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Data Sufficiency is a game as much as it’s a “problem.” Look at the statistics in the Veritas Prep Question Bank and you’ll see that most Data Sufficiency questions are created with a particular trap answer in mind and that at least 1-2 answer choices are rarely-if-ever chosen.

For example, look at these graphs:

In the first graph, the answer is E but the author desperately wants you to pick C. In the second, the answer is B but the author is baiting you hard into picking C. And in the third, the answer is C but the author is tempting you with E. In any of these cases, the strategy behind the question is as important – if not more important – than the math itself. Because it’s usually fairly easy for an average (or below) student to eliminate 1-2 answer choices on Data Sufficiency questions, the authors have to “get their odds back” through gamesmanship, by showing you a statement (or two) that look one way (sufficient or not) but that act counterintuitively. And to understand how to play this game well, it may be helpful to see Data Sufficiency through the lens of another popular game, the card game Hearts.

In Hearts, the goal of the game is to avoid getting “points”, and you get points when you end up with any hearts (one point each) or the Queen of Spades (13 points) after having taken a trick. And like with Data Sufficiency, there are really two ways to play: the way you’d play with a middle-schooler who’s learning the game, and the way you’d play with a group of adults who are each trying to win.

Playing Hearts with kids is like doing Data Sufficiency questions below the 550 level – you pretty much just play it straight. In Hearts, that means that when you don’t have any cards of the suit that was led, you try to get rid of your highest point-value cards immediately. If clubs are led and you don’t have clubs, you either get rid of the Queen of Spades if you have it, or you pick your highest heart and unload that. Your goal is to get rid of high cards and point cards quickly so that you end up with as few points as possible.

But if you’re playing with adults, you have to consider the possibility that someone may be trying to “Shoot the Moon” – getting *all* of the points cards in which case they get 0 points and every other player gets 26. What might seem like a counter-intuitive strategy to a 12-year old is often quite necessary when you suspect an opponent may be trying to shoot the moon: even though you may have a chance to get rid of your king-of-hearts, you might hold on to it because you want a high heart in case you need to “win” one of the last tricks to stop the opponent from getting all of the hearts. When you’re playing with adults (or attempting Data Sufficiency questions in the high 600s and into the 700s), you need to see the game with more nuance and develop an instinct for when to avoid the “obvious” play to save yourself from a more-catastrophic outcome.

This is especially true when you notice something suspicious from your opponent; if in one of the first few hands an opponent leads with, say, the jack of hearts, that’s a suspicious play. Why would she fairly-willingly open herself up to taking four points? Or if the first time a heart is played, an opponent swoops in with a high card of the suit that was led, but you know they probably have a lower card that would have let them avoid taking the heart, you again should be suspicious. In either of these cases, an astute player will make a mental note to hold back a high card or two just in case shooting-the-moon is in play. Playing hearts as an adult, you’re often playing the opponent as much as you’re playing the cards.

How does this apply to Data Sufficiency?

Consider this question:

Is a > bc?

(1) a/c > b

(2) c > 3

Playing “middle school hearts”, many test-takers will run through this progression:

Step one: If you multiply both sides by c, you get a > bc so this looks sufficient*. The answer, then, would be A or D.

Step two: Forget everything you learned about statement 1 since you’ve already made your decision about it. Statement 2 is clearly insufficient on its own, so the answer must be A*.

(*we know the math here is slightly flawed; demonstration purposes only!)

But here’s how you’d play the game as an adult, or as a 700-level test-taker:

Step one: Same thing – if you multiply both sides by c you’ll get a > bc, so this one looks sufficient.

Step two: Wait a second – statement 2 is absolutely worthless. And statement one wasn’t *that* hard or interesting. Maybe the author of this question is “shooting the moon”…

Step three: Look at both statements together, reconsidering statement 1 by asking myself if statement 2 matters. If statement 2 is true and c is, say, 10, then a/10 > b would mean that a > 10b, so this still holds. But what if c is -10, and statement 2 is not true. a/(-10) > b would mean that when I multiply both sides by -10 I have to flip the sign, leaving a < -10b. This time it’s not true. Statement 2 *seems* worthless but in actuality it’s essential. Statement 1 is not sufficient alone; as it turns out I need statement 2.

What’s the difference between the two methodologies?

The 500-level, “middle school hearts” approach – NEVER consider the statements together unless they’re each insufficient alone – leaves you vulnerable to the author’s bait. On hard questions, authors love to shoot the moon…that’s their best chance of tricking savvy test-takers.

The 700-level, “playing hearts with grownups” approach seems counterintuitive, much like saving your king of hearts and knowingly accepting points in a hearts game would seem strange to a seventh-grader. But it’s important because it saves you from that bait. On a question like this, it’s easy to think that statement 1 is sufficient; abstract algebra is great at getting your mind away from numbers like negatives, zero, fractions… But statement 2′s worthlessness (ALONE) functions two ways: it’s a trap for the unsuspecting 500-level types, and it’s a reward for those who know how to play the game. That worthless statement 2 is akin to the author leading a high heart early in the game – the novice player sees it as a freebie; the expert considers “why did she do that?” and re-examines statement 1 by asking specifically “what if statement 2 weren’t true; would that change anything?”.

Remember, when you’re taking the GMAT you’re playing against other very-intelligent adults, and so the authors of these questions have a responsibility to “shoot the moon”. While the rules of the game dictate that you don’t want to consider the statements together until you’ve eliminated A, B, and D, there’s a caveat – if you have reason to believe that the author of the question is trying to trick you (which is very frequently the case on 600+ level questions), you have to consider what one statement might tell you about the other; you have to play the game.

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Before we get started, be sure to take a look at Part I of this article. Number properties concepts come across as pretty easy, theoretically, but they have some of the toughest questions. Today let’s take a look at some properties of prime numbers and their sum. Note that don’t try to “learn” all the takeaways you come across for number properties – it will be very stressful. Instead, try to understand why the properties are such so that if you get a question related to some such properties, you can replicate the results effortlessly.

To start off, we would like to take up a simple question and then using the takeaway derived from it, we will look at a harder problem.

Question 1: Which of the following CANNOT be the sum of two prime numbers?

(A) 19

(B) 45

(C) 68

(D) 79

(E) 88

Solution: What do we know about sum of two prime numbers?

e.g. 3 + 5 = 8

5 + 11 = 16

5 + 17 = 22

23 + 41 = 64

Do you notice something? The sum is even in all these cases. Why? Because most prime numbers are odd. When we add two odd numbers, we get an even sum.

We have only 1 even prime number and that is 2. Hence to obtain an odd sum, one number must be 2 and the other must be odd.

2 + 3 = 5

2 + 7 = 9

2 + 17 = 19

Look at the options given in the question. Three of them are odd which means they must be of the form 2 + Another Prime Number.

Let’s check the odd options first:

(A) 19 = 2 + 17 (Both Prime. Can be written as sum of two prime numbers.)

(B) 45 = 2 + 43 (Both Prime. Can be written as sum of two prime numbers.)

(D) 79 = 2 + 77 (77 is not prime.)

79 cannot be written as sum of two prime numbers. Note that 79 cannot be written as sum of two primes in any other way. One prime number has to be 2 to get a sum of 79. Hence there is no way in which we can obtain 79 by adding two prime numbers.

(D) is the answer.

Now think what happens if instead of 79, we had 81?

81 = 2 + 79

Both numbers are prime hence all three odd options can be written as sum of two prime numbers. Then we would have had to check the even options too (at least one of which would be different from the given even options). Think, how would we find which even numbers can be written as sum of two primes? We will give the solution of that next week. So the takeaway here is that if you get an odd sum on adding two prime numbers, one of the numbers must be 2.

Question 2: If m, n and p are positive integers such that m < n < p, is m a factor of the odd integer p?

Statement 1: m and n are prime numbers such that (m + n) is a factor of 119.

Statement 2: p is a factor of 119.

Solution: First of all, we are dealing with positive integers here – good. No negative numbers, 0 or fraction complications. Let’s move on.

The question stem tells us that p is an odd integer. Also, m < n < p.

Question: Is m a factor of p?

There isn’t much information in the question stem for us to process so let’s jump on to the statements directly.

Statement 1: m and n are prime numbers such that (m + n) is a factor of 119.

Write down the factors of 119 first to get the feasible range of values.

119 = 1, 7, 17, 119

All factors of 119 are odd numbers. So (m + n), a sum of two primes must be odd. This means one of m and n is 2. There are many possible values of m and n e.g. 2 and 5 (to give sum 7) or 2 and 15 (to give sum 17) or 2 and 117 (to give sum 119).

Note that we also have m < n. This means that in each case, m must be 2 and n must be the other number of the pair.

So now we know that m is 2. We also know that p is an odd integer. Is m a factor of p? No. Odd integers are those which do not have 2 as a factor. Since m is 2, p does not have m as a factor.

This statement alone is sufficient to answer the question!

Statement 2: p is a factor of 119

This tells us that p is one of 7, 17 and 119. p cannot be 1 because m < n < p where all are positive integers.

But it tells us nothing about m. All we know is that it is less than p. For example, if p is 7, m could be 1 and hence a factor of p or it could be 5 and not a factor of p. Hence this statement alone is not sufficient.

Answer (A)

Something to think about: In this question, if you are given that m is not 1, does it change our answer?

Key Takeaways:

- When two distinct prime numbers are added, their sum is usually even. If their sum is odd, one of the numbers must be 2.

- Think what happens in case you add three distinct prime numbers. The sum will be usually odd. In case the sum is even, one number must be 2.

- Remember the special position 2 occupies among prime numbers – it is the only even prime number.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

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The ACT Science Test is a source of anxiety for many students, and it’s easy to understand why. After all, three or four years’ worth of high school science is a lot to review! How well do you really remember that lab activity about springs from ninth grade physics? Fortunately, the test doesn’t test the content of high school science nearly as much as it tests a very narrow set of skills developed in high school math and science classes. Here are five things you need to know in order to succeed on on the ACT Science test:

1. This Is Not A (Science) Test

In fact, the ACT Science test might be more accurately described as a hybrid of a data analysis and reading comprehension test. A student’s ability to deal with graphs and tables is far more important to success on the test than her ability to label a diagram of a cell or balance a chemical equation. Any formulae needed to answer questions (and there are very few) are provided in the passage, and you won’t need to learn any complicated scientific concepts on the spot.

On the flip side, don’t skip over the data in the passage and start answering questions because you think that you already know all there is to know about acids and bases. Just like in the ACT Reading test, make sure you’re only using the information provided.

2. Manage Time Wisely

The Science test consists of 40 questions in 35 minutes, and pacing is a big issue for many students. The most efficient way to work through the test is to complete the data analysis passages first, followed by the research summaries, and leave the opposing viewpoints passage for last. You might notice that this list moves from figure-heavy to text-heavy passage types. That’s because it’s easier to pull information quickly from a table or a graph than from a chunk of text. Because all of the questions are weighted equally, starting with the quicker, figure-heavy passages makes the best use of your time.

3. Don’t Read The Passage Until You Know What You’re Reading For

Because you’re not terribly concerned with gaining a deep understanding of the scientific topic discussed by the passage (and don’t have time to, even if you wanted to), you need to use the questions to guide our use of the passage. Lots of questions will tell you exactly which data representation or experiment to look at with phrases like “According to Figure 2…”

4. Don’t Let The Jargon Get You Down

When many students skim a science passage and see something like “Helicobacter pylori,” panic sets in. So, what the heck is Helicobacter pylori? For the purposes of the ACT, it’s “the thing measured by the graph with an axis labeled ‘Helicobacter pylori,’” plus any plain-English description given by the passage (for the purposes of the real world, it’s both a bacterium found in the stomach and two decent Scrabble words). The good news is that the only things you’ll need to know about Helicobacter pylori to knock out the passage are 1) whatever the passage explicitly tells you about it and 2) whatever information the graphs and tables present about it.

5. If You’re Doing Complicated Math, You’re Doing Something Wrong

You’re not allowed to use a calculator for ACT Science, which is fine because there’s absolutely no reason to use one. The only math you’ll have to do is very simple, and the numbers will all come directly from the data provided by the passage.

With these tips you’ll be ready to rock ACT Science (even if geology isn’t your area of expertise)!

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Harvey Mudd College, located in Claremont, California, is one of the top science and engineering liberal arts colleges in the country and comes in at #25 on the Veritas Prep Elite College Rankings. It is one of the five colleges of the Claremont College Consortium, known as the 5Cs, along with Claremont McKenna, Scripps College, Pomona College, and Pitzer College.

The coeducational Harvey Mudd College was founded in 1955 at a time when the nation was turning its attention to the space race and encouraging students to focus their educations on math and science. The university presents a unique approach of focusing on the humanistic aspects of becoming elite scientists, mathematicians, and engineers. Their Clinic Program, which originated in 1963, allows teams of undergraduate students to address difficult social problems posed by the non-profit sector, business and industry, and even the government, through scientific and technological research. The result has been one of the highest rates in the country of graduates who go on to earn PhDs in their fields.

Harvey Mudd academics takes a three-pronged approach to education. They set a foundation in math and science with their Common Core STEM program. In order for students to understand the interconnectedness of STEM disciplines, all students must take classes in each – biology, chemistry, computer science, engineering, math, and physics. In keeping with their philosophy that the purpose of science and technology is to serve humanity, students also complete rigorous coursework in humanities, social sciences, and the arts.

At Harvey Mudd, they believe that “technology divorced from humanity is worse than no technology at all.” It is from this perspective, which marries STEM with liberal arts, that students are fundamentally prepared to then move on to the third prong – a chosen major where they develop competence and achieve excellence in a single area of expertise. Majors at Harvey Mudd include biology, chemistry, computer science, engineering, mathematics, physics, and dual majors chemistry and biology, math and computer science, and mathematical and computational biology. Since there is no graduate school, the rigorous curricula are delivered in a highly personal environment where fewer than 800 students are enrolled.

The stereotypical personality of a “Mudder” is extraordinarily talented in math, science, and technology, politically liberal and socially conscious, with an insane work ethic, and an uncanny ability to throw a great party. Men outnumber women by almost 4:1, but all-female Scripps College is adjacent to the campus, and all five campuses are within walking distance of one another. The Mudd student population is open-minded, diverse racially and ethnically, and actively supportive of gender equality.

Greek life on campus is non-existent, but over-the-top parties – some complete with artificial waterfalls, are common and draw students from across the consortium of colleges. Add in the nice weather year-round, late night college sponsored pizza, carne asada nights in the dining hall, and numerous clubs, intramural sports, and campus activities, and you have a recipe for a great college experience. Just be sure you are mentally prepared to also work harder than you have ever worked in your life.

In the 5-college Claremont College Consortium, Claremont McKenna, Harvey Mudd College, and Scripps College form a single NCAA Division III athletic program. They belong to the South California Intercollegiate Athletic Conference. The athletic program is represented by 21 intercollegiate teams, both men and women. Student-athletes demonstrate the same dedication to excellence in sports as they do in academics. Fourteen of their twenty-one teams have finished seasons in top ten positions in the country since their inception in 1958. However, due to the rigor of the college’s academics, the only way to excel at a sport too is to be highly disciplined, self-motivated, and organized. The school also has a strong commitment to intramurals, with inner tube water polo being one of the most popular.

Maybe one of the coolest and oldest of Harvey Mudd traditions is The Foster’s Run. Unicycles became a thing on campus in the 1970s. Before long, a unicycle club called Gonzo Unicycle Madness was formed, which still thrives today. The Foster’s Run is a 9.6 mile unicycle ride to what was once Foster’s Donuts and is now the Donut Man donut shop. The reward for making the trip is delicious strawberry donuts. Once riders return to campus, they participate in “the shakedown,” which is essentially getting the kinks out after a nearly twenty-mile unicycle ride.

The student group, Increasing Harvey Mudd’s Traditional Practices, was organized to revive and preserve this and other on-campus traditions, which also include Wednesday Nighters, Friday Nooners, 5-Class Competition, and occasionally pranking Cal-Tech. If you’re ready to add two parts rigor and one part amusement, this could be your school.

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In 2016 the SAT will undergo major changes for the first time in over a decade. Test takers can expect a less obscure vocabulary, more focused math sections, and a shorter test overall. Our own Shaan Patel was quoted in an article from U.S. News: “My opinion is this test will be easier than the current SAT and the College Board is betting on more students taking the SAT because of that.”

Experts and thought leaders in education are weighing the pros and cons of these upcoming changes but, right now your best strategy is to be prepared. Take a look at the infographic below for a comprehensive break down of the key changes coming with the new SAT. Key points include:

- No More Required Essay

- Presenting Evidence

- More Varied Text

- Less Obscure Vocabulary

- More Focused Math Sections

We’ve broken it all down for in this infographic that details key differences between the old and new SAT, as well as the history of the SAT since its launch more than a century ago:

(Click on the infographic below to enlarge it!)

Want more details about the SAT changes coming in 2016? We’ve put more info, including a detailed FAQ devoted to the new SAT, here!

ForumBlogs - GMAT Club’s latest feature blends timely Blog entries with forum discussions. Now GMAT Club Forums incorporate all relevant information from Student, Admissions blogs, Twitter, and other sources in one place. You no longer have to check and follow dozens of blogs, just subscribe to the relevant topics and forums on GMAT club or follow the posters and you will get email notifications when something new is posted. Add your blog to the list! and be featured to over 300,000 unique monthly visitors

We get lots of questions from applicants about the best time to return to business school. While this is certainly an individual assessment, and one size does not fit all, from an admissions perspective, there are good seasons and bad seasons to maximize your odds of acceptance.

Six years ago, during a time none of us will soon forget, the air was let out of the economy. A similar rushing sound was also heard in the country’s top business schools, but it was the sound of people rushing in. Applications peaked the next year as everyone ran to hide from old man recession for two years in hopes the job market would come back while they were getting smarter and checking the graduate school box.

Simple mathematics demonstrated even to the casual observer that as the applicant pool increased, the odds of getting in went down. We saw the competitive nature of an already very competitive arena turn quickly into a bloodbath: GMAT scores rose, applications overwhelmed admissions officers, and thousands of perfectly qualified applicants were denied their dream schools. As Wharton said, “Everyone who applies here is qualified, so we have the luxury of being choosy.” Suddenly every unsuspecting 27 year old applicant was competing with top performing bankers, accountants and consultants from the best firms, many of whom were laid off in the throes of the great recession. Business schools had the proverbial pick of the litter.

Things did not get better quickly in the economy, so for about four years, or two full time MBA cycles, this phenomenon perpetuated. As the frozen GDP began to thaw, however, and people began to find jobs again, the applications to MBA programs started to wane. Suddenly everyone seemed to value their jobs, ostensibly from either having lost it temporarily, or come close enough to feel the fear of losing it, so giving it up voluntarily to return to grad school did not seem as appealing. Running and hiding was no longer necessary. So goes the business school admissions cycle.

Fast forward to now, and consumer sentiment is back on track, the stock market is humming and jobs are finally fairly easy to get. And guess what? Your odds of getting into your dream school are better. It’s always tough to be a contrarian. Swimming upstream is never easy, but things in life worthwhile rarely are. If you have the stomach to sit out a couple of years of percolating markets, you might just find your way into a school where a couple of years ago, you would not have had a chance. Business cycles are pretty reliable, and the chances of another big downturn so soon after the last one are slim. This means two or three years from now, the job market will likely be even stronger, which would time out well with your graduation. So dust off your GMAT score and get back on the horse. The time to apply is nigh!

If you want to talk to us about how you can stand out, call us at 1-800-925-7737 and speak with an MBA admissions expert today. Click here to take our Free MBA Admissions Profile Evaluation! As always, be sure to find us on Facebook and Google+, and follow us on Twitter!

Scott Bryant has over 25 years of professional post undergraduate experience in the entertainment industry as well as on Wall Street with Goldman Sachs. He served on the admissions committee at the Fuqua School of Business where he received his MBA and now works part time in retirement for a top tier business school. He has been consulting with Veritas Prep clients for the past six admissions seasons.

ForumBlogs - GMAT Club’s latest feature blends timely Blog entries with forum discussions. Now GMAT Club Forums incorporate all relevant information from Student, Admissions blogs, Twitter, and other sources in one place. You no longer have to check and follow dozens of blogs, just subscribe to the relevant topics and forums on GMAT club or follow the posters and you will get email notifications when something new is posted. Add your blog to the list! and be featured to over 300,000 unique monthly visitors

The essay begins the SAT and it is easy to feel overwhelmed by the prospect of writing a five paragraph essay in 25 minutes, but there are a few steps that can make the essay a piece of cake!

1. Make An Essay Template

The time spent figuring out how to structure an essay on the SAT is time wasted. This may sound counter intuitive as structure is a big part of what the SAT graders are evaluating, but it is this reason exactly that makes the structure of the essay the first thing that can be systematized and recycled. The essential make up of a five paragraph essay is simple. There is an introduction which presents the topic, states the thesis, acknowledges the opposition, and lays out how the essay will argue its point, three body paragraphs which use examples to support the thesis, and a conclusion which restates the thesis and briefly reminds the reader what it has just read.

This is all a five paragraph essay is! Because it is so formulaic in its structure, and because the topics are always essentially taking a side on some issue, the majority of the essay can be “written” beforehand in the form of a template. By plugging in this formula, it is easy to essentially create a template for what to say. Here is an example introduction using a template:

“The notion that [Assignment] has been demonstrated in numerous contexts to be [true/false] (Thesis). Though there are some who would argue that [whatever opposition might say], this perspective does not adequately reflect the intricacies and complexities of [topic] (Acknowledgment of opposition). ([General statement about why topic is important or why thesis is true]). Three demonstrations of [thesis] are [Example One], [Example Two], and [Example Three] (How Thesis Will Be Defended).

The entire essay can essentially be sketched out in advance. By determining the structure in advance, more time can be dedicated to showing how the examples demonstrate the thesis.

2. Skip The Directions

This is true throughout the SAT, but is especially important on the essay because time is so limited. There is a lot of information on the directions page, but it can all be boiled down into the instructions “write an argumentative essay”, which will not change from test to test and can thus be assumed. EVERYTHING ON THE PAGE, the quote, the parameters for writing a good essay, the amount of time given for writing, all that stuff is a waste of time EXCEPT the ASSIGNMENT. The assignment is the question that is being asked and it is the only thing necessary to attack the essay. If the assignment asks “Should people care about others outside their own borders?”, this is all that is necessary to make the template a real introduction. Given this assignment, the template becomes:

“The notion that people should care for those outside of their own borders has been demonstrated in numerous contexts to be true. Though there are some who would argue that it is most important to concentrate resources on those who are closest in proximity and in culture, this perspective does not adequately reflect the intricacies and complexities of the interconnected world of today. Neglecting those outside of our own borders, not only isolates people from analyzing and alleviating hardships that they themselves might one day face, but also creates a feeling of isolation which runs counter to the current trend toward a more interdependent and interconnected world. Three demonstrations which exemplify the importance of caring for people outside of our own borders are the Indian independence movement, the American revolution, and the events of the second world war.”

As is demonstrated above, there is still some room to word things creatively, but any excess time and brain power can be allocated to making these variable sections really pop rather than on reading directions.

3. Study Potential Examples Beforehand

Because of how broad the questions on the SAT can be, virtually any topic with a lot of substance to it can be applied to a number of potential SAT questions. Questions like “Is it more important to be decisive or to consider things carefully?” or “Should people put family first?” can be argued using nearly any work of literature or historical event that involves decisions or family (which pretty much includes all of history and literature).

To “study” for this section pick ten or so potential examples (books, historical events, current events) that have already been covered in classes and review the main points of these examples. The better these examples have been prepared, the more uses for them will present themselves.

4. Relate Topic Sentences Back To The Thesis

At the beginning of each body paragraph, each topic sentence should relate back to the thesis. Here is what not to do:

“The Indian Independence movement showed the resilience of the Indian people in the face of the tyrannical British.”

This isn’t factually false, but it doesn’t show the thing that is being argued. Every paragraph exists to argue for the thesis; therefore each topic sentence should relate the example to the thesis.

“The power of the international community in aiding the Indian Struggle for independence clearly illustrates of the importance of caring for those outside of one’s own borders.”

This second example states what the example is and that it demonstrates the thesis. Now all the test taker needs to do is to explain how this example shows the importance of caring for people abroad. The test taker will likely state how foreign pressure was instrumental in forcing the British to reduce their brutal repression of non violent protests and in humanizing the Indian people in their struggle. With this stated, he or she can just rinse and repeat this same strategy with the other two body paragraphs.

5. Keep The Conclusion Simple

The work is done by the time of the conclusion. The introduction shows the structure of the essay, the body paragraphs show how the student’s knowledge can be applied to argue a point of view, but the conclusion is really just to restate the thesis and what has already been argued. Keeping with the topic of caring for those outside of one’s own country, a sample conclusion might look something like this:

“The importance of caring for those outside one’s own borders is clearly demonstrated in both the Indian and African American struggles against tyranny and in the global struggle of World War Two. It is in conflict with the demands of the interconnected world that humans inhabit to attempt to cloister oneself away and not participate in the struggles of those outside of one’s own country. Only by caring for those in faraway lands can people hope to create a world where all human beings are safe and able to pursue their own happiness.

This conclusion isn’t trying to do too much. It essentially parrots back the stuff that was stated in the introduction and relates all the examples back to the thesis. This is all the conclusion needs to do. When students get into their desired college and they write their honors thesis they can use their conclusions to draw important connections between examples and put their own unique spin on all the information already provided, but for the SAT, keep it simple.

By using these steps students can ace the SAT essay and start off the test feeling successful. All the work for the SAT essay is done in advance so study some examples and write a few essays to develop a template that works. Happy studying!

David Greenslade is a Veritas Prep SAT instructor based in New York. His passion for education began while tutoring students in underrepresented areas during his time at the University of North Carolina. After receiving a degree in Biology, he studied language in China and then moved to New York where he teaches SAT prep and participates in improv comedy. Read more of his articles here, including How I Scored in the 99th Percentile and How to Effectively Study for the SAT.

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When answering sentence correction problems on the GMAT, it’s very common to use your ear as a barometer of how the answer choice sounds. Particularly for native English speakers, this is often the number one way they approach any given sentence. The problem with this strategy is that sentence correction is often much more about the meaning than about the grammar. By extension, the test makers of the GMAT know they can fool many students by simply making the correct answer choice unappealing to the students’ ears (Won’t get fooled again!).

Anything that makes a sentence sound more awkward than it should is fair game to try and get test takers to pick the wrong answers. Some strategies come back more often than others, and today I want to discuss these types of errors as it pertains to the timeline of a sentence. Students often have preconceived notions hammered in during high school that a sentence must always be in the same tense, no matter what. While this is a nifty rule of thumb, it doesn’t have to be the case in every sentence.

As an example, consider a student studying for the GMAT. The student could say “I have studied for the GMAT” or “I will study for the GMAT”. Both of these options make sense. What about “I will be ready for my GMAT next week because I have been studying for months”? This sentence is also fine, even though one verb tense is in the future and the other is in the present perfect continuous. As long as the phrase makes logical sense and what is being described in the past took place in the past, the sentence is valid.

The trick on the GMAT that gets students confused is that you have to pick one sentence out of the five answer choices. However, none of them might be exactly what you’re expecting. In other words, if given “carte blanche”, I could rewrite this sentence in a much clearer way than any of these five middling choices. That’s half the difficulty, though, because you have to pick the sentence from among the choices that contains no grammatical mistake, even though you don’t necessarily like everything in the answer choice.

Let me highlight this with a sentence correction question that regularly gives students fits:

A 1999 tax bill changed what many wealthy taxpayers and large corporations are allowed to deduct on their tax returns.

(A) changed what many what many wealthy taxpayers and large corporations are allowed to deduct on their tax returns

(B) changed wealthy taxpayers’ and large corporations’ amounts that they have been allowed to deduct on their tax returns

(C) is changing wealthy taxpayers’ and large corporations’ amounts that they have been allowed to deduct on their tax returns

(D) changed what many wealthy taxpayers and large corporations had been allowed to deduct on their tax returns

(E) changes what many wealthy taxpayers and large corporations have been allowed to deduct on their tax returns

Going through the answer choices, it seems fairly clear that 1999 is in the past. Whether it’s 2014 or 2015 or whenever, you would not reference 1999 in the future (unless you’re Prince). As such, we can eliminate answer choices C and E because both use the present tense and make it sound like this bill is happening right now and not 15+ years ago.

Similarly, answer choice B changes the meaning of the sentence. The sentence is saying the bill will change what people are allowed to deduct, whereas answer choice B modifies the meaning to just the amount they are allowed to remove. There is a significant difference between deducting 500$ for school expenses or 700$ for school expenses versus deducting school expenses or capital gains expenses. There is a niche corner case where the two may have exactly the same meaning, but the original sentence has a much broader definition and thus can’t be pigeonholed into answer choice B.

This leaves the two most common answer choices: A and D. If you’re going by sound, you likely think that answer choice D is the correct answer. However, even though answer choice D sounds like what you’d expect to hear, it creates an illogical timeline. Let’s look at a sample timeline and determine when the changes were made:

Answer choice D uses the past perfect continuous (had been allowed) which can only be used if the event described happened before another event in the past. Example: By the time I took the GMAT in 2007, I had been studying for over two months. You cannot have a tax code change in 1999 that affects the years prior to 1999. Otherwise everyone would have to resubmit their taxes for the past 6 years to reflect the change. Any tax code change can only come change future tax years.

Answer choice D meaning, with the period having been changed in red.

Answer choice A, even though it uses the present tense (are) can be considered grammatically correct here as long as the law wasn’t repealed. Since there is no indication of the law having been changed, answer choice A is a valid (although awkward sounding) way of rewriting this sentence.

I would like to further illustrate this point using the Stampy example of the seminal Simpsons episode where Bart gets an elephant. In the episode (titled Bart gets an elephant), Homer realizes that he can make money off the elephant, and decides to charge people 1$ to see the elephant and 2 $ to ride the elephant. Upon realizing that he’s still losing money, he updates his prices to 100$ and 500$ respectively. Since this drives away all of his business, Homer visits the homes of his friends and tells them:

Homer: “Millhouse saw the elephant twice and rode him once, correct?

Millhouse’s dad: “Yes, but we already paid you the 4$”

Homer: “That was under our old price structure. Under our new price structure, you owe me 700$”

Millhouse’s dad: “Get out of my house”

Hopefully this little skit helped drive home the point I’m trying to make. You cannot retroactively change what people can deduct. You can only change things going forward. Answer choice A may not be the preferred way to rewrite this sentence (Example: “changed what people would be allowed to deduct” would have been clearer), but there is no grammatical error contained within it. When it comes to sentence correction, make sure you understand the logic of the sentence and don’t depend on your ear as your only line of defense.

Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.

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As coincidence would have it, within a couple weeks after the College Board announced major changes to the SAT (coming in 2016), I was already planning on taking the SAT at Lower Merion High School (which, as Kobe Bryant likes to point out, is the high school he and I went to…). Sure, I am 44-years old, but I take the SAT often to stay up on trends and changes to the test and to show students and parents that their tutor is capable of a 99th%-ile score in any of the three sections.

The “redesign” announcement has inspired fear in the hearts of students, parents, counselors and some tutors (not me, of course). With that angst lurking, it was more important than ever that I take last month’s test. Sure, 2016 is 2 years away, but that didn’t stop the College Board from somewhat jumping the gun on the changes. Last month’s SAT featured major changes consistent with the College Board’s announced plans.

Before I inadvertently cause more fear, let me be very clear: the changes appeared in the SAT’s “Experimental” section (which never counts towards a student’s score). However, the College Board (administrator and writer of the test) never publicizes which of the 10 sections is the unscored experimental section. Usually, it blends in very well with the rest of the scored sections and therefore provides the College Board with useful data about how students perform on that section under high-stakes conditions.

However, for a professional like myself, the “new SAT” experimental section stood out like a sore thumb. For example, one reading passage had only one single question (I’ve never seen a test in the last several years with fewer than two questions). Another reading passage had a whopping 16 questions (the most questions to follow a reading passage has been capped at 13 questions over the last several years). In the 16-question reading passage, there were two instances of questions that asked for specific evidence regarding one’s answer to the previous question. This was consistent with what the College Board announced regarding the reading sections (I’ll provide a list of changes later in this post). Others reported changes in the math and writing sections.

What does this mean for your child’s upcoming appointment with the SAT? It won’t likely mean much if your child is in the 10th grade or higher. But if your child is in 9th grade, he or she will be in the first class to take the Next-Generation SAT. How should you prepare? Quite frankly, there isn’t too much you should do differently. The questions will still largely be similar, and most of the techniques in the Veritas Prep curriculum will not change substantially. There will be some math sections where calculator use is prohibited, so I advise you to cutback your child’s calculator dependency. And there will certainly be changes and updates to specific sections of any good SAT course to narrowly tailor to the redesigned test. But the same study habits, thought processes, and content emphases over the next few years of your child’s high school career will continue to pay off.

Some of the announced changes include:

– The vocabulary section will be made more relevant to modern language (i.e., words that have a chance of making it into a conversation this decade. See you later “capricious!”)

– The reading questions will emphasize the use of evidence from the passage just read (as discussed above, this change was on full display in my experimental section).

– The essay will no longer affect one’s overall score.

– Students should answer every question, as there will be no longer be a penalty for wrong answers. This should come as welcome news to almost everyone, as the “incorrect guess” penalty has long been a source of stress for students and teachers.

– The new exam will narrow its mathematical focus to a few areas, including algebra, deemed most needed for college and life afterward. A calculator will be allowed only on certain math questions, instead of on the entire math portion at current (again – as a parent you may want to wean your child off of calculator reliance over the next few years)

– The overall score will be reverted back to its original level of 1600, instead of the current 2400. The reading and writing sections will be collapsed into one section worth 800.

We’ll continue to post updates as these changes are previewed publicly or appear on experimental questions. In the meantime, those taking the “current” SAT should avoid any undue stress about changes that will not affect their own tests or scores, and those who will take the redesigned test should continue to excel in high school (and junior high) and build sound math, reading, and English skills while they wait for redesigned SAT curriculum to help harness it toward that 1600 score!

Steve Odabashian received his BA in Economics from the University of Virginia and then went on to receive his JD at Villanova. He has worked in Tokyo as a foreign attorney, done pro bono work for the Committee of Seventy in several Philadelphia elections, and he is a well known pianist and comic entertainer in Philadelphia. Steve has been teaching for Veritas Prep since 2004.

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What makes the GMAT difficult? For most examinees, the time pressure is arguably the biggest factor; given unlimited time, most 700-level aspirants could get most problems right, but with that clock ticking and time of the essence we’re all vulnerable to silly mistakes, mental blocks, and the need to give up on hard questions.

So how can you overcome the pressures of pacing? Try this three-step method:

1) Take Your Time

This may seem a bit counter-intuitive if you’re pressed for time, but the GMAT scoring algorithm so heavily punishes you for missing “easy” questions that you can’t afford to fall victim to silly or careless mistakes. Most test-takers could finish between 32-34 quant questions and 36-38 verbal questions in the 75 allotted minutes, but it’s that 37 quant / 41 verbal question allocation that forces examinees to budget time. If, for example, on quant you’d be great if you could average 2:20 per question instead of the allotted 2:05, that extra 15 seconds you’d like per question may well be your Achilles’ heel if, in your haste to get down closer to 2 minutes per question, you fall victim to:

-Silly calculation mistakes

-Setting up an equation incorrectly

-Leaving a problem one step short and picking the trap answer

-Answering “the wrong question” (e.g. they asked for y, you solved for x)

These mistakes, as you’ve likely seen in your practice tests and homework sets, are quite common, so make sure that you’re aware of them and know to slow down to avoid them. Double check your work, which can largely go wrong in the first 20-30 seconds of a problem (setting up a problem incorrectly) or the last 20-30 seconds (answering the wrong question, skipping a calculation step because it looks like you’ll get right to one answer choice). Know your common mistakes and spend that extra 10-15 seconds double-checking for them. Too many examinees, knowing that they’d need 10% more time than they have, do a “90% job on 100% of questions” (a lot of wrong answers) instead of a “100% job on 90% of questions” (making sure that when they can get a question right, they do. As we say often on the GMAT, your floor is more important than your ceiling – missing easy questions hurts you much more significantly than correctly answering hard question helps you. So step 1 on pacing – make sure that you take the time you need to successfully finish problems on which you’ve done most of the work right.

2) Plan to Guess

Here’s where you get the time back. If you still know that the above strategy – take the time that you need – will leave you 5-6 minutes short of where you’d need to be to finish the section, then save that time by knowing that up to 3-4 times per section you’ll just guess early on a problem to bank that time for when you really need it. Why does this work? If you’re doing well on a section by successfully answering most of those questions within your ability level, you’re going to see some extremely difficult questions as your “reward” based on the adaptive algorithm. You WILL get questions wrong, and the key is to not invest too much time in questions that you were probably going to get wrong anyway. The problem with guessing is much more psychological than real – when you get stuck on a problem and “have to” guess, you get that panic feeling in your mind and it shakes your confidence for future questions. Plus you’ve probably spent up to your average pace-per-question (if not more) by that point, so you’re doubly worried…time is ticking away *and* you just had to blow a guess.

The remedy? Give yourself up to 4 “free passes,” questions on which you’ll just guess in the first 20-25 seconds if you realize that it’s probably beyond you and/or it will probably sap a lot of time. (For example, plenty of 750+ scorers have admitted that “hard to start” geometry problems fall into this category for them…geometry with detailed figures can be very time-consuming, so if they don’t see the path early on they know to just save that time for something more concrete) By consciously using a “free pass” instead of nervously venturing a guess, you own the guess as a strategy and not a cop-out, and you’ll save that time for when you need it and can best use it for correct answers.

3) Have a Pacing Plan

How do you know when you need to guess? Segment each section into approximate quarters and have benchmarks for where you’ll want to be. Since the clock ticks down from 75 minutes, have those benchmarks in mind the way you’ll see them:

Quant

After 10 questions —- 53 minutes remaining*

After 20 questions —- 33 minutes remaining

After 30 questions —- 14 minutes remaining

(which leaves about 2 minutes per question)

Verbal

After 10 questions —- 55 minutes remaining*

After 20 questions —- 36 minutes remaining

After 30 questions —- 18 minutes remaining

(which leaves about 1:40 per question)

(*You can adjust these benchmarks to your liking; here we’re using a little more time in the initial 10 questions, not because “they’re more important” as the myth goes, but more because you can’t use any additional time at the end of the section, so if you’re going to err on pacing it’s better to get the early questions right and hustle a little later than it is to make silly mistakes early, banking time that won’t help you later.)

Whenever you’re more than a minute or so behind your desired pace, that’s when you’ll want to look at using a “free pass” within the next 4-5 questions to get back on track. By having a plan to check every 10 questions, you’ll avoid that pressure (and wasted time) that comes from calculating your pace-per-question frequently throughout the test (seriously, people do this – they’re so worried about not having enough time that they waste valuable time doing extra, irrelevant math problems!!) and you’ll have a contingency plan in place so that you’re not panicked if you are a little behind. If you’re behind, you have a “free pass” in your back pocket to help get back to where you want to be.

Pacing on the GMAT is tricky for everyone – that’s a major factor of what makes it “the GMAT.” But if you follow this process, you can make the best out of that limited time and maximize your chance of success. Remember, 75 minutes per section is hard for just about everyone, so even if you’re not comfortable with the pacing but you have a better plan for how to use that scarce resource, pacing can be your competitive advantage.

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Continuing our discussion on numberproperties, today we will discuss how factorials affect the behavior of odd and even integers. Since we are going to deal with factorials, positive integers will be our concern. Using a question, we will see how factorials are divided.

Question: If x and y are positive integers, is y odd?

Statement 1: (y+2)!/x! = odd

Statement 2: (y+2)!/x! is greater than 2

Solution: The question stem doesn’t give us much information – just that x and y are positive integers.

Question: Is y odd?

Statement 1: (y+2)!/x! = odd

Note that odd and even are identified only for integers. Since (y+2)!/x! is odd, it must be a positive integer. This means that x! must be equal to or less than (y+2)!

Now think, how are y and y+2 related? If y+2 is odd, y+1 is even and hence y is odd. If y+2 is even, by the same logic, y is even.

(y+2)! = 1*2*3*4*…*y*(y+1)*(y+2)

x! = 1*2*3*4*…*x

Note that (y+2)! and x! have common factors starting from 1. Since x! is less than or equal to (y+2)!, x will be less than or equal to (y+2). So all factors in the denominator, from 1 to x will be there in the numerator too and will get canceled leaving us with the last few factors of (y+2)!

If the division of two factorials is an integer, the factorial in the numerator must be larger than or equal to the factorial in the denominator.

So what does (y+2)!/x! is odd imply? It means that the leftover factors must be all odd. But the leftover factors will be consecutive integers. So after one odd factor, there will be an even factor. If we want (y+2)!/x! to be odd, we must have either no leftover factors (such that (y+2)!/x! = 1) or only one leftover factor and that too odd.

If we have no leftover factor, it doesn’t matter what y+2 is as long as it is equal to x. It could be odd or even. If there is one leftover factor, then y+2 must be odd and hence y must be odd. Hence y could be odd or even. This statement alone is not sufficient.

Statement 2: (y+2)!/x! is greater than 2

This tells us that y+2 is not equal to x since (y+2)!/x! is not 1. But all we know is that it is greater than 2. It could be anything as seen in examples 2 and 3 above. This statement alone is not sufficient.

Both statements together tell us that y+2 is greater than x such that (y+2)!/x! is odd. So there must be only one leftover factor and it must be odd. The leftover factor will be the last factor i.e. y+2. This tells us that y+2 must be odd. Hence y must be odd too.

Answer (C)

Takeaways: Assuming a and b are positive integers,

- a!/b! will be an integer only if a >= b

- a!/b! will be an odd integer whenever a = b or a is odd and a = b+1

- a!/b! will be an even integer whenever a is even and a = b+1 or a > b+1

Think about this: what happens when we put 0 in the mix?

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

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In most high schools in the United States, juniors and seniors naturally tend towards either the ACT or the SAT, depending on the region. In the Bay Area, for instance, far more college-bound students take the SAT than the ACT, for no apparent reason besides the fact that most of their peers are taking the SAT. In Southern states, the ACT is more dominant. Region, however, should not be the determining factor in choosing between these two tests; their subject matter, style, and requirements differ in important ways that many students don’t consider.

I’ve taken both. Only my SAT score, however, was sent along with my college applications. (My ACT score was released long after my college acceptance). I originally took the SAT instead of the ACT just because everyone I knew was taking the SAT, and because the SAT was offered on a more convenient day in my schedule. Looking back, I realize this was a poor decision on my part. If I had done my research, I would have quickly realized that I as a student was far better suited to the ACT than to the SAT, and would have saved myself quite a lot of worry. Here are the things I should have considered:

1. The ACT has a science section.

This is arguably the most famous difference between the tests. In high school, I liked reading much more than I liked science, so I originally dismissed the ACT entirely. My mistake: I didn’t realize that the ACT doesn’t actually require test-takers to know any complicated science concepts. In fact, it’s more like a reading test than a science test. As long as test-takers are able to read simple graphs and tables, they need only know some basic scientific vocabulary and concepts. Even those are often defined and explained within ACT passages themselves.

2. The SAT tests complicated vocabulary and focuses more on reading comprehension.

Students who lack confidence in their reading comprehension skills or who do not want to deal with complicated vocabulary should strongly consider taking the ACT instead.

3. The ACT tests more complicated math.

Conversely, students who are not comfortable with trigonometry should consider opting for the SAT.

4. The ACT lets you skip the essay.

The SAT essay is mandatory, while the ACT essay is optional. I recommend writing the essay if you take the ACT, but in the interest of making informed choices, you should be aware that the section is not required.

5. The SAT is longer.

If you have trouble sitting still for more than three hours, the ACT might be a better option for you.

6. The ACT was designed as an achievement test, while the SAT was designed as a reasoning test.

In other words, material on the ACT will more closely resemble the work that most high school students do in daily classes, while the SAT will challenge them to approach familiar subjects in less conventional ways.

7. US colleges accept both the SAT and the ACT, and treat the tests equally.

Choosing one over the other will not necessarily make your application more or less impressive to an admissions office. Take whichever test you believe suits you better.

8. Practice tests and questions are available for both the SAT and the ACT.

These are available on the official SAT and ACT websites and in test prep books and courses. Instead of guessing which you might perform better on, you can sample each and compare your scores.

9. If you still have trouble deciding, you have the option of taking both tests.

This will likely involve more study and more test fees, but will allow you the freedom to try both options and submit whichever test score is higher.

The choice between the ACT and the SAT offers students a valuable chance to play to their strengths, and to play down their weaker subject areas. Taking advantage of that opportunity can save time, effort, stress, and test prep money. Trust me; your future self will thank you for it.

Courtney Tran is a student at UC Berkeley, studying Political Economy and Rhetoric. In high school, she was named a National Merit Finalist and National AP Scholar, and she represented her district two years in a row in Public Forum Debate at the National Forensics League National Tournament.

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Benefactor Johns Hopkins willed $7 million to start up the University that now bears his name. It was founded in 1876 on the premise that research and discovery were of equal importance to teaching and learning. Johns Hopkins University’s other ideal is not to limit education to its students, but rather to lift the collective education of society at the same time. With these principles in mind, Johns Hopkins has become a preeminent model for other research universities across the globe, and is #26 on the Veritas Prep Elite Colleges Rankings.

Johns Hopkins reputation for excellence and leadership is built primarily on its health and sciences focus, which includes JHU School of Medicine, Bloomberg School of Public Health, and JHU School of Nursing graduate programs associated with the renowned Johns Hopkins Hospital. The University is also widely recognized as a leader in international studies, and its proximity to Washington, D.C. offers students coveted internships and jobs in the nation’s capital. The graduate program in the School of Advanced International Studies has three campuses – Washington, D.C., Bologna, Italy, and Nanjing, The People’s Republic of China.

The first graduating class of fifteen at JHU School of Medicine was in 1897. One of its first major research contributions came in the 1920s when researchers at the University developed the process of chlorination to purify drinking water, which is universally used by municipalities and industries today. Johns Hopkins boasts twenty-two Nobel laureates affiliated with the University. Countless government officials and public servants including President Woodrow Wilson, Vice President Spiro Agnew, Secretary of State Madeleine Albright, NYC Mayor Michael Bloomberg, and others have passed through the University.

Over sixty notable academicians, mathematicians, and scientists have made their marks on the world like NASA’s Michael Griffin, physicist Frank Oppenheimer, and John Dewey, education reformer. The list goes on and on for business, literature, art, and media. Johns Hopkins vision has played out in the world through the work of its faculty and graduates in ways he probably couldn’t have imagined. If you are looking for a place to grow and make your impact on the world, Johns Hopkins University could be the school for you.

Johns Hopkins University is an urban school whose primary Homewood campus lies on a former 140-acre private estate in northern Baltimore, Maryland. It is home to 5,800 undergrads, the Krieger School of Arts & Science and the Whiting School of Engineering grad programs. The Federalist architecture, marked by red brick buildings, is organized into quads with plenty of open green spaces. Freshmen and sophomores are required to live on campus, with freshmen living in the older Alumni Memorial Residences (ARMS) on the freshmen quad, or student-athletes living in Wolman located just off campus on the other side of St. Paul Street, which runs parallel to campus. The student housing is the hub of social living and offers regularly scheduled social events.

Many sophomores live in McCoy, a dorm similar to Wolman in age and architecture. Charles Commons is the newest of the dorms, putting its suite-style living with various amenities and the Nolan dining hall, in high demand. Bradford and Homewood offer apartment-style student housing with studios, one, two, three, and four bedroom units. They are less social by design and offer more privacy for upperclassmen. Fraternities also offer housing for their members, and regularly host parties. Greek life for both male and female students is evident on campus, but doesn’t dominate the social scene by any means.

The consensus among students is that Johns Hopkins is not the best college choice for a thriving social scene. Many students complain about the lack of social opportunities on campus and the strain of seriousness that permeates the atmosphere. There are opportunities to socialize through over 100 student clubs and organizations, and resourceful students are able to find plenty of options in the city of Baltimore for fun activities. Some residents of Bradford and Homewood throw house parties, if you happen to make the right relationships. This university has a seriously competitive academic atmosphere that doesn’t leave a lot of room for typical college socializing, so if partying is essential to your college experience, this might not be the school for you.

The Johns Hopkins Blue Jays are an NCAA Division III program in the Centennial Conference. Their water polo team is consistently one of the top D3 teams in the nation, often competing against D1 schools. The men’s and women’s basketball and soccer programs, along with men’s baseball, are also competitive. The pride of JHU sports, however, are the men’s and women’s lacrosse teams, which play Division I and have amassed 44 national titles – nine of them Division I. Their primary rival is Duke lacrosse, and home games on Homewood Field draw large crowds of cheering students who sit in The Nest (student section), wearing black or blue.

Traditions at JHU include the men’s lacrosse season home-opener, which generates a lot of enthusiasm, homecoming weekend, fall festival, the lighting of the quads for the holidays, and the popular and heavily attended spring fair. A newer tradition is High Table, where freshmen share a formal dinner with deans and faculty. If you’re looking to make your mark in medicine, research, science, or on the political world stage, JHU is an excellent choice to develop your platform.

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When students, even those who consider themselves strong in math, get to the final two problems of the SAT, many begin to sweat like they are about to embark on some epic journey from which they may never return. The hard probability problem makes students very uncomfortable, but in reality most harder math problems simply require one or two more steps than less difficult problems. Probability questions are actually some of the simplest to solve.

The “Hard” Probability Question

All probability questions are the same. The definition of probability is the number of desired outcomes divided by the total number of outcomes (sometimes this is multiplied by 100 to come up with a percent probability). Every problem requires finding the total number of possibilities, the desired possibilities, or both.

Here is an example:

“Four paintings, A, B, C, and D are being hung in four adjacent display cases. What is the probability that paintings A and D will be either first or last?”

At first glance this may seem intimidating, but it is just like any other probability question. The first step is to figure out the total number of possible outcomes. This is solved by thinking about this like a counting problem. If there are four “slots” that these painting could occupy let’s imagine that each slot has a number of possible paintings that could occupy it.

For the first slot, the number of possible paintings that could occupy it is four since there are four paintings. For slot two, the number of possible paintings is three because one painting is already in slot one. This may seem difficult to understand at first, but remember that these are the number of possibilities to choose from in each slot, so there will be one less painting to choose from in slot two because one possibility is gone. For slot three, it is one fewer painting still, and there will be just one painting left by the time the fourth is selected. In order to find the total number of possible outcomes, the possibilities in each slot must simply be multiplied together: 4 x 3 x 2 x 1 = 24. This process is the same for any problem where possibilities are calculated based non repeating possibilities on discreet “slots”.

Half the problem is done (total number of possibilities); now, all that is left is to find the number of desired possibilities. For this part of the problem we can calculate desired possibilities in a very similar way as we did above. The only tricky part is this calculation will require two steps.

Imagine painting A is selected to go in slot one (one possibility for the desired outcomes). If A is in slot one, then in order for A and D to be either first or last, D will have to go in the last slot. This leaves just 2 paintings to choose from in slot two and only one in slot three. The number of possibilities when A is 1st then is 1 x 2 x 1 x 1 = 2. The other possibility is for D to be first which leads to similar constraints. Painting A must be last, two possibilities are left for slot two, and just one is left for slot three. The total number of possibilities when D is first is also 1 x 2 x 1 x 1 = 2. The total desired outcome is all the possibilities when A is first (2) added to all the possibilities when D is first (also 2), which leaves a total desired outcomes of four.

With the total desired outcomes and the total possible outcomes found, the final step is to create a fraction with desired outcomes on the top and total outcomes on the bottom or 4/24 which reduces to 1/8.

Hard probability problems and counting problems, like most hard problems on the SAT, are not really “hard”. The main thing to keep in mind is the technique that is discussed above for calculating possibilities in different contexts. If students can master this technique and remember the definition of probability (desired outcomes over total outcomes) the hard probability problem becomes a piece of cake. Happy Studying!

David Greenslade is a Veritas Prep SAT instructor based in New York. His passion for education began while tutoring students in underrepresented areas during his time at the University of North Carolina. After receiving a degree in Biology, he studied language in China and then moved to New York where he teaches SAT prep and participates in improv comedy. Read more of his articles here, including How I Scored in the 99th Percentile and How to Effectively Study for the SAT.

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Many students feel that the GMAT is only necessary to get into business school, and otherwise serves no real purpose in their everyday lives. I, as a GMAT enthusiast (and overall math nerd), see a lot of real world applications in the concepts being tested on this exam. It’s actually somewhat surprising how often splitting the cheque at a restaurant or calculating investment returns requires me to delve into my GMAT knowledge. Such an instance just happened the other weekend, and it’s the kind of story I’d like to use to illustrate how pervasive GMAT knowledge is in daily life.

After celebrating Easter lunch, the family enjoyed dessert and spirited conversation (yelling) for a few hours. When it was time to leave, like many Mediterranean families, everyone felt the need to kiss everyone else goodbye (this is a great way to spread disease, by the way). While people were busy lining up to wish each other farewell, my GMAT brain took over. I asked myself: if there were 14 people gathered there, and everyone had to say goodbye to everyone else, how many embraces would that encompass in total?

The first idea that came to mind was 14! I quickly dismissed this idea, as this is an astronomical number. I know 10! Is about 3.5 million, so 14! Is well into the billions (87 billion and change, according to the calculator). If this were the case, we’d still be saying goodbye until 2015. However my brain instinctively went that direction for a reason. I thought about a little more.

Every person had to say goodbye to the 13 other people there. This means that I would have to say goodbye to the 13 other people. Similarly, every other person there would have to say goodbye to the 13 others as well. This leads to 14 x 13, and explains why I initially thought of factorials. However there is no need to keep multiplying by 12 and 11 and so on. 14 x 13 is essentially the answer, as every person there would get to say goodbye to everyone else. You can solve this little equation fairly quickly, especially if you know that 14 x 14 is 196 and then you drop 14 to 182.

However, 182 would not be the correct answer, because I am double counting all the goodbyes. For instance, I have counted saying goodbye to my mother, and I have also counted her saying goodbye to me. This is clearly the same event, so I should only count it once. This will be true of all the salutations, which means I must take my overall total of 182 and divide it by two. The actual answer should thus be 91.

I was confident that I had the correct answer, but surely there was a better way of solving this than going though the logic person-by-person (there is a better way, and don’t call me Shirley). In essence, this is a problem about combinatorics. I’m taking 14 individuals and making groups of 2s where the order doesn’t matter. This is a combination of 14 choose 2. Remembering that the formula for this kind of problem is n!/k!(n-k)!

Replacing the n by 14 and the k by 2, I’d get all the unordered pairings of people at my family gathering.

14!/2!(14-2)!

Which becomes

14!/2!(12!)!

Simplifying the 12! That’s common to both the numerator and denominator:

14*13/2!

Which ultimately yields 182/2 or just:

91

Now that we’ve solved my Easter farewell dilemma, let’s see if we can apply this same logic to actual GMAT problems:

If 10 people meet at a reunion and each person shakes hands exactly once with each of the other participants, what is the total number of handshakes?

(A) 10!

(B) 10 * 10

(C) 10 * 9

(D) 45

(E) 36

Given that this is the same principle as the issue above, we can even see where the trap answers come into play. Answer choice A is the tempting factorial option, but it’s important to note the order of magnitude of this choice. Answer choice B essentially lets you make everyone shake hands with everyone, including the nonsensical option of shaking hands with yourself (Hello Ron, nice to meet you Ron). Answer choice C removes the self-adulation, but still does not provide the correct answer because it double counts the handshakes.

Using logic, we can validate that answer choice D is correct because everyone shakes hands with the 9 others but the handshakes are double counted. Using the mathematical formula yields

n!/k!(n-k)!

Where n is 10 and k is 2:

10!/2!(10-2)!

Which then becomes

10!/2!(8)!

And then simplifies to

10*9/2!

Or just

45

We can also see that answer choice E would be correct if we decided that n should be 9 instead of 10 (possibly because we’re on a wicked bender). As is often the case, the GMAT test makers do not pick four arbitrary values for their other four answers, but rather choices you could realistically get to on this problem. Be wary not to fall into the traps laid out for you by combining your knowledge of the formula with your use of logic.

One takeaway I really like from this question is that this is the type of problem you can solve in 30 seconds or less (like a really fast pizza). If you understand what is going on here, it’s really just a question of taking n, multiplying it by n-1 and dividing by 2. This applies to any round-robin style tournament, which is the colloquial term for a tournament where everyone meets every other team.

As such, if you have a round-robin tournament of 16 teams, then you’ll just have 120 games to watch over (16 x 15 / 2). This might help to explain why the March Madness tournament is done as an elimination tournament, because otherwise the 64 teams would be playing well into the summer. Having certain question types that you understand ahead of time will help you succeed on the GMAT, and hopefully at your next gathering you’ll have good news to share with everyone before saying your goodbyes.

Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.

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Carleton College is ranked #28 on the Veritas Prep Elite College Rankings; it’s a smaller liberal arts college located in Northfield, Minnesota. Founded in 1866, this is a private college on 880 acres of land with three ecosystems – prairies, a forest, and wetlands. A tranquil campus that is only thirty minutes from the bustling twin cities. Carleton offers 37 majors with 15 different concentrations in humanities, the arts, natural sciences, and social sciences. This is an ideal college for students who are looking for a close-knit campus community with a focus on excellence. From sustainability to community service projects, Carleton is the perfect place for those who want to have an expanded college experience. As the third oldest college in the state of Minnesota they believe that “human knowledge is the real frontier.”

At Carleton the curriculum is designed to focus on problem solving, creativity, critical thinking, flexibility, and communication giving you a well-rounded foundation. Their academic program is revered by many and noted as one of the best in the world, and on almost every notable list of elite colleges. Carleton has distinguished faculty that are among leading scholars, artists, researchers, and scientists in their respective fields; but it’s their commitment to education that makes them invaluable. Often times, they’ll open their homes or share a lunch hour to better their students’ educations. Carleton uses a three term academic period to allow students to focus on classes and engage in a variety of diverse interests throughout each year. Great study abroad options, community service, research projects, and seminars are just a few of the ways this college excels at providing excellent post-secondary educational experiences.

Roughly ninety percent of the students live on campus, with three housing options: residence halls, campus townhouses, and shared interest houses. There are three dining facilities, kitchenettes in each residence hall, and vending machines throughout the campus. The campus is dedicated to sustainability, as evidenced through the use of a wind turbine for energy, locally grown food, and the installation of an energy-saving roof. There are more than 230 student organizations, with new ones like Random Acts of Kindness or The Gender Neutral Cheerboys that are created each year. If you need a break and space to clear your mind take a walk through “The Arb,” which is the arboretum campus, or head thirty minutes to the Twin Cities where you can lose yourself in the culture and energy of city life. There are also plenty of things to do in nearby Northfield, such as plays, lectures, concerts, films, and sporting events.

Carleton is known for its academic prowess, but it is also home to eighteen NCAA Division III varsity athletic teams in the Minnesota Athletic Intercollegiate Conference. The college offers organized club sports and the Intramural Sports League that allows both students and faculty a chance to participate. If you are not interested in team sports and just want to get out and enjoy nature while working up a sweat, there are many areas on campus to walk, run, bike, and cross country ski. For those who prefer the indoors there are yoga classes, bosu, meditation, massage, racquetball, a fitness center, and a climbing wall. No matter what physical activity interests you, Carleton has what you need.

If you appreciate Midwestern values with a global perspective, then Carleton College in Minnesota may be the ideal school for you. They value tradition and are dedicated to providing an enriching college experience for every student. Have you ever dreamed of painting a water tower? This is one of Carleton’s more unusual traditions; one year they painted the tower with a Clinton theme and he ended up being elected President that year. They have an annual “Late Night Trivia” game show every winter term that is hosted by their college radio station where students compete by guessing songs and answering crazy questions. If you dream of a college that is dedicated to your success, and a place where you build authentic and lasting friendships, then Carleton College is the college for you.

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If you’ve studied for the GMAT for a while, you likely have a decent understanding of the answer choices:

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;

(C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient;

(D) EACH statement ALONE is sufficient to answer the question asked;

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed

And you probably have a device to help you both remember these answer choices and use process of elimination. Some like “AD/BCE” (make your decision on statement 1 and cross out one side), others like “1-2-TEN” (1 alone, 2 alone, together, either, neither). But, ultimately, remembering the answer choices (which are always attached to the question on test day anyway) and understanding how to use process of elimination is just the “price of entry” for actually solving these problems correctly. For true Data Sufficiency mastery and a competitive advantage, you should think of the answer choices this way:

___D___

A_____B

___C___

___E___

Why?

As an added bonus it’s helpful for process of elimination (like the other tools) but as a strategic thought process it can be instrumental in using your time wisely and avoiding trap answers. Because what these answers really mean is:

___D___ — Each statement alone is sufficient

A_____B — One statement alone is sufficient; the other is not

___C___ — Both together are sufficient, but neither alone is sufficient

___E___ — The statements are not sufficient, even together

And since most Data Sufficiency questions are created with one of these constructs:

*One answer seems fairly obvious but it’s a trap

*One statement is clearly sufficient; the other is a little tricky

*One statement is clearly insufficient, but gives you a clue as to something you need to consider on the other

The above chart tells you how to better assess the answer given the answer that looks most promising. Consider a question like:

Set J consists of terms {2, 7, 12, 17, a}. Is a > 7?

(1) a is the median of set J

(2) Set J does not have a mode

For most, statement 1 looks very sufficient, as if a is the “middle number” then it would go between 7 and 12 on the list {2, 7, a, 12, 17}. That would mean that on this chart, you’re at A, as statement 2 is pretty worthless on its own:

___D___

A_____B

___C___

___E___

You can very confidently eliminate B and probably E, too, but if you’re sitting on a “probable A,” you’ll want to consider one level above and one level below your answer on the chart. Why? Because if the answer is, indeed, trickier than your first-30-seconds-assessment, the options are that either:

*The statement you thought was sufficient was close, but there’s a little hiccup (you thought A, but it’s C)

*The statement you thought was not sufficient was actually really cleverly sufficient had you just worked a little harder to reveal it (you thought A, but it’s D)

This is what Veritas Prep’s Data Sufficiency book calls “The Reward System” – many questions are created to reward those examinees who dig deeper on an “obvious” answer via critical thinking, and to “punish” those who leap to judgement and fall for the sucker choice. If A is the sucker choice, the answer is almost always D or C, so you know what you have to do…check to make sure that statement 2 is not sufficient, and then check (often using statement 2) to make sure that you haven’t overlooked a unique situation that would show that statement 1 is actually not sufficient. And here, further review shows this:

If a = 7, a is still the median of the set, but 7 is NOT greater than 7, so that answer would be “no” – there’s a way that a is not greater than 7, so we actually need statement 2. If there is no mode, then a can’t be 7 (that would be a duplicate number, making 7 the mode). So the answer is C, and the Reward System thinking can help make sure you streamline your thought process to help you identify that. If you picked A you’re not alone – many do. But if you picked A and then considered the chart:

___D___

A_____B

___C___

___E___

You should have spent that extra 30 seconds making sure that the answer wasn’t C or D, and that may have given you the opportunity to reap the rewards of thinking critically via the Data Sufficiency question structure.

So remember – merely knowing what the answer choices are is an elementary step in Data Sufficiency mastery; learning to use those to your advantage via the Reward System will help you avoid trap answers and stake your place among those being rewarded.

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Once you have covered your fundamentals, we suggest you to practice advanced questions and jot down your takeaways from them. Sometimes students wonder how to find that all important “takeaway”. Today, let’s discuss how to elicit a takeaway from a question which seems to have none.

What is a takeaway? It is a small note to yourself which you would do well to remember while going for the exam. Even if you don’t remember the exact property you jotted down, knowing that such a property exists is enough. You can always try it on a couple of numbers in the test to recall the exact content.

The question we will discuss today serves another purpose – it discusses properties of squares of odd and even integers so in a sense is a continuation of our advanced number properties discussion.

Question: Given x and y are positive integers such that y is odd, is x divisible by 4?

Statement 1: When (x^2 + y^2) is divided by 8, the remainder is 5.

Statement 2: x – y = 3

Solution: As of now, we don’t know any specific properties of squares of odd and even integers. However, we do have a good (presumably!) understanding of divisibility. To recap quickly, divisibility is nothing but grouping. To take an example, if we divide 10 by 2, out of 10 marbles, we make groups of 2 marbles each. We can make 5 such groups and nothing will be left over. So quotient is 5 and remainder is 0. Similarly if you divide 11 by 2, you make 5 groups of 2 marbles each and 1 marble is left over. So 5 is the quotient and 1 is the remainder. For more on these concepts, check out our previous posts on divisibility.

Coming back to our question,

First thing that comes to mind is that if y is odd, y = (2k + 1).

We have no information on x so let’s proceed to the two statements.

Statement 1: When (x^2 + y^2) is divided by 8, the remainder is 5.

The statement tell us something about y^2 so let’s get that.

If y = (2k + 1)

y^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 4k(k + 1) + 1

Since one of k and (k+1) will definitely be even (out of any two consecutive integers, one is always even, the other is always odd), k(k+1) will be even. So 4k(k+1) will be divisible by 4*2 i.e. by 8. So when y^2 is divided by 8, it will leave a remainder 1.

When y^2 is divided by 8, remainder is 1. To get a remainder of 5 when x^2 + y^2 is divided by 8, we should get a remainder of 4 when x^2 is divided by 8. So x must be even. If x were odd, the remainder when x^2 were divided by 8 would have been 1. So we know that x is divisible by 2 but we don’t know whether it is divisible by 4 yet.

x^2 = 8a + 4 (when x^2 is divided by 8, it leaves remainder 4)

x^2 = 4(2a + 1)

So x = 2*√Odd Number

Square root of an odd number will be an odd number so we can see that x is even but not divisible by 4. This statement alone is sufficient to say that x is NOT divisible by 4.

Statement 2: x – y = 3

Since y is odd, we can say that x will be even (Since Even – Odd = Odd). But whether x is divisible by 2 only or by 4 as well, we cannot say. This statement alone is not sufficient.

Answer (A)

So could you point out the takeaway from this question?

Note that when we were analyzing y, we used no information other than that it is odd. We found out that the square of any odd number when divided by 8 will always yield a remainder of 1.

Now what can you say about the square of an even number? Say you have an even number x.

x = 2a

x^2 = 4a^2

This tells us that x^2 will be divisible by 4 i.e. we can make groups of 4 with nothing leftover. What happens when we try to make groups of 8? We join two groups of 4 each to make groups of 8. If the number of groups of 4 is even, we will have no remainder leftover. If the number of groups of 4 is odd, we will have 1 group leftover i.e. 4 leftover. So when the square of an even number is divided by 8, the remainder is either 0 or 4.

Looking at it in another way, we can say that if a is odd, x^2 will be divisible by 4 and will leave a remainder of 4 when divided by 8. If a is even, x^2 will be divisible by 16 and will leave a remainder of 0 when divided by 8.

Takeaways

- The square of any odd number when divided by 8 will always yield a remainder of 1.

- The square of any even number will be either divisible by 4 but not by 8 or it will be divisible by 16 (obvious from the fact that squares have even powers of prime factors so 2 will have a power of 2 or 4 or 6 etc). In the first case, the remainder when it is divided by 8 will be 4; in the second case the remainder will be 0.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

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Rice University is ranked #30 on the Veritas Prep Elite College Rankings. This is a small research university located right in the heart of Houston, Texas, the fourth largest city in the United States. Student will find themselves immersed in a campus devoted to diversity and elite educational opportunities. Rice University is passionate about teaching and developing leadership ideals in each of their students.

Their high values, comprehensive and well recognized residential college system, and devotion to giving each student the tools to rise to their own greatness is what makes them an elite University. They take pride in turning their already outstanding students into exceptional leaders in any field they choose. This is a University for students who want to impact the world in a grand way.

There are six areas of study offered at Rice: architecture, humanities, music, natural sciences, engineering, and social sciences. Each school offers a strict curriculum, created to make each student more than capable to become leaders in their desired fields. With more than fifty majors to choose from across the six schools coupled with complimentary minors, unique interdisciplinary, and professional programs to choose from, they provide the ultimate opportunities in higher education. Rice not only offers an excellent curriculum, but also superb research and collaboration opportunities to further each student’s skills.

Extensive curriculum options are just the beginning of student success at Rice; the advising program that was put in place to correctly guide each student in the right direction also plays a key role. In the first two years students will focus on their general education, using the second portion of sophomore year as the time to decide a major. Trained faculty and student guides help students declaring a major; students’ major guides stick with them throughout the remaining years making sure they are on track with their major. They also assist students in locking down research opportunities, internships, and other professional programs to reach their education goals.

Campus life at Rice University is unique; before starting day one on campus each student is randomly assigned to one of the eleven residential colleges. These are subdivisions within the University, with equal parts diversity in each one. The process is designed to give students the chance to enhance peer interaction among students, faculty, and staff. This system promotes the development of strong relationships and strengthens intellectual achievements. Within these residential colleges are student run governments, each with their own responsibilities within the college. To enjoy free time on campus and connect with fellow students, everyone is encouraged to join one of the more than 200 different clubs on campus. Along with excellent social clubs there is a wide variety of events featured on campus from theater arts to politics. Students can enjoy a multitude of lectures offered on a wide range of subjects throughout the year. The amenities are abundant on the Rice campus, the cinema, art gallery, and media center are just a few of the exceptional places to spend and few hours after classes.

Athletics and physical education are top notch at Rice University, with over 70% of students participating in one way or another. Along with sixteen Division I teams, there are intercollegiate club sports, college sports, and intramural sports offered at Rice. Every student gets free tickets to all home varsity sporting events, making it easy to show your school spirit and cheer on the infamous Owls. The Barbara and David Gibbs Recreational and Wellness Center is an exceptional part of Rice University. A forty-one million dollar facility that has an assortment of indoor and outdoor courts from basketball to racquetball as well as a dance studio, pools, weight-lifting areas, and more. Not only does this University boast state-of-the-art equipment, an array of athletic teams to participate in, but it also offers a multitude of outdoor activities, where you can sign up to go camping or kayaking among many others. Rice University might be known for their awesome academics, but it’s clear that they take just as much pride in their students’ health as well.

To attend Rice University is to belong to a school that takes pride in their traditions. Be ready to wear the blue and gray, and stand behind “Sammy” the Owl while singing the Rice Fight Song. Each semester offers a new tradition to take part in; during the fall students will be enthralled with the tradition of Welcome Back Week, as well as a wide range of events from homecoming to the activities fair. Nothing compares to the spring tradition of Willy Week, where you’ll enjoy festivities such as Beer Debates and International Beer Night. Rice University is about as well-rounded as they come paying equal attention to academics, athletics, community, and of course fun!

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One of the biggest tricks the SAT uses is to confuse students is putting a question in theoretical terms instead of in practical terms. This simply means the questions on the SAT will sometimes reference a general term, for example an even integer, rather than giving a concrete number that fits that description, such as two or four.

The good news is that it is easy to correct this by simply plugging in concrete numbers when the question gives general terms.

Here is an easy example:

An even and an odd integer are multiplied together. Which of the following could not be the square of their product?

(A) 36

(B) 100

(C) 144

(D) 225

(E) 400

One way to approach this problem is to start with an even and an odd integer and plug them in to the parameters set by the problem. If we begin with two and three, we see that the product is six and the square of the product is thirty six.

(2)(3) = 6 6² = 36

Similarly we can see that two and five, three and four, and four and five all give us possible answer choices.

(2)(5) = 10 10² = 100

(4)(3) = 12 12² = 144

(4)(5) = 20 20² = 400

Answer choice (C) is also a perfect square, but if we take the square root of it, we see that the result is fifteen, which is not divisible by an even and an odd number. Thus the only answer that could not be the squared product of an even and odd integer is answer choice (C).

Here is a slightly more difficult question.

A right triangular fence is y inches on its smallest side and z inches on its largest side. If y and z are positive integers, what represents the formula for the area of the fenced in region in square feet?

(A) √(z² – y² ) (y)

(B) 24 (z² – y² )

(C) √(z² – y² ) (y/12)

(D) √(z² – y² ) (y/24)

(E) (½) √(z² – y³)/ 12

At first glance, this problem may seem complex, but we can simply plug in real numbers into this problem and solve by seeing which answer choice gives the same response as the answer we derive. This is a right triangle, so if z is five and y is 3, then the third side, which is also the height, would be four. The total area in inches would then be one half base times height. To convert inches to feet we would have to divide the area by twelve.

y = 3

z = 5

H = 4

1/2 (3)(4) = 6 in²

6/12 = 1/2 ft²

Only answer choice (D) gives the correct answer of one half when the numbers we chose are plugged into the equation. We can also see that, if multiplied by 12 to account for the change to feet, answer choice (D) is essentially the formula for the area of a triangle with √(z² – y² ) as the height.

It is easy to get frustrated when given a theoretical problem, but when real numbers are inserted for the theoretical ones, the problem becomes surprisingly simple. So throw some real numbers into the mix and see what happens. The only thing to be wary of is that in certain contexts, it may be necessary to plug in different combinations of numbers that fit the given parameters to make sure that the general equation works with different sets of specific numbers. Even with this caveat, with a little practice, this technique can make even very confusing problems seems quite simple. Happy studying!

David Greenslade is a Veritas Prep SAT instructor based in New York. His passion for education began while tutoring students in underrepresented areas during his time at the University of North Carolina. After receiving a degree in Biology, he studied language in China and then moved to New York where he teaches SAT prep and participates in improv comedy. Read more of his articles here, including How I Scored in the 99th Percentile and How to Effectively Study for the SAT.

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