# Quarter Wit, Quarter Wisdom: The Power in Factorials

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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# Quarter Wit, Quarter Wisdom: The Push and Pull Around One

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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# Quarter Wit, Quarter Wisdom: Beyond the Rule of Thumb

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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# Quarter Wit, Quarter Wisdom: An Intellectual Exercise in TSD

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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# Quarter Wit, Quarter Wisdom: A Remainders Post for the Geek in You!

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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# Quarter Wit, Quarter Wisdom: Knocking Off the Remaining Remainders

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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# Quarter Wit, Quarter Wisdom: Divisibility Applied to Remainders Part II

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.

Question: When positive integer n is divided by 3, the remainder is 2. When n is divided by 7, the remainder is 5. How many values less than 100 can n take?

(A) 0
(B) 2
(C) 3
(D) 4
(E) 5
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# Quarter Wit, Quarter Wisdom: Divisibility Applied to Remainders

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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# Quarter Wit, Quarter Wisdom: Divisibility Applied on the GMAT

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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# Quarter Wit, Quarter Wisdom: Divisibility Unraveled

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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# Quarter Wit, Quarter Wisdom: Don’t Get Mixed-Up in Mixtures

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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# Quarter Wit, Quarter Wisdom: Zap the Weighted Average Brutes

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.

John and Ingrid pay 30% and 40% tax annually, respectively. If John makes \$56000 and Ingrid makes \$72000, what is their combined tax rate?

a. 32%
b. 34.4%
c. 35%
d. 35.6%
e. 36.4%

If we do not use weighted averages concept, this question would involve a tricky calculation. Something on the lines of:

Total Tax = (30/100)*56000 + (40/100)*72000

Tax Rate = Total Tax / (56000 + 72000)

But we know better! The big numbers – 56000 and 72000 are just a smokescreen. I could have as well given you \$86380 and \$111060 as their salaries; I would have still obtained the same average tax rate! What is important is not the actual values of the salaries but the relation between the values i.e. the ratio of  their salaries. Let me show you.

We need to find their average tax rate. Since their salaries are different, the average tax rate is not (30 + 40)/ 2. We need to find the ‘weighted average of their tax rates’. In the last post, we discussed

w1/w2 = (A2 – Aavg) / (Aavg – A1)

The ratio of their salaries w1/w2 = 56000 / 72000 = 7/9

7/9 = (40 – Tavg) / (Tavg – 30)

Tavg = 35.6%

Imagine that! No long calculations! In the last post, when we wanted to find the average age of boys and girls – 10 boys with an average age of 17 yrs and 20 girls with an average age of 20 yrs, all we needed was the relative weights (relative number of people) in the two groups i.e. 1:2. It didn’t matter whether there were 10 boys and 20 girls or 100 boys and 200 girls. It’s exactly the same concept here. It doesn’t matter what the actual salaries are. We just need to find the ratio of the salaries.

Also notice that the two tax rates are 30% and 40%. The average tax rate is 35.6% i.e. closer to 40% than to 30%. Doesn’t it make sense? Since the salary of Ingrid is \$72,000, that is, more than salary of John, her tax rate of 40% ‘pulls’ the average toward itself. In other words, Ingrid’s tax rate has more ‘weight’ than John’s. Hence the average shifts from 35% to 35.6% i.e. toward Ingrid’s tax rate of 40%.

Let’s now look at PS question no. 148 from the Official Guide which is a beautiful example of the use of weighted averages.

If a, b and c are positive numbers such that [a/(a+b)]*20 + [b/(a+b)]*40 = c and if a < b, which of the following could be the value of c?

(A) 20

(B) 24

(C) 30

(D) 36

(E) 40

Let me tell you, it isn’t an easy question (and the explanation given in the OG makes my head spin).

First of all, notice that the question says: ‘could be the value of c’ not ‘is the value of c’ which means there isn’t a unique value of c. ‘c’ could take multiple values and one of those is given in the options. Secondly, we are given that a < b. Now how does that figure in our scheme of things? It is not an equation so we certainly cannot use it to solve for c. If you look closely, you will notice that the given equation is

(20*a + 40*b) / (a + b) = c

Does it remind you of something? It should, considering that we are doing weighted averages right now! Isn’t it very similar to the weighted average formula we saw in the last post?

(A1*w1 + A2*w2) / (w1 + w2) = Weighted Average

So basically, c is just the weighted average of 20 and 40 with a and b as weights. Since a < b, weightage given to 20 is less than the weightage given to 40 which implies that the average will be pulled closer to 40 than to 20. So the average will most certainly be greater than 30, which is right in the middle of 40 and 20, but will be less than 40. There is only one such number, 36, in the options. ‘c’ can take the value ‘36’ and hence, (D) will be the answer. Elementary, isn’t it? Not really! If you do not consider it from the weighted average perspective, this question can torture you for hours.

These are just a couple of many applications of weighted average. Next week, we will review Mixtures, another topic in which weighted averages are a lifesaver!

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 in Detroit, Michigan, and regularly participates in content development projects such as this blog!

# Quarter Wit, Quarter Wisdom: Heavily Weighted Weighted Averages

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.

The first question is – What is Weighted Average? Let me explain with an example.

A boy’s age is 17 years and a girl’s age is 20 years. What is their average age?

Simple enough, isn’t it? Average age = (17 + 20)/2 = 18.5

It is the number that lies in the middle of 17 and 20. (Another method of arriving at this number would be to find the difference between them, 3, and divide it into 2 equal parts, 1.5 each. Now add 1.5 to the smaller number, 17, to get the average age of 18.5 years. Or subtract 1.5 from the greater number, 20, to get the average age of 18.5 years. But I digress. I will take averages later since it is just a special case of weighted averages.)

Now let me change the question a little.

There are 10 boys and 20 girls in a group. Average age of boys is 17 years and average age of girls is 20 years. What is the average age of the group?

Many people will be able to arrive at the following:

Average Age = (17*10 + 20*20)/(10 + 20) = 19 years

Average age will be total number of years in the age of everyone in the group divided by total number of people in the group. Since the average age of boys is 17, so total number of years in the 10 boys’ ages is 17*10. Since the average age of girls is 20, the total number of years in the 20 girls’ ages is 20*20. The total number of boys and girls is 10 + 20. Hence you use the expression given above to find the average age. I hope we are good up till now.

To establish a general formula, let me restate this question using variables and then we will just plug in the variables in place of the actual numbers above (Yes, it is opposite of what you would normally do when you have the formula and you plug in numbers. Our aim here is to deduce a generic formula from a specific example because the calculation above is intuitive to many of you but the formula is a little intimidating.)

There are w1 boys and w2 girls in a group. Average age of boys is A1 years and average age of girls is A2 years. What is the average age of the group?

Average Age = (A1*w1 + A2*w2)/(w1 + w2)

This is weighted average. Here we are not finding the average age of 1 boy and 1 girl. Instead we are finding the average age of 10 boys and 20 girls so their average age will not be 18.5 years. Boys have been given less weightage in the calculation of average because there are only 10 boys as compared to 20 girls. So the average has been found after accounting for the weightage (or ‘importance’ in regular English) given to boys and girls depending on how many boys and how many girls there are. Notice that the weighted average is 19 years which is closer to the average age of girls than to the average age of boys. This is because there are more girls so they ‘pull’ the average towards their own age i.e. 20 years.

Now that you know what weighted average is and also that you always knew the weighted average formula intuitively, let’s move on to making things easier for you (Tougher, you say? Actually, once people know the scale method that I am going to discuss right now (It has been discussed in our Statistics and Problem Solving book too), they just love it!)

So, Average Age, Aavg = (A1*w1 + A2+w2)/(w1 + w2)

Now if we re-arrange this formula, we get, w1/w2 = (A2 – Aavg)/(Aavg – A1)

So we have got the ratio of weights w1 and w2 (the number of boys and the number of girls). How does it help us? Knowing this ratio, we can directly get the answer. Another example will make this clear.

John pays 30% tax and Ingrid pays 40% tax. Their combined tax rate is 37%. If John’s gross salary is \$54000, what is Ingrid’s gross salary?

Here, we have the tax rate of John and Ingrid and their average tax rate. A1 = 30%, A2 = 40% and Aavg = 37%. The weights are their gross salaries – \$54,000 for John and w2 for Ingrid. From here on, there are two ways to find the answer. Either plug in the values in the formula above or use the scale method. We will take a look at both.

1. Plug in the formula

w1/w2 = (A2 – Aavg)/(Aavg – A1) = (40 – 37)/(37 – 30) = 3/7

Since A1 is John’s tax rate and A2 is Ingrid’s tax rate, w1 is John’s salary and w2 is Ingrid’s salary

w1/w2 = John’s Salary/Ingrid’s Salary = 3/7 = 54,000/Ingrid’s Salary

So Ingrid’s Salary = \$126,000

It should be obvious that either John or Ingrid could be A1 (and the other would be A2). For ease, it a good idea to denote the larger number as A2 and the smaller as A1 (even if you do the other way around, you will still get the same answer)

2. Scale Method

On the number line, put the smaller number on the left side and the greater number on the right side (since it is intuitive that way). Put the average in the middle.

The distance between 30 and 37 is 7 and the distance between 37 and 40 is 3 so w1:w2 = 3:7 (As seen by the formula, the ratio is flipped).

Since w1 = 54,000, w2 will be 126,000

So Ingrid’s salary is \$126,000.

This method is especially useful when you have the average and need to find the ratio of weights. Check out next week’s post for some 700 level examples of weighted average.

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 in Detroit, Michigan, and regularly participates in content development projects such as this blog!

# Quarter Wit, Quarter Wisdom: Cracking the Work Rate Problems

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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# Quarter Wit, Quarter Wisdom: Applications of Ratios in TSD

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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# Quarter Wit, Quarter Wisdom: The Sorcery of Ratios

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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# Quarter Wit, Quarter Wisdom: Simple Interest and the Not-So-Simple One

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.

Starting at age 25, if you put \$200 every month for 40 years at 10% per annum, you will have more than 1.25 million at the age of 65.

Starting at age 40, if you put \$200 every month for 25 years at 10% per annum, you will have only \$265,000 at the age of 65.

You might have reservations about the fact that in the first case, you are investing more, after all! It’s not just about the longer time period. So look at it another way.
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# Quarter Wit, Quarter Wisdom: Mark Up, Discount, and Profit

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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# Quarter Wit, Quarter Wisdom: A Simple Approach to Percentages

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!).

Let me begin by saying that a percentage is a fraction. A fraction where the denominator is always 100, but just a fraction nevertheless. 50% means 50 per ‘cent’ (cent being 100) or 50/100 or 50 out of every 100.
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# Quarter Wit, Quarter Wisdom: The Holistic Approach to Mods – Part II

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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# Quarter Wit, Quarter Wisdom: The Holistic Approach to Mods

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.

A Complicated Question: If y = ||x – 5| – 10|, for how many values of x is y = 1? (I remember once someone said, “I think I would rather eat spinach than try such questions.”)

Easy Pickings:

Graph of y = x is a line passing through the center with slope 1.

Graph of y = |x| is as shown below. Modulus cannot be negative so all negative values of y are flipped to positive. (The red line shows the original position for reference.)

To get the graph of y = |x – 5|, shift the above graph 5 units to the right on the x axis. This is so because in the graph above, y = 0 when x = 0. But in the required graph, y should be 0 when x = 5. Hence the point at (0, 0) shifts to (5, 0). Since the slope of the line is 1, it makes an intercept of 5 on the y axis.

The graph of y = |x – 5| – 10 is just the above graph shifted down by 10 units because now y is 10 less than every previous value of y.

Now we need to take the modulus of the equation above to get y = ||x – 5| – 10|. Since a modulus is never negative, whatever part of the graph is below the x axis i.e. in quadrants III and IV, gets reflected above the x axis in quadrants I and II.

This is the graph we wanted. We see that the line y = 1 (shown in green below) intersects this graph at 4 points. So y = 1 when x = -6 or -4 or 14 or 16.

Put these values in the given equation if you want to cross check. Once you get the hang of it, you can arrive at this graph in under a minute! Such tricky questions can be elegantly handled using this approach. In fact, we can add many more levels of complexity and still easily arrive at our answer. For shakes, try out the graphs of y = |||x – 5| – 10| -5| and y = |||x| – 3| – x|!

Now that you have lost your sleep, I think I will sleep easier!

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 in Detroit, Michigan, and regularly participates in content development projects such as this blog!

# Quarter Wit, Quarter Wisdom: Of Pounds and Ounces

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!

And then it struck me. It was showing me my weight in pounds instead of the usual kilograms I like to see it in (weight in double digits is so much more reassuring!). The constriction in my chest relaxed and I breathed again. I remembered I had converted it to pounds to check the weight of my check-in luggage before leaving for my vacation. I have half a mind to sue those weighing scale manufacturers – the confusion caused by offering the feature of both units is hazardous to the health of people, an outcome which is in direct contrast to the intended use of the instrument.
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# Quarter Wit, Quarter Wisdom: Do What Dumbledore Did!

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 absolute value (perhaps, not to the same extent!). So I can imagine what you feel when you come across a DS question with both absolute value 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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# Quarter Wit, Quarter Wisdom: Bagging the Graphs – Part III

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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# Quarter Wit, Quarter Wisdom: Bagging the Graphs – Part II

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.
Continue reading “Quarter Wit, Quarter Wisdom: Bagging the Graphs – Part II”

# Quarter Wit, Quarter Wisdom: Bagging the Graphs

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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# Quarter Wit, Quarter Wisdom: Factors of Perfect Squares

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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# Quarter Wit, Quarter Wisdom: Writing Factors of an Ugly Number

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.
Continue reading “Quarter Wit, Quarter Wisdom: Writing Factors of an Ugly Number”

# Quarter Wit, Quarter Wisdom: A Prelude

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.
Continue reading “Quarter Wit, Quarter Wisdom: A Prelude”