# mathematics shortcuts

Document Sample

```					 We are going to begin with my favorite number 5 and how things get easy
with it.

Fun with Number 5
Rules: This is normally used for multiplication and in this case both the
units in digit i.e. the last numbers are 5. Also, the beginning numbers
are same. You will understand this as you see some examples.

25 x 25 = 6 | 25 (This one's easy, right?)
Explanation: As you see both the ending numbers are 5 and also they are
beginning with 2. So we have satisfied both the rules. Now how do we get
625. The first step is that you write down 25 on the right hand side. Why
we do so? Do not think about that. Just write 25 on the right hand side.
Then we take the number on the left hand side (of 25), in this case 2.
And we multiply it with the next higher digit number i.e. 3. Therefore we
get the answer 2 x 3 = 6, which we write down on the right hand side. So
we finally get the answer as 625.

45 x 45 = 20 | 25
Explanation: As this one satisfies the rule too. Both the left numbers
are same - 4, and the ending numbers are 5. So without thinking the first
thing we do is write 25 on the right hand side. Then we take the left
hand number 4 and multiply it with 5, hence we get 4 x 5 = 20 (the left

135 x 135 = 182 | 25 (slightly tough)
Explanation: Both the rule has been satisfied. We write down 25 on the
right hand side. We multiple 13 with the next higher number and get 13 x
14 = 182. And there we have our answer 18225. Easy right?

Slight Modification to the Number 5 method
Rule: The ending number should add up to 10. Infact in the above case too
the ending number 5 + 5 added upto 10. And the left hand side number
should be same. This is very similar, so lets look at some example.

23 x 27 = 6 | 21 (I am sure that,not everyone finds this simple).
Explanation: Ending numbers 3 + 7 add upto 10. So the first rule is
satisfied. And both the number on left hand side are same i.e. 2. First
thing you do is, you multiply 3 x 7 = 21. And then like earlier you
multiply 2 with the next higher digit number i.e. 3. And we get 2 x 3 = 6
on the left hand side. Final answer is 621.

62 x 68 = 42 | 16
Explanation: Both the rule have been satisfied. We multiply 8 x 2 as the
add upto 10 and we get the left hand side of the answer 16. Then we
multiply 6 with the next higher digit number 7 and get 6 x 7 = 42

Fun with number 11
Now we try the tricks which can make calculation by 11 easy. We will also
learn how to multiply by 22, 33, 44, 55, 66, etc. As it is nothing but 11
x 2, 11 x 3, 11 x 4, 11 x 5, 11 x 6 respectively.

Steps involved:
1. To multiply a number by 11 put a 0 in the beginning and at the end
of the number.
2. Starting from the digit’s place add two consecutive digit. Remember
to go from right to left.

Examples

342 x 11 = 03420 = 0+3 /4 + 3/2 + 4/0 + 2 = 3762
Explanation: We added 0 at the end and the beginning of 342 and got
03420. Now, as you see we broke the number by taking each consecutive
number and added them. The first part is 0 + 2 = 2, the first number of
our answer, then we took 2+4 and got 6, second number of our answer and
so on.

4932 x 11 = 049320 = 0 + 4/4 + 9/3 +9/3 + 2/0+2 = 54252
Explanation: In this case when we add 3 + 9 we write down 2 and carry
forward 1, we again get 4 + 9 + 1 we write down 4 and carry forward 1. We
add 4 + 1 and get 5
In the above case if we need to multiply by 22. Then we first multiply by
2 and then by 11.

312 x 22 = 624 (312 x2) x 11 = 06240 = 0 +6/6 + 2/4 + 2/0 + 4 = 6864
Explanation: Here we are trying to multiply with 22. Hence we first
multiply the main number with 312 with 2 and get 312 x 2 = 624. And then
multiply the same number with 22. Rest of the steps remain the same.

312 x 22 = 03120 x 11 x 2 = (0+3 / 3 + 1 /2 + 1 /0 + 2) x 2 = 3432 x 2 =
6864
Explanation: This is the exact same sum as above. In this case you
multiply the end answer by 2 and get the exact same answer.

Similarly we can calculate any number into any multiple of 11. Even 11 x
11. We just repeat the process twice.

Fun with number 9

Part A (Original number and the number of 9 is same, for example 234 x
999 but we cannot use this method for 234 x 9999 or 234 x 99)
Rules: As usual there is a left hand part of the answer and the right
hand part of the answer. When multiplying by 9 we subtract one from the
original number and write it down on the left hand side. Then we subtract
each number on the left hand side from 9 and write it down on the right
hand site (this will become clearer with the examples that follow).

234 x 999 = 233 | 766
Explanation: We subtract one from 234 and   write it down on the left hand
side i.e 234 - 1 = 233. Then we take each   individual number and subtract
that number from 9 in the order - left to   right. So we subtract 9 - 2 =
7, then we subtract 9 - 3 = 6 and again 9   - 3 = 6. Hence, we get the
right part of our answer i.e 766

Common Mistake: As you see from the above example we subtract 9 from each
individual number, but we did so from the left hand side of our answer
and not the original number. In simple words, we should subtract
individually from 233 and not from 234. This is a common mistake I have
found and would advice you to remember this crucial point.

948224243 x 999999999 = 948224242 | 051775757
Explanation: Wow! I can beat the calculator. Most of the calculator
cannot perform calculation more than 12 digits and I doubt even your
mobile phones can do that. Here is how its done, we subtract 1 from
948224243 and get 948224242 . Then we subtract 9 - 9 get 0, then 9 - 4 =
5, then 9 - 8 and so on. I am sure by now you get the idea.

But now here is the question what if the number of 9's are more than the
number which we going to multiply, for example,
273 x 99999 ( we cannot do this as of now)

Part B (Original number is less then the number of 9's, for example 234 x
99999)
Rules: Similar to above method, but just imagine 0's in front of the
original number. But make sure the number of 9's are more and not less or
equal.)

383 x 9999 will become 0383 x 9999 = 0382 | 9617
Explanation:It is simple, we imagine the number to be 0383. Subtract one
from it and get 0382. Then as usual we calculate 9 - 0 = 9, then 9 - 3 =
6 and so one.

383 x 999999 will become 000383 x 999999 = 000382 | 999617
Explanation: In this case we had 3 extra 9's. So we imagine 3 extra 0's
in front of 383. Therefore our first case will be 000383 - 1 = 000382,
which becomes the left part of the answer. And like before we subtract 9
- 0 and get 9, then again 9 - 0 = 9 and again 9 - 0 = 9, then 9 - 3 = 6
and so on.

Part C (Original number is more then the number of 9's, for example 234 x
9, this method is tough to undestand so pay close attention to this one)
Rules:I will directly goto the example in this case, as the rules will
confuse you at this level.

14 x 9 = 12 | 6
Explanation: From the number 14 we choose 1(the number in the ten's
digit), increase it by 1 i.e. 2 or we add + 1. Subtract that from 14 and
we get 12. Next we subtract 4 from 10 and and get 6 i.e. 10 – 4 = 6. The
method is confusing but once a person gets a hang of it, it’s pretty
easy.

24 x 9 = 21/6

Explanation: From the number 24 we choose 2(ten's digit); increase it by
1 i.e. 3 or we add + 1 (2 + 1). Subtract that from 24 and we get 21 (24 -
3). Next we subtract 4 from 10 and get 6 i.e. 10 – 4 = 6.

47 x 9 = 42/3
Steps involved: From the number 47 we choose 4(ten's digit); increase it
by 1 i.e. 5. Subtract that from 47 and we get 42 (47 - 5). Next we
subtract 7 from 10 and get 3 i.e. 10 – 7 = 3.

112 x 99 = 110/88

Steps involved: From the number 112 we choose 1; increase it by 1 i.e. 2
or we can say we add + 1 (1 + 1). Subtract that from 112 and we get 110
(112 - 2). Next we subtract 99 from 12 and then add 1 i.e. 99 – 12 + 1 =
88.

112 x 9 = 100/8

Steps involved: From the number 112 we choose 11; increase it by 1 i.e.
12 or we can say we add + 1 (11 + 1). Subtract that from 112 and we get
100 (112 - 12). Next we subtract 9 from 2 and then add 1 i.e. 9 – 2 + 1 =
8

So we have learned some simple methods of multiplying with 5 , 11 & 9.
But that was all about scratching the surface. We have much more to
cover. Today it might be slightly tuff.

Rules: This method is based on selecting a proper base and then using the
Vedic maths method to calculate. Selecting a proper base is one of the
most important things in this method, which makes major calculations very
simple. See the example of different bases and make sure you use your
wits to choose a proper base while calculation. It works in the following
manner.

9 x 9
9 – 1 [ We assume the base as 10 ]
9 – 1
8 /1
Steps involved:-
We select the base as 10.
Then we check the difference for each number as compared to 10. In this
case the difference is -1. So we write -1 on the right on side of this
number.
Then we multiply the right hand side. In this case -1 x -1. So we get 1
Then we do cross addition/subtraction. In this case we would subtract -1
from the number 9 and then we get the answer as 8. And the final answer
as 81.

9 x 8 [ We assume the base as 10 ]
9 – 1
8 – 2
7 /2
Steps involved:-
We select the base as 10.
Then we check the difference for each number as compared to 10. In this
case the difference is -1 & -2. So we write -1 & -2 on the right on side
of these numbers respectively.
Then we multiply the right hand side. In this case -1 x -2. So we get 2
Then we do cross addition/subtraction. In this case we would subtract -2
from the number 9 or we subtract -1 from the number 8 and then we get the

11 x 12 [ We assume the base as 10 ]
11 + 1
12 + 2
13 /2
Steps involved:-
We select the base as 10.
Then we check the difference for each number as compared to 10. In this
case the difference is +1 & +2. So we write +1 & +2 on the right on side
of these numbers respectively.
Then we multiply the right hand side. In this case +1 x +2. So we get 2
Then we do cross addition/subtraction. In this case we would add +2 from
the number 11 or we add +1 from the number 12 and then we get the answer
as 13. And the final answer as 132.
It is simple, isn't it?

91 x 91 [ We assume the base as 100 ]
91 – 9
91 – 9
82/81
Steps involved:-
We select the base as 100.
Then we check the difference for each number as compared to 100. In this
case the difference is -9 & -9. So we write -9 & -9 on the right on side
of these numbers respectively.
Then we multiply the right hand side. In this case -9 x -9. So we get 81
Then we do cross addition/subtraction. In this case we would subtract -9
from the number 91 and then we get the answer as 82. And the final answer
as 8281.
Now, we have moved on to higger numbers too.

You may practice higher numbers like 120 x 102 or 105 x 108. But remember
so far our base has only been 10 & 100. We will move on to slightly
complex task tomorrow. You may check the next few examples. Solution is
not given, but by now you should be used to doing these simple
calculation mentally.

888 x 998 [ We assume the base as 1000 ]
888 – 112
998 – 002
886/224

598 x 998 [ We assume the base as 1000 ]
598 – 402
998 – 002
596/804
Rules: This method is based on selecting a proper base and then using the
Vedic maths method to calculate. Selecting a proper base is one of the
most important things in this method, which makes major calculations very
simple. See the example of different bases and make sure you use your
wits to choose a proper base while calculation. It works in the following
manner.

Importance of choosing the RIGHT BASE.

25 x 98 [ We assume the base as 100 ]
25 – 75
98 – 2
24/50
Steps involved:-
We select the base as 100.
Then we check the difference for each number as compared to 100. In this
case the difference is -2 & -75. So we write -2 & -75 on the right on
side of these numbers respectively.
Then we multiply the right hand side. In this case -2 x -75. So we get
150 as the answer. But since the base is 100 we can only use 2 up to 2
digit and we will carry forward 1 to the left hand side
Then we do cross addition/subtraction. In this case we would subtract -2
from the number 25 or we subtract -75 from the number 98 and then we get
out to be 24. And the final answer as 2450.

So far we have looked at numbers where after addition/subtraction from
the base, both the numbers were either positive or negative. Now lets
consider the other way round.

12 x 8 [ We assume the base as 10 ]
12 + 2
8 – 2
10/-4 = 96
Steps involved:-
We select the base as 10.
Then we check the difference for each number as compared to 10. In this
case the difference is +2 & -2. So we write +2 & -2 on the right on side
of these numbers respectively.
Then we multiply the right hand side. In this case +2 x -2. So we get -4
Then we do cross addition/subtraction. In this case we would subtract -2
from the number 12 or we add +2 from the number 8 and then we get the
However since the we have – 4 on the right hand side we subtract it from
the base and -1 from the left hand side.
And we get the answer 96

It is slightly confusing but you will get the hang of it pretty soon.

108 x 97 [ We assume the base as 100 ]
108 + 8
97 – 3
105/-24 = 104/76
Steps involved:-
We select the base as 100.
Then we check the difference for each number as compared to 100. In this
case the difference is +8 & -3. So we write +8 & -3 on the right on side
of these numbers respectively.
Then we multiply the right hand side. In this case +8 x -3. So we get -24
Then we do cross addition/subtraction. In this case we would subtract -3
from the number 108 or we add +8 from the number 97 and then we get the
However since we have – 24 on the right hand side we subtract it from
the base and -1 from the left hand side.
And we get the answer as 10476

We have also been using the base such as 10, 100 & 1000. Now, lets try
some different base. Play close attention to the difference as it will

49 x 49 [ We assume the base as 50 ]
49 – 1
49 – 1
48/01
24/01
Steps involved:-
We select the base as 50.
Then we check the difference for each number as compared to 50. In this
case the difference is -1 & -1. So we write -1 & -1 on the right on side
of these numbers respectively.
Then we multiply the right hand side. In this case -1 x -1. So we get 1
Then we do cross addition/subtraction. In this case we would subtract -1
from the number 49 or we subtract -1 from the number 49 (same thing in
this case) and then we get the answer as 48.
But since the base is 50 which is nothing but 100/2. So we divide 48 by 2
And we get the answer 2401.

Similar Example:
46 x 46 [ We assume the base as 50 i.e 100/2 ]
46 – 4
46 – 4
42/16
21/16

23 x 23 (BASE AS 20 i.e. 10 x 2)
23 + 3
23 + 3
26/9
52/9

Steps involved:-
We select the base as 20.
Then we check the difference for each number as compared to 20. In this
case the difference is +3 & +3. So we write +3 & +3 on the right on side
of these numbers respectively.
Then we multiply the right hand side. In this case +3 x +3. So we get 9
because here the base is 10 x 2. That means the underlying base still
remains 10. And that’s why we just write one number on the right hand
side i.e. 9
Then we do cross addition/subtraction. In this case we would subtract +3
from the number 23 or we add +3 from the number 23 (same thing in this
case) and then we get the answer as 26.
But since the base is 20 which is nothing but 10 x 2. So we multiply the
base 26 x 2.
And we get the answer 529.

23 x 23 (BASE AS 20 i.e. 100/5) [ Notice the above sum has the same
numbers]
23 + 3
23 + 3
26/09
5.20/09
5/29

Steps involved:-
We select the base as 20.
Then we check the difference for each number as compared to 20. In this
case the difference is +3 & +3. So we write +3 & +3 on the right on side
of these numbers respectively.
Then we multiply the right hand side. In this case +3 x +3. So we get 9
as the answer. NOTE: We write the base as 09 as our basic base or main
base is 100/5. The whole idea is if the basic base or main base is a
multiple or division of 100 then we will have 2 digits on the left hand
side. And when the basic base of main base is a multiple of 10 then we
take only one digit on the left hand side, rest we carry forward to the
left hand side after doing the division.
Then we do cross addition/subtraction. In this case we would subtract +3
from the number 23 or we add +3 from the number 23 (same thing in this
case) and then we get the answer as 26.
But since the base is 20 which is nothing but 100/5. So we divide 26 by 5
Now whenever we get answer in decimal form after division we transfer it
to the right hand side by adding it. Therefore we add 20 to right hand
side and get the answer as 29
And we get the final answer as 529.

GOLDEN RULES FOR ADD SUBTRACT METHOD.

Choose the base according to your common sense so that the calculation
becomes easy.

If the base is 100 there will be 2 digits on the right hand side, if the
base is 10 there will be 1 digit on the right hand side, if the base is
1000 there will be 3 digits on the right hand side and so on. And the
extra digit you carry over to the left hand side.
Remember to always divide the left hand side or multiply it first, if
required. And then minus right hand side (if negative) or carry forward
from left hand side (if positive).

carry forward -2 to the left hand side. Then your answer will be 22/-6.
And then you subtract. This will give you the answer 21/4. However, you
can do it directly too, since you know by subtraction of 24 by 3 will
give you 21 and you will be able to carry 30 to the left hand side and
then you will be able to get your answer easily. i.e. 30 – 26 = 4.

26 x 24 =

317 x 22 =

125 x 125 =

999 x 294 =

99999 x 93 =

112 x 99 =

98 x 97 =

110 x 95 =

49 x 49 =

23 x 23 =

Basic Multiplication But Mentally and Faster.
The reason we are learning this method is because it is not possible to
do all calculation using the add and subtract method. There are many
other methods involved but this is simple and easy.
Rules: This method simplifies our normal method of multiplication. Just
remember the following diagram when you multiplying 2 digit number with a
2 digit number.
Multiply

Conventional Calculation
54
x 84
216
4320
4536

By simply Calculation
54
x 84
40/52/16
40/53/6
45/3/6
Steps involved:
We follow conventional method but we try to skip the extra work to reduce
time and effort.
First we multiply the right hand side number 4 x 4 (straight line). Write
Then we multiply 5 x 4 and add it with 8 x 4 i.e. 5 x 4 + 8 x 4 (the
cross). We get 52 as our answer.
Then we multiply the left hand side number 5 x 8 (straight line). Write
Then we carry forward 1 from 16 to the left hand side as we will have
only one digit. So we get 53 in the centre.
Then we carry forward 5 from 53 to the left hand side as we again will
have only one digit in the centre. So we get 3 in centre and on the left
hand side we get the answer 45

27
x 35
6/31/35
6/34/5
9/4/5

Steps involved:
First we multiply the right hand side number 5 x 7 (straight line). Write
Then we multiply 5 x 2 and add it with 3 x 7 i.e. 5 x 2 + 3 x 7 (the
cross). We get 31 as our answer.
Then we multiply the left hand side number 2 x 3 (straight line). Write
Then we carry forward 3 from 35 to the left hand side as we will have
only one digit. So we get 34 in the centre.
Then we carry forward 3 from 34 to the left hand side as we again will
have only one digit in the centre. So we get 4 in centre and on the left
hand side we get the answer 9.

67
x   4

This is

67
x 04
0/24/28
0/26/8
2/6/8

Steps involved:
First we multiply the right hand side number 7 x 4 (straight line). Write
Then we multiply 4 x 6 and add it with 0 x 7 i.e. 5 x 2 + 0 x 7 (the
cross). We get 24 as our answer.
Then we multiply the left hand side number 6 x 0 (straight line). Write
Then we carry forward 2 from 28 to the left hand side as we will have
only one digit. So we get 26 in the centre.
Then we carry forward 2 from 26 to the left hand side as we again will
have only one digit in the centre. So we get 6 in centre and on the left
hand side we get the answer 2.

Now, if we want to multiply a 3 digit number with a 3 digit number we
keep in mind the following diagram.

342
x 425
12/22/31/24/10
12/22/31/25/0
12/22/33/5/0
12/25/3/5/0
14/5/3/5/0

Steps involved:

1. First we multiply the right hand side number 2 x 5 (straight line).
Write down 10 as our answer
2. Then we multiply 4 x 5 and add it with 2 x 2 i.e. 4 x 5 + 2 x 2
(the first cross). We get 24 as our answer.
3. Next we multiple 3 x 5 add it to 4 x 2 and again add that to 4 x 2
i.e. 3 x 5 + 4 x 2 + 4 x 2 (the star). And we get the answer 31.
4. Then we multiply 3 x 2 and add it with 4 x 4 i.e. 3 x 2 + 4 x 4
(the second cross). We get 22 as our answer.
5. Then we multiply the left hand side number 3 x 4 (straight line).
Write down 12 as our answer.
6. Then we carry forward 1 from 10 to the left hand side as we will
have only one digit.

-     We keep carrying on each number to the left hand side and we leave
one number in each case and hence we get the number 14/5/3/5/0

1. Our final answer is 145350.

862
x   35

we can add a 0 to 35 and then solve as following

862
x 035
0/24/58/36/10
0/24/58/37/0
0/24/61/7/0
0/30/1/7/0
3/0/1/7/0
Steps involved:

1. First we multiply the right hand side number 2 x 5 (straight line).
Write down 10 as our answer
2. Then we multiply 6 x 5 and add it with 3 x 2 i.e. 6 x 5 + 3 x 2
(the first cross). We get 36 as our answer.
3. Next we multiple 8 x 5 add it to 0 x 2 and again add that to 6 x 3
i.e. 8 x 5 + 0 x 2 + 6 x 3 (the star). And we get the answer 58.
4. Then we multiply 3 x 8 and add it with 0 x 6 i.e. 3 x 8 + 0 x 6
(the second cross). We get 24 as our answer.
5. Then we multiply the left hand side number 8 x 0 (straight line).
Write down 0 as our answer.
6. Then we carry forward 1 from 10 to the left hand side as we will
have only one digit.

-     We keep carrying on each number to the left hand side and we leave
one number in each case and hence we get the number 3/0/1/7/0

We have already covered a lot and today seems to be one of my favourite
topics and I am sure you are going to enjoy all the neat tricks I am
going to teach. Have fun!

Let’s learn square using the add and subtract method.

In this method we again choose a base. This method is nothing but the
above method in a more simplified manner for calculating squares.

7 x 7 = (7-3)/3 x 3 = 4/9
By subtract and add method. Taking the base 10
7 – 3
7 – 3
(7-3)/ (-3 x -3) = 49

Look at the following examples

12 x 12 = (12 + 2)/2 x 2 = 14/ 4

14 x 14 = (14 + 4)/4 x 4 = 18/16 = 19/6 (remember the base is 10)

91 x 91 = (91 – 9)/9 x 9 = 82/81

95 x 95 = (95 – 5)/5 x 5 = 90/25

108 x 108 = (108 + 8)/8 x 8 =116/64

988 x 988 = (988 – 12)/12 x 12 = 976/144 (The base is 1000. Therefore we
do not need to carry 1 of 144 on the left hand side.

45 x 45 = (45 – 5)/5 x 5= 40/25 = 20/25 (The base is taken as 50 (100/2).
Therefore divide by 40 by 2)
45 x 45 = (45 – 5)/5 x 5 = 40/25 = 200/25 = 202/5 (The base is taken as
50 (10 x 5). Therefore we multiply 40 x 5 = 200 and the carry forward 2
from left hand side to right hand side as the base is 10)

Squares using the formula:
(a + b) (a + b) = a2 + 2ab + b2
(a - b) (a - b) = a2 - 2ab + b2

This is simple enough to understand.

97 x 97 = (100 -3)(100-3) = 10000 – 600 + 9

108 x 108 = (100 + 8) (100 + 8) = 10000 + 1600 + 64

29 x 29 = (30 – 1) (30 – 1) = 900 -60 + 1 = 841

49 x 49 = (50 -1) (50 – 1) = 2500 – 100 + 1 = 2401

78 x 78 = (80 – 2) (80 – 2) = 6400 – 320 + 4 = 6084

Let’s try a different method to find squares now:

Method:

1. Square the units digit.
2. Tens digit x units digit x 2
3. Square of tens digit

Examples

42 x 42 = 16/16/4 = 1764 (we carry forward 1 from the centre calculation
to the left hand side)

Method:

1. Square the units digit. We get 4
2. Tens digit x units digit x 2. We get 16
3. Square of tens digit is 16.

23 x 23 = 4/12/9 = 5/2/9 (we carry forward 1 from the centre calculation
to the left hand side)

Method:

1. Square the units digit - 9
2. Tens digit x units digit x 2 - 12
3. Square of tens digit - 4

62 x 62 = 36/24/4=38/4/4

Method:
1. Square the units digit - 4
2. Tens digit x units digit x 2 - 24
3. Square of tens digit – 36

73 x 73 = 49/42/9 =53/2/9

26 x 26 = 4/24/36= 6/7/6

(We carry forward 3 to 24. That makes it 27. We put down 7 and carry
forward to the left 2. Thus we get 6 on the extreme left. And our answer
is 676)

34 x 34 = 9 / 24 / 16 = 1156

42 x 42 = 16/16/4 = 1764

63 x 63 = 36/36/9 = 3969

Consider base 10

Cube for 13
13 = (13 + 3 x 2)/(9 x 3)/3 x 3 x 3
= 19/27/27 (But since the base is 10 only one number will remain in
each left hand side.
= 21/9/7.

Steps involved

1. First we double the excess number from the base 10 and add to the
original number i.e. 13- 10 = 3. Then 3 x 2(double the excess) + 13
(original number) = 19
2. Then we subtract 13 from base, get 3, which we multiply with 9 (
the excess number of the left hand side number minus the base i.e. 19 -10
). So basically - excess from original number from 10 x excess from the
left number which we got in step 1. We get 3 x 9 = 27.
3. Then we take the cube of the unit digit of the original number.
4. And doing the required calculation for the base 10, we get the

Cube for 15
15 = (15 + 5 x 2)/(15x5)/5 x 5 x 5
= 25/75/125
=25/87/5
=33/7/5

Steps involved

1. First we double the excess number from the base 10 and add to the
original number i.e. 15- 10 = 5. Then 5 x 2 + 15 (original number) = 25
2. Then we subtract 15 from base, get 5, which we multiply with 15 (
the excess number of the left hand side number minus the base i.e. 25 -
10.) So basically, 15 x 5 = 75.
3. Then we take the cube of the unit digit of the original number.
4. And doing the required calculation for the base 10, we get the

97 = (97 – 2 x 3)/(-3 x -9)/-3 x -3 x -3
= 91/27/-27
=91/26/73

Steps involved

1. First we double the excess number from the base 100 and subtract to
the original number i.e. 100 - 97 = -3. Then -3 x 2 -3 (original number)
= -9
2. Then we subtract 97 from base, get -3, which we multiply with -9 (
the excess number of the left hand side number minus the base i.e. 100 -
91.). So basically, -3 x -9 = 27.
3. Then we take the cube of 100 – 97.
4. And doing the required calculation for the base 100, we get the

In similar manner,

52 = (52 + 2 x 2)/(2 x 6)/08
= 56/12/08
= 14/06/08

104 = (100 + 3 x 4)/3 x 4 x 4/4 x 4 x 4 = 112/48/64

This is just a introduction to this method and advance method for this
has not been taught in this lesson.

Straight Division

4096/ 64

4       /  940   6
6   /  4      1 .
6    4

Steps involved

1. First separate the divisor into two parts, write the first number
down and then above it write the second number. Therefore, we write 6
down and then 4 above it.
2. Then we manually check the closest number divisible by 6 from 4096,
as we would do in our normal division, if we were dividing 4096.
3. Therefore, we divide 40 by 6; write down the quotient 6 down, and
the remainder below the next number, in this case we write 4 below 9.
4. Now, we multiply 4 into our quotient - 6. The 4 we are using is of
64 or the one which we wrote above 6.
5. This we subtract from 49 i.e. 49 – 24 = 25
6. Next we divide 25 by 6 and get our quotient 4, which we write
below, and remainder below the next number, in this case 1 below 6.
7. We again repeat the procedure. We multiply 4 with the next quotient
i.e. 4, therefore 4 x 4 and we subtract it from 16.
8. We get the answer 0. Therefore our answer is complete which is 64

996/64

4 / 9    96
6 /    3 5
1 5 .6

Or
4 / 9    9       6
6 /        3    5
1       5 36

Where 15 is quotient and 36 the remainder.

Steps involved

1. First separate the divisor into two parts, write the first number
down and then above it write the second number. Therefore, we write 6
down and then 4 above it.
2. Then we manually check the closest number divisible by 6 from 996,
as we would do in our normal division, if we were dividing 996.
3. Therefore, we divide 9 by 6; write down the quotient 1 down, and
the remainder below the next number, in this case we write 3 below 9.
4. Now, we multiply 4 into our first quotient - 1. The 4 we are using
is of 64 or the one which we wrote above 6.
5. This we subtract from 39 i.e. 39 – 4 = 35
6. Next we divide 35 by 6 and get our quotient 5, which we write
below, and remainder below the next number, in this case 5 below 6.
7. We again repeat the procedure. We multiply 4 with the next quotient
i.e. 5, therefore 4 x 5 and we subtract it from 56.
8. We get the answer 36. Now either we can leave it as it, because
this will become the remainder. Or we can divide this by 6 to get the
answer in decimal form. Therefore our answer is complete which is 15.6

5100 / 25
5 / 3 1           0 0
2/    1          2 2
1 2          4 . 0

Therefore we get the answer as 124

Steps involved

1. We first divide 3 by 2, get quotient 1, which we write down and
carry over the remainder 1 below the next number of 3100, which would be
1
2. Now we subtract 11 from 5 x 1. Get 6 which we should divide by 2
and write down the quotient as 3.
3. But we don’t do this. Reason being, if we take quotient 3 our
remainder will be 0 and we won’t be able to continue with the
calculation.
4. Therefore, when ever writing down the quotient, we should check the
next calculation should not give us a answer 0 or negative.
5. Hence, we write down 2 us our quotient and carry forward 2.
6. Now we again repeat the procedure, 20 - 5 x 2 (quotient) = 10
7. 10 we divide by 2, since again our next number should not be 00, we
take the quotient as 4 and write the remainder as 2
8. And 20 – 5 x 4 = 0
9. Therefore our answer is 124.

Remember the last number is always the remainder. If the last number is 0
that means it is perfectly divisible.

80742 / 53
3 / 8 0     7 4 2
5 /    3    2 2 3
1 5    2 3 23
Therefore our   answer is 1523 with the remainder 23

Steps involved

We first divide 8 by 5 get the quotient 1, which we write down and
carry the remainder below 0
* Next we subtract 3 (3 x 1) from 30 = 27
* Divide 27 by 5, write down the quotient 5 below and the
remainder 2 below 7.
* 27 – 15 (3 x 5) = 12
* We divide 12 by 5, write down the quotient 2 and carry
forward the remainder 2 below 4.
* 24 – 6(3 x 2) = 18
* 18 divide by 5, gives the quotient 3 and the remainder 3.
* 32 – 9 (3 x 3) = 23, which becomes our remainder.
We are going to begin with my favorite number 5 and how things get easy
with it.

Fun with Number 5
Rules: This is normally used for multiplication and in this case both the
units in digit i.e. the last numbers are 5. Also, the beginning numbers
are same. You will understand this as you see some examples.

25 x 25 = 6 | 25 (This one's easy, right?)
Explanation: As you see both the ending numbers are 5 and also they are
beginning with 2. So we have satisfied both the rules. Now how do we get
625. The first step is that you write down 25 on the right hand side. Why
we do so? Do not think about that. Just write 25 on the right hand side.
Then we take the number on the left hand side (of 25), in this case 2.
And we multiply it with the next higher digit number i.e. 3. Therefore we
get the answer 2 x 3 = 6, which we write down on the right hand side. So
we finally get the answer as 625.

45 x 45 = 20 | 25
Explanation: As this one satisfies the rule too. Both the left numbers
are same - 4, and the ending numbers are 5. So without thinking the first
thing we do is write 25 on the right hand side. Then we take the left
hand number 4 and multiply it with 5, hence we get 4 x 5 = 20 (the left

135 x 135 = 182 | 25 (slightly tough)
Explanation: Both the rule has been satisfied. We write down 25 on the
right hand side. We multiple 13 with the next higher number and get 13 x
14 = 182. And there we have our answer 18225. Easy right?

Slight Modification to the Number 5 method
Rule: The ending number should add up to 10. Infact in the above case too
the ending number 5 + 5 added upto 10. And the left hand side number
should be same. This is very similar, so lets look at some example.

23 x 27 = 6 | 21 (I am sure that,not everyone finds this simple).
Explanation: Ending numbers 3 + 7 add upto 10. So the first rule is
satisfied. And both the number on left hand side are same i.e. 2. First
thing you do is, you multiply 3 x 7 = 21. And then like earlier you
multiply 2 with the next higher digit number i.e. 3. And we get 2 x 3 = 6
on the left hand side. Final answer is 621.

62 x 68 = 42 | 16
Explanation: Both the rule have been satisfied. We multiply 8 x 2 as the
add upto 10 and we get the left hand side of the answer 16. Then we
multiply 6 with the next higher digit number 7 and get 6 x 7 = 42

Fun with number 11
Now we try the tricks which can make calculation by 11 easy. We will also
learn how to multiply by 22, 33, 44, 55, 66, etc. As it is nothing but 11
x 2, 11 x 3, 11 x 4, 11 x 5, 11 x 6 respectively.

Steps involved:

1. To multiply a number by 11 put a 0 in the beginning and at the end
of the number.
2. Starting from the digit’s place add two consecutive digit. Remember
to go from right to left.
Examples

342 x 11 = 03420 = 0+3 /4 + 3/2 + 4/0 + 2 = 3762
Explanation: We added 0 at the end and the beginning of 342 and got
03420. Now, as you see we broke the number by taking each consecutive
number and added them. The first part is 0 + 2 = 2, the first number of
our answer, then we took 2+4 and got 6, second number of our answer and
so on.

4932 x 11 = 049320 = 0 + 4/4 + 9/3 +9/3 + 2/0+2 = 54252
Explanation: In this case when we add 3 + 9 we write down 2 and carry
forward 1, we again get 4 + 9 + 1 we write down 4 and carry forward 1. We
add 4 + 1 and get 5
In the above case if we need to multiply by 22. Then we first multiply by
2 and then by 11.

312 x 22 = 624 (312 x2) x 11 = 06240 = 0 +6/6 + 2/4 + 2/0 + 4 = 6864
Explanation: Here we are trying to multiply with 22. Hence we first
multiply the main number with 312 with 2 and get 312 x 2 = 624. And then
multiply the same number with 22. Rest of the steps remain the same.

312 x 22 = 03120 x 11 x 2 = (0+3 / 3 + 1 /2 + 1 /0 + 2) x 2 = 3432 x 2 =
6864
Explanation: This is the exact same sum as above. In this case you
multiply the end answer by 2 and get the exact same answer.

Similarly we can calculate any number into any multiple of 11. Even 11 x
11. We just repeat the process twice.

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