# (Vedic Maths )Reference material to CRACK MATHS

#### kartik

##### Kartik Raichura
Staff member

Eg: 500
+995
----
A: 1495

Tip : 50/0
99/5

crack the numbers (0+5)=5 , add the rest (50+99)=149. thus the answer will be 1495.

Another example : 6789
+4622
--------
A: 11411

Tip : 67/89
46/22
------
114/11

add 89+22=111 . key in the last 2 digits and carry forward 1 . add 67+46=113
and add the 1 which was carry fwd.
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Mutiplying Big nos :

Eg: 254* 50

Tip 25/4
X 50
-----
12700

Crack the 1st no . Multiply each by the second no. Ie. 50 *4 = 200 , 50 *25= 12500.
Add the two [12500 + 200 ].

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Multiplying by 5

suppose the no is 300 . divide the no by 2 = 150 and add a "zero". So the ans= 1500

another example: 244 * 5 = 1220
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mutiplying a number by 11

Tip : for 2 digit nos Simply add the first and second digits and place the result between them.

Eg. 24 * 11 = 264

another eg : 79*11 = units place =9 , here 7+9= 16 so write 6 in middle..
carry fwd 1 and add to 7 so the ans will be 869

__________________________________________________________________________________________-

Cube root trick (works only for ans of cube roots from 1-99)

Eg. Find cube root of 148877 ( Needs memorsing last digits of cubes from 1-9)

here is the list :

1-------->1
2-------->8
3-------->7
4-------->4
5-------->5
6-------->6
7-------->3
8-------->2
9-------->9

Can be memorised as These are easily memorized. 1 and 9 (at the extremes) are
"self-enders", as are the 4, 5, and 6 (in the center).
The others involve "a sum of 10": 2 ends in 8, 8 ends in 2, 3 ends in 7, and 7 ends in 3.

To instantly determine his original number, do the following:

1. Drop the last three digits and find the largest cube contained in 148. This is 5^3 = 125, so the tens-digit is 5.
(This is why you had to memorize the cubes of the digits 1 through 9)
2. Now go back to the last three digits. Look at the last digit, 7. That's the same ending as 33, so your units-digit is 3.
(This is why you had to memorize the "endings" of the cubes for digits 1 through 9)

so cube root of 148877 is 53 .

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Sq root trick with nos above 100

Needs memorising of last digits of squares from 1-9

Here is the list

1---------->1
2---------->4
3---------->9
4---------->6
5---------->5
6---------->6
7---------->9
8---------->4
9---------->1

suppose the no is 169, then take the hundreds value (over here 100 and key in its sq rt ie 10)

now observe the last digit . Thats the same ending for 3 and 7 . so the no is either 10+3=13
Or 10+7= 17 . To determine the correct answer , take average of the sq of extreme digits ie
over here (10'2 + 20'2) /2 = 250. In our case 169<250 . Therefore the answer will be 13 .

If here the problem number was greater than 250 then the answer would have been 17 .

----------------------------------------------------

Method for multiplying numbers where the first figures are same
and the last figures add to 10

Eg: 42 x 48 = ???
figures (2 and 8) add up to 10.

just multiply 4 by 5 (the next number up)
to get 20 for the first part of the answer.

And we multiply the last figures: 2 x 8 = 16 to
get the last part of the answer

another eg : 33 * 37 = 1221

-------------------------------------------------------------------------------------------
Method for multiplying numbers where the first figures add up 10
and the last figures are same

Eg : 44X64

Here first figures are 4 and 6 and their add up 10 and unit figures of both number are same
Just multiplying the last figures 4x4=16 Put it at right hand side
Again multiplying the first figures and add common digit

(4x6 )+4=24+4=28 ,
put it at left hand side
Now we get required answer 2816

Similarly 36x76 , 6X6 =36 right hand side , (3x7)+6= 21+6=27 left hand side

NOTE If multiplication of last figures is less than 10 add zero before unit digit
Ex 81x21 , 1x1=01,( 8x2)+1= 16+1=17 Required answer 1701

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Method for multiplying numbers where the first number"s add upto 10 and and the second
number's digits are same
Eg : 46X55

Here first number's add up is 10 and second number "s digits are common i.e 5

Just multiplying last figures of both numbers 6x5 =30 put it at right hand side
Again multiplying first figures of both numbers and add common digit of second number

(4x5)+5 =20+5 =25 put it left hand side

Required answer is 2530 ( If multiplication is in unit in first step add zero before it)

--------------------------------------------------------------------------------------------

Multiplying numbers just over 100.

108 x 109 = 11772 14042

Similarly 107 x 106 = 11342

The answer is in two parts: 117 and 72,

117 is just 108 + 9 (or 109 + 8),
and 72 is just 8 x 9.

Another eg: 118 * 119 = [ 18 * 19 = 342,so key in the last 2 digits and c/f 3,
now, 118+19 =137.we add the c/f 3 to 137 = 140 ]

So ans is 14042
--------------------------------------------------------------------------------------------

If you know ne more methods then feel free to hit a reply to this !!. :wink:

#### kartik

##### Kartik Raichura
Staff member
Part 2

1. If you want to multiply any number with 9, 99, 999,...just use
this rule: eg. 999x343=342657 (second number -1)(subtract 9 from each
of these new digits)ie. 342 and (9-3)(9-4)(9-2)

2. If you want to square a number that is closer to 10, 100,
1000, ...: eg. 988^2=976144 (base number-(nearest '0' number - the
base number))(square of the inner difference) ie. 988-(1000-988)and
(1000-988)^2. I am at loss of words here, hope you get the idea.

3. If you want to square a number (that is not closer to 10,
1000,...): eg. 43^2=1849 (first digit x (the base number + second
digit))(square of second digit) ie. (4 x (46))(3^2)=>1849

4. The square of any number ending in a five: (x5)^2 = [(x)(x+1)]25
Examples: 15^2 = ((1)(1+1))25 = 225 95^2 = (9(9+1))25 = 9025

5. The result of one over any number ending in a 9 can be computed in
two very simple ways. I will give the easier example. 1/x9 ->

a. Take the digit x. Increment it by one.

c. Multiply that number by (x+1) and write that number to the left of
it.

d. Now take this new number. Multiply this number by (x+1).

e. Write the last digit of the result to the left of the result so
far.

f. Any carry over, remember it.

g. Take the last digit of result entered. Multiply it with (x+1) and
add the carry over remembered in step f.

h. Write the last digit of the result to the left of the result so
far.

i. Any carry over, remember it.

j. Repeat the steps g, h, and i untill the result pattern starts
repeating or you hit a zero as the result digit..

k. In most cases the pattern will repeat after every (x9 - 1) digits
of result.

6. Here are examples, again for numbers closer to 10, 100, 1000... If
you want to multiply two of these numbers, here are two cases: (Since
folks are getting confused with syntax, I am just giving examples)

a. 95 x 97 = <97-(100-95)><(100-95)x(100-97)> = <92><5x3> = 9215 It
is actually much easier when you write it on a piexe of paper.

b. if one of the numbers is over the '0' number: 94 x 104 = <104-(100-
94)><(100-94)x(100-104)>. Here is where you use your intuition. The
next step is (9800)-(24)=9776

This logic above cannot be used for numbers that are not close to 10,
100,1000 etc.

#### kartik

##### Kartik Raichura
Staff member
Re

Today, I will be describing how to find square of a digit near 100.

Say you want to find Square of 98.

98 is 2 away from 100. So what?

The square will be given by
98-2/square of 2 = 9604

Say you want to find Square of 97

97 is 3 away from 100. Square of this will be given by
97-3/square of 3 = 9409

Let us take one example which is more than 100.

Say you want to find square of 102
102 is 2 away from 100, its 2 more than 100 therefore we will add 2
more to it and the square will be given by
102+2/square of 2 = 10404

After going through these three examples you should be able to
understand the technique. The technique is to reduce or add as much
as it is away from 100 and put square of difference on it. This is
how we do it. We will have something more on this next week.

#### kartik

##### Kartik Raichura
Staff member
Today I am going to describe divisibility of a number by 13. In
schools and coaching institutes students are taught divisibility of a
number by 2, 3, 4, 5, 6, 8, 9 and 11. I hope you are all aware of
this.

A number will be divisible by 2, if it is an even number.A number
will be divisible by 3, if sum of the digits is divisible by 3 and so
on and so fourth... But what about 7, 13, 17, 19 and other odd
numbers?

Our contemporary mathematics is silent on this. In Vedic we have
solutions for all these (surprised?)

Let me introduce 13. [Operator of 13 is 4 How?** ]

Find whether 1001 is divisible by 13?

By seeing it is difficult to say.

Do the following operations:

1001= 100+1(last digit)*4(operator)= 104= 10+ 4 (last digit)*4
(operator)=26

Since 26 is divisible by 13 therefore, 1001 is divisible by 13.

What have we done here?

We seperated the last digit of the number, multiplied it by the
operator and added the result to the rest. This we did until we found
a managable number. Go ahead and test this!

#### kartik

##### Kartik Raichura
Staff member
To Multiply two digit numbers:
Remember:
1. The digits on the left hand should be same.
2. On adding the right hand digits you must get 10.

Example:
65 x 65 or 36 x 34

The Magical Method:
65 x 65 = 4225

What did we do?
1. We first multiplied the right hand side digits (5 x 5) and wrote
25.
2. We added 1 to the left digit 6 and got 7.
3. We then multiplied 6 by 7 and wrote the answer 42 to the left of
25.
4. We thus arrived at the answer 4225.

Let's try another example:
46 x 44
1. 6 x 4 = 24
2. 4 + 1 = 5
3. 4 x 5 = 20

Note:
What if we have to multiply 69 x 61?

But why the extra 0?
Because the right hand side of the answer should always have two
digits.
:x

#### kartik

##### Kartik Raichura
Staff member
How to find Square of a number near 50?

Do you face difficulty in finding square of a number near 50? Try
this method.

Square of 51 = square of 5 + 1/ square of 01= 25 + 1/ 01 = 2601

What is the meaning of the above notation?

It means that LHS of the answer is given by square of 5 + 1 and RHS
is given by square of the difference of the number from 50.

One thing you will be required to keep in mind that we require two
digits on the RHS.

After learning this can you find out Square of 52?

Square of 52 = Square of 5 + 2 / Square of 02 = 25 + 2 / 04 = 2704

And Square of 53= 25 +3/ Square of 03 = 2809

And Square of 54 = 25 + 4 / Square of 4 = 2916

Try out other digits near 50 like this and find out whether they are
correct. Can any body guess what will you do incase of digits less
than 50?

#### divyajot

##### New member
Vedic Maths

Hi all

Just adding on to kartik's posts:

There are certain easy techniques in vedic mathematics, which
facilitate fast computation. These techniques are especially applied
to multiplication of the number and thus reduce the time for
calculation. Vedic mathematics also gives techniques for division of
division.

Multiplication of number :-

Method 1 :-
In this method we define the base. The base is generally in power of
10. ie 10¹, 10²,. etc. There are special cases when we change the
base which we will discuss later.
Lets take for example, the multiplication of number say 9 x 7.
Here since the number are close to 10. We take the base as 10. write
the deviation of these
9 / - 1 number from the base
x 7 / - 3 as in this case it is - 1, 2 -3
resp,
6 / 3

Multiply the numbers which appear in the right hand side of the slash
by considering their sign as well. ie (-3) x (-1) = 3. This will be
the digit in the units place.
Now cross add the digits ie [9 + (-3)] or [7 + (-1)] {Here either of
the cross addition would give the same answer} ie 6 in this case. So
Note that since in this case the base was 10¹, the number of digits
to the right hand side of the slash should be one. If there are two
digits on the right side of the slash when the base is 10. Then add
the digit in the ten's place to the digits on left side of the slash.
Ex:- 14 / + 4
13 / + 3
17 / 12 ie 182

If the number on the Right hand side of slash of the answer has a
negative sign then substract that number from the number on ( LHS x
10 )
(since the base is 10 in this case)
Ex:- 14 / + 4
6 / - 4
10 / - 16
100
- 16
84

Now we will consider some examples, when base is 10².
Ex:- 92 / -8
98 / -2
90 / 16 Therefore, the answer is 9016
Note that since the base is 10² ie 100. The number of digits on RHS
of slash should be two.

Special Cases :-
This method is useful when the two quantity we multiply
are close to the base value. But what if the number deviates very
much from the base ?
For ex:- 53 x 57.
Here we follow the change of base. The base should be
changed to 50. This change can be done in 2 ways
(1) consider the parent base as 10 and then follow the rules of
multiplication by considering the base 10, but the deviation is
considered from shifted base.
After the result is obtained,
Multiply the LHS of the slash by x because of shift in base ie 10 × X
= Xo.
Ex:- 53 / + 3
57 / + 7
60 / 21 Here since shifted base is
50 Multiply LHS by 5
× 5 Therefore, the answer is
3021.
300 / 21

(2) Consider the parent base as 100 and then follow the rules of
multiplication by considering parent base as 100, but the deviation
is considered from the shifted base.
After the result is obtained, divide the LHS of the slash by X ,
because of shift in base.
The X is such that 100 = Xo
X

Ex :- 53 / + 3
57 / + 7 Since the base is 50
60 / 21 Therefore, X = 100 = 2

50

Therefore, 60 / 21 ie 3021 is the answer.
2

Ex:- 997 / - 3 Since the base is 10³.
× 992 / - 8 3 digit should appears on
RHS of slash.
989 / 024

Let us consider another example for practice.
Ex:- 82 / + 2 Shifted base = 80
76 / - 4 parent base = 10
78 / (-8) Therefore, X = 8
× 8

624 / (-8) ie 6240
- 8

(1) The computation time is reduced drastically.
(2) This method is particularly useful when the numbers under
consideration are close to base.
(3) Very useful in finding the squares.

The disadvantage of this method is that the numbers under
consideration should be very close to each other. If there is a large
duration in the numbers from each other then it is very difficult to
fix the base for ex:- 32 × 78 etc.

Method II :-
The disadvantage of method I can be overcome by another
method of multiplication used in vedic multiplication. It eliminates
the base consideration.
Here the multiplication is broken into smaller multiplication just as
we do in conventional multiplication.

Let us try to understand this by an example.
Ex :- 324 × 629

324 Step1:- The
multiplication is performed as (i) 6 × 3
629 (ii) (3 × 2) + (6 × 2)
(iii) (3 × 9) + (6 × 4)
11523 (iv) (9 × 2) + ( 2 × 4)
(v) 9 × 4.
88566
203796

Step 2 :- The digit in tens place arising out of this multiplications
is written on the upper row while the digit in units place is written
on the lower with a shift by one to the right side. This is carried
out for the series of multiplication as indicated in step 1.

Step 3 :- The results is obtained by adding the two rows.

Here the order of multiplication is important
for ex : a b c The multiplications are
carried out as follows
x y z (i) a × x
(ii) (a × y) + bx
(iii) az + cx + by
(iv) bz + cy
(v) cz

Similarly for a 4 digit multiplication it would become
a b c d (i) a × w
w x y z (ii) ax + bw
(iii) ay + cw + bx
(iv) az + wd + cx + by
(iv) bz + xd + cy
(v) cz + dy
(vii) dz

Let us do this with the help of example
3 2 9 1
4 3 5 2
1 1 5 4 5 2 -
2 7 7 7 2 3 2
1 4 3 2 2 4 3 2

Apart from these technique of multiplication there are other
techniques which are used for certain specific calculation.
(i) when we are asked to calculate the area when the lengths and
breadth are given in two units for e.g feets & inches.
Then going by conventional rules we first convert the units in one
unit. Calculate the area & then again convert back to the two units
(if required).
Let us consider an example :-
Calculate the area of wooden piece with a length 5ft 6in and breadth
4ft 3m & express the answer in sq.ft & sq.inches.
Soln :- Going by the conventional rules we would first convert either
feet into inches or convert back to the required conditions.
But instead of doing this, vedic maths suggest a simple way of
doing this
(1) Express the length & breadth into simultaneous equation & perform
the multiplication
ie 5ft 6in ---- 5x + 6y
4ft 3in ---- 4x + 3y
20x² + 39 xy + 18 y²

(2) Split the middle term ie xy term, such that one part is a factor
of 12 (since 1ft = 12 inches)
(3 × 12) xy + 3xy = 39xy.
(3) Consider the term (3 × 12) xy. Add this 3 to the x² term. The
square feet term becomes
20 + 3 = 23 sq.ft
(4) The remaining term 3xy multiply it by 12 (since 1ft = 12 inches)
& add this to the coefficient of
y2 ie 3 × 12 + 18 = 54
This will be the sq. inches term.

Thus our final answer will be 23 sq.ft 54 sq.inches

Note:- If the Sq.inches term is > 144
then add one sq.ft & correspondingly substract from sq.
inches term.
as in the example 5ft 11m × 3ft 9m
Therefore, 5x + 11y
3x + 9y
15x² + 78xy + 99y² = 15x² + [(12 ×6) +
6 ] xy + 99y²
ie (15 +6) sq.ft & [(12 ×6) + 99 ] sq
inches
21 sq.ft 171 sq
in but 1sq.ft = 144 sq.in
Real answer would be 22 sq.ft ,
27 sq.in

This rule of multiplication can be applied to any unit by taking care
of proper conversion factor.
There are certain other rules derived from vedic maths. One such
is used for finding the square. This is derived from the method I
described earlier. (were it is necessary to define base)
Ex :- 14² = ( 14 + 4)/ 4² = 18/6 = 196
12² = ( 12 + 2)/ 2² = 14/4 = 144.
(24)² = ( 24 + 4)/ 4² = 28/16 = (28×2) +
1/6 = 576
(22)² = (22 + 2)/ 2² = (24)/ 4
×2
484

There are other simple techniques which are applied for certain
specific class.
(1) Square of a number with 5 in its units place.
Therefore (x5)² = (x) (x + 1) / 25
ie (35)² = 3 × 4 / 25 = 1225
(65)² = 6 × 7/ 25 = 4225
(115)² = 11 × 12/ 25 = 13225

(2) Square of a number between 50 to 59.
(5x)² = (25 + x) / x² when x = 0, 1, .......9.
take for ex :- (54)² = (25 + 4) / 4² = 2916
similarly (59)² = (25 + 9) / 9² = 3481

#### DIVINEPASSION

##### New member
)) hi ppl out here, i would b glad 2 explore the opportunuity to keep in touch with u
ppl, plz guide me out here as i m a newbie, im doin my bms 2yr

pls do help as i need all ur help n support to get tips 4 my projects n mba entrance tests,

cya n hope 2 get a positive reply soon

8) divinepassion

#### kartik

##### Kartik Raichura
Staff member
DIVINEPASSION said:
)) hi ppl out here, i would b glad 2 explore the opportunuity to keep in touch with u
ppl, plz guide me out here as i m a newbie, im doin my bms 2yr

pls do help as i need all ur help n support to get tips 4 my projects n mba entrance tests,

cya n hope 2 get a positive reply soon

8) divinepassion
D00d .. we wil try to help u in d best way possible :-? ... Chillax.. njoy bms .. n keep in touch..

#### gaurav200x

##### Gaurav Mittal
My 2 cents to this topic:

Multiplication of any no. by

eg. if i want to multiply a number 567321 by 11

i start from the right most digit and bring down 1... Now consider the two rightmost digits - 1 & 2 (the 1st pair) add them -> 3 and write it to the left of 1

So, we have written 31 till now...

Similarily consider the 2nd last and the 3rd last digits,, i.e. 3 & 2 (next pair)
Their total is 5 . Write it besides 3

Taking the next pair of 7 ans 3... u get 10. So write 0 and carry 1. This 1 should be added to the next sum obtained....

Following like this u exhaust all the pairs (u reach till 5 & 6). Next u consider 5 and 0 (0567321) and write its sum and if u have any carry.... add to it...

The procedure is extremely simple once u have tried it...

u will get ur final answer as 6240531

Similarily, try for any number. The technique holds good for any number of any length.....

Cheers

Gaurav

#### gaurav200x

##### Gaurav Mittal
Multiplication by 12:

I hope the above method for multiplication by 11 must have been pretty clear.

Now for multipying by 12, all we need to do is to multiply the left-side digit by 2 before adding it to the right-side digit. The method is similiar to that of the above except for the multiplication part.

eg. if we need to multiply 45632 by 12

Take 2 and multiply it by 2 and write 4 below.

Next take 3 and multiply it by 2. You get 6 and add it to the right-side digit 2
i.e
(3 2)
| ^
V |
X +
2= 6
Take the next two digit, i.e. 6 and 3 and repeat the same. You will notice, u get a carry here, when u multiply 6 with 2 and add 3. Keep the carry and add it to the next pair 5 & 6

This way, u will get ur final answer as 45632 * 12 = 547584

p.s. remember that 45632 should actually be read as 045632 and hence the operation must be applied to the last pair, i.e. 0 and 4.

Regards
Gaurav

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