## Finding the Product of Factors on GMAT Questions

*Posted on on August 24, 2015*

We have discussed how to find the factors of a number and their properties in these two posts:

We have discussed how to find the factors of a number and their properties in these two posts:

Today, we look at the relative rate concept of work, rate and time – the parallel of relative speed of distance, speed and time.

But before we do that, we will first look at one fundamental principle of work, rate and time (which has a parallel in distance, speed and time).

Say, there is a straight long track with a red flag at one end. Mr A is standing on the track 100 feet away from the flag and Mr B is standing on the track at a distance 700 feet away from the flag. So they have a distance of 600 feet between them. They start walking towards each other. Where will they meet? Is it necessary that they will meet at 400 feet from the red flag – the mid point of the distance between them? Think about it – say Mr A walks very slowly and Mr B is super fast. Of the 600 feet between them, Mr A will cover very little distance and Mr B will cover most of the distance. So where they meet depends on their rate of walking. They will not necessarily meet at the mid point. When do they meet at the mid point? When their rate of walking is the same. When they both cover equal distance.

Let’s take a look at a very tricky GMAT Prep critical reasoning problem today. Problems such as these make CR more attractive than RC and SC to people who have a Quantitative bent of mind. It’s one of the “explain the paradox” problems, which usually tend to be easy if you know exactly how to tackle them, but the issue here is that it is hard to put your finger on the paradox.

We have discussed these three concepts of statistics in detail:

– Arithmetic mean is the number that can represent/replace all the numbers of the sequence. It lies somewhere in between the smallest and the largest values.

– Median is the middle number (in case the total number of numbers is odd) or the average of two middle numbers (in case the total number of numbers is even).

They say Mathematics is a perfect Science. There is a debate over this among scientists but we can definitely say that Mathematical methods are not perfect so we cannot use them blindly. We could very well use the standard method for some given numbers and get stranded with “no solution.” The issue is what do we do when that happens?

Today, we want to take up a conceptual discussion on expressions and equations and the differences between them. The concept is quite simple but a discussion on these is warranted because of the similarity between the two.

An **expression** contains numbers, variables and operators.

A few weeks back we discussed the kind of questions which beg you to think of the process of elimination – a strategy probably next only to number plugging in popularity.

Today we discuss the kind of questions which beg you to stay away from number plugging (but somehow, people still insist on using it because they see variables).

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Sometimes, to solve some tough questions, we need to make inferences. Those inferences may not be apparent at first but once you practice, they do become intuitive. Today we will discuss one such inference based high level question of an official GMAT practice test.

**Question**: In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x – y is:

Today we will discuss the flip side of “do not assume anything in Data Sufficiency” i.e. we will discuss “go ahead and assume in Problem Solving!”

Problem solving questions have five definite options, that is, “cannot be determined” and “data not sufficient” are not given as options. So this means that in all cases, data is sufficient for us to answer the question. So as long as the data we assume conforms to all the data given in the question, we are free to assume and make the problem simpler for ourselves. The concept is not new – you have been already doing it all along – every time you assume the total to be 100 in percentage questions or the value of n to be 0 or 1, you are assuming that as long as your assumed data conforms to the data given, the relation should hold for every value of the unknown. So the relation should be the same when n is 0 and also the same when n is 1.

There is something about factors and divisibility that people find hard to wrap their heads around. Every advanced application of a basic concept knocks people out of their seats! Needless to say, that the topic is quite important so we are trying to cover the ground for you. Here is another post on the topic discussing another important concept.

Before we start today’s discussion, recall a previous post on joint variation. A question arose some days back on the applicability of this concept. This official question was the case in point:

We know that we are often tested on parallelism on the GMAT. The logically parallel entities should be grammatically parallel. But today, we need to talk about circumstances where you might be tempted to employ parallelism but it would be incorrect to do so.

For example, look at this sentence:

A few days back, a student of ours asked me this question – in which cases is symmetry useful to us? Honestly, I don’t think I can create an exhaustive list of the topics where it could be useful. The first thing that comes to mind is of course, Geometry. Circles/equilateral triangles/squares/cubes are symmetrical figures. Symmetry helps us simplify questions which are based on these figures. We have also seen the uses of symmetry in dice throwing. In arrangements too, symmetry helped decrease our work substantially.

Sometimes, you come across some seriously interesting questions in Combinatorics. For example, this question I came across seemed like any other Combinatorics question, though it was a little cumbersome. But when I saw the answer, it got me thinking – it couldn’t have been a coincidence. There had to be a simpler logic to it and there was! I just wish I had thought of it before going the long route. So I must share it with you; you never know what might come in handy on test day!

As noticed in the first post of Alphametics, a data sufficiency alphametic is far more complicated than a problem solving alphametic. An alphametic can have multiple solutions and establishing that it does not, is time consuming. Hence, it is less likely that you will see a DS alphametic in the actual exam.

Last week, we looked at alphametics involving addition and subtraction. The logic becomes a little more involved when the alphametic involves multiplication. When a two digit number is multiplied by another two digit number, the process of finding the result is composed of multiple levels. Today, let’s see how to handle those multiple levels. The question involves quite a few steps and observations using number properties. Hence, you are unlikely to see such a question in actual GMAT but you might see a simpler version so it’s good to be prepared.

Today, let’s learn how to solve alphametics. An alphametic is a mathematical puzzle where every letter stands for a digit from 0 – 9. The mapping of letters to numbers is one-to-one; that is, the same letter always stands for the same digit, and the same digit is always represented by the same letter.

We have discussed simple and compound interest in a previous post.

We saw that simple and compound interest (compounded annually) in the first year is the same. In the second year, the only difference is that in compound interest, you earn interest on previous year’s interest too. Hence, the total two year interest in compound interest exceeds the two year interest in case of simple interest by an amount which is interest on year 1 interest.

On the GMAT, most sentence correction questions involve compound/complex sentences with multiple phrases, clauses and modifiers. Hence it is very likely that you will see some run-on sentences on your test. In the complicated sentences that we get on the GMAT, it is very easy to overlook that we are dealing with run-on sentences.

Confess it – while watching Harvey Specter and Mike Ross on ‘Suits’, many of you have wondered how ‘cool’ it would be to be a lawyer. It’s surprising how they question every assumption, every reason and come up with an innovative solution which looks as if the magician just pulled a rabbit out of a hat.

Last week, we looked at the basics of how to handle function questions. Today, let’s look at a couple of questions. We will start with an easier one and then go on to a slightly tougher one.

Let’s discuss how to handle functions today. People usually perceive functions as an advanced topic mainly because of the notation. But actually, the function questions are very simplistic and can be solved with a simple process. If we ask you the value of 5x^3 where x = 3, would you be worried about what to do? We assume you won’t be. Then there should be no problem with “given f(x) = 5x^3, what is the value of f(3)?”

We know that Combinatorics and Probability are tricky topics. It is easy to misinterpret questions of these topics and get the incorrect answer – which, unfortunately, we often find in the options, giving us a false sense of accomplishment.

In many questions, we need to account for different cases one by one but we don’t really see such questions on the GMAT since we have limited time. Also, we don’t tire of repeating this again and again – GMAT questions are more reasoning based than calculation intensive. Usually, there will be an intellectual method to solve every GMAT question – a method that will help you solve it in seconds.

Today, we would like to discuss with you one of our most debated critical reasoning questions. It is an absolutely brilliant question – not just because the correct option fits in beautifully but because the other four options are also very well thought out. It is easy to write the incorrect four options such that the student community will be split between 2 options – the correct one and one of the four incorrect ones but when the jury is split between 4 or all 5 options, that’s when we know that we have come up with an absolute masterpiece. Of course, in such questions, a lot of effort is needed to convince everyone of the correct answer but it is well worth it.

Today, let’s look in detail at a relation between arithmetic mean and geometric mean of two numbers. It is one of those properties which make sense the moment someone explains to us but are very hard to arrive on our own.

When two positive numbers are equal, their Arithmetic Mean = Geometric Mean = The number itself

We all know about the role of pre-thinking in Critical Reasoning and how anticipating the answer can be supremely beneficial in not just the physical aspect of saving time in analyzing options but also the psychological aspect of promoting our self-confidence – we were thinking that the answer should look like this and that is exactly what we found! Pre-thinking puts us in the driver’s seat and we feel energized without consuming any red bull!

We often tell you that if you are short on time, you can guess intelligently on a few questions and move on. Today we will discuss what we mean by “intelligent guessing”. There are many techniques – most of them involving your reasoning skills to eliminate some options and hence generating a higher probability of an accurate guess. Let’s look at one such method to get values in the ballpark.

We know that speed is important in GMAT. We have about 2 mins per question and we always have questions in which we get stuck, waste 3-4 mins and probably still answer incorrectly. So we are always trying to go faster, rush, complete the easy ones in less time! In our bid to save time, sometimes we sacrifice accuracy. We should know that accuracy is most important. No point running through questions and completing all of them before time if at the end of it all, most of our answers are incorrect – there are no bonus points for completing the test before time, after all!

Those of you who have seen the previous version of our curriculum would know that we had tips and tricks under the heading of ‘Lazy Genius’. These used to discuss innovative shortcuts for various questions – the way very smart people would solve the question – without putting in too much effort!

As promised last week, we will look at another question which involves finding the last two digits of the product of some random numbers. In this question, along with the concepts discussed last week, we will assimilate the concept of negative remainders too discussed some weeks ago.

Let’s continue the discussion of last two digits we started last week. We discussed the concept of pattern recognition and how it can help us determine the last two digits in case of numbers raised to some powers. Today we look at what happens when there is no pattern to determine! What if we are asked to determine the last two digits of the product of a bunch of numbers. We know that getting the last digit in this case is very easy – just multiply the last digits of the numbers together. But last TWO digits would seem much more complicated.

We all know how to find the last digit using cyclicity when we are given a number raised to a power. Last digit of a number depends only on the last digit of the base. You must be quite familiar with something like this –

Last Digit of Base:

0 – Last digit of expression with any power will be 0.

You must have come across questions which you thought tested one concept but later found out could be easily dealt with using another concept. Often, crafty little mixture problems belong to this category. For example:

Mark is playing poker at a casino. Mark starts playing with 140 chips, 20% of which are $100 chips and 80% of which are $20 chips. For his first bet, Mark places chips, 10% of which are $100 chips, in the center of the table. If 70% of Mark’s remaining chips are $20 chips, how much money did Mark bet?

We have discussed weighted averages in detail here but one thing we are yet to talk about is how you decide what the weights will be in weighted average problems. It is not always straight forward to identify the weights. For example, in a question such as this one,

A few weeks back, we wrote a post busting some Sentence Correction myths. Let’s continue from where we left. We discussed how we can have pronouns referring to different antecedents in different clauses of the same sentence. Let’s take another example illustrating that principle. Also, we learn how to use ‘being’ correctly in GMAT.