In the previous two posts, we took a look at the first two building blocks to increase your score and percentile on CAT Quant — Accuracy & Question Selection. In this post, we will look at the third building block — Speed.
Increasing your speed by at least 2x is a function of changing three things in the way you execute a solution:
- increasing your number-crunching muscle
- reducing your dependence on writing extensively
- viewing problems through an alternative lens
Increase your number-crunching muscle
Find (1² + 3² + 4² + 5² + 7² + 8² + 9² + 11² + 12² + 13² + 15² + 16² +…….49²) / 1249 if it is an integer. (TITA)
This is a question from a SimCAT and most people would have let the question go. Upon reading the solution, they would have become doubly sure that it was a good leave.
But as we have often seen in T20 cricket, guys with huge reach and immense power can send balls, which can be potential dot balls to other batsmen, sailing into the crowd.
I decided to take on the above problem using some number-crunching muscle
Firstly, I knew that this numerator was the sum of the first 49 squares minus the even squares— we just need to substitute n = 49 in the formula, n(n+1)(2n+1) /6 and minus the sum of the missing squares.
49*50*99/6 can be reduced to 49*25*33 which can be approximated to be 50*25*33 = 1250*33 = 12*3 = 36 so a value greater than 36000, also this has to end in a 5 since it is 49*25*33.
If I ignore the values that need to be subtracted — the even squares — and look at the denominator 1249, I know the answer has to be 35 since the value is greater than 36000 and has to end in a 5, 1249 has to be multiplied by a number ending in 5 to get a value ending in 5.
But the catch is that some values are missing from the sum of squares, which I promptly listed:
2² + 6² + 10² + 14² + 18² + 22² + 26² + 30² + 34² + 38² + 42² + 46²
We can quickly estimate the last digit of the sum of these terms — 4,6,0,6,4,4,6,0,6,4,4,6 — 0, so even after subtracting the value has to end in 5 and hence the answer has to end in 5, so if without subtracting it is 35, then after subtracting it has to be 25, 15 or 5.
I set about quickly estimating the value of the above terms by starting from the right end since the squares of the larger number will make up most of the value and the smaller ones can be effectively ignored.
I am writing the approximation as I processed it mentally:
- 42² + 46² = Two 40s, more than 1600, 1600, so around 3500
- 30² + 34² + 38² = Three 30s = more than 1000, 1000, 1000 = around 3500
- 22² + 26² = 400, 600 = more than 1000
- 2² + 6² + 10² + 14² + 18² = less than 1000
So from a value greater than 36000 I need to subtract around 10000 so it will be around 26000 and hence value greater than 26000 ending in 5 when divided by 1249, to give an integer, has to yield 25 since 12*2 = 24 and 12*3 = 36
Thinking is always faster than writing so the actual time that I took for this crunching process was under 2 minutes.
Given that the CAT is a test of speed, what matters is not knowing how to solve but how quickly you can reach from knowing how to solve to the final answer.
This can only happen if you have some serious number-crunching muscle — extensive conceptual clarity cannot compensate for this muscle. You need to ask your self
- when the clock is ticking and you are executing a solution, can you break the numbers down or will you buckle under their load?
If your answer is sometimes I can crack, sometimes I buckle or most often I buckle, then you are not yet in shape to take the CAT. But you have three months to build some serious muscle by memorizing all of these:
- all squares from 2 to 30
- all cubes from 2 to 12
- all powers of 2 from 1 to 12
- all powers of 3 from 1 to 6
- all fraction and equivalent percentages from 1/2 to 1/11
- tables from 2*10 to 20*10
Most of you want to do an MBA so that you can do high-quality, high-paying work. If that is the case then you should approach the 180 minutes of the CAT in a such a way that we do only quality work during the 180 minutes of the test.
- Do you want to be calculating 29², if it is the answer to a TITA question?
- Do you want to be calculating the value of 2 raised to 8 by starting with 2 raised to 5 in the middle of a problem?
All of these values should already be fed into and so deeply embedded in the system, that there is no gap between retrieving and executing the solution.
In short, you need to be the calculating-equivalent of a T20 big-hitting beast.
Reduce your dependence on writing extensively
A few years back a student came to have a chat with me, he was retaking the CAT and needed a plan. He said he was really good at Math but could not solve more than 22 problems on CAT 2016, which as you know had an easy and hence high-scoring QA. He was working with a good Consulting firm, had a good profile and was thus looking at premier schools that needed high percentiles.
My first instinct was to give him a piece of paper and a problem just to watch how he solved it. He wrote down around 10 steps on the paper and solved the problem and that was the exact reason why he was only able to attempt 22 questions.
Writing and executing the entire solution of a problem is the biggest speed-breaker or decelerator in front of you. Each one of you will have varying degrees of dependence on writing. Sometimes, the more diligent the aspirant, the more steps he or she will write (systematically and without clutter) while solving a question.
The problem is that this method will result in fewer than 15 attempts in 60 minutes and when coupled with a few mistakes will end up in a percentile that is perennially hovering in the 85 range.
So if you are among those who write diligently then you need to drastically change your approach to increase your percentiles and understand than
- you are used to writing because you are used to submitting homework
- you are used to writing because missing steps can mean fewer marks being awarded
- you are used to writing neatly because so far good, clutter-free writing fetched you higher marks
None of the things listed above applies to CAT Math — just like none of the rules of Test cricket applies to T20 Cricket — no marks for handwriting, no marks for steps
How do you decrease the amount of writing you do? Start with the following steps
- do not duplicate information from the screen on to your rough sheet
- a man does a piece of work in 20 days, then do not write 20 or t = 20 on your paper, you are just executing robot-like steps without getting any closer to solving the problem
- do not write what you need to execute
- if you need to calculate the average of five numbers then there is no way you are writing 5 numbers with plus sign between them, drawing a line underneath them and writing a five; you need to just get adding and dividing without writing anything
- start to skip writing down each step by executing intermediate steps mentally
- if you have to solve 1/(x+1) + 1/(x+2) = 2, then maybe the next step you should write is 2x + 3 = 2x² + 6x + 4
In short, write only if you cannot process the information in your head!
This is a question in one of this year’s SimCATs that I solved purely mentally when an aspirant who is working with us in the content team came to me and asked me to solve a paper live to see how I approach a section to learn selection and speed.
The function F is defined as F(k) = 2k³ – 3k² – 5k + 7 and the function G is defined as G(k) = 2k³ + 1k² + 7k + 15. Find the product of all values of ‘k’ for which F(k) and G(k) are equal.
I knew that I had to equate the two since that is when they will become equal, the cubes cancel themselves out and if I take the remaining terms to one side, it becomes a quadratic with 4,12, and 8, which gets reduced to 1,3, and, 2, whose roots will be 1 and 2, making their product 2. This took about 60 seconds or fewer mentally.
Even if you take more time, the key is that there is no need to write. Some of you might say that if I do it this way, I’ll be making mistakes, it is like saying if I drive faster I will crash, but then you won’t win the race — it is about driving the fastest you can without crashing.
This is easier said than done since we have been conditioned over the years to equate writing with thinking and solving, so like the legendary dog of Pavlov, we start writing the moment we start reading a problem.
To practice the above steps consciously you should try a few special practice sessions.
- Take a section test or an area test or the Quant section of a take-home SimCAT with your hands folded or your palms locked in front of you.
- Have a pen and paper handy on the table to write only if the need arises
- Force yourself to execute a few steps mentally
- Do not be bothered about time running out, do not be bothered about the score
If you do not try out these things during practice you will keep doing the same thing over and over again and keep expecting different results, which alas is a non-sequitur (CR enthusiasts should check what this means).
Viewing problems through an alternate lens
The above heading might seem as if there is a standard lens and there is an alternate lens. In fact, nothing could be farther from the truth. What always matters is the most optimal solution to a problem. Why do we not find the optimal solution?
Reject the formula-first approach
Our gut reaction to solving a problem is to try to immediately fit it to a formula. In fact, when faced with a question we immediately ask ourselves — what formula do I know that can help me solve this.
A formula is only one of the tools that you will use to reach the solution that you have devised; they are similar to a surgeon’s or a mechanic’s tools. A surgeon does not decide on the kind of surgery to be done based on the tools he has, neither does mechanic decide on how he is going to fix a vehicle based on the tool he has!
So do not make formula-fitting your first step.
Do not algebraify a problem by force
If we do not go to a formula, we then start taking the first thing we encounter as X and we try to form an equation.
We feel that if we can convert English into Algebra, we have done our job but Algebra is just another language like English. What you have to convert it into is a logical language, which can still be in English but uses words or Algebra that uses symbols.
Do not convert all problems into Algebra, especially the Arithmetic ones.
Move to the question first approach
Put the question and what is finally asked as the first and most important thing. Work backwards from there to determine what you really need instead of trying to build towards the answer from the first bits of information. Information is never given sequentially and usually, the most important piece of information is given right at the end and that might make your whole job easier.
Do not treat the given information passively
On most good questions, the given information itself holds more than meets the eye, provided you are willing to at least turn it over in your palm to see a small latch that you can pull.
If you take it just the way it is and do not try to even squeeze a wee bit or cut it, a lemon is as good or bad as a stone or softball (in fact not even as good as those).
The methods we know of are a function of the teachers we have had
We are more or less a function of what we have been taught and made to do; there are always a few who can see things by themselves but there are a great many who can do much better than what they if someone points the way.
How many really high-quality teachers have we had (and I do not mean by good because of their nature; I will prefer a horrible person who teaches stuff with fresh eyes to a good human being who teaches stuff in the most mundane of ways; there a few who combine the best qualities of the two types and I have had the pleasure of knowing two of them) ?
I started looking at problems differently after I encountered my colleague pulling off some amazing solutions (go through all of them); it did not feel like it was way above my league since it was more a question of approach and attitude rather than some genius intuition (which he does have) because it was something that was easily understood.
I figured that it was more about my ability to let go of my conditioned responses and less about my ability to find alternate solutions.
And surely enough over a short period of time looking for the fastest route to goal by looking at questions through an alternate lens became second nature to me as well.
With Quant, it was almost like it was with cricket for me. I have never thought of myself as a natural sportsman (like I never thought of myself as a natural QA guy) and got by with grit and competitiveness. So while I opened the batting for my department team, it was more a function of my ability to stay at the crease and maximise whatever I could with my limited but effective range of strokes I had at my disposal and the limited talent in our department rather than high-quality skill.
We managed to play almost everyday (back in the day we had annual exams so we could afford to chill all year and study at the end) and it is only at the end of three years and that too during a practice session that I stepped out of my crease and tonked the bowler (a gentle medium pacer) over his head for six. It had taken me close to 10 years of playing cricket to do it (and I can still step back into that feeling).
To find a new road, you have to first get off the old one, you need to step out of the crease as well during practice.