Square Roots and Cube roots.

Square roots of Numbers which are not perfect squares :

To find the square root of 500:
 
Let us guess that the square root is 20.

Divide 500 by 20 to get the quotient 25.

Take the average of the factor 20 and the quotient 25 which is 22.5.

This 22.5 is closer to the actual root of 500 than the initial estimate of 20.

Repeating the above process:

500/22.5 = 22.2222

Average of 22.5 and 22.2222 is 22.3611.

For more accuracy, we can repeat the step once again to get the next estimate as 22.36068.

The actual square root of 500 is 22.36068.

Cube Root of Numbers which are not perfect Cubes :

To find the cube root of 78654

Let the initial guess be 40.

Step 1: 78654 / 40 = 1966.35

Step 2: 1966.35 / 40 = 49.15875

The average of 40, 40 and 49.15875 is 43.05292.

You can repeat the above process with the starting number as 43 (No need to start with 43.05292).
Actual cube root of 78654 is 42.84567.

Even if you start with a very wild initial guess, you will only need a few more iterations to reach the answer.

Finding Square root of Perfect square numbers

For finding the square roots of perfect squares we need to observe the last digit of the square of the number.

1 ^2 = 1

2^2  = 4

3^2 = 9

4^2 = 6 (Forget 1 before 6 , we are only interested in the last digit after squaring a number)

5^2 = 5 (Forget 2 before 5 , we are only interested in the last digit after squaring a number)

6^2 = 6 (Forget 3 before 6 , we are only interested in the last digit after squaring a number)

7^2 = 9 (Forget 4 before 9 , we are only interested in the last digit after squaring a number)

8^2 = 4 (Forget 6 before 4 , we are only interested in the last digit after squaring a number)

9^2 = 1 (Forget 8 before 1 , we are only interested in the last digit after squaring a number)

0^2 = 0

Now the obvious question which must come to your mind is why do we find / observe the last digit ?

The answer is simple. We observe the squares of numbers to estimate the units digit of the square root of the  number.( We are now concerned only with finding the unit's digit of the number ) Estimating the unit's digit will enable us to find the other digits.

 Example : Find the unit's digit of the square root of 1024

Observe that the last digit is 4 ( hence from the observation we see that both 2^2 and 8^2 results in 4) So the units digit of the root so formed can be either 2 or 8.

 Example : Find the unit's digit of the square root of 625

From the observation table we know that the unit's digit will be 5 only when the number 5 is squared.So the units digit of the root is 5


 Example : Find the unit's digit of the square root of 676

From the observation table we know that the unit's digit will be 6 only when the number 6 or 4 is squared. So the units digit of the root is either 4 or 6



Now when we know how to find the unit's digit of any number let's proceed to find the Hundredth's and Tenth's digit.

Lets understand it with the help of an example :


Find the square root of 361
 

Step 1 : Here the first thing we have to do is separate the numbers in pairs from right to left 

( For 3 digit number separate it as 2 numbers from left side , For 4 digit number separate is 2 from the left and 2 from the right ,For 5 digit number cut out 2 digits from the right and leave 3 numbers in the left , for 6 digit numbers cut out evenly)

We separate the digits as follows

3 61

Step 2 : Next from our observation we can say that the units digit of the square root of the number is 1 or 9 ( Concept which we have learned earlier)

61
Sow can get an approximate value of  the units digit as either 1 or 9

Step 3 : Now when the units digit have been estimated we must find the Hundredth's and/or the Tenth's digit.

From the above  observation we have to find the number ( We have to find the number , not the square of the number) whose highest square which can be taken away from 3

Here from the table of squares which we know  the highest square which can be taken away from 3 is 1 and the corresponding number is 1.

Hence the square root of the number is either 11 or 19.

 Now the question is which one is the answer. Simply take any one and square it.

11^2 =121

19^19 = 361 ( You can guess it also , The highest square above 361 is 400  which is nothing but square of 20 , so your answer will be slightly less than 20 )

 Lets take one more example.

Find the square root of 12321

Step 1 : Separate the digits 123 21

Step 2: Estimate the units digit Either 1 or 9

Step3 : From 123 the highest square which can be taken away from it is 121 , which corresponds to the square of 11

Hence the answer is either 113 or 119.

Check out by multiplying either 113 or 119

Comparing Fractions

This topic is of extremely important for Data Interpretation and other areas of Quants Paper. It is highly recommended for aspirants of various exams to internalize the concepts .

Type 1 : When the numerators are same

When the numerators are same and denominators are different , the fraction with the highest denominator is the smallest.

Example : 3/5 , 3/7 , 3/13 , 3/8

Here the smallest fraction is 3/13 and the largest fraction is 3/5

Type 2 : When the numerators are different but the denominators are all same

The fraction with the largest denominator is the largest.

Example : 7/5 , 9/5 , 4/5 , 11/5

 Here the smallest fraction is 4/5 and the largest fraction is 11/5

Type 3 :Fraction with the largest denominator and smallest numerator is the largest
Example : 19/16 , 24/11 , 17/13 , 21/14 , 23 /15

Here 24/11 is the largest fraction.

Type 4 :When the difference between the numerator and the denominator is same ( When the value of the fraction is less than 1)

When the difference between the numerator and the denominator is same the fraction with the largest value of numerator and denominator is the largest.


Example : 31/37 , 23/29 , 17/23 , 35/41 , 13/19

Here 35/41 is the largest fraction and 13/19 is the smallest fraction.



Type 5 :When the difference between the numerator and the denominator is same ( When the value of fraction is more than 1)

When the difference between the numerator and the denominator is same the fraction with the smallest value of numerator and denominator is the largest.


Example :31/27 , 43/39 , 57/53 , 27/23 , 29/25

Here the largest fraction is 27/23


These fraction comparisons are extremely useful for quick DI calculations .

Hope you find them useful.

Happy preparations.......

Squaring Numbers above 25 in seconds

There are some cool tricks to square numbers above 25 in seconds :

Squaring numbers from 51 - 100

Step 1 : Subtract the number you want to square from 100. (100 - x)

Step 2 :  Take this difference and subtract it from the number you want to square. (x - (100 - x)). This is the first 2 digits of your answer.

Step 3 :  Square the difference, the answer from step 1. This is the last 2 digits of your answer.

Example : 96²

100 - 96 = 4.
96 - 4 = 92. You're halfway there. 92__.
4² = 16. There's your last 2 digits. Thus, your answer: 9,216. 

Squaring Numbers from 50-59

Square the tens digit and add to that the ones digit of your original number. That's your first two numbers. Second, square your ones digit. That's your last two numbers.
Step 1 : 5²+ 4 = 29. That's our first two numbers.

Step 2: 4² = 16. That's our last two numbers. Thus, the answer 2916

 Squaring numbers greater than 100

Step 1: Subtract 100 from the number you want to square (x - 100 )


Step 2:Take the difference and add it to the number you want to square. (x + (x - 100)). This is the first 3 digits of your answer.


Step 3: Square the difference, the answer from step 1. This is the last 2 digits of your answer.


Example: 112²

112 - 100 = 12
112 + 12 = 124. 124__.
12² = 144. Carry the 1 over the 4.
__144
124__
12,544

<sup>Squaring Numbers around 1000

It follows the same rule as the others described above.</sup>


996²

1000 - 996 = 4
996 - 4 = 992. 992___ First 3 digits.
4² = 16. Last 3 digits.
Answer: 992,0161007²


1007²



1007 - 1000 = 7
1007 + 7 = 1014. 1014___ First 4 digits.
7² = 49. Last 3 digits.
Answer: 1,014,049

I hope everyone is aware of the division rules , still I am posting them here some of the divisibility rules:


Divisibility by 2: If its units digit is any of 0,2,4,6,8.


Divisibility by 3: If the sum of its digits is divisible by 3.


Divisibility by 4: If the number formed by the last two digits is divisible by 4

Divisibility by 5: If its units digit is either 0 or 5.


Divisibility by 6: If it is divisible by both 2 & 3.



Divisibility by 8: If the last three digits of the number are divisible by 8.


Divisibility by 9: If the sum of its digit is divisible by 9.


Divisibility by 10: If the digit at units place is 0 it is divisible by10.


Divisibility by 11: If the difference of the sum of its digits at odd places and sum of its digits at even places, is either 0 or a number divisible by 11.


Divisibility by 12: A number is divisible by 12 if it is divisible by both 4 and 3.

Divisibility by 14: If a number is divisible by 2 as well as 7.

Divisibility by 15: If a number is divisible by both 3 & 5.

Divisibility by 16: If the number formed by the last 4 digits is divisible by 16.


Divisibility by 24: If a number is divisible by both 3 & 8.

Divisibility by 40: If it is divisible by both 5 & 8.

Divisibility by 80: If a number is divisible by both 5 & 16.

Number	Method	Example
7	Subtract 2 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 7, the original number is also divisible by 7	Check for 945: : 94-(2*5)=84. Since 84 is divisible by 7, the original no. 945 is also divisible
13	Add 4 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 13, the original number is also divisible by 13	Check for 3146:: 314+ (4*6) = 338:: 33+(4*8) = 65. Since 65 is divisible by 7, the original no. 3146 is also divisible
17	Subtract 5 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 17, the original number is also divisible by 17	Check for 2278:: 227-(5*8)=187. Since 187 is divisible by 17, the original number 2278 is also divisible.
19	Add 2 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 19, the original number is also divisible by 19	Check for 11343:: 1134+(2*3)= 1140. (Ignore the 0):: 11+(2*4) = 19. Since 19 is divisible by 19, original no. 11343 is also divisible
23	Add 7 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 23, the original number is also divisible by 23	Check for 53935:: 5393+(7*5) = 5428 :: 542+(7*8)= 598:: 59+ (7*8)=115, which is 5 times 23. Hence 53935 is divisible by 23
29	Add 3 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 29, the original number is also divisible by 29	Check for 12528:: 1252+(3*8)= 1276 :: 127+(3*6)= 145:: 14+ (3*5)=29, which is divisible by 29. So 12528 is divisible by 23
31	Subtract 3 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 31, the original number is also divisible by 31	Check for 49507:: 4950-(3*7)=4929. Since 492-(3*9) is divisible by 465:: 46-(3*5)=31. Hence 49507 is divisible by 31
37	Subtract 11 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 37, the original number is also divisible by 37	Check for 11026:: 1102 - (11*6) =1036. Since 103 - (11*6) =37 is divisible by 37. Hence 11026 is divisible by 31
41	Subtract 4 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 41, the original number is also divisible by 41	Check for 14145:: 1414 - (4*5) =1394. Since 139 - (4*4) =123 is divisible by 41. Hence 14145 is divisible by 41
43	Add 13 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 43, the original number is also divisible by 43. *This process becomes difficult for most of the people because of multiplication with 13.	Check for 11739:: 1173+(13*9)= 1290:: 129 is divisible by 43. 0 is ignored. So 11739 is divisible by 43
47	Subtract 14 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 47, the original number is also divisible by 47. This too is difficult to operate for people who are not comfortable with table of 14.	Check for 45026:: 4502 - (14*6) =4418. Since 441 - (14*8) =329, which is 7 times 47. Hence 45026 is divisible by 47

Quick Addition

Addition and Subtraction Concepts are easy to learn and are very helpful.Unfortunately after trying a number of times I failed to show it here due to formatting issues.

I am hence going to provide you with a link to download the files and have a look.The tutorial files are comprehensive and easy to learn.




In case of any doubt feel free to ask your query.

http://www.ziddu.com/download/11129676/VedicMathematicsaddition.doc.html

http://www.ziddu.com/download/11129666/VedicMathematicssubtraction.doc.html

Squares of 2

1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 11 121 12 144 13 169 14 196 15 225 16 256 17 289 18 324 19 361 20 400 21 441 22 484 23 529 24 576 25 625                     Cubes of 2
1 1 2 8 3 27 4 64 5 125 6 216 7 143 8 512 9 729 10 1000 11 1331 12 1728
Powers of 2
2 ^1 2 2^2 4 2^3 8 2^4 16 2^5 32 2^6 64 2^7 128 2^8  256 2^9 512 2^10 1024 2^11 2048 2^12 4096

Powers of 3

Number	Value
3^1	3
3^2	9
3^3	27
3^4	81
3^5	243
3^6	729

Reciprocal of Fractions

1/1                     1

1/2                   0.5

1/3                   0.33

1/4                   0.25

1/5                   0.20

1/6                   0.1666....  

1/7                   0.142857..........

1/8                   0.125

1/9                   0.11111.........

1/10                 0.10 

1/11                 0.0909.....

1/12                 0.8333........

1/13                 0.076293 

1/14                 0.0714285.......

1/15                 0.066....... 

1/16                 0.0625 

1/17                 0.058823

1/18                 0.055...........

1/19                 0.052631 

1/20                 0.05

With a little practice, it's not hard to recall
the decimal equivalents of fractions up to 10/11!

First, there are 3 you should know already:

1/2 = .5
1/3 = .333...
1/4 = .25

Starting with the thirds, of which you already know one:

1/3 = .333...
2/3 = .666...

You also know 2 of the 4ths, as well, so there's only one new one to learn:

1/4 = .25
2/4 = 1/2 = .5
3/4 = .75

Fifths are very easy. Take the numerator (the number on top),
double it, and stick a decimal in front of it.

1/5 = .2
2/5 = .4
3/5 = .6
4/5 = .8

There are only two new decimal equivalents to learn with the 6ths:

1/6 = .1666...
2/6 = 1/3 = .333...
3/6 = 1/2 = .5
4/6 = 2/3 = .666...
5/6 = .8333...

What about 7ths? We'll come back to them
at the end. They're very unique.

8ths aren't that hard to learn, as they're just
smaller steps than 4ths. If you have trouble
with any of the 8ths, find the nearest 4th,
and add .125 if needed:

1/8 = .125
2/8 = 1/4 = .25
3/8 = .375
4/8 = 1/2 = .5
5/8 = .625
6/8 = 3/4 = .75
7/8 = .875

9ths are almost too easy:

1/9 = .111...
2/9 = .222...
3/9 = .333...
4/9 = .444...
5/9 = .555...
6/9 = .666...
7/9 = .777...
8/9 = .888...

10ths are very easy, as well.
Just put a decimal in front of the numerator:

1/10 = .1
2/10 = .2
3/10 = .3
4/10 = .4
5/10 = .5
6/10 = .6
7/10 = .7
8/10 = .8
9/10 = .9

Remember how easy 9ths were? 11th are easy in a similar way,
assuming you know your multiples of 9:

1/11 = .090909...
2/11 = .181818...
3/11 = .272727...
4/11 = .363636...
5/11 = .454545...
6/11 = .545454...
7/11 = .636363...
8/11 = .727272...
9/11 = .818181...
10/11 = .909090...

As long as you can remember the pattern for each fraction, it is
quite simple to work out the decimal place as far as you want
or need to go!

Oh, I almost forgot! We haven't done 7ths yet, have we?

One-seventh is an interesting number:

1/7 = .142857142857142857...

For now, just think of one-seventh as: .142857

See if you notice any pattern in the 7ths:

1/7 = .142857...
2/7 = .285714...
3/7 = .428571...
4/7 = .571428...
5/7 = .714285...
6/7 = .857142...

Notice that the 6 digits in the 7ths ALWAYS stay in the same
order and the starting digit is the only thing that changes!

If you know your multiples of 14 up to 6, it isn't difficult to,
work out where to begin the decimal number. Look at this:

For 1/7, think "1 * 14", giving us .14 as the starting point.
For 2/7, think "2 * 14", giving us .28 as the starting point.
For 3/7, think "3 * 14", giving us .42 as the starting point.

For 4/14, 5/14 and 6/14, you'll have to adjust upward by 1:

For 4/7, think "(4 * 14) + 1", giving us .57 as the starting point.
For 5/7, think "(5 * 14) + 1", giving us .71 as the starting point.
For 6/7, think "(6 * 14) + 1", giving us .85 as the starting point.

Practice these, and you'll have the decimal equivalents of
everything from 1/2 to 10/11 at your finger tips!

If you want to demonstrate this skill to other people, and you know
your multiplication tables up to the hundreds for each number 1-9, then give them a
calculator and ask for a 2-digit number (3-digit number, if you're up to it!) to be
divided by a 1-digit number.

If they give you 96 divided by 7, for example, you can think,
"Hmm... the closest multiple of 7 is 91, which is 13 * 7, with 5 left over.
So the answer is 13 and 5/7, or: 13.7142857

The reciprocal charts should be memorized by heart . For quick calculation of Percentages , Data Interpretation Questions and other questions requiring quick conversion.

Subsequent posts will show you the practical application of these forms.