CAT 2013: Go for 3-step process to crack Quant section

How should CAT Aspirants approach preparation for quant section? What is the mantra of problem solving in Quant section? What is the key to solving different level of questions in Quant?
Mr Rajesh Balasubramanian, an IIT, IIM alumnus, & 2011, 2012 CAT Topper and the author of CAT books published by Access Publishing India, suggests a 3-step process that you must go for each topic in Quant section.
The CAT Expert & Author who will guides CAT Aspirants through a series of expert articles on MBAUniverse.com over the next 120 days left for CAT, shares insight on preparation strategy for Quant section in his first prep article:

For each topic in Quant section, you must break the process into three steps-

1. Learn the theory from first principles: Make a conscious effort to learn theory from basic ideas. We have a tendency to jump steps when we get familiar with a topic. Resist the temptation to use plug-and-play formulae and learn the theory with an open mind. Let me illustrate this with a simple example

Consider a class of 40 students with an average score of 60 marks. Two students leave this class and the average mark increases by x. If no student can take a negative mark, and x is a positive integer, how many values can the average of the two students take? One can write equations and solve this. However, someone who has learnt from basic ideas will follow a more intuitive approach.

Let us look at this approach. Let us start with a simple case where the average does not change when these two students leave the class. In this case, the two students should have scored 120 marks overall (average of 60). Now, since the average has increased, these two together should have scored less than 120.

If the average increases by 1, then they should have scored 38 marks less than 120. If average increases by 1, total should have increased by 38, or the two that left the class should have taken out 38 less than 120. So, if they had scored 120 38 = 82, the average would have fallen by 1.

If they had scored 120 - 38 x 2 = 44, average would fall by 2. If they had scored 120 38 x 3 = 6, average would have fallen by 3. Three different possibilities exist. The intuitive method scores over the formulaic method in questions like these. It is very important to retain the more intuitive approach and not get consumed by the algebra.

2. Do the grind post this: Do lots of practice. Intuition is important, but the ability to power through lots of questions is extremely vital. Major portions of the processes should be on autopilot. Fatigue is an important factor in CAT and it is nearly impossible to be switched on for 140 minutes continuously. When confronted with a tricky question, if you pick the right approach after may be 60 seconds of thinking, the remaining 100 seconds of actually solving the question should be done in mechanical fashion; in a manner that does not take much out of you.

More the processes on autopilot, the less tired you get, and therefore better the odds of cracking the key idea behind the next question. To give a parallel, think about the time you tried to learn to drive a car (or ride a bike). In the first few instances, when the gear-clutch-brake mechanisms were still new, the experience would have been very tiring. Driving for even 30 minutes would have been exhausting. When the mind is processing that level of data, fatigue creeps in rapidly. But once you have done 200 kms on a vehicle, muscle-memory kicks in, the left hand and left leg move automatically, without having to be told. Now, there is even bandwidth to speak on the phone while driving!

Again, let us see this with an example relevant to CAT Given 2x + 5y = 1001, and x, y are positive integers, what is the sum of all possible values of x?
Now, let us start with trying to find one pair of values that satisfies this equation. If we substitute x = 1, y is not an integer, x cannot be 2 either. When x = 3, 5y = 995; y = 199 works. Now, 5y = 1001 2x. or, 1001 -2x should be a multiple of 5. If x = 3 works, then the next value of x that works should be 8.

Till this point one might have spent bandwidth figuring out the method. From now on, the mechanical part should kick in. x = 3, 8, 13 work. The highest value x can take is 498. x cannot be more than 500 as that would not leave room for positive y. Now, we need to find 3 + 8 + 13 + 18 .+ 498. This is an AP with first term 3, common difference 5, nth term 498. This part should be mechanically (but carefully) solved.

3. Handle tougher questions by revisiting theory: Once you have learnt from first principles and done the grind, you would have learnt all standard templates. Now the knowledge base is set, it is time build the ability to discover newer frameworks. Ask more fundamental questions and work with proofs and properties. Again, to see a few examples

We know the test of divisibility by 9, but have you thought about why this works? That is a very interesting idea. Let us take a four-digit number abcd; this can be expanded as 1000 + 100b + 10c + d. Now, 999a + 99b + 9c is a multiple of 9. If we subtract this from the 4-digit number, we get a + b + c + d. So, if a + b + c + d were a multiple of 9, so would 1000a + 100b + 10c + d. This is the basis for test of divisibility by 9. If you had spent time thinking about this proof, a question involving base 6 math would appear far simpler.

Now, think about the basis of test of divisibility by 11.
Area of a triangle is rs, where s is the semi-perimeter and r is the inradius. Why does this work? Etc.

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