## Quarter Wit, Quarter Wisdom: Tackling Roots on the GMAT

*Posted on on August 1, 2011*

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I thought I will take up “Roots” today but the rules of exponents and roots can get pretty monotonous. So let’s take a break today and do something more interesting. I keep telling my students that GMAT questions do not involve long calculations. If you find yourself dividing a three digit number with a two digit number, it means you have missed the point of the question. Today, I will depict what I mean with the help of a couple of examples. I start with one my favorite questions from the Veritas book.

Question 1: A certain portfolio consisted of 5 stocks, priced at $20, $35, $40, $45 and $70, respectively. On a given day, the price of one stock increased by 15%, while the price of another decreased by 35% and the prices of the remaining three remained constant. If the average price of a stock in the portfolio rose by approximately 2%, which of the following could be the prices of the shares that remained constant?

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As promised, in this post, I will discuss the questions I gave you in the last post. Let’s apply the rules of exponents that we have learned. I will recap all the rules first and then we will proceed to the questions.

Rule 1: a^{^m} × a^^{n} = a^{^(m + n)}

Rule 2: a^{^m} / a^^{n} = a^{^(m – n)}

Rule 3: (a^^{m})^^{n} = a^^{mn}

Rule 4: For any number a, a^{^0 }= 1

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No matter what score you are aiming for, knowing how to deal with exponents is crucial. Generally, questions based solely on exponents lie in the range of 600-650 and are one of the favorite topics of GMAT (but that’s just my opinion). It is also a topic which is very easy once you get the ‘hang of it’ but generates instant dislike until you don’t. I am sure many of you hate the sordid looking expressions/equations such as

(2^^{a} × 4 × 3^^{-4} × 3^{^b} )/(3^{^4} × 2^{^2}) = 8^^{-4} × 729

But of course, I don’t blame you. Sadly, exponents and roots are basics that we should be experts in – whether we are working on Number Systems, Algebra or even Geometry! So let’s roll!

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Last week, we started discussing some number properties. Let’s continue that discussion and dive into some more of those. In my opinion, it is the single most important topic on GMAT and one in which the smartest people slip easily. Think of this as a relatively easy way to earn another (or save) 20 or 30 points on your total GMAT score!

Let me show you the concept we will discuss today right away:

QUESTION: If 2^k is a factor of (10!), what is the greatest possible value of k?

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Continuing the discussion of some stand alone topics, let’s discuss an important number property today. It is not only useful to know for GMAT but, if understood well, will also help you a lot during your MBA, especially if you are keen on subjects in Finance since these subjects use a lot of ratios — e.g., Financial Leverage, P/E etc. You will often come across a situation where you will need to compare ratios. Say, you have a given ratio N/D. Now, a number ‘A’ is subtracted from both N and D. Is the new ratio (N – A)/(D – A) greater than or less than N/D? The answer depends on the original value of N/D.

The main concept is as follows:

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A couple of days back, during a class, we got into a discussion on “If you have ‘n’ variables, you need at least ‘n’ equations to solve for each variable.” It centered on the cases where you cannot apply this rule of thumb. Let me discuss a couple of those cases in detail today:

1. A question such as this:

Question: What is the value of (x + y + z)?

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Okay, let’s move away from “Divisibility and Remainders” on the GMAT, at least temporarily, and let’s focus our attention on another topic. If you have read some of my previous posts, I guess you know that I like to solve questions “logically.” I like to avoid making equations. Instead, I try to make myself “figure it out.”

A few days back, I came across a Time-Speed-Distance problem which was a perfect example of how you could “figure stuff out” without dealing with any equations. Actually, you can do that with a majority of GMAT questions (and save yourself loads of time!) but what was special about this question was that a couple of instructors of one of our competitor (I am not at liberty to disclose exactly who this competitor is!) had told their students that it is not possible to solve it logically. That got me thinking that perhaps, logical thinking is not as widely utilized as we at Veritas would like to believe. (I think our love for logical thinking is also apparent in the way we teach Sentence Correction!) Anyway, I thought of sharing the question and its logical solution with you! Here goes…

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In this post, I would like to focus on a particular type of remainder questions and how to solve them in a particular way. For the type of questions I am going to discuss today, I like to use “Binomial Theorem.” You might be tempted to run away right now and save yourself some precious time if you are not a Math geek but wait! We will just use an application of Binomial which I will explain in very simple language. I am quite certain that you will be comfortable with the method if you just give it a chance.

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For the past few weeks, we have been focusing on Divisibility and Remainders. There are some more ‘types’ of remainder questions. Let’s take them one by one so that by the end of it all, you are an expert in everything related to remainders. In this post I will start with a question similar to what you mind find in the *Official Guide for GMAT Review*.

Question: If a and b are positive integers such that a/b = 97.16, which of the following cannot be the remainder when a is divided by b?

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Let’s continue on our endeavor to understand divisibility and remainders in this post. Last week’s post focused on situations where the remainders were equal. Today, let’s see how to deal with situations where the remainders are different.

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Let’s continue our discussion of Divisibility and Remainders. If you have been preparing for GMAT for a while, I am sure you would have come across a question of the following form:

Question: When positive integer n is divided by 3, the remainder is 1. When n is divided by 7, the remainder is 5. What is the smallest positive integer p, such that (n + p) is a multiple of 21?

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Let’s continue from where we left the last post on divisibility problems on the GMAT. I will add another level of complexity to the last question we tackled in that post.

Question: A number when divided by 3 gives a remainder of 1. How many distinct values can the remainder take when the same number is divided by 9?

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Today, I will start the topic of Divisibility. We will discuss what divisibility is at a very basic level this week and then move on to remainders in the coming weeks.

So the first question is –- what is division? Don’t tell me what to do to divide a number by another, tell me why you do it. What is it that you are achieving by dividing one number by another? Let me tell you what I think –- I like to think that division is grouping. Not happy? Let’s look at an example then.

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I visit the GMAT Club forum regularly and discuss some ideas, some methodologies there. The weighted averages method I discussed in my previous two posts is one of my most highly appreciated inputs on the forum. People love how easily they can solve some of the most difficult questions by just drawing a scale or using a ratio. If you are not a Quant jock, I am sure you feel a chill run down your spine every time you see a mixtures problem. But guess what, they are really simple if you just use the same weighted average concepts we discussed in the previous two posts. Let’s look at a mixtures question in detail*:*

Mixture A is 20 percent alcohol, and mixture B is 50 percent alcohol. If the two are poured together to create a 15 gallon mixture that contains 30 percent alcohol, approximately how many gallons of mixture A are in the mixture?

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Let me start today’s discussion with a question from our Arithmetic book. I love this question because it is very crafty (much like actual GMAT questions, I assure you!) It looks like a calculation intensive question and makes you spend 3-4 minutes (scribbling furiously) but is actually pretty straight forward when understood from the ‘weighted average’ perspective. We looked at an easier version of this question in the last post.

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Today, I will delve into one of the most important topics (ubiquitous application) that are tested on GMAT. It is also one of the topics that will appear time and again during MBA e.g. in Corporate finance, you might be taught how to find ‘*Weighted Average* Cost of Capital’. So it will be highly beneficial if you have a feel for weighted average concepts.

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Being comfortable with common ratios can save you a lot of time on the GMAT. Last week we covered distance/rate problems. Another great application of ratios is work rate problems. An important relation that helps us solve work rate problems is:

Work Done = Rate * Time

This relation will lead a perceptive observer to draw a parallel with another very popular relation most of us have come across:

Distance = Speed * Time

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Let’s start with the applications of ratios today. Before you go on to an actual question, there is a relation between variables that you need to understand:

Distance = Speed × Time

Now let’s say my driving speed is a cool 100 mph. If I have to travel 100 miles, how much time will it take me? An hour, simple! Alright. If I have to travel 200 miles, how long will it take me? 2 hrs, you say? That is correct. What if I have to travel 500 miles? How long will it take me? 5 hrs, of course.. (now I am wasting your time, I know, but bear with me) When I hold my speed at a steady 100 mph, do you see a relation between Distance and Time? Can I say that if my distance doubles, my time taken doubles too? Can I say that if the distance that I have to travel on two different days is in the ratio 1: 5 (100 miles and 500 miles respectively), then the time I take on these two days will also be in the ratio 1:5 (1 hr and 5 hrs respectively)?

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Two trains, A and B, traveling towards each other on parallel tracks, start simultaneously from opposite ends of a 250 mile route. A takes a total of 3 hours to reach the opposite end while B takes a total of 2 hours to reach the opposite end. When train A meets train B during the journey, how far is train A from its starting point?

This question is not difficult but when I give it to my students, I see long, winding solutions, solutions that I find difficult to follow and after looking at a couple, my head starts to spin. Though, admittedly, most of them have the correct answer at the end. But you see, the correct answer alone is not enough. I like to see correct answers along with intuitive, intellectually stimulating methods. The most satisfying is seeing a blank rough sheet and the correct answer together. Rather than relying on the scratch pads and the markers, try and rely on your beautiful minds!

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I am sure you have heard of the phenomenal “power of compounding.” Elders love to preach about the wisdom of starting a savings account at the age of 25 (when you don’t have any money left from your pay check after your not-so-sensible Vuitton/Gucci/Chanel escapades!) rather than at the age of 40 (when you have 2 mortgages, 2 kids and a high maintenance Cadillac). Let’s crunch some numbers to see if they are right.

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Mark Up, Discount and Profit questions confuse a lot of people. But, actually, most of them are absolute sitters — very easy to solve — a free ride! How? We will just see. Let me begin with the previous post’s question.

Question: If a retailer marks up an article by 40% and then offers a discount of 10%, what is his percentage profit?

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Today, I will take a topic I briefly introduced in the previous post on percentages. Let me start with the question I posted there.

What does a 20% sale with an additional 25% off on the $85 sweater that you have your eye on mean to you?

It means a big rebate. Let’s see how much:

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Today, I will take up a relatively simple topic – Percentages. It is extremely relevant for GMAT and your everyday life. For the critics amongst you, let me give an example: What does a 20% sale with an additional 25% off on the $85 sweater that you have your eye on mean to you? Rather than flipping open your HP12C, blink your eyes and the answer will swim in front of you… Uhh… I mean, after I tell you what you have to do in that blink (There is always a catch!).

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There is one more important concept in Modulus that I want to discuss. Once it is done, we will bid farewell to Mods (for the time being at least), I promise. The concept involves dealing with multiple Mod terms (I will explain in just a minute). Before I start with the discussion, let me point out that it is relevant only if you are looking for a 50/51 in Quant and if you are looking for a 50/51 in Quant, then it is definitely relevant (I remember seeing a mean Modulus question in my GMAT a while back). But remember, don’t waste time on advanced Modulus questions if you are uncomfortable with Number Properties or other such high-weightage topics. Only when you are above 48 consistently in Quant, should you spend time on the two posts titled ‘Holistic Approach to Mods’. That said, everyone is welcome to read the posts and elicit his/her own takeaways.

Let’s start.

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I have a dream… A dream that one day, I will see my students making a bonfire of all their pens and pencils… that I will see them lost in thought in my Quant class, occasionally drawing lines and curves on a drawing sheet with colorful crayons… I will see them coming up with innovative logical solutions, just like that… But I know that no dream of mine is realized until and unless I keep my nose to the grindstone (I am not waiting with bated breath to achieve that elusive target weight.) So on this particular sleepless night, I will write a post with some more figures, figures that make complicated questions look like easy pickings. Let me explain using step by step approach.

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The other day, I stepped on the weighing scale after a long time (which included a 10-day-pigging-out vacation so I was a little apprehensive in the first place). When I saw the figure displayed, my head spun for a minute. What the…? How is it possible? I got down from the scale, tapped it here and there a bit — willing it to be sensible – and with my heart drumming in my ears, got on again. Still the same darned figure! Really now!

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I can say with reasonable certainty that until and unless you are a quant jock, you hate Inequalities (especially when they present themselves in DS questions). I can also say the same thing about Modulus (perhaps, not to the same extent!). So I can imagine what you feel when you come across a DS question with both Modulus and Inequalities! I am also certain that a small part of you, a very small part indeed, is secretly thrilled to see such a question because it implies that you are doing well in the exam and the software is getting jittery and trying to give you harder and harder questions. But wouldn’t it be something if you could crack the question in a minute and then say, “What else you got?”

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If you haven’t read parts I and II of this topic, I strongly suggest you read them first: Bagging the Graphs – Part I, Bagging the Graphs – Part II

While introducing this concept in Part I of Graphs, I had mentioned: “the one thing that I would suggest to increase speed in Co-ordinate Geometry and Algebra is Graphs”

Did you wonder why I included “Algebra” here? If yes, then this post will answer your question. In part 2, I gave an example of a Geometry question that can be easily solved using Graphs. In this post, I will take up an Algebra question for which you can do the same.

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If you haven’t read part I of this topic, I strongly suggest you read it first: Bagging the Graphs – Part I

Would you say it is easy breezy to draw a line if two points through which it passes are given? Sure. Plot the points approximately and join them! There you have your required line.

**A quick method of drawing the line represented by an equation: find two points through which it passes, plot the points and join them.**

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One thing that I would like to suggest to increase speed in Arithmetic is Multiplication Tables. Much to my dad’s chagrin, I am still a little lost when confronted with 16×7 or 17×8 or 18 × 7 (I know the last two are 126 and 136 but in what order, I am not sure) but rest I can pretty much manage. And many a times, while solving little toughies, I have blessed my dad for his incessant reproach regarding tables in days yonder.

Now, the one thing that I would like to suggest to increase speed in Coordinate Geometry and Algebra is learning how to draw graphs. Know how to draw a line from its equation in under ten seconds and you shall solve the related question in under a minute. For now, take my word for it and go ahead.

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Let’s start this thread from where we left the previous one. (If you haven’t read my previous post on this topic, I strongly suggest you read it first: Writing Factors of an Ugly Number.)

What happens if the total number of factors of a number is odd?

Let us take the example of N = 100. Break it down into its prime factors.

100 = 10 × 10 = 2^^{2} × 5^^{2}

How many factors will it have? (2 + 1)(2 + 1) = 9. Let us write them down:

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*This is the second installment in a new occasional series on our blog, authored by Karishma, one of our star GMAT prep instructors in Detroit, Michigan. From time to time Karishma will share some of her unique insights into how to maximize your potential on the GMAT. Enjoy!*

Let’s start with a very interesting and important topic that GMAT loves to test you on – Factors.

Factors are the divisors of a number. 1 is a factor of every number and the number itself is also a factor of that number. For the sake of giving the definition, let me say: A positive integer x is a factor of a positive integer N when there exists another positive integer y such that x × y = N. In other words, when N is divided by x, it doesn’t leave any remainder.

Now, let’s cut to the chase and go on to real business.

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*Today we introduce a new occasional series on our blog, authored by Karishma, one of our star GMAT prep instructors in Detroit, Michigan. From time to time Karishma will share some of her unique insights into how to maximize your potential on the GMAT. Enjoy!*

First up, I have a confession to make. I stumbled on the name for this section — “Quarter Wit, Quarter Wisdom” — while watching Spongebob Squarepants. If you are wondering what the heck was I doing watching the mind-numbing sanity-poaching children’s cartoon in the first place, then let me explain. My very stubborn two-year-old refuses to put a bite of food in her mouth until and unless Spongebob, the annoying little yellow porous sponge, and Patrick, the woefully unintelligent pink sea star, are on the television.

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