Addition shortcuts
Addition Of Numbers Close To Multiple Of 10
In this article you are going to learn addition of those numbers which are close to Multiple of 10.
Numbers, which are close to multiple of 10, are rounded off. For example if the number is 38 then it is rounded off to 40. i.e. (40-2)
Now let’s take an example.
Method 1:
52 + 38
Here we are rounding 38 off. And writing it as (40 – 2).
So we can write
52 + (40 – 2)
Open the parenthesis.
52 + 40 – 2
Now it is easy to add 40 in 52. This addition gives us 92 as a relult.
So we can write
92 – 2
Subtracting 2 from any number is very easy.
So the answer is: 90.
Method 2:
In this method, 52 is rounded off and you can write it as (50 + 2).
And the rest of the procedure is same.
(50+2) + 38
Open the parenthesis.
50 + 2 + 38
Now adding 38 into 50 is easy and results in 88. So now we have
88 + 2
So the answer is: 90
Addition By Double Column Method
This method is used to add big numbers which are greater than two digits.
Let’s learn this method by taking an example.
Let’s say we want to add 345 and 567.
First of all we are going to divide these numbers in double column.
Then add right most column.
Here we have 45 + 67. So we can add it like 45 + (60 + 7)
This gives us 105 + 7. And finally we have 112.
But the column is of only two digits.So Left most ‘1’ is carried forward and ‘12’ is last two digits of the answer.
So now we have
Another Example
Let’s say the numbers are 1234 + 4567+ 7890 + 8765.
Divide them into double column.
This way you can add any numbers.
Multiplication shortcuts
Multiplication of Number With 11
If the number is multiplied by 11 then the following procedure is to be followed to get the answer quickest.
Let’s understand these methods by taking an example.
345 × 11
Method 1:
Write 345 as 0345
Now multiply each and every digit by 1 and add it to its right neighbor.
Start from the right side.
Step 1:
Multiply 5 with 1 and add it to its right neighbor. 5 have no right neighbor so add the multiplication to 0. That is (5 × 1) + 0 = 5. This is right most digit of our answer.
0345 × 11 = xxx5
Step 2:
Multiply 4 with 1 and add it to its right neighbor 5. That is (4 × 1) + 5 = 9. This is second right digit of the answer.
0345 × 11 = xx95
Step 3:
Multiply 3 with 1 and add it to its right neighbor 4. That is (3 ×1) + 4 =7. This is third right digit of the answer.
0345 × 11 = x795
Step 4:
Multiply 0 with 1 and add it to its right neighbor. That is (0 × 1) + 3 = 3. This is leftmost digit of the answer.
0345 × 11 = 3795
So the answer is 3795.
You can notice that all these steps can be calculated mentally.
Method 2:
Write 11 as (10 + 1)
345 × (10 + 1)
= 345 × 10 + 345 × 1
= 3450 + 345
= 3795
Multiplication Of Number With 12 To 19
If the number is multiplied by 12 to 19 then following method is to be followed.
Let’s understand this method by taking an example.
345 × 12
Step 1:
Write 345 as 0345.
Step 2:
Multiply each and every digit with 2 and add it to its right neighbor. This process is started from right. Multiply 5 with 2 and add it to its right neighbor. 5 has no right neighbor so add the multiplication to 0. This gives 10 as a result. So 0 is written down and 1 is carried forward.
Step 3:
Multiply 4 with 2 and add it to 5. Carry 1 is added to the result. This gives 14 as a result.
4 is written down and 1 is carry forwarded.
Step 4:
Multiply 3 with 2 and add it to 4. Carry 1 is added to the result. This gives 11 as a result.
1 is written down and 1 is carry forwarded.
Step 5:
Multiply 0 with 2 and add it to 3. Carry 1 is added to the result. This gives 4 as a result.
You can take this procedure as standard procedure.
Now, if the number is multiplied by 13 then multiply each and every digit with 3 instead of 2. And rest of the procedure is same.
If the number is multiplied by 14 then multiply each and every digit with 4 instead of 2. And rest of the procedure is same.
If the number is multiplied by 15 then multiply each and every digit with 5 instead of 2. And rest of the procedure is same.
If the number is multiplied by 16 then multiply each and every digit with 6 instead of 2. And rest of the procedure is same.
If the number is multiplied by 17 then multiply each and every digit with 7 instead of 2. And rest of the procedure is same.
If the number is multiplied by 18 then multiply each and every digit with 8 instead of 2. And rest of the procedure is same.
If the number is multiplied by 19 then multiply each and every digit with 9 instead of 2. And rest of the procedure is same.
Multiplication Of Two Digit Numbers
This is method for multiplying two tow digit numbers. Let us say universally that the two numbers are AB and CD.
If you want to multiply them directly then following method is to be followed.
Write both numbers as shown below.
A B
C D
Step 1: B × D
Step 2: (A × D) + (B × C)
Step 3: A × C
Let’s take an example.
67 × 56
Step 1: 7 × 6 = 42
Step 2: (6×6) + (5×7) = 71 +4(carry) = 75
Step 3: 6 × 5 = 30 + 7(carry) = 37
Multiplication Of Three Digit Numbers
This method is for multiplying two three digit number.
Let us say the numbers are ABC and DEF.
Write them as shown below.
A B C
D E F
Step 1: C × F
Step 2: (B × F) + (C ×E)
Step 3: (A × F) + (C × D) + (B × E)
Step 4: (A × E) + (B × D)
Step 5: A × D
Let’s take an example to understand this method.
789 × 123
Step 1: 3 × 9 = 27
Step 2: (8×3) + (9×2) = 42 + 2(Carry) = 44
Step 3: (7 × 3) + (1 × 9) + (8 × 2) = 46 + 4(Carry) = 50
Step 4: (7 × 2) + (8 × 1) = 22 + 5(Carry) = 27
Step 5: 7 × 1 = 7 + 2(Carry) =9
So 789 × 123 = 97047
Division Shortcut Methods
Division is most frequently used in solving competitive exams problems. By learning shortcut methods of division, you can save crucial time in the exam. Here are some shortcut methods for division.
In division, numerator is called Dividend and denominator is called Divisor.
Method 1: Division using the factors of the divisor
This method is also called Double Division.
Here you can directly divide 75 by 15 and the answer would be 5. But, to understand this method, we are not doing like that.
Now, factories divisor.
So we can write 15 as 5 × 3.
Now we can divide 75 by 3 which gives us 25.
Now we can divide 25 by 5 and that gives 5 as answer.
So the answer is 5.
Method 2: Division By Parts
Write 75 as 45 + 30
So we can write
=
So the answer is 5.
Method 3: Division by 10.
Just move the decimal point one place to the left side.
So the answer is 1.256
Method 4: Division by 100
Just move the decimal point two places to the left side.
So the answer is 0.1256
Method 5: Division by 5
Divide the dividend by 100 and multiply by 20.
So the answer is 40
Method 6: Division by 50
Divide the dividend by 100 and multiply it by 2.
So the answer is 4.
Method 7: Division by 25
Divide the dividend by 100 and multiply it by 4.
So the answer is 8
Divisibility Rules (Tests)
Divisibility Rules or Divisibility Tests are used or easily test if the number can be divided by divisor or not.
The number is said to be divisible by another if the result of division is a whole number.
Here are the rules which will make your process easy.
Divisible By: / If / Examples2 / The number is even. Or The last(units) digit of the number is 0, 2, 4, 6 or 8 / 84 is divisible by 2.
85 is not divisible by 2.
3 / The sum of the digits of number is divisible by 3. / 1248 is divisible by 3.
(1 + 2 + 4 + 8 = 15)
346 is not divisible by 3.
(3 + 4 + 6 = 13)
4 / The last two digits of the number is divisible by 4. And the numbers having two or more zeros at the end. / 23456 is divisible by 4.
(56 is divisible by 4)
13000 is divisible by 4.
(Two or more zeros at the end.)
5 / The numbers having 0 or 5 at the end. / 12345 is divisible by 5.
(5 is there at the end)
1234 is not divisible by 5.
(0 or 5 is not there at the end)
6 / The number is divisible by both 2 and 3. / 5358 is divisible by 6.
(It is divisible by both 2 and 3)
6782 is not divisible by 6.
(It is divisible by 2 but not divisible by 3)
7 / The difference between twice the units digit and the number formed by other digits is either 0 or divisible by 7. / 861 is divisible by 7.
[86 – (1 × 2)) = 84 which is divisible by 7]
21 is divisible by 7.
[2 – (1 × 2)) = 0]
868 is divisible by 7.
[86 – (8 × 2)) = 70 which is divisible by 7]
8 / The number formed by last three digits is divisible by 8. And the numbers having three or more zeros at the end. / 2056 is divisible by 8.
(056 is divisible by 8)
13000 is divisible by 8.
(Three zeros at last)
9 / The sum of all the digits is divisible by 9. / 5301 is divisible by 9.
(5 + 3 + 0 + 1 = 9 which is divisible by 9)
10 / The number ends with zero. / 467590 is divisible by 10.
(It ends with zero)
11 / The difference between the sum of digits at even places and sum of digits at odd places is 0 or divisible by 11. / 10538 is divisible by 11.
[(1 + 5 + 8) – (0 + 3) = 11 which is divisible by 11]
724867 is divisible by 11.
[(7 + 4 + 6) – (2 + 8 + 7) = 0]
12 / The number is divisible by both 3 and 4. / 5472 is divisible by 12.
( The number is divisible by both 3 and 4)
5475 is not divisible by 12.
(The number is divisible by 3 but not divisible by 4)
13 / Method: Multiply last digit of the number by 4 and add it to the remaining number. Continue this process until two digit number is achieved. If this two digit number is divisible by 13 then the number is divisible by 13. / 182 is divisible by 13.
(Multiply last digit by 4 i.e 2×4 = 8.
Add it to the remaining number i.e 18 + 8 = 26.
26 is divisible by 13 so 182 is divisible by 13)
2145 is divisible by 13
( Multiply last digit by 4 i.e 5×4 = 20.
Add it to the remaining number i.e 214 + 20 = 234 which is not a two digit number so repeat the process.
Multiply last digit of 234 by 4 i.e. 4×4= 16.
Add it to the remaining number i.e 23 + 16 = 39 which is divisible by 13 so 2145 is divisible by 13)
14 / The number is divisible by both 2 and 7. / 7966 is divisible by 14.
(The number is divisible by both 2 and 7)
15 / The number is divisible by both 3 and 5. / 3525 is divisible by 15.
(The number is divisible by both 3 and 5)
16 / The number formed by last four digits is divisible by 16. / 41104 is divisible by 16.
(The number formed by last four digits is divisible by 16)
17 / Method: Multiply last digit with 5 and subtract it from the remaining number. If the result is divisible by 17 then the original number is also divisible by 17. Repeat this process if required. / 4029 is divisible by 17
( Multiply last digit by 5 i.e. 9×5= 45.
Subtract it from the remaining number.
402 – 45 = 357 which is divisible by 17 so 4029 is divisible by 17)
18 / The number is even and divisible by 9. / 4428 is divisible by 18.
(It is even and divisible by 9)
19 / Method: Multiply last digit with 2 and add it to the remaining number. If the result is divisible by 19 then original number is also divisible by 19. Repeat this process if required. / 1235 is divisible by 19.
(Multiply last digit with 2 i.e 5×2 = 10.
Add it to the remaining number 123 i.e 123 + 10 = 133 which is divisible by 19 so 1235 is also divisible by 19)
Squares Shortcut Methods
General Shortcut Method For Finding Squares
You can find square of any number in the world with this method.
Let’s say the number is two digit number. i.e. AB.
So B is units digit and A is tens digit.
Step 1: Find Square of B
Step 2: Find 2×A×B
Step 3: Find Square of A
Let’s take an example.
We want to find square of 37.
Step 1: Find square of 7.
Square of 7 = 49.
So write 9 in the answer and 4 as carry to the second step.
Step 2: Find 2×(3×7)
2 × (3 × 7) = 42.
42 + 4(Carry) = 46.
Write 6 in the answer and 4 as a carry to the third step.
Step 3: Find square of 3
Square of 3 = 9
9 + 4(Carry) = 13.
Write 13 in the answer.
So the answer is 1369.
Now, If the number is of three digit i.e. ABC
Here C is unit’s digit, B is ten’s digit and A is hundredth digit.
Step 1: Find Square of C
Step 2: Find 2 × (B × C)
Step 3: Find 2 × (A × C) + B2
(NOTE: You may observe that in odd number of digit case, we are multiplying end two digits with 2 (here: A and C) and squaring single digit (here B).
Step 4: Find 2 × (A × B)
(NOTE: You may observe that whenever there are double digits, we are multiplying it with 2. And whenever there is single digit, we are squaring it.)
Step 5: Find square of A
(NOTE: Here is single digit, so we are squaring it.)
Let’s take an example.
Find square of 456.
Step 1: Find square of 6.
Square of 6 = 36.
So write 6 in the answer and 3 as a carry to the second step.
Step 2: Find 2 × (5 × 6)
2 × (5 × 6) = 60
60 + 3(Carry) = 63
Write 3 in the answer and 6 as a carry to the third step.
Step 3: Find 2 × (4 × 6) + 52
2 × (4 × 6) + 52 = 73
73 + 6(Carry) = 79
Write 9 in the answer and 7 as a carry to the fourth step.
Step 4: Find 2 × (4 × 5)
2 × (4 × 5) = 40
40 + 7 = 47
Write 7 in the answer and 4 as a carry to the fifth step.
Step 5: Find square of 4
Square of 4 = 16
16 + 4(Carry) = 20
Write 20 in the answer.
So 4562 = 207936.
Now, If the number is of four digit i.e. ABCD
Here D is unit’s digit, C is ten’s digit, B is hundredth digit and A is thousands digit.
Step 1: Find Square of D
Step 2: Find 2 × (C × D)
Step 3: Find 2 × (B × D) + C2
(NOTE: You may observe that in odd number of digit case, we are multiplying end two digits with 2 (here: B and D) and squaring single remaining digit (here C).
Step 4: Find 2 × (A × D) + 2 × (B × C)
(NOTE: You may observe that where ever there is even digits, we are multiplying end two digits with 2 + remaining two digits with 2.)
Step 5: Find 2 × (A × C) + B2
(NOTE: You may observe that in odd number of digit case, we are multiplying end two digits with 2 (here: A and C) and squaring single remaining digit (here B).
Step 6: Find 2 × (A × B)
(NOTE: You may observe that here even digits so we are multiplying them with 2, and no remaining digits so we are not adding anything.)
Step 7: Find square of A
Let’s take an example.
Find square of 1234
Step 1: Find Square of 4
Square of 4 = 16
So write 6 in the answer and 1 as a carry to the second step.
Step 2: Find 2 × (3 × 4)
2 × (3 × 4) = 24
24 + 1(Carry) = 25
Write 5 in the answer and 2 as a carry to the third step
Step 3: Find 2 × (2 × 4) + 32
2 × (2 × 4) + 32 = 25
25 + 2(Carry) = 27
Write 7 in the answer and 2 as a carry to the fourth step.
Step 4: Find 2 × (1 × 4) + 2 × (2 × 3)
2 × (1 × 4) + 2 × (2 × 3) = 20
20 + 2(Carry) = 22
Write 2 in the answer and 2 as a carry to the fifth step.
Step 5: Find 2 × (1 × 3) + 22
2 × (1 × 3) + 22 = 10
10 + 2(Carry) = 12
Write 2 in the answer and 1 as carry to the sixth step.
Step 6: Find 2 × (1 × 2)
2 × (1 × 2) = 4
4 + 1(Carry) = 5
Write 5 in the answer
Step 7: Find square of 1
Square of 1 = 1
There is no carry so write 1 in the answer.
So, 12342 = 1522756
Square Of The Numbers With Unit Digit As 5
This method is to find square of the numbers which has unit’s digit as 5. i.e.: 25, 45, 65, etc.
You can find square of these numbers by three easy steps.
Step 1: Multiply ten’s digit with its next number.
Step 2: Find square of unit’s digit. i.e.: Square of 5.
Step 3: Write answers of step 1 and step 2 to together or side by side.
Let’s take examples.
Find square of 35
Step 1: Multiply ten’s digit with its next number.
3 × ( 3 + 1 ) = 3 × 4 = 12
Step 2: Find square of unit’s digit. i.e.: Square of 5.
Square of 5 = 25
Step 3: Write answers of step 1 and step 2 to together.
Answer = 1225
Find square of 65
Step 1: Multiply ten’s digit with its next number.
6 × ( 6 + 1 ) = 6 × 7 = 42
Step 2: Find square of unit’s digit. i.e.: Square of 5.
Square of 5 = 25
Step 3: Write answers of step 1 and step 2 to together.
Answer = 4225
Find square of 95
Step 1: Multiply ten’s digit with its next number.
9 × ( 9 + 1 ) = 9 × 10 = 90
Step 2: Find square of unit’s digit. i.e.: Square of 5.
Square of 5 = 25
Step 3: Write answers of step 1 and step 2 to together.
Answer = 9025
Find square of 115
Step 1: Multiply ten’s digit with its next number.
11 × ( 11 + 1 ) = 11 × 12 = 132
(Note: We are taking whole 11 as a ten’s digit.)
Step 2: Find square of unit’s digit. i.e.: Square of 5.
Square of 5 = 25
Step 3: Write answers of step 1 and step 2 to together.
Answer = 13225
Find square of 215
Step 1: Multiply ten’s digit with its next number.
21 × ( 21 + 1 ) = 21 × 22 = 462
(Note: We are taking whole 21 as a ten’s digit.)
Step 2: Find square of unit’s digit. i.e.: Square of 5.
Square of 5 = 25
Step 3: Write answers of step 1 and step 2 to together.
Answer = 46225
Square Of The Numbers In 50s
This method is used to find square of the numbers in 50s i.e. numbers from 51 to 59.
You can find square of these numbers in three simple steps.
Step 1: Add 25 to the unit’s digit.
Step 2: Square the unit’s digit.
Step 3: Write the answers of step 1 and step 2 together or side by side.
Let’s take an example.
Square of 56
Step 1: Add 25 to the unit’s digit
6 + 25 = 31
Step 2: Square the unit’s digit
62 = 36
Step 3: Write the answers of step 1 and step 2 together.
Answer = 3136
Square of 59
Step 1: Add 25 to the unit’s digit
9 + 25 = 34
Step 2: Square the unit’s digit
92 = 81
Step 3: Write the answers of step 1 and step 2 together.
Answer = 3481
Square of 53.
Step 1: Add 25 to the unit’s digit.
3 + 25 = 28
Step 2: Square the unit’s digit.
32 = 9
Step 3: Write answers of step 1 and step 2 together.
Answer: 2809
(NOTE: Whenever square of unit’s digit is on only single digit then we are adding 0 before it.)
Square of 52
Step 1: Add 25 to the unit’s digit
2 + 25 = 27
Step 2: Square the unit’s digit
22 = 4
Step 3: Write the answers of step 1 and step 2 together.
Answer = 2704
(NOTE: Whenever square of unit’s digit is on only single digit then we are adding 0 before it.)
Square The Number If You Know Square Of Previous Number
This method is to find square of the number if you know square of the previous number.
You can find answers in three simple steps.
Step 1: Find square of the previous number which is known.
Step 2: Multiply the number being squared by 2 and subtract 1.
Step 3: Add Step 1 and Step 2
Let’s take some examples.
Find square of 31.
Step 1:Find square of previous number (30) which is known.
302 = 900
Step 2: Multiply the number being squared (31) by 2 and subtract 1.
(31 × 2) – 1 = 62 – 1 = 61
Step 3: Add Step 1 and Step 2
900 + 61 = 961
Find square of 26.
Step 1:Find square of previous number (25) which is known.
252 = 625
Step 2: Multiply the number being squared (31) by 2 and subtract 1.
(26 × 2) – 1 = 52 – 1 = 51
Step 3: Add Step 1 and Step 2
625 + 51 = 676
Find square of 81.
Step 1:Find square of previous number (80) which is known.
802 = 6400
Step 2: Multiply the number being squared (31) by 2 and subtract 1.
(81 × 2) – 1 =162 – 1 = 161
Step 3: Add Step 1 and Step 2
6400 + 161 = 6561
Shortcut Method For Finding Cube
This method is to find cube of two digit numbers.
Let’s understand this method by taking some examples.
Find cube of 14.
Step 1: Find cube of tens digit and write it down as first digit .
Step 2: 1 and 4 are in the ratio of 1:4 So write next three numbers in the ratio of 1:4.
Step 3: Write double of second and third digit below them.
Step 4: Add both these rows as shown below.
- Starting from right, write 4 of 64 in the answer and 6 as carry.
- Add 6 + 16 + 32 = 54.
Write 4 in the answer and 5 as carry.
- Add 5 + 4 + 8 = 17
Write 7 in the answer and 1 as carry.
- Add 1 + 1 = 2
Write 2 in the answer.
Answer is 2744.
Find cube of 48.
Step 1: Find cube of tens digit and write it down as first digit .
Step 2: 4 and 8 are in the ratio of 1:2 So write next three numbers in the ratio of 1:2.
Step 3: Write double of second and third digit below them.
Step 4: Add both these rows as shown below.
- Starting from right, write 2 of 512 in the answer and 51 as carry.
- Add 51 + 256 + 512 = 819.
Write 9 in the answer and 81 as carry.