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Maths Session 3

Synopsis of 2 sutras in Vedic mathematics

Lets get down to business. Let us see some magic of vedic mathematics to make our life simple. In the following sessions, we will try to cover 16 sutras of Vedic mathematics which will eventually help you in calculating faster. This is very essential for CAT quants & DI section.

1. All from 9 and the last from 10

For example 1000 - 357 = 643

We simply take each figure in 357 from 9 and the last figure from 10.

So the answer is 1000 - 357 = 643

And that is all there is to it! This always works for subtractions from numbers consisting of a 1 followed by noughts: 100; 1000; 10,000 etc.

Similarly 10,000 - 1049 = 8951







Vertically and crosswise

Suppose you need 8 x 7

8 is 2 below 10 and 7 is 3 below 10. Think of it like this:






The answer is 56. The diagram below shows how you get it.





You subtract crosswise 8-3 or 7 - 2 to get 5, the first figure of the answer. And you multiply vertically: 2 x 3 to get 6, the last figure of the answer.

Suppose you want to multiply 88 by 98. You can give the answer immediately, using the same methods above. Both 88 and 98 are close to 100.88 is 12 below 100 and 98 is 2 below 100.

You can imagine the sum set out like this:







As before the 86 comes from subtracting crosswise: 88 - 2 = 86 (or 98 - 12 = 86: you can subtract either way, you will always get the same answer).And the 24 in the answer is just 12 x 2: you multiply vertically. So 88 x 98 = 8624

103 x 104 = 10712

The answer is in two parts: 107 and 12, 107 is just 103 + 4 (or 104 + 3), and 12 is just 3 x 4.

Similarly 107 x 106 = 11342 : 107 + 6 = 113 and 7 x 6 = 42

Look at another example:






Multiply crosswise and add to get the top of the answer: 2 x 5 = 10 and 1 x 3 = 3. Then 10 + 3 = 13. The bottom of the fraction is just 3 x 5 = 15. You multiply the bottom number together.

So:






Subtracting is just as easy: multiply crosswise as before, but the subtract:






More in next session. Till then, do write me for any clarification…


By:
Arjun Pal
(Knowledge Cell - Globsyn Business School)

Tackling English

Well, it is inevitable to compare CAT with cricket as there are so many analogies that can be drawn. As even at the last few minutes hard working & luck to find out the easy questions during the exam can be compared with the last ball of the last over in perfectly competitive match.
When you look at the English section of a CAT paper, there are two variables which help you maximize your score viz Attempts and Accuracy. Neatly juggling these two parameters will help one score the marks that would be required to get a good score in English.
For an English section, it is prudent to go slow in a paper which has more easy questions as the accuracy percentage in an easy paper (for the easy questions) would be higher and hence taking unnecessary risks (there will still be difficult questions) in the paper would reduce this overall accuracy thereby making your score relatively lower than others' scores.If the English section is difficult, then try to maximize the attempts and if it is easy, then concentrate on accuracy.
The broad classification for the English section can be viewed as RC (Reading Comprehension) & Non-RC sections. A good balance between the two can be made to create an impressive score in this section.
Attempting RC based questions will be much easier as we can easily guess the meaning of unknown words. Also, with the help of regular practice, speed reading can be adapted so that we can make ourselves strong enough in solving RC. Much space is there for guess work in RC.
Questions in Non-RC section are based on English grammar, though IFJ (Inference-Fact-Judgment) or Puzzle based questions are also common. These are binary type questions as you can answer these only if you know it. Guess work is not possible or proved to be highly risky.
Following table shows the trend for this section:



CAT 2007 had 25 questions in this section. A good attempt of around 15 questions had helped people to reach the cut off.
The clear fact arises is that RC, on an average contributes to 50% of total questions in CAT. The expected number of RC in this year will be 2. We advice our readers that they should try to attempt both RC, if possible.
Jumbled paragraphs will be an area of interest for the paper setters as this is frequently used as a speed breaker. Practice this section to get sure shot marks in these questions.
Try to go through major English newspapers daily as this will improve your ability of speed reading. Take a target of solving 3 RC a day in 15 minutes.
Best of Luck !!!

By:
Arjun Pal
Knowledge Cell - Globsyn Business School

Improvement in Reading Speed and Comprehension (Verbal Ability)

Quite often a CAT aspirant must be wondering as to how one manages to answer so many questions in such a short span of time. There is no doubt that one needs to be very well prepared through regular practice and covering wider and wider range of sample problems. But there is one additional way by which one can help the cause – by improving his speed of reading. This may sound incredible to start with but if we look at the facts, the real picture will emerge.

An average person can read 150 to 200 words in a minute with a comprehension level of 60% by concentrating hard and his performance can be termed satisfactory. According to experts, an average reader is nearly five times slower than a good reader and if we consider reading efficiency, the gap is even wider. Reading efficiency is defined as reading speed multiplied by comprehension rate called efficient words per minute (ewpm). For an average reader, this is typically about 120 ewpm while for top readers this can be as high as 850 ewpm. A reasonably good reader can achieve a reading efficiency of about 300 ewpm – i.e., 2.5 times the speed of an average reader. Let us now examine the impact this has on a person sitting for CAT. The average time taken to read a question by the average reader is 15 – 20 seconds while this will come down to 6 – 8 seconds for s good reader. Thus a person can get nearly 15 minutes extra for answering the entire examination.

It therefore pays to improve the speed of reading. But how does one do it? To start with, you may take a on line test to measure your current status by visiting http://www.readingsoft.com/ or http://www.freereadingtest.com/

There are important areas for improvement like in the areas of Eye movement, Eye span, etc.

Some good tips regarding how to improve your reading speed are available at http://www.mindtools.com/speedrd.html and free software (trial basis) are also available for download at http://www.rocketreader.com/download/RocketReaderDownload.html. There are many websites which help in improvement in English vocabulary, comprehension and reading speed.

By:

Prof. Bikramjit Sen

IIM C Batch 08

How to Speed Read?

CAT has a section called Reading Comprehension. Reading Comprehension tests your ability to read a document and take out the most important points in a very short time.

Reading Comprehension section expects you to read a particular prose and answer some questions based on that prose. Advantage of this section is that if you can read and understand that particular prose you can answer a number of questions at one go. This helps you to increase your number attempted in the test. Downside is if you waste too much time reading and understanding then the time taken for answering each question may become disproportionately high.

Therefore to successfully crack Reading Comprehension (RC) you need to speed read. Speed Reading is a technique by which you can pick up the salient points in a prose without reading each and every word. Practicing speed reading will help you to scan a prose fast and comprehend the basic points the author is trying to convey.

To practice speed reading you first need to read the entire prose word by word. Then mark the relevant words in the text which you feel are most important and without which you would be unable to comprehend the paragraph. Then read those words only. See whether you are able to answer all the questions by only reading the underlined words. Once you have done this a few number of times you would start automatically read the relevant section of the text passage and your speed reading competency would increase. Slowly reduce number of words you need to read in a passage till such time you hit your basic minimum. Remember speed reading is a very individualistic exercise. The number of words you need to read to understand the passage may vary from your friend.

Practice the above mentioned technique for next 1 month and you will see a drastic improvement in RC scores.

By:

Prof. Debdutta Choudhury

IIM C – batch 37

Maths Session 2

Let’s Get Back to History

Before dipping into the vast ocean of mathematics, let us take some deep breath and visit our country’s rich heritage of mathematics. You can be proud to know the fact that this branch of science took birth in this country.In all early ages of civilizations, the first expression of mathematical understanding appears in the form of counting systems. Number Systems in early societies were typically represented by groups of lines, though later different numbers came to be assigned specific numeral names and symbols (as in India) or were designated by alphabetic letters (such as in Rome).

Mathematical Activity in the Vedic Period

In the Vedic period, proceedings of mathematical activity are mostly to be found in Vedic texts associated with ritual activities. Arithmetic operations (Ganit) such as addition, subtraction, multiplication, fractions, squares, cubes and roots are enumerated in the Vishnu Purana by Ved Vyas (pre-1000 BC).Pythagoras - the Greek mathematician and philosopher who lived in the 6th C B.C was familiar with the Upanishads and learnt his basic geometry from the Sulva Sutras. An early statement of what is commonly known as the Pythagoras theorem is to be found in Baudhayana's Sutra - The chord which is stretched across the diagonal of a square produces an area of double the size. His Sutra also contains geometric solutions of a linear equation in a single unknown. Examples of quadratic equations also appear.Philosophical formulations concerning Shunya - i.e. emptiness or the void may have facilitated in the introduction of the concept of zero. While the zero (bindu) as an empty place holder in the place-value numeral system appears much earlier, algebraic definitions of the zero and it's relationship to mathematical functions appear in the mathematical treatises of Brahmagupta in the 7th C AD. Tangible evidence for the use of the zero begins to proliferate towards the end of the Gupta period.

The Indian Numeral System

Although the Chinese were also using a decimal based counting system, the Chinese lacked a formal notational system that had the abstraction and elegance of the Indian notational system, and it was the Indian notational system that reached the Western world through the Arabs and has now been accepted as universal. Several factors contributed to this development whose significance is perhaps best stated by French mathematician, Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. It's simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions."

Influence of Trade and Commerce, Importance of Astronomy

The growth of trade and commerce, particularly lending and borrowing demanded an understanding of both simple and compound interest which probably stimulated the interest in arithmetic and geometric series. Brahmagupta's description of negative numbers as debts and positive numbers as fortunes points to a link between trade and mathematical study. Knowledge of astronomy - particularly knowledge of the tides and the stars was of great import to trading communities who crossed oceans or deserts at night. The science of astronomy was also spurred by the need to have accurate calendars and a better understanding of climate and rainfall patterns for timely sowing and choice of crops.The world did not have to wait for the Europeans to awake from their long intellectual slumber to see the development of advanced mathematical techniques. India achieved its own scientific renaissance of sorts during its classical era, beginning roughly one thousand years before the European Renaissance. Probably the most celebrated Indian mathematicians belonging to this period was Aaryabhata, who was born in 476 CE.In 499, when he was only 23 years old, Aaryabhata wrote his Aaryabhatiiya, a text covering both astronomy and mathematics. With regard to the former, the text is notable for its awareness of the relativity of motion. This awareness led to the astonishing suggestion that it is the Earth that rotates the Sun. He argued for the diurnal rotation of the earth, as an alternate theory to the rotation of the fixed stars and sun around the earth. He made this suggestion approximately one thousand years before Copernicus, evidently independently, reached the same conclusion.With regard to mathematics, one of Aaryabhata's greatest contributions was the calculation of sine tables, which no doubt was of great use for his astronomical calculations. He also developed methods of solving quadratic and indeterminate equations using fractions. He calculated pi to four decimal places, i.e., 3.1416. In addition, Aaryabhat.a invented a unique method of recording numbers which required perfect understanding of zero and the place-value system.Bhaskar I continued where Aryabhatta left off, and discussed in further detail topics such as the longitudes of the planets; conjunctions of the planets with each other and with bright stars; risings and settings of the planets; and the lunar crescent. Again, these studies required still more advanced mathematics and Bhaskar I expanded on the trigonometric equations provided by Aryabhatta, and like Aryabhatta correctly assessed pi to be an irrational number. Amongst his most important contributions was his formula for calculating the sine function which was 99% accurate. He also did pioneering work on indeterminate equations and considered for the first time quadrilaterals with all the four sides unequal and none of the opposite sides parallel.

Applied Mathematics, Solutions to Practical Problems

In the 9th C, Mahaviracharya ( Mysore) wrote Ganit Saar Sangraha where he described the currently used method of calculating the Least Common Multiple (LCM) of given numbers. He also derived formulae to calculate the area of an ellipse and a quadrilateral inscribed within a circle.In the late 9th C, Sridhara (probably Bengal) provided mathematical formulae for a variety of practical problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns. Some of these examples involved fairly complicated solutions and his Patiganita is considered an advanced mathematical work.The study of mathematics appears to slow down after the onslaught of the Islamic invasions and the conversion of colleges and universities to madrasahs. But this was also the time when Indian mathematical texts were increasingly being translated into Arabic and Persian. Although it appears that original work in mathematics ceased in much of Northern India after the Islamic conquests, Benaras survived as a center for mathematical study, and an important school of mathematics blossomed in Kerala. Madhava (14th C, Kochi) made important mathematical discoveries that would not be identified by European mathematicians till at least two centuries later. His series expansion of the cos and sine functions anticipated Newton by almost three centuries. Important discoveries by the Kerala mathematicians included the Newton-Gauss interpolation formula, the formula for the sum of an infinite series, and a series notation for pi.Why, one might ask, did Europe take over thousand years to attain the level of abstract mathematics achieved by Indians such as Aaryabhatta? The answer appears to be that Europeans were trapped in the relatively simplistic and concrete geometrical mathematics developed by the Greeks. It was not until they had, via the Arabs, received, assimilated and accepted the place-value system of enumeration developed in India that they were able to free their minds from the concrete and develop more abstract systems of thought. This development thus triggered the scientific and information technology revolutions which swept Europe and, later, the world. The role played by India in the development is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of it's greatest contributions to world civilization.

By:

Arjun Pal

Knowledge Cell - Globsyn Business School

Some Tips for Success in Quantitative Methods

The IIM-s have announced their programme for selection of students aspiring for admission to their flagship programme of Post Graduate Diploma in Management – christened as Post Graduate Programme or PGP by the IIM-s. Last time around 2.3 lakh students wrote the examination and the number is sure to go up this year.

The Common Admission Test (CAT) conducted by the IIM-s has been one of the toughest of the admission tests for admission to B-Schools, all over the world and many aspiring students must now be preparing very hard for the examinations scheduled to be held on the 16th of November 2008. If we have a look at the announcements made, the tests will be of two and half hour duration and will test the quantitative, verbal, logical and data interpretive abilities of the candidates. The criteria for short listing and ultimate selection vary between different IIM-s.

Since the tests typically contain a large number of questions (75 in CAT 2007), it is often not possible to answer all the questions which should not, by itself, be construed by any aspiring student as a failure. This is because the tests are competitive in nature and tests the comparative qualities of the competing students.

Thus success or failure will depend solely on the relative performance which is impossible to predict before hand and can be known only after all the candidates have performed in the test on that day. It is to be remembered that the performance on that day is what matters and not what the student is actually capable of or what has been his performance on previous occasions. Thus it is advisable to maintain one’s cool during the test and be very cautious in approach – wasting time over any question is to be avoided at any cost until all the questions have been read through and the easier ones answered. It is to be remembered that there are no special consideration or marks awarded for attempting or answering a difficult question and it is best to avoid any difficult question.

Remember that speed and accuracy both are essential for being successful in CAT. Since there are negative marking for wrong answers, guess work should be avoided. Considering that there are 75 questions and a total time of 150 minutes available, the average time available for answering each question is 2 minutes. Thus the key to success lies in the level of accuracy and/or strike rate and in selecting questions properly.

Regarding preparations, there are several institutions which provide coaching and guidance through class room lectures, tutorials and mock tests. There are also some websites and distance learning facilities, which are useful. Similarly, number of books (question banks) are also available which are very useful for practice purpose. The previous question papers will also be extremely useful for such preparations. The question paper for CAT 2007 examination can be downloaded from the website of CAT.

One major area of concern is the questions for testing the quantitative abilities, particularly for students from non-quantitative background. One point should be kept in mind that the tests are based on the syllabus for high school mathematics only and is not much difficult. But considering the large number of questions required to be answered the speed of answering needs to be improved. Among the standard books available, the books by NCERT for classes VII to X are very helpful and cover the entire range. Here I would like to suggest that the candidates can make use of Vedic Mathematics – a system of mathematics presented during the later half of twentieth century by Swami Bharati Krishna Tirtha which consists of a list of 16 basic sütras, or aphorisms. Although there has been controversies regarding the claims made by its presenter, the sütras have been found to be extremely useful for solving problems in mathematics and algebra, within the education system. These can be applied in a number of ways to calculation methods for faster problem solving. In fact few schools in the UK have started courses on Vedic Mathematics for their students. Interested candidates can visit the website www.hinduism.co.za where there are names of some of the useful books available in the market. More importantly, there are tutorials available on line (8 in all) to help the students have a real glimpse of the system followed. I am sure students will find this extremely helpful provided they follow the system properly and allow necessary time to grasp the essentials. A number of CAT aspirants in the recent past have found this extremely useful.

By:
Prof. Bikramjit Sen
IIM C Batch 08