More Advanced Fact: As a general rule, if you wish to solve for the values of n different variables, you need at least n different equations.  (If n = 3, this could possibly come into play on 700+ questions on GMAT Quantitative.)

A Boon for Data Sufficiency!

Just this one mathematical fact, the first one, has powerful implications for hundreds of possible DS questions.  Consider this template DS question:

1) In blah blah blah scenario, x is blah blah blah and y is blah blah blah.

Statement #1: 2x + y = 6

Statement #2: x – y = 6

Statement #1: two variables, one equation, not sufficient by itself.  Statement #2: two variables, one equation, not sufficient by itself.  Combined: two equations, two variables, can be solved, sufficient.  Done.  Answer = C.

Notice, we had to do a minimum of math to solve this question.  Just this one fact can make short work of any of a number of seeming complicated DS questions.  Often, notice that the “equations” are given in verbal form: one first has to translate, but once you realize the verbal information constitutes an equation, you don’t even need to find the question: you can just use this logic to power through to an answer.

Strategy for Problem Solving

For DS, all you have to determine is whether you can find an answer.  On Problem Solving, and on the occasional Two-Part Analysis in the Integrated Reasoning section, actually have to find the answer.

Your Algebra Two teacher probably taught you two different ways to solve these equations.  For simplicity, I am just going to review one method, the one that is most useful in solving the systems of equations you will see on the GMAT.  If you remember the other method, and prefer that, by all means use it.

The method I am going to review is sometimes called “elimination” or “linear combination.”  Here is the strategy.

Step #1 — Multiply one or both equations so that the coefficients of the same variable are opposites of each other (e.g. +7 and –7).

Step #2 — Add the two equations

Sometimes you get very lucky because in the equation, as given, coefficients of one variable already are opposites, so you can bypass Step #1 and proceed immediately to Step #2.

I’ll demonstrate with the two equations in my hypothetical question above.  Suppose, in a PS question we were given those two equations (2x + y = 6 and x – y = 6) and had to solve for either x or y.  Here, we are very lucky —- the coefficients of y are opposites, all we have to do is add the two equations:

Once we know x = 4, we can plug into either equation to find that y = –2.

A Slightly More Challenging Example

Consider this question:

2) The symphony sells two kinds of tickets: orchestra, for $40, and upper tiers, for $25.  On a certain night, the symphony sells 90 tickets and gets $2625 in revenue from the sales.  How many orchestra tickets did they sell?

1. 25
2. 35
3. 45
4. 55
5. 65

Notice, in verbal form, this is a two-equations-two variables problem.  The two variables are x = the number of orchestra tickets, and y = the number of upper tier tickets.

One equation we get is x + y = 90, for the total number of tickets sold.  That’s one equation.  If we sell x orchestra tickets, we get $40 for each, so we get 40x in total revenue from all of the orchestra tickets.  Similarly, we get 25y in total revenue from all the upper tier tickets.  Thus, the total revenue is 40x + 25y = 2625.  That’s our second equation.

The first equation is incredibly convenient to multiply.  I would rather multiply by 40 than by 25, so I make the coefficients of x match.  I’ll multiply the first equation by 40, and the second equation by –1.

Now, at first blush, that might look like an ugly division problem awaiting us, 975 divided by 15.  Let’s break it down a bit.   I know 900/3 = 300, and 75/3 = 25, so if I divide both sides by 3, I get

5y = 975/3 = 325

Now, 100/5 = 20, so three times that is 300/5 = 60.  Of course, 25/5 = 5, so

y = 325/5 = 65

We want x, so x + (65) = 90 x = 25, answer = A.

BTW, if these steps totally elude you, remember you can always backsolve from the numerical answers as a backup strategy.

Practice Question

Here’s a practice question: http://gmat.magoosh.com/questions/335

If you can solve these, you are a master of what is, by some counts, the fourth most commonly tested concept on GMAT Quantitative.

This formula, called “the difference of two squares” formula, is a favorite of standardized test writers.  A simple enough pattern: see if you can detect where it shows up in the following challenging problems.

1)  =

(A)

(B)

(C) 

(D) 

(E)

2) What is the sum of a and b?

(1)

(2)

1. Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
2. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
3. Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
4. Each statement alone is sufficient to answer the question.
5. Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

3)  In the diagram above, ∠A = ∠ABC, ∠CBD = ∠BDC, and ∠CBE =90°.  If AE = 16 and DE = 4, what is the length of BE?

(A) 7

(B) 8

(C) 9

(D) 10

(E) 11

Practice Problem Solutions

1) This involves a relatively sophisticated trick known as “multiplying by the conjugate.”  When we have an expression of the form a+sqrt(b), the “conjugate” of this is a-sqrt(b).  When we multiply a radical expression by its conjugate, we employ the difference of two squares.  For example:

This is answer B.  BTW, the trick of multiplying by the conjugate is at the very outside edge of what you might be expected to do the hardest GMAT math problems.

2) The prompt of this DS problem is straightforward.

Statement #1 tells us a = 4, but we have no idea of b, so, by itself, this is not sufficient for finding the sum.

Statement #2 gives us a value for an algebraic expression that lends itself nicely to simplification.

Thus, the sum is 7.  Statement #2, by itself is sufficient.  Answer = B.

3) This is a tricky one.  Remember, it’s a diagram drawn to scale, so if all else fails, you can estimate (see this post).  But, let’s solve this with math.  The fact that ∠A = ∠ABC tells us triangle ABC is isosceles, with AC = BC.  The fact that ∠CBD = ∠BDC tells us triangle BCD is isosceles, with BC = CD.  The fact that ∠CBE = 90° means that (BC)² + (BE)² = (CE)².  This means

(substitutions from the two isosceles triangles)

Answer = B

Here’s a further practice question.

http://gmat.magoosh.com/questions/118

Australia, a land of contrasting landscapes: lush tropical rainforests and barren red deserts, modern cosmopolitan cities and the ancient out-back towns, home to some of the most deadly, venomous snakes and spiders in the world. But how does Australia fare as a top destination for higher business education?

Australia’s booming economy, close proximity to Asia and exceptionally high living standards make it an attractive business education destination. In the current climate, studying in an English speaking country that offers genuine career prospects when you graduate coupled with the ‘out-door lifestyle’ and endless beaches make Australia an attractive proposition.

Here’s some thoughts from BusinessBecause.com members on why they chose to study down-under…

EAST COAST – AGSM MBA Programs, Sydney

The Australian Graduate School of Management (AGSM), certainly benefits from its excellent location in Sydney, on Australia’s south-east coast. In 2011, it’s MBA program was ranked first in Australia and 36th in the world by the Financial Times and fourth in Asia Pacific by the QS Global 200 Business Schools Report. The school also has received both AACSB and EQUIS accreditation.

Matthew Barnett - AGSM MBA Alumni

When asked ‘Where was the best country you’ve visited this year?’ Matthew Barnett, an AGSM MBA alumni, answered…

“The best country I’ve visited this year, well I’ve actually just stayed in Australia studying and also setting up a business, so it would have to be here! That said I spend a lot of my free time exploring, and have travelled all over Australia; the fact its bigger than Europe and pretty much the same size as the USA, you don’t need to go overseas to experience a wealth of alternative landscapes and climates. The north is tropical and deadly, where crocodiles, snakes and spiders run rife, but beautiful and wild too, from the rainforests to the barrier reef. The west is classic red desert Australia with the big red kangaroos, Wedge tailed eagles, 50,000 year old aboriginal heritage and rolling deserts. The south hosts Melbourne and Adelaide, the more cultural cities, with temperate climates, home to some of the best surf spots in Australia, and the arts capitals of the country.”

Read the full story with quotes from other b-school students around the world here.

Elad Sherf  – AGSM MBA Alumni

Elad Sherf, an Israeli lawyer, moved from Tel Aviv (where he’d been born and raised) to Sydney Australia to do the AGSM MBA…

Despite obvious cultural differences, Sherf didn’t experience any great cultural shock. “Sydney in a way is very similar to Tel Aviv. And the language isn’t such a huge barrier, as we’ve been studying English in Israel since fourth grade!” he says.

His AGSM MBA days “have to be valued at a number of levels”; says Sherf, but the “diversified experience” he had was the highlight.

He says: “We had 60-plus people from more than 30 countries, and it was very interesting to experience Australia and that part of the world. It was awesome. I’ve made friends that will last for life.”

Read Elad’s full story here.

WEST COAST – Edith Cowan University, Perth

Western Australia has a booming economy; rich in natural resources such as iron-ore, gold, natural gases and wheat. If Western Australia were a country of its own it would be in the top 50 economies in the world.

Western Australia is a territory making up a third of the continent but with only 10% of the population. Economists are warning that skill shortages coupled with wage and price increases could damage the economic growth. The country is calling out for skilled workers, businessmen and women to maintain growth.

Along with white beaches and a thriving metropolitan area, Perth is also home to Edith Cowan University’s Perth Graduate School of Business.

Michelle Slater – ECU MBA Alumni

Studying in Australia means that a student will have access to a quality education while having the opportunity to work with fellow students from diverse cultures and backgrounds.

I think this is a key advantage in the global economy and goes hand in hand with an MBA – the ability to be flexible and work productively with anyone, no matter the cultural differences. I am a firm believer that valuing diversity in a team bringing about better results, and I think diversity in the student body does a similar thing for your education and skill set.

I was drawn to ECU because of the flexibility – I worked full time while I was studying so it was important for me to be able to attend classes at night. The intensive summer and winter school units on weekends were handy as well.

Read Michelle’s full interview here.

Minh Phan – ECU MBA student

Studying for an MBA at Edith Cowan University made such an impression on Vietnamese styudent Minh Pham, that she says would chose Edith Cowan University again if she had to reapply to business school.

Minh Pham applied to the ECU MBA on the recommendation of a friend in Hanoi, who had loved his time on the Edith Cowan MBA. She was attracted to apply because of its location, its high standards of teaching and especially its affordability.

Edith Cowan University also provides a lot of little perks for its students that don’t go unnoticed. For example, if you’re working late at the library, the campus security service will escort you to the bus stop. The ECU Student Village, where Minh lives, is very safe and a great place to make friends from all over the world.

Watch Minh’s full interview here.

About the author: Sian Fleming Jones is a director at BusinessBecause.com – professional network for business students – helping you make connections before, during and after your MBA. You will find many more stories like the ones mentioned in this article by searching the website as well as useful information on MBA rankings, MBA jobs, and fresh daily editorial such as the Why MBA  series.

Do you spot the error in the preceding sentence?  This error is common in casual spoken English, but it will cost you on the GMAT Sentence Correction.  In that sentence, the word “if” is incorrect: it should be replaced by the word “whether.”

When to use “if”

The word “if” is used for clauses that specify conditions or speculate on something hypothetical.

1.) Condition: “If you finish your peas, you can have dessert.”

2.) Hypothetical: “If I regularly ate my vegetables, I probably would be healthier.”

In formal logic, the clause following the “if” clause would begin with the word “then”: that perfectly acceptable grammatically, but not at all necessary.  For example, in both of those sentence, the word “then” could be inserted right after the comma, and would add a bit of emphasis to the logical relationship, if that were something that needed underscoring.

The last clause of the previous paragraph highlights a particular category of conditional statements, those using the subjunctive.  For more on the subjunctive mood, see this post.  The GMAT loves “if”-clauses involving the subjunctive.

When to use “whether”

The word “whether” is a relative pronoun, which means it introduces a relative clause.  A “whether” clause is always about the uncertainty in a choice or alternative, and the clause itself may stand apart from the sentence, the way an “if” clause does, or may act as a noun.  When it stands apart, it is like an “if” clause in which the definite causal nature has been replaced with uncertainty or irrelevance.  When it acts as a noun, the clause may act as the subject of the sentence, or as the object of an epistemological verb (to know, to wonder, etc.)  or a volitional verb (to care, to prefer, etc.)

Stands apart:

3.) Whether you study French or Spanish, you will encounter an unfamiliar language in Japan.

4.) Whether or not I get the raise, I am going to buy that new car.

Notice, in either of those: if we removed the uncertainty of the choice, we could replace the word whether with the word “if” to get a more definitive conditional statement.  Without making those changes, the word “if” would be wrong.

Subject of sentence:

5.) Whether you like jazz will influence your opinion of this new club.

6.) Whether I walk on her left or right side matters a great deal to her.

Object of an epistemological or volitional verb:

7.) I don’t know whether there is intelligent life elsewhere in the Universe.

8.) He doesn’t care whether you serve broccoli or Brussels sprouts with dinner.

In sentence #5-8, the word “if” would be 100% incorrect.   The GMAT Sentence Correction loves to test that particular mistake.

Whether . . . or not

The word “whether” implies a choice, at least a pair of alternatives.  Sometimes that choice is made explicit (as in sentences #6 and #8), and sometimes it is implicit (as in sentences #5 & #7).  When the choice is implicit, is it grammatically correct to add the words “or not” after whether?

When the “whether” clause acts as a noun, the words “or not” add absolutely nothing to the sentence. Consider:

5a.) Whether you like jazz will influence your opinion of this new club.

5b.) Whether or not you like jazz will influence your opinion of this new club.

The meaning of both sentences is exactly the same.  The second sentence adds two more words that contribute zilch to the overall meaning of the sentence.  What is GMAC’s opinion of tossing in extra words that lengthen the sentence and contribute bupkis to the meaning?  As you may well guess, they frown on these.  Don’t expect to see “whether or not” in any correct GMAT SC answer choice when the clause is used as a noun.

When the clause stands apart, as in sentences #3 & #4, that’s another matter.  In that construction, the alternative must be made explicit.  In #3 there already was an explicit comparison of the two languages, but in #4 we absolutely must include the words “or not” after the word “whether”: the grammatical construction demands it.  This is the only case in which the words “whether or not” could be correct on GMAT sentence correction.

Whether or not you like it, knowing the correct use of “whether” and “if” is important for GMAT Sentence Correction.  If you can master these distinctions, you will perform well on a question that that befuddles many.

Two relevant SC questions in the GMAT Official Guide, which appear as:

a.) #34 & #75 in OG12e, and

b.) #34 & #78 in OG13e

1a) After I get my next paycheck, I am seriously thinking about buying a jet ski, I am seriously thinking about treating my friends to dinner, and I am seriously thinking about putting some money away in savings.

Obviously, that sentence is screaming for the simplification that parallel structure brings:

1b) After I get my next paycheck, I am seriously thinking about buying a jet ski, treating my friends to dinner, and putting some money away in savings.  (Whew!)

Notice the words in 1a that were eliminated in 1b: the repeated phrase “I am seriously thinking about.”  The repetition of that phrase is precisely what makes 1a sound hideously redundant.  These words, the words that would be repeated in each piece of the parallel structure, are called the common words.

Cut the Repeated Common Words

One of the guiding principles of parallelism, one might even say the very point itself, is to streamline by eliminating repetitions of the common words.  A common GMAT Sentence Correction wrong answer choice is of the form

[common words] A, B, and [common words]C

When A, B, and C are not single words, but rather long complicated phrases, it can be confusing to track the overall structure, and in its typical incorrect SC choices, the GMAT loves to “interrupt” the parallel structure by repeating some or all of the common words further down the list.  This can be particularly tricky if the parallel structure begins before the underlined section and ends within the underline section.  Here’s an example from the OG (one of the 20 new questions in the OG 13):

79) Ryunosuke Akutagawa‘s knowledge of the literatures of Europe, China and that of Japan were instrumental in his development as a writer, informing his literary style as much as the content of his fiction.

1. that of Japan were instrumental in his development as a writer, informing his literary style as much as
2. that of Japan was instrumental in his development as a writer, and it informed his literary style as well as
3. Japan was instrumental in his development as a writer, informing both his literary style and
4. Japan was instrumental in his development as a writer, as it informed his literary style as much as
5. Japan were instrumental in his development as a writer, informing both his literary style in addition to

The parallel structure in this sentence is among the three cultural regions cited: Europe, China, and Japan.  The common words are “knowledge of the literatures of”, and these words apply equal to all three terms.  Notice that, in answer choices (A) and (B), the GMAT supplies us with the classic mistake structure described above.  We have [common word] Europe, China, and [common word] Japan.  What’s particularly confusing is that the words “that of” is a very typical GMAT SC turn of a phrase, often appearing in correct answer choices.   Here, though, (A) & (B) have the classic mistake structure.  I’ll discuss the rest of this question at the end of this post.

But Keep Some Common Words

This is the rule that adds nuance to the previous principle.  When each of the three parallel elements is a single word, as is the case in the above sentence about Ryunosuke Akutagawa, then it’s appropriate to drop all repeated common words.  Often, though, especially on the GMAT SC, the parallel elements are not single words but rather long complicated phrases or clauses.  In that case, repeating a single common word, such as a preposition, can be crucial as a “signpost” for the parallel structure.  One of the more inspiring examples of this comes from a talented young writer of another hemisphere:

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.

In these famous words, the brilliant writer Thomas Jefferson alerts us to the parallel structure simple by introducing each new element with the word “that.”  That single word univocally illuminates the parallel structure of the sentence.  Another even loftier example:

In the midst of the candlesticks one like unto a son of man, clothed with a garment down to the foot, and girt about at the breasts with a golden girdle; and his head and his hair were white as white wool, white as snow; and his eyes were as a flame of fire; and his feet like unto burnished brass, as if it had been refined in a furnace; and his voice as the voice of many waters, and he had in his right hand seven stars: and out of his mouth proceeded a sharp two-edged sword: and his countenance was as the sun shineth in his strength.

In this richly poetic passage from the Book of Revelation (RSV, 1:13-16), the repeated word “and” acts as the signpost which guides us through the complex parallel structure.  Admittedly, the word “and” is an unlikely choice as the single repeated word on GMAT SC, and divine eschatological revelation is an exceedingly unlikely topic, but I hope this gives you a sense of the diversity of possibilities for this grammatical structure.

Bundling

Because of the balance of the previous two principles, every writer has a certain amount of discretion about how many of the common words are repeated, especially when there are only two term.  Some parallel structures “bundle” the parallel items with a set of preceding signal words: for example, “both X and Y”, “neither J nor K”, “not only P but also Q” (right there are three of the GMAT’s favorite parallelism templates!)

The rule for this situation is a little more complex.  Any common word that is not repeated must precede the first word of the signal words.  Any word that appears after the first signal word must also appear after the second signal word.  One outside is correct, and both inside is also correct, but an inside/outside combination is incorrect.  For example, consider this faux sentence completion question.  (I felt a somewhat less exemplary theme was in order after the foregoing examples!)

3) The senator bought a Valentine’s Day card for both his wife as well as for his mistress.

1. for both his wife as well as for his mistress.
2. for both his wife and also his mistress.
3. both for his wife as well as for his mistress
4. for both his wife and for his mistress.
5. both for his wife and for his mistress

The first thing we have to know: the correct structure recognized by the GMAT is “both A and B.”  The GMAT views the structures “both A and also B” and “both A as well as B” as redundant and incorrect.  Knowing this, we can immediately eliminate (A) & (B) & (C).  Choice (B) would have been perfect without the word “also”, but as is, it’s wrong.  Choice (D) falls foul of the bundling rule: the first “for” is before the word “both”, which should make it apply equally to both, but then a second “for” crops up in front of “his mistress.”  One outside, one inside: the classic mistake format for this particular structure.  Only (E) gets everything correct: it has the proper “both … and” structure, and the word “for” appears twice, once in front of each term.  Another correct choice would have been “for both his wife and his mistress”, but that was not offered among these five answer choices.

An Addendum on the Ryunosuke Akutagawa question

I promised I would resolve that question from the OG.  From parallelism, we have already eliminated (A) & (B), as discussed above.  The subject of the sentence, “knowledge”, is singular, so the verb must be singular: “was”, not “were.”  That eliminates (E).  This leaves (C) and (D), which have many similarities.  One difference is the ending.  (C) ends simply with “and”, correctly completing the “both … and” structure.  (D) avoids the word “both”, and instead ends with “as much as.”  The phrase “as much as” is a comparative phrase  —– “the teacher like me as much as she like you!” — but in this context, we are not performing a comparison.  The two items in question are Akutagawa’s literary style and the content of his fiction.  These both were informed by his vast knowledge of literature, but there’s nothing in the sentence that suggests a comparison is in order.  (D) also has that awkward phrase “as it informed”, instead of the shorter and more direct “informing” in (C).  For these reasons, (D) is incorrect, and (C) is by far the best answer choice.

Why does the GMAT ask CR?  Why does it matter?

You are preparing for the GMAT, which ostensibly means you are planning on attending business school, which in turn suggests that you are anticipating a management career in some aspect of the business world.  The entire business world runs on buying and selling: even if you are not a salesperson yourself, the success of your business, in a sense the raison d’etre of the business, is the money it makes from sales.

Well, in its essence, every sale is an argument.  If I want to sell you sometime, I have to present a case in some form to convince you to buy it.   If I make a wonderfully cogent argument, I may well generate the sale.  If my argument is faulty, and I repeat this pattern, that can only mean bad things for the long-term financial well-being of my business.  Arguments are important in business, and the skill of evaluating arguments is one that every manager should cultivate.  That’s precisely why business schools want you to bone up on it, which is why the GMAT asks about it in CR questions.

The 8 Types of CR Questions

Nearly all of the CR questions fall into one of the following eight categories.

1.) Weaken the Argument

2.) Strengthen the Argument

3.) Find the Assumption

4.) Draw a Conclusion/What Can Be Inferred?

5.) What is the Structure of the Argument?

6.) What is the Flaw in the Argument?

7.) Paradox Questions

8.) Evaluate the Conclusion

As I explain in this post, finding the assumption can help not only with question type #3, but also with either strengthening or weakening the argument.

1.) Last night, I walked my dog.

2.) I have walked Bucky every night for the last two years.

In the first sentence, I am doing the action, ‘walk’, only once. In the second sentence, I am describing something that has taken place on a number of occasions in the past and continues on till today (meaning tonight I will most likely walk Bucky).

The first tense is the simple past (if you look at my description it is very simple). The perfect tenses, on the other hand, aren’t so simple. To show you what I mean, let’s compare the present and the past.

1) Before I moved to California, I had walked Bucky in the mornings, not at nights.

2) Since moving to California, I have walked Bucky every evening.

The first sentence is an example of the past perfect tense. Notice, like the present perfect, that we have the verb ‘have’ coupled with another verb (which we call the participle):

Present Perfect: Has/have + Participle

Past Perfect: Had + Participle (plus another verb in the Simple Past)

Why use one tense versus the other? Well, if you notice in the first sentence, I am talking about two events that happened in the past: my walking Bucky and my moving to California. Whenever you are dealing with two events in the past, one of which started or happened before the other, you must use the past perfect tense to describe the event that started first.

First Event: I walked Bucky in the morning = Past Perfect Construction

Second Event: I moved to California = Simple Past

Another way to think of the past perfect is with specific dates. Let’s say I moved to California in 1984. I walked Bucky every morning from 1981 to 1984. The sentence implies that once I moved to California I no longer walked Bucky in the morning. That is, an event that happened repeatedly in the past stopped when another event happened. That interrupting event uses the simple past.

Now let’s try a couple of practice questions:

1) The corporation suffered/had suffered from consecutive quarterly losses until it hired/had hired a new CEO.

2) Every Christmas, the CEO granted/has granted employees three days off to celebrate the holidays.

In the first sentence, the event that happened first is the corporation suffering. So we want the past perfect tense: had suffered. The more recent action, the hiring of a new CEO, should be in simple past: hired.

For the second sentence, we want to describe an event that started in the past and continues in the present. So we need to use the present perfect tense: has granted.

Key Points

#1: Present Perfect: Has/Have + Participle = describes action/event that happened in the past and continues in the present.

#2: Past Perfect: Had + Participle = describes an action/event in the past that happened before another action in the past.

#3: Whenever we use the past perfect, we must also have another verb in the sentence that is in the simple past.

The two special triangles are right triangles with special angles and side.  Like all right triangles, they satisfy the Pythagorean Theorem.  These two triangles are “special” because, with just a couple pieces of information, we can figure out all their properties.  The GMAT-writers love this about these two triangles, so these special triangles are all over the place on the GMAT Quantitative Section.  In what follows, do your best to understand the argument: remembering what you understand is far more effective than simple memorizing.

The 45-45-90 Triangle

Let’s start with the square, that magically symmetrical shape.  Assume the square has a side of 1.  Cut the square in half along a diagonal, and look at the triangle that results.

We know ∠C = 90º, because it was an angle from the square.  We know AC = BC = 1, which means the triangle is isosceles, so ∠A = ∠B = 45º.  Let’s call hypotenuse AB = x.  By the Pythagorean Theorem,

The sides have the ratios .  We can scale this up simply by multiplying all three of those by any number we like: .

So, the three “names” for this triangle (which are useful to remember, because they summarize all its properties) are

1) The Isosceles Right Triangle

2) The 45-45-90 Triangle

3) The Triangle

The 30-60-90 Triangle

Let’s start with an equilateral triangle, another magically symmetrical shape.  Of course, by itself, the equilateral triangle is not a right triangle, but we can cut it in half and get a right triangle.  Let’s assume ABD is an equilateral triangle with each side = 2.  We draw a perpendicular line from A down to BD, which intersects at point C.  Because of the highly symmetrical properties of the equilateral triangle, the segment AC (a) forms a right angle at the base, (b) bisects the angle at A, and (c) bisects the base BD.

So, in the triangle ABC, we know ∠B = 60º, because that’s the old angle of the original equilateral triangle.  We know ∠C = 90º, because AC is perpendicular to the base.  We know ∠A = 30º, because AC bisects the original 60º angle at A in the equilateral triangle.  Thus, the angles are 30-60-90.  We know AB = 2, because that’s a side from the original equilateral triangle.  We know BC = 1, because AC bisects the base BD.  Call AC = x: we can find it from the Pythagorean Theorem.

The sides are in the ratio of .  This can be scaled up by multiplying by any number, which gives the general form: .

So, the three “names” for this triangle (which are useful to remember, because they summarize all its properties) are

1) The Half-Equilateral Triangle

2) The 30-60-90 Triangle

3) The Triangle

Summary

Math is not a spectator sport.  Now that you have seen these arguments, see if you can re-create the entire argument for the properties without looking at this post.  If you can remember, or even half remember, the entire argument, that will be enormously beneficial in remembering the properties themselves.

Here are some free practice questions:

1) http://gmat.magoosh.com/questions/1017

2) http://gmat.magoosh.com/questions/1016

The Mean

The mean is just the ordinary average: add up all the items on the list, and divide by the number of items.  As a formula, that’s

Notice, we can rewrite that as: (sum of items) = average*(number of items).  Rewritten in that form, it becomes one of the most powerful formulas on the GMAT: see this post [http://magoosh.com/gmat/2012/gmat-averages-and-sums-formulas/] for more details.

The Median

When we put the list in ascending order, the median is the middle.  If there are an odd number of items on the list, the middle item equals the median: for example, in the seven-element set {3, 5, 7, 9, 13, 15, 17}, the median is the fourth number, 9.  If there are an even number of items on the list, then the median is the average of the two middle numbers; for example, in the eight-element set {3, 5, 7, 9, 13, 15, 17, 17}, the median is 11 (the average of the fourth & fifth entries, 9 and 13).  Notice: when the number of items on the list is even, the median can equal a number not on the list.  Numbers above and below the median can be equal to the median, and that doesn’t change the fact that it’s a median; for example, the median of the set {1, 3, 3, 3, 3, 3, 74, 89, 312} is just 3, the fifth number of that nine-element set.

Range

The GMAT loves this one, because it’s so simple.  The range is the difference between the maximum value and the minimum value.  In the set {3, 5, 7, 9, 13, 15, 17}, the range = 17 – 3 = 14.  In the set {1, 3, 3, 3, 3, 3, 74, 89, 312}, the range = 312 – 1 = 311.

Standard Deviation

The range is a measure of the spread from the highest to the lowest value, but it doesn’t “feel” the numbers in between.  The standard deviation is also a measure of spread, that is to say, an indication of how far apart the numbers on the list are from each other.  Like the mean and unlike the range, the standard deviation “feels” every number on the list.   It has a technical definition that we can forego here; the majority of appearances of standard deviation on the GMAT revolve around a few simple ideas about it.

a) If all the entries of the list are equal, the standard deviation = 0.  In other words, they don’t deviate at all, because they’re all the same.

b) If you add/subtract a constant to/from every number on a list, that doesn’t change the standard deviation at all.  It’s just like taking the batch of data points and sliding them up or down the number line: that process doesn’t change how far apart they are from each other.

c) If you multiply/divide a list by a constant, then you also multiply/divide the standard deviation by this constant.

I’ll add an additional rule that really could only come into play in a very difficult upper-700s question:

d) If all the entries are the same distance from the mean, that distance is the standard deviation.  For example, in the set {3, 3, 3, 7, 7, 7}, the mean = 5, and every number “deviates” from the mean by exactly two units, so the standard deviation = 2.

If you master these simple ideas, you will dominate on Descriptive Statistics in the GMAT Quantitative section.  Here are some practice questions:

1) http://gmat.magoosh.com/questions/112

2) http://gmat.magoosh.com/questions/938

3) http://gmat.magoosh.com/questions/937

1) Shakespeare wrote fifteen comedies (including the so-called “romances”), ten histories, and twelve tragedies.  If a summer Shakespeare festival always has one comedy, one history, and two tragedies, how many different combinations of plays can the festival host?

As you see, this “counting” is a little more challenging than the kind of “counting” you learned in your salad days.  I would like to convince you, though, that you are quite capable of solving problems like this.

The Fundamental Counting Principle

This one big idea will give you a lot of mileage on any of the problems where the GMAT asks you to count things.

If option #1 has P alternatives and option #2 has Q alternatives (assuming that the two sets of alternatives have no overlap), then total number of different pairs we can form is P*Q.  For example: Shakespeare wrote fifteen comedies and ten histories.  If we want to select one comedy and one history, the total number of possible pairs is 15*10 = 150.

The FCP easily extends from two choices to three or any higher number.  However many collections of alternatives there are, you simply multiple the number of alternatives in each set to produce the total number of combinations.  For example: Shakespeare wrote fifteen comedies, ten histories, and twelve tragedies.  If we are going to pick one of each kind, and ask how many different trios of plays can we create, the total number is simply 15*10*12.  BTW, figuring that out without a calculator is not so hard.  First, on the 15*12, use the “doubling & halving” trick —– 15 times 2 is 30, and 12 divided by 2 is 6, so 15*12 = 30*6 = 180, and last, the easiest of all, multiplying by ten: 15*12*10 = 1800.

Permutations and Combinations

Before we can answer the original question posed, we have to clarify some terminology about counting.  A permutation is a set in which order matters —- AABC is a different permutation from BACA.  A combination is a set in which order does not matter —- AABC and BACA are the same combination of four letters.

This distinction is important in counting because we have to know whether to include the sets that repeat elements in different order.  In question #1 above, the question explicitly asks for combinations.  In other words, if we pick Hamlet and then King Lear, that’s will be considered the same as picking King Lear and then Hamlet.

This post: http://magoosh.com/gmat/2012/gmat-permutations-and-combinations/ considers permutations and combinations in greater detail.  For the purposes of this post, we just have to be careful to consider what we are counting.

How Many Pairs of Tragedies?

Shakespeare wrote twelve tragedies, and we want to pick a pair from these.  How many different pairs can we pick?  We will use the FCP.

Let’s break the task into a first choice and a second choice.  For the first choice, we can choose any of the 12 tragedies, so we have 12 choices.  Suppose, for the sake of argument, we pick Romeo and Juliet on the first choice.  Now, for the second choice, notice that we don’t still have 12 choices, because we don’t want to pick Romeo and Juliet again on the second choice.  In general, on the second choice, you have one fewer choice than you had on the first choice, because you don’t want to duplicate whatever element was chosen the first time.  Thus, on the second choice there are eleven choices.  By the FCP, it would seem the total number of possible pairs of would be 12*11.

If we were interested in permutations, then 12*11 would be the correct answer.  Here, though, we are interested in combinations, not permutations.  Again, if we pick Romeo and Juliet, then Macbeth, we will count that as the same pair as Macbeth first, followed by Romeo and Juliet.  The figure of 12*11 automatically counts each permutations, and so counts every pair twice, as AB and then as BA.  Thus, we have to divide 12*11 by two.  Thus, 12*11/2 = 6*11 = 66 is the number of possible pairs of tragedies.

The Whole Shebang

Now, we can answer the entire question.  We want the number of combinations of four plays consisting of one comedy, one history, and two tragedies.  By the FCP, that’s 15*10*66.  Notice the math tricks to multiply easily without a calculator.

Split 66 back into 6*11 —- 15*10*6*11

Switch the order —– 15*6*11*10

Use doubling-halving on the 15 & 6 —- (15*6)*11*10 =  (30*3)*11*10 = 90*11*10

Then 90*11 —- 90*11*10 = 990*10

Then, the easiest, multiplying by ten — 990*10 = 9900

Thus, the festival can come up with 9900 combinations consisting of one comedy, one history, and two tragedies.

Practice Question

Here’s a practice question on this very topic: https://gmat.magoosh.com/questions/845