Before we get started, be sure to take a look at Part I of this article. Number properties concepts come across as pretty easy, theoretically, but they have some of the toughest questions. Today let’s take a look at some properties of prime numbers and their sum. Note that don’t try to “learn” all the takeaways you come across for number properties – it will be very stressful. Instead, try to understand why the properties are such so that if you get a question related to some such properties, you can replicate the results effortlessly.
Tag Archives : Quarter Wit Quarter Wisdom
Don’t worry, we are not going to discuss (Even + Even = Even) and (Odd + Odd = Even) type of basic number properties in this post. What we have in mind for today is something based on this but far more advanced. Often, people complain that they thoroughly understand the theory but have difficulties applying it and hence are stuck at a score of 600. They look for practice questions and tend to ignore concepts since they already “know” them. We often ask them to go back to concepts since we believe that a strong foundation of concepts is necessary for ‘score increase’. Mind you, when we do that, we don’t mean to ask them to review the basic concepts again, we mean to ask them to deduce and work on advanced concepts. Let’s show you with the help of a question.
I am no fan of formulas, especially the un-intuitive ones but the one we are going to discuss today has proved quite useful. It is for a concept tested on GMAT Prep so it might be worth your while to remember this little formula.
When two items are sold at the same selling price, one at a profit of x% and the other at a loss of x%, there is an overall loss. The loss% = (x^2/100)%
Recall the important property that we discussed about the relation between the areas of the two similar triangles last week – if the ratio of their sides is ‘k’, the ratio of their areas will be k^2. As mentioned last week, it’s an important property and helps you easily solve otherwise difficult questions. The question I have in mind today also brings in focus the Pythagorean triplets.
Our Geometry book discusses the various rules we use to recognize similar triangles such as SSS, AA, SAS and RHS so we are assuming that we needn’t take those up here.
We are also assuming that you are comfortable with the figures that beg you to think about similar triangles such as
Today, we will discuss the question we left you with last week. It involves a lot of different concepts – remainder on division by 5, cyclicity and negative remainders. Since we did not get any replies with the solution, we are assuming that it turned out to be a little hard.
I could have sworn that I had discussed negative remainders on my blog but the other day I was looking for a post discussing it and much as I would try, I could not find one. I am a little surprised since this concept is quite useful and I should have taken it in detail while discussing divisibility. Though we did have a fleeting discussion of it here.
I came across a discussion on one of our questions where the respondents were split over whether it is a strengthen question or weaken! Mind you, both sides had many supporters but the majority was with the incorrect side. You must have read the write up on ‘support’ in your Veritas Prep CR book where we discuss how question stems having the word ‘support’ could indicate either strengthen or inference questions. I realized that we need a write up on the word ‘suspect’ too so here goes.
We pick up this post from where we left the post of last week in which we looked at a few properties of absolute values in two variables. There is one more property that we would like to talk about today. Thereafter, we will look at a question based on some of these properties.
We have talked about quite a few concepts involving absolute value of x in our previous posts. But some absolute value questions involve two variables. Then do we need to consider the positive and negative values of both x and y? Certainly! But there are some properties of absolute value that could come in handy in such questions. Let’s take a look at them:
In the last three weeks, we discussed a couple of strategies we can use to solve max-min questions: ‘Establishing Base Case’ and ‘Focus on Extremes’. Now try to use those to solve this question:
Question: A carpenter has to build 71 wooden boxes in one week. He can build as many per day as he wants but he has decided that the number of boxes he builds on any one day should be within 4 off the number he builds on any other day.
(A) What is the least number of boxes that he could have build on Saturday?
(B) What is the greatest number of boxes that he could have build on Saturday?
Continuing our discussion on maximizing/minimizing strategies, let’s look at another question today. Today we discuss the strategy of establishing a base case, a strategy which often comes in handy in DS questions. The base case gives us a starting point and direction to our thoughts. Otherwise, with the number of possible cases in any given scenario, we may find our mind wandering from one direction to another without reaching any conclusions. That is a huge waste of time, a precious commodity.
We haven’t dealt with maximizing/minimizing strategies in our QWQW series yet (except in sets). The reason for this is that the strategy to be used varies from question to question. What works in one question may not work in another. You might have to think up on what to do in a question from scratch and you have only 2 mins to do it in. The saving grace is that once you know what you have to do, the actual work involved to arrive at the answer is very little.
Last week we discussed the properties of terminating decimals. We also discussed that non-terminating but repeating decimals are rational numbers.
Last week, we discussed the basics of terminating decimals. Let me review the important points here:
- To figure out whether the fraction is terminating, bring it down to its lowest form.
We know the basics of decimals and rational numbers.
- Decimals can be rational or irrational.
- Decimals which terminate and those which are non-terminating but repeating are rational. They can be written in the form a/b.
- Decimals which are non-terminating and non-repeating are irrational such as ?2, ?3 etc.
This is hard to confess publicly but I must because it is a prime example of how GMAT takes advantage of our weaknesses – A couple of days back, I answered a 650 level question of weighted averages incorrectly. Those of you who have been following my blog would understand that it was an unpleasant surprise – to say the least. I know my weighted averages quite well, thank you! For this comedown, I blame the treachery of GMAT because it knows how to get you when you become too complacent. The takeaway here is – no matter how easy and conventional the question seems, you MUST read it carefully.
While eagerly awaiting the kick off of season 3 of BBC’s Sherlock, let’s put our time to good use. Though we have already spent a lot of it speculating over what really happened to Sherlock (HOW did he come back?!), perhaps we can take a leaf out of his book and learn to notice little things in whatever is leftover. There is a good reason to do that – there are little clues in some questions that the test maker unwittingly leaves to bring clarity to the question. If we understand those clues, a seemingly mysterious problem could be easily unraveled. Let us show you with an example.
When faced with an unusual quadratic equation, some people waste a lot of time while trying to ‘split the middle term’. The common refrain is ‘I am just not good at it.’ Actually it has little to do with intuition and a lot to do with understanding how numbers work. If I am looking at a quadratic equation and am unable to find the required factors, I will go back to check my quadratic to see if it is correct rather than try to use the esoteric quadratic formula.
Coming back to Integrated Reasoning question types, let’s discuss a cumulative graph today. They are usually a little trickier than your usual line/pie/bar graphs since you have to focus on not the data points but ‘the change’ from one data point to another. Every subsequent data point will be either above or at the same level as the previous data point.
I know I promised that I will bring you some tricky Integrated Reasoning questions this week, but I am really irked by the ‘which’ vs ‘that’ debate and would like to put it to rest once and for all. Hence, in this post I would like to talk about restrictive and non-restrictive clauses, about ‘which’ and ‘that’, about when to use a comma and some other such things.
Now that we have seen some basic Integrated Reasoning question types, we will look at some tricky questions but not this week. This week, we would like to discuss a Critical Reasoning question. This question is simple and straight forward but still many people falter in it. The reasons for this are not hard to find. Let’s analyze this question in detail.
For the past couple of weeks, we have been talking about integrated reasoning. Today we will continue with that and take up a multi-source reasoning question. These questions often include substantial data and require you to make inferences based on it. They test your logical and reasoning aptitude so don’t get lost in the data. Review the given information and then jump on to the questions. Then come back to the relevant part of the given information and peruse it in detail.
Continuing our discussion on IR, let’s look at a graph today and learn how to infer from the data given in it. You may not need to do too many calculations because the options in the drop down menu may allow you to approximate i.e. the options may be quite far apart. Also, you will need to segment the graph into regions using imaginary vertical lines e.g. number of household spending less than 2 hrs at the mall in the graph given below.
Starting today, for the next few weeks we would like to focus on the ‘Integrated Reasoning’ section of the GMAT. The 1.5 yr old section of the GMAT has been giving jitters to many people. We have come across people with 48+ Quant scores but a 2/3 on the IR section. In my opinion, that’s a little strange. If you have strong reasoning skills, there is no reason you cannot apply those to this section as well.
Continuing our scrutiny of interesting standalone questions with important takeaways, let’s discuss today how too much knowledge can actually let you down. We often come across people wondering whether they should learn up the many formulas in permutation/combination, co-ordinate geometry etc. Our take on the question is a flat ‘No’. Formulas won’t take you far in GMAT, perhaps up to 600 but certainly not further. In fact, until and unless you have an eidetic memory or a Math PHD, chances are that knowing too many formulas will be a disadvantage. Let me show you why:
Some days back I came across a question which was a slight twist on a regular question type. The usual active voice of the sentence had been changed to passive but in such a way that the meaning had been altered. It was a lesson in DS as well as SC – read every word carefully. One word could change a 600 level to a 750 level one, a mundane everyday question to a smart question. We often see this interesting transformation in P&C questions but for that to happen in algebra was quite a delight. Let’s discuss that particular question today.
Many of us are hooked on to using algebra in Quant questions. The thought probably is that how can it be a Quant question if one did not need to take a couple of variables and make a couple of equations/inequalities. We, at Veritas Prep, love to harp on about how algebra is time consuming and unnecessary in most cases. But today we will go one step further and discuss how indiscriminate use of algebra can actually result in incorrect answers. Surprised, eh?
One of the basic things you need to know before you start your GMAT preparation is how to solve quadratic equations i.e. factorize the quadratic and equate each factor to 0 to get the possible values that x can take. Today we will discuss how you can solve a third degree equation.
Say an equation such as x^3 – 6x^2 + 11x – 6 = 0.
Last week we looked at a question whose solution was quite hard to explain. This week we will look at a question in which the question itself is hard to explain (so no point worrying about the difficulty in explaining the solution as of now!) So why are we discussing such a question? Because it is certainly not out of GMAT-scope. It uses the concepts of relative speed and GMAT could give you some pretty intimidating questions at higher levels. So what should be your strategy when you come across a question which takes a minute or more to sink in? After you understand the question, first of all you should congratulate yourself that the toughest part is already over. If the question is hard to understand, the solution would be cake walk (well, at least it will feel like it).
I have to admit that probability is confusing. The problem is not so much that students find it hard to understand as that teachers find it hard to explain. There are subtle points in a probability question that make all the difference in the world and it takes a ton of ingenuity to explain them in a manner that others understand. You either get the point right away, or you don’t.
By now, you know that we like to discuss visual approaches to problems. A visual tool that we have used before for solving inequality and modulus questions is the number line. The number line is also useful in helping us solve many number properties questions.
A few things to keep in mind when dealing with number line:
A few months back, one of our posts talked about knowing which numbers to plug-in in case you want to use the number–plugging method. To be more exact, we discussed that you need to find the transition points i.e. the points where the two sides of the inequality become equal. The transition points tend to reverse the relation between the two sides. For a detailed discussion of this concept, revisit this post.
Most people who plan to take GMAT seriously take a few prep tests, practice tests or mock tests, whatever you may like to call them. Usually, the tests are taken to gauge one’s current level i.e. to get an approximate idea of what one would score if one were to take GMAT that day. Of course, they have other uses too – practice in timed environment, build stamina, identify strengths and weaknesses etc. Usually, these tests are fairly accurate (with an error of up to 40-50 points in the total score). A recent phenomena has been much lower score (especially verbal) compared to the prep test scores (not among Veritas Prep students though – I will explain the reason for this soon).
Let’s look at a question today which encompasses most of what we have discussed in this topic. This will be the last post on this topic for a while now. We assume that after going through these posts thoroughly, if you come across any question on ‘this inscribed in that’, you should be able to handle it. Just a reminder, keep in mind the symmetry of the figures you are handling.
Last week we looked at questions on polygons inscribed in a circle. This week, let’s look at questions on circles inscribed in regular polygons. As noted earlier, it’s important to keep in mind that regular polygons are symmetrical figures. You need very little information to solve for anything in a symmetrical figure.