ARITHMETIC + – * /

Add, Multiply & Divide with Ease

You need to look at numbers from 1 to 100 first

Make an attempt to learn the half and double of each of these numbers. Practice. Write down in an excel file and print it out and paste it on the wall and read it once in a while while you pass by the wall. Practice. 

It will be to your great advantage if you can find the 100 complement of each of these numbers. Again an excel file will help. 

Further we will find easy ways to learn the tables, multiply any two 2- digit numbers and square any two digit number with ease.

We will also learn easy ways to multiply and divide any number with 5 and 25

​Finally we will learn a very easy method to divide any number with a 2-digit number e.g. 99795 / 79

PRIME NUMBERS

Any number that can be divided only by 1 and itself is a Prime Number. It starts with 2 which is the only Even Prime Number. 1 is not considered Prime Number. 

So the Prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 

There are certain odd numbers between 1 & 100 which seem like Prime numbers. It is useful to learn their factors and familiarize oneself with them. Let us look at some of them and their factors

39 = 3×13
49 = 7×7
57 = 19×3
69 = 3×23
87 = 3×29

ADDITION

Add 10 packets first. For this you need to know all the 10 packets viz: 
1+9, 2+8, 3+7, 4+6, 5+5

You also need to learn the addition tables of all single digit numbers. As we did with 10 above let us reckon all the combinations possible with 2 digit numbers

11 comprises 2+9; 3+8; 4+7; 5+6

12: 3+9, 4+8, 5+7, 6+6

13: 4+9, 5+8, 6+7

14: 5+9, 6+8, 7+7

15: 6+9, 7+8

16: 7+9, 8+8

17: 8+9

18: 9+9

Add two 2-digit numbers
first the tens then the units e.g. 78+23: 90, 11, 101
or 58+69: 110, 17, 127
69+77=130, 16, 146
This is very very simple and must be learnt. 
you can even extend to three digit numbers 

137+233= 300, 60, 10, 370
89+923= 900, 100, 12, 1012
67+767= 700, 120, 14, 834


SQUARING TWO DIGIT NUMBERS
The simplest numbers to square: those ending with 5

Squaring Any two digit number

As shown above, this method can be used to square any two digit number
Step 1: Write down the squares of the two digits
Step 2: Write down the product of the two digits x 2
Step 3: Add the two lines and voila, you have the square

Numbers around 50 or 100: Square them easily
The method is add the distance from the boundary here the difference from 100, which is the boundary and add the same difference again and append the square of the difference next. The examples above show you the way. 

Since here the numbers are below the boundary, we reduce further by the same difference and append the square of the difference. Simple. 

Picture

Below we show squaring numbers around 50 and since 50 squared gives us 2500, the difference from the boundary we have to add to 25 and as usual append the square of the difference from the boundary


Below we subtract the boundary difference from 25 and append its square at the end. Follow along and practice. 
MULTIPLYING
WITH 5 & 25

x5
divide the number by 2 and append zero. This is because 5 is half of 10. 
28×5=28/2: 14, append zero, so 140

when the last digit is odd,
append 5 instead of 0
e.g. 39×5=39/2: 19 rem 1, append 5: 195

x25
divide the number by 4 and append two zeroes because 100/4 = 25

so, 28×25=28/4: 700

when it is an odd number, notice the remainder: it will be 1 or 2 or 3
Accordingly append 25 or 50 or 75
and you are done

39×25: 39/4: 9 rem 3: 975 is the answer
or 147×25: 147/4: 36 rem 3, so 3675
44×25: 1100
69×25: 1725

DIVISIBILITY RULES

5 is the easiest: Any number ending in 5 or 0
2: any number ending with multiple of 2
4: last two digit of the no. divisible by 4
8: last three digits divisible by 8
3 & 9: sum of the digits divisible by 3 or 9
6: divisible by both 2 and 3

Divisibility Rules for 7, 11, 13 
Step 1: Write down the number as clusters of 3 digits starting from the right, e.g.

18’053

Step 2: Now subtract the left cluster from the right one 
i.e. bring down 18 below 053 & subtract
the result is 35 which is divisible by 7. Thus the original no. IS divisible by 7. 

Let us take another example
234’689
bring down 234 below 689
 689
-234
689-234=455 
455 is divisible by 7 as well as by 13
so 234689 is divisible both by 7 & 13

Simple, isn’t it ?

TABLES

Picture

2 digit no. x 1 digit no.
say 19×7: first do 10×7=70, then 9×7=63 & add: 133
or 17×8: 80, 56, 136 is the answer: just say these three numbers. let us try another
45×9: 360, 45,405 or 23×9: 180,27,207

Tables from 11×11 to 19×19
(1) Add unit digit of one with the other number
(2) Append a 0 at the end i.e. multiply with 10
(3)
Multiply the unit digits
(4) Add to (1)
e.g.14×17: 14+7: 21: 210; 28; 238, 

16×19: 250,54,304 
13×15: 180,15,195

MULTIPLY
2 digit number x 2 digit number

this is an extremely useful rule
the rule is IXI. See the diagram below 


Step 1: 7×9…write the carry over down
Step 2: do the cross i.e. 2×9 + 7×3
Step 3: dont forget to add the carry, 6
Step 4: 2×3
and voila, you have the answer: 1053

1: do 8×3
2: do the cross: write it on the side
3: do 9×2
and there you have it

Please practice a bit. This is extremely useful and easy to use


BOUNDARY DIVISION
a fascinating method to divide easily with any two digit no.

Above we are dividing by 30 and that is what makes it very very easy. The difference with 29 is 1, so we need to adjust by multiplying the quotient each time with the difference, 1 as shown by the RED DIGITS. Follow along & it should be very very easy. 

Above, at one point while adding the product of the quotient & the difference, 2, divisor becomes more than 30, so we divide it again and show it as a RED digit 1, below the earlier quotient. And finally add it in. 
In this example we do not show all the steps. As we multiply with 40 to get the quotient, the result we don’t write, only the difference.

Divide with 5 & 25

Divide with 5
Multiply with 2 and shift the point by 1place leftwards for dividing with 10

66/5: 132: 13.2
69/5: 138, so 13.8

Divide with 25
Multiply by 4 and shift the decimal point leftwards by 2places since 25=100/4, so dividing by 25 is tantamount to multiplying by 4/100

e.g. 78/25:
first multiply by 4: 280, 32, 312,
now shift the point, so 3.12 is the answer

or 565/25:
first multiply by 4: 
2000, 240, 20, 2260,
then shift leftwards by two points
so 22.6 is the answer

Vignettes from Nature