In this chapter, I will give you some instant tips that you can use for competitive exams. In exams like CAT, SAT, GMAT, GRE, UPSC, Railways, Defence, Bank PO and many other exams, a lot of emphasis is laid on quick calculation and accurate estimation of numerical problems. In many cases, we have observed that the average time per question is less than 60 seconds and so it becomes extremely important to develop the skill to quickly tackle questions.
Let us have a look at a few such handy techniques:
(A) Mental Calculation of Numbers
In school we are taught to do all calculations from right to left. Suppose you want to add up 4639 + 1235, here’s how you would do the calculation.
4639+1235
First you would begin from the extreme right and add up 9 + 5 is 14. Then you would write 4 and carry over 1. Next you would add (3 + 3) is 6 plus 1 carried over is 7 and so on…
This technique of going from right to left works fine on paper. However, whenever you are forced to do mental calculations, a much better strategy is to move from left to right instead of right to left
Here’s how you should do the above calculation.
Keep the first number 4639 as it is in your mind. Break up the second number 1235 as 1000 + 200 + 30 + 5. Now add all the numbers one after another
4639 + 1000 = 5639
This 5639 + 200 = 5839
5839 + 30 = 5869
And 5869 + 5 = 5874
Thus your final answer is 5874.
At first glance, it appears cumbersome. But once you get habituated to left-to-right calculations, you will be able to calculate with amazing speed!The same technique can be applied for other mathematical calculations also. Let us see subtraction.
(Q) Subtract 4142 from 7580.
Now you have two possible options to solve this question. One is the traditional right-to-left method where you try to subtract 2 from 0 and realize that it is not possible. So you ask 0 to borrow from its neighbor and so on…
However, this method is unnecessarily cumbersome. Instead, if we represent the same number in left-to-right manner, we notice it becomes easy to solve it.
7580 minus 4142 (4000 + 100 + 40 +2)
So first we have 7580 – 4000 = 3580
Next, 3580 – 100 = 3480
3480 – 40 = 3440
And finally, 3440 – 2 = 3438
When I was a young school boy, I always hated learning multiplication tables. However, I was equally fascinated by an
old uncle in my neighbourhood who always boasted that he knew all multiplication tables from 1 to 100. You could ask him, what is 76 times 7? In a jiffy, he would answer that it is 532. Or if you would ask him what is 83 times 8, within a fraction of a second he would tell you the answer 664. During my early school years, I was deeply impressed by the fact that he had memorized so many multiplication tables. But, a few years later, he revealed to me that he never knew multiplication tables up to 100; he only knew tables up to 10. What he was doing was breaking the number from left to right (as I just mentioned above). Here’s how he used to do it:
Suppose you would ask him what is 76 times 7.
He would mentally break up the number 76 as 70 + 6 and then multiply each of these values by 7
70 + 6
7 + 7
•70 x 7 is 490
•6 x 7 is 42
•490 + 42 is same as adding 490 + (10 + 32)
•So you first add 490 + 10 and get a convenient number of 500 and then simply add 32 to get 532
Suppose you have to mentally multiply 83 with 8
•First you multiply 80 with 8 and get the answer 640
•Then you multiply 3 with 8 to get 24
•The final answer (640 + 24) can be instantly written as 664.
So, please remember, whenever you have to make mental calculations, avoid the traditional method of going from right to left and instead go from left to right to get the answer quickly.
(B) Estimation of Imperfect Square Roots
Now we move on to the second technique in this series called ‘Estimation of Square Roots.’ Kindly note that I have used the word ‘estimation’ of square roots and not used the word ‘calculation of square roots.’ The reason is very simple. This technique will only give you a rough idea (estimation) of the answer and not the accurate answer.
Let us see how this technique works:
(Q) Find the square root of 70.
•First we have to find a perfect square root less than 70. So this is how we do it. We start counting downwards from 70 and come to 64 and we know that 64 is a perfect square whose square root is 8.
•Now we divide 70 by this 8 and we will get the answer (70 divided by 8) is 8.75.
•We now take the average of the two numbers 8 and 8.75 to get the answer 8.37. Thus 8.37 is the approx. square root of 70 (take only 2 decimal places).
(Q) Find the square root of 150.
•The perfect square just below 150 is 144 whose square root is 12.
•We divide 150 by 12 to get the answer 12.5.
•Finally, we take the averages of 12 and 12.5 to get 12.25 which is the approx. square root of 150.
(Q) Find the square root of 8200.
•The perfect square just below 8200 is 8100 whose square root is 90.
•8200 divided by 90 gives 91.11.
•The average of 90 and 91.11 = 90.55 which is our approx. answer.
(C) Fractions, Percentages and Decimals
I have travelled across many countries of the world to conduct my workshops. During these tours I have noticed a vast difference in the mathematical aptitude of people. In some countries I have seen modestly educated shopkeepers easily calculating percentages and decimals on their fingertips while in some places I have seen people perspiring (literally!) to mentally calculate 186 plus 146.
In this sub-section, I will urge the reader to memorize this key of very important standard fractions which will help him in calculating such questions with ease.
The fractional and decimal value with numbers 2,3,4,5 are very easy and almost everybody knows them. The ones with 6 will require a little learning. The values with 7 repeat themselves in cyclical order. And, as can be seen by mere observation, the values with 8 increase by 0.125 as we move on and the values with 9 move by 0.11 as we move on.
It is essential for those giving competitive exams to remember this key. (With number 7, you don’t have to remember up to 6 decimal places, even 2 decimal places will be sufficient.)
So if someone asks you what is 40% of 250, then, immediately on hearing the term 40% the term 2/5 should flash in your mind and you must mentally do the calculation (250 x 2 is 500 and 500 divided by 5) to get the answer 100.
If someone asks you what is the net price after 3/5th discount on 1600, then it should immediately flash in your mind that 3/5th is 60% and 60% of 1600 is 960 (because 16 times 6 is 96). And then, instead of subtracting 960, you can first subtract 1000 and then add back 40.
So, 1600 minus 1000, is 600 and when you add back 40 to
600 you get 640.
Or better still, balance amount after deducting 3/5th discount from 1600 should be same as 2/5th of 1600. And 2/5th of 1600 is 40% of 1600 which is 640 (because 16 * 4 = 640).
Have you seen the way Sachin Tendulkar plays a super fast delivery of Shoaib Akhtar or Brett Lee? The speed-o-meter says that an express delivery from someone like Shoaib Akhtar takes less than one second to reach Sachin Tendulkar’s bat. Now imagine the situation—within one second Tendulkar’s eyes have to see the ball, his hands and legs have to adjust themselves and his bat must come down at the right time to hit the delivery. Technically, it sounds difficult but ace batsmen like Sachin Tendulkar, Brian Lara and others have developed an almost ‘reflex action’ sort of response and so their bodies and minds work very fast to respond to the express deliveries.
The secret to cracking math and numerical questions in competitive exams is to develop this similar reflex-action sort of behavior. The moment you see a question related to numbers, within a second the numbers and calculations must start happening in your mind. For example, suppose you are solving a question where you have to find the average of 75,72 and 70. The traditional method requires you to first add 75+72+70 and then divide the total by 3 to get the average. However, if you want to crack competitive exams, you need to have a more instinctive, reflex-action sort of approach. This is how your mind should work:
‘Hmm, so I have to find the average of 75, 72 and 70 Now if 75 gives 2 to 70, the numbers will be 73, 72 and 72
If I divide this 1 extra by number 3, it should yield 0.33 Thus the final average is 72.33’
Although I have taken more than 5 lines to explain this question, in your mind, the entire procedure must take place in 2 to 3 seconds.
It must be remembered that the success rate in competitive exams is sometimes as low as 2 to 3%. In other words, out of every 100 students appearing for an exam, only 2 or 3 will make it to the next level. So in order to beat the competition, not only should your performance be super good it must also be super fast!
This is precisely the reason why I am insisting that instead of sitting and calculating each number in the traditional, sequential, line-by-line manner, you must slowly cultivate the art of jumbling around with the number and get the answer in a jiffy!
Happy Solving!
