvement.

  1. 93
  2. 124
  3. 186
  4. 372

17.1  A

Explanation: The probability is .32 for a 6th grader, .29 for a 7th grader, .24 and for an 8th grader. Since the three grades have the same number of students, the probability for a student of any grade is the average of the three probabilities.

.32 +.29 +.24 = .283

Round to .28.

17.2  B

Explanation: Since there is an equal number of students in each grade level, there are 400 students in grade 8.

400 x .31 = 124

Question 18

If ↺m↻ = m² – m, what is the value of ↺-⅔↻ + ↺⅔↻?

  1. – ²/₉
  2. – ⅔
  3. ⁴/₉
  4. ⁸⁄₉


Explanation: Substitute ⅔ and -⅔  individually for m in the defined operation.

↺⅔↻ = ⁴/₉ – ⅔ = ⁴/₉ – ⁶/₉ = – ²/₉

↺-⅔↻ = ⁴/₉ + ⅔  = ⁴/₉ + ⁶/₉ = ¹⁰⁄₉

Then, add the two results together: -²/₉ + ¹⁰⁄₉ = ⁸⁄₉.

Question 19

If u + v = w, and if u – v = z, then u =

  1. ½ (w + z)
  2. w – z
  3. w + z
  4. ½ (w – z)
  5. ½ wz


Explanation: Add the two equations to find the answer:

u + v = w
u – v = z
2u = w + z

2u = w + z simplified is u = ½ (w + z)

Question 20

In the given figure, the centers of all three circles lie on the same line. The radius of the medium-sized circle is twice that of the smallest circle. If the radius of the smallest circle is 2 inches, the boundary of the shaded area is _____ inches.

  1. 12
  2. 12π
  3. 12π


Explanation: The medium circle has a radius of 4. Therefore, the diameter of the largest circle must be 12, which makes its radius 6.

The arc of a semicircle is half the circle’s circumference or πr. The length of the boundary of the shaded region can be found by adding the arcs of the three semicircles.

2π + 4π + 6π = 12π