Math - Aptitude Tricks Part 1

Aptitude Tricks - PART- 6

LCM and HCF Tricks

About LCM and HCF

LCM i.e. least common multiple is a number which is multiple of two or more than two numbers. For example: The common multiples of 3 and 4 are 12,24 and so on.

Therefore, l.c.m.is smallest positive number that is multiple of both. Here, l.c.m. is 12.. HCF i.e. highest common factor are those integral values of number that can divide that number. LCM and HCF problems are very important part of all competitive exams.

Some important l.c.m. and h.c.f. tricks

1) Product of two numbers = Their h.c.f. * Their l.c.m.

2) h.c.f. of given numbers always divides their l.c.m.

3) h.c.f. of given fractions =     h.c.f. of numerator / l.c.m. of denominator
                                           
4) l.c.m. of given fractions =    l.c.m. of numerator  / h.c.f. of denominator
                                         
5) If d is the h.c.f. of two positive integer a and b, then there exist unique integer m and n, such that
     d = am + bn

6) If p is prime and a,b are any integer then P / ab ,This implies   P / a or  P / b
                                                                                         
7) h.c.f. of a given number always divides its l.c.m.

Most important points about l.c.m. and h.c.f. problems

1) Largest number which divides x,y,z to leave same remainder = h.c.f. of y-x, z-y, z-x.

2) Largest number which divides x,y,z to leave remainder R (i.e. same) = h.c.f of x-R, y-R, z-R.

3) Largest number which divides x,y,z to leave same remainder a,b,c  = h.c.f. of x-a, y-b, z-c.

4) Least number which when divided by x,y,z and leaves a remainder R in each case = ( l.c.m. of x,y,z) + R

HCF and LCM questions

Problem 1: Least number which when divided by 35,45,55 and leaves remainder 18,28,38; is?

Solution:

i) In this case we will evaluate l.c.m.
ii) Here the difference between every divisor and remainder is same i.e. 17.
Therefore, required number = l.c.m. of (35,45,55)-17 = (3465-17)= 3448.

Problem 2: Least number which when divided by 5,6,7,8 and leaves remainder 3, but when divided by 9, leaves no remainder?

Solution:

l.c.m. of 5,6,7,8 = 840
Required number = 840 k + 3
Least value of k for which (840 k + 3) is divided by 9 is 2
Therefore, required number = 840*2 + 3
                                            = 1683

Problem 3: Greater number of 4 digits which is divisible by each one of 12,18,21 and 28 is?

Solution:

l.c.m. of 12,18,21,28 = 254
Therefore, required number must be divisible by 254.
Greatest four digit number = 9999
On dividing 9999 by 252, remainder = 171
Therefore, 9999-171 = 9828.

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Number Series Methods Tricks

Important info

Number Series shortcut tricks are very important thing to know for your exams. Time takes a huge part in competitive exams. If you manage your time then you can do well in those exams. Most of us miss that part. Few examples on number series shortcuts is given in this page below. These shortcut tricks cover all sorts of tricks on Number Series. We request all visitors to read all examples carefully. These examples will help you to understand shortcut tricks on Number Series.

First of all do a practice set on math of any exam. Choose any twenty math problems and write it down on a page. Solve first ten math problems according to basic math formula. You also need to keep track of timing. After solving all ten math questions write down total time taken by you to solve those questions. Now practice our shortcut tricks on number series and read examples carefully. After finishing this do remaining questions using Number Series shortcut tricks. Again keep track of the time. The timing will be surely improved this time. But this is not enough. If you need to improve your timing more then you need to practice more.

Math section in a competitive exam is the most important part of the exam. It doesn’t mean that other topics are not so important. But if you need a good score in exam then you have to score good in maths. You can get good score only by practicing more and more. All you need to do is to do math problems correctly within time, and only shortcut tricks can give you that success. But it doesn’t mean that you can’t do math problems without using any shortcut tricks. You may have that potential to do maths within time without using any shortcut tricks. But other peoples may not do the same. For those we prepared this number series shortcut tricks. Here in this page we try to put all types of shortcut tricks on Number Series. But it possible we miss any. We appreciate if you share that with us. Your little help will help so many need .

What is Number Series ?

Number series is a form of numbers in a certain sequence, where some numbers are mistakenly put into the series of numbers and some number is missing in that series, we need to observe first and then find the accurate number to that series of numbers.

Anything we learn in our school days was basics and that is well enough for passing our school exams. Now the time has come to learn for our competitive exams. For this we need our basics but also we have to learn something new. That’s where shortcut tricks are comes into action.

In competitive exams number series are given and where you need to find missing numbers and mistakenly put into the series numbers. The number series are come in different types. At first you have to decided what type of series are given in papers then according with this you have to use shortcut tricks as fast as you can.

Perfect Square Series

Perfect square series is a arrangement of numbers in a certain order, where some numbers Series are based on square of a number which is in same order you need to place one square number that is missing in that given series, we need to observe and find the accurate number to the series of numbers.
This type of problem are given in Quantitative Aptitude which is a very essential in banking exam. It is simple to work on perfect square root numbers you can easily obtain the result of perfect square number. How you get easily by Perfect square numbers missing term by memorize square and square root numbers shortcut tricks .
The square of same number and the square result of a number which is equal to the square of another same element. In mathematical world, a square number or perfect square is number of an integer positive integer that is the square of an same integer number always and the numbers are non-negative.
In other words, we say it is the result of product of multiplication of some positive integer numbers with itself always. For example, we consider 4 is a result of square numbers, since it as 2 × 2 in normal way.
The normal representation of square numbers is n² and that is similar with products of n × n, but it is similar with exponentiation of n² ,
In Square numbers are positive number. So we can explain it that a positive number is a square number, where its square roots are always integers positive numbers. so For example, √4 = ±2, so 4 is a square number.

Perfect Square Series
Here we see the some examples that how the perfect square are arranged how the missing square series are arranged.
Example 1: 441, 484, 529, 576, ?,
Answer: 441 = 21², 484 = 22², 529 = 23², 576 = 24² ,625 = 25².
Example 2: 121, 144, 169, ?, 225
Answer: 121 = 11², 144 = 12², 169 = 13², 196 = 14², 225 = 15².
Example 3: ?, 2116, 2209, 2304, 2401, 2500
Answer: 2025 = 45², 2116 = 46², 2304 = 48², 2401 = 49², 2500 = 50²

Example 4: 961, 1024, ?, 1156, 1225
Answer: 961 = 31², 1024= 32², 1089 = 33², 1156 = 34², 1225 = 35².
Example 5: 36, ?, 64, 81, 100, 121
Answer: 36 = 6², 49 = 7², 64 = 8², 81 = 9², 100 = 10², 121 = 11².
Example 6 : 121 , 169 , ? , 289 , 361
Answer : 11² = 121 , 13² = 169 , 15² = 225 , 17² = 289 , 19² = 361.
Example 7 : 121 , 484 , 1089 , 1936 , ? , 4356
Answer : 11² = 121 , 22² = 484 , 33² = 1089 , 44² = 1936 , 55² = 3025 , 66² = 4356.
Example 8 : 961 , 1024 , 1089 , ? 1225
Answer : 31² , 32² , 33² , 34² , 35²
Example 9 : 1849 , ? , 2025 , 2116 , 2209
Answer : 43² , 44² , 45² , 46² , 47²
Example 10 : 2500 , 2401 , 2304 , ? , 2116 , 2025
Answer : 50² , 49² , 48² , 47² , 46² , 45²

Perfect Cube Series

Perfect cube series is a arrangement of numbers in a certain order,where some numbers this Types of Series are based on cube of a number which is in same order and one cube number is missing in that given series.we need to observe and find the accurate number to the series of numbers. This type of problem are given in Quantitative Aptitude which is a very essential paper in banking exam.Under below given some more example for your better practice.
All numbers are arranged in sequent order. we need to observe and find the accurate number to this type series of numbers. Here we learn the perfect cube series of Example.
This type of problem are given in Quantitative Aptitude which is a very essential in banking exam. Under below given some more example for your better practice.
In perfect cube series number is a combination of cube number are arranged. In example 1) 1331, 1728, 2197, ? where you need to count them in a one step or two step calculation for obtain the difference common result according with the series of ratio numbers .
At first you can calculate missing number in ratio series and that you place the actual missing number in the ? or missing place. Be prepared when you calculate differences because it is either one or two step calculation. So when you calculate and get result of two difference numbers you need follow some step wise.
At first calculate the first number cube value and second number cube value if all number are maintain a sequential order cube value then follow same steps which is carry up to last and after that you get actual missing number by finding the common value when you put the missing number you have noticed that all series numbers are common difference in between them.
This kind of missing series calculation you go thorough some common calculation shortcut tricks using cube and cube shortcut tricks, or you memorize the 1 to 30 cube series number value.
In this type series example questions, it is sounds hard, but it really isn’t. Get it? Once you have done this, by practice with more example then you just easily can do in your way as well competitive and as in bank exam also . So, each of our examples are given below.
Perfect Cube Series:
Example 1 : 1331 , ? , 35937 , 85184 , 166375
Answer : 11³ , 22³ , 33³ , 44³ , 55³
Example 2 : 125, ?, 343, 512, 729, 1000
Answer : 125 = 5³ , 216 = 6³, 343 = 7³, 512 = 8³, 729 = 9³, 1000 = 10³.
Example 3 : 1 , 9 , 125 , 343 , ? , 729
Answer : 1³ , 3³ , 5³ , 7³ , 8³ , 9³
Example 4 : 125, ?, 343, 512, 729, 1000
Answer: 125 = 5³, 216 = 6³, 343 = 7³, 512 = 8³, 729 = 9³, 1000 = 10³.
Example 5 : 8 , 64 , ? , 512 , 1000 , 1728
Answer : 2³ , 4³ , 6³ , 8³ , 10³ , 12³
Example 6 : 4096, 4913, 5832, ?, 8000
Answer: 4096 = 16³, 4913 = 17³, 5832 = 18³, 6859 = 19³, 8000 = 20³.
Example 7 : 1331 , ? , 29791 , 68921 132651
Answer : 11³ , 21³ , 31³ , 41³ , 51³
Example 8 : 1331, 1728, 2197, ?
Answer: 1331 = 11³, 1728 = 12³, 2197 = 13³, 2744 = 14³.
Example 9: 1728, 1331, ?, 729, 512
Answer: 1728 = 12³, 1331 = 11³, 1000 = 10³, 729 = 9³, 512 = 8³.
Example 10 : 1000 , 8000 , ? ,64000 , 125000
Answer : 10³ , 20³ , 30³ , 40³ , 50³
Example 11 : 125000 , 64000 , ? , 8000 , 1000
Answer : 50³ , 40³ , 30³ , 20³ , 10³

Ratio and Proportion Methods shortcut tricks

You all know that math portion is very much important in competitive exams. That doesn’t mean that other sections are not so important. But only math portion can leads you to a good score. A good score comes with practice and practice. All you need to do is to do math problems correctly within time, and only shortcut tricks can give you that success. But it doesn’t mean that without using shortcut tricks you can’t do any math problems. You may have that potential that you may do maths within time without using any shortcut tricks. But so many people can’t do this. Here we prepared ratio and proportion shortcut tricks for those people. Here in this page we try to put all types of shortcut tricks on Ratio and Proportion. But we may miss few of them. If you know anything else rather than this please do share with us. Your little help will help so many needy.

• What is Ratio?
  A ratio is a relationship between two numbers by division of the same kind. The ration of a to b is written as a : b = a / b, In ratio a : b, we can say that a as the first term or antecedent and b the second term or consequent.

Example : The ratio 4 : 9 we can represent as 4 / 9 after this 4 is a antecedent and, consequent = 9

• Rule of ration : In ratio multiplication or division of each an every term of a ratio by the same non- zero number does not affect the ratio.

Different type of ratio problem are given in Quantitative Aptitude which is a very essential topic in banking exam. Under below given some more example for your better practice.
Anything we learn in our school days was basics and that is well enough for passing our school exams. Now the time has come to learn for our competitive exams. For this we need our basics but also we have to learn something new. That’s where shortcut tricks and formula are comes into action.

• What is Proportion?
  The idea of proportions is that two ratios are like equal.
  If a : b = c : d, we write a : b : : c : d,
  Ex. 3 / 15 = 1 / 5
  a and d called extremes, where as b and c called mean terms.

• Proportion of quantities
  the four quantities like a, b, c, d we can say proportion then we can express it
  a : b = c : d
  Then a : b : : c : d <–> ( a x d ) = ( b x c )
  product of means = product of extremes.
  
  If there is given three quantities like a, d, c of same like then we can say it proportion of continued.
  a : d = d : c , d is called mean term. a and c are called extremes.

Geometric Series

Examples 1: 5, 45, 405, 3645, ?
Answer: 5 x 9 = 45, 45 x 9 = 405, 405 x 9 = 3645, 3645 x 9 = 32805.
Examples 2: 73205, 6655, 605, 55, ?
Answer: 5 x 11 = 55, 55 x 11 = 605, 605 x 11 = 6655, 6655 x 11 = 73205.

Examples 3: 25, 100, ?, 1600, 6400
Answer: 25 x 4 = 100, 100 x 4 = 400, 400 x 4 = 1600, 1600 x 4 = 6400.

Examples 4: 9, 54, ?, 1944, 11664
Answer: 9 x 6 = 54, 54 x 6 = 324, 324 x 6 = 1944, 1944 x 6 = 11664.

Mixed Series

Examples 1:

111, 220, 438, ?, 1746
Answer:
from 111 to 220 we get using this 111 x 2 = 222 – 2 = 220,similarly we follow next steps
from 220 to 438 we get using this 220 x 2 = 440 – 2 = 438,
from 438 to ? we get using this 438 x 2 = 876 – 2 = 874,
from 874 to 1746 we get using this 874 x 2 = 1748 – 2 = 1746.
So the missing number is 874

Examples 2:

24, ?, 208, 622, 1864
Answer:
from 24 to ? we get using this 24 x 3 = 72 – 2 = 70, Similarly we follow next steps
from 70 to 208 we get using this 70 x 3 = 210 – 2 = 208,
from 208 to 622 we get using this 208 x 3 = 624 – 2= 622,
from 622 to 1864 we get using this 622 x 3 = 1866 – 2 = 1864.
So the missing number is 70

Examples 3:

11, 24, 50, 102, 206, ?
Answer:
11 x 2 = 22 +2 = 24,
24 x 2 = 48 + 2 = 50,
50 x 2 = 100 + 2 = 102,
102 x 2 = 204 + 2 = 206,
206 x 2 = 412 + 2 = 414.
So the missing number is 414.

Example 4:

0, 6, 24, 60, 120, 210, ?
Answer :The given series is : 1³ – 1, 2³ – 2, 3³ – 3, 4³ – 4, 5³ – 5, 6³ – 6,
So the missing term = 7³ – 7 = 343 – 7 = 336 .

Example 5:

11, 14, 19, 22, 27, 30, ?
Answer :
The pattern is + 3, + 5, + 3, + 5, …………
So the missing term is = 30 + 5 = 35 .

Example 6:

6, 12, 21, ? , 48
Answer :
The pattern is + 6, + 9, + 12, +15 ………..
So the missing term is = 21 + 12 = 33 .

Example 7:

18, 22, 30, ? ,78, 142
Answer :
The pattern is +4, +8, +16, +32, +64
So the missing term is = 30 + 16 = 46 .

Example 8:

589245773, 89245773, 8924577, 924577, ?
Answer :
The pattern is The digits are removed one by one from the beginning and the end in order alternately, So to obtain the subsequent terms of the missing series is = 92457 .

Example 9:

8, 35, ? , 143, 224, 323
Answer :
The pattern is (3² – 1), (6² – 1),………., (12² – 1), (15² – 1), (18² – 1)
So the missing term is = (9² – 1 ) = 81 – 1 = 80 .

Example 10:

3, 7, 23, 95, ?
Answer :
The pattern is ( x 2 + 1 ),( x 3 + 2) , ( x 4 + 3 ) , ……….
So the missing term is = 95 x 5 + 4 = 479 .

Cube and Cube Root Shortcut Tricks - Basic Math Tricks Part 6

Basic Tricks - PART- 6

Cube & Cube Root Shortcut Tricks

CUBE up to 30

13=1	113=1331	213=9261
23=8	123=1728	223=10648
33=27	133=2197	233=12167
43=64	143=2744	243=13824
53=125	153=3375	253=15625
63=216	163=4096	263=17576
73=343	173=4913	273=19683
83=512	183=5832	283=21952
93=729	193=6859	293=24389
103=1000	203=8000	303=27000

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Find Five digit Cube and cube root Tricks

Find Five digit Cube and cube root Tricks

Need to remember 1 to 10 cube and this so easy for any one.Which will help in obtaining cube and cube root numbers.
1³=1
2³=8
3³=27
4³=64
5³=125
6³=216
7³=343
8³=512
9³=729
10³=1000

Example 1:
³√13824 = ?
Answer :
Step 1: Last digit of cube number from right side is 4 that we consider 64 = 4³ we put down 4. Then
Step 2: Take the number whose cube is nearest to 13.
That is 13 is nearest to 2³ and 3³ we take small one cube digit that is 2.³√13824
So the answer is 24.

Example 2:

³√15625= ?
Answer :
Step 1: Last digit of cube number from right side is 5 that we consider 125 = 5³ we put down 5. Then
Step 2: Take the number whose cube is nearest to 15.That is 15 is nearest to 2³ and 3³ we take small one cube digit that is 2.
So the answer is 25.

Example 3:

³√42875= ?
Answer :
Step 1: Last digit of cube number from right side is 5 that we consider 125 = 5³ we put down 5. Then
Step 2: Take the number whose cube is nearest to 42.That is 42 is nearest to 3³ and 4³ we take small one cube digit that is 3.
So the answer is 35.

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Find Six digit Cube and cube root Tricks

Find Six digit Cube and cube root Tricks

Need to remember 1 to 10 cube and this so easy for any one.Which will help in obtaining cube and cube root numbers.
1³=1
2³=8
3³=27
4³=64
5³=125
6³=216
7³=343
8³=512
9³=729
10³=1000

Example 1:

³√166375 = ?
Answer :
Step 1: Last digit of cube number from right side is 5 that we consider 125 = 5³ we put down 5. Then
Step 2: Take the number whose cube is nearest to 166.That is 166 is nearest to 5³ and 6³ we take small one cube digit that is 5.
So the answer is 55.

Example 2:

³√185193
Answer :
Step 1: Last digit of cube number from right side is 3 that we consider 343 = 7³ we put down 7. Then
Step 2: Take the number whose cube is nearest to 185.That is 185 is nearest to 5³ and 6³ we take small one cube digit that is 5.
So the answer is 57.

Example 3:

³√274625
Answer :
Step 1: Last digit of cube number from right side is 5 that we consider 125 = 5³ we put down 5. Then
Step 2: Take the number whose cube is nearest to 274.That is 274 is nearest to 6³ and 7³ we take small one cube digit that is 6. So the answer is 65.

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Find Seven digit Cube and cube root Tricks

Need to remember 1 to 20 cube and this so easy for any one.Which will help in obtaining cube and cube root numbers.

13=1	113=1331
23=8	123=1728
33=27	133=2197
43=64	143=2744
53=125	153=3375
63=216	163=4096
73=343	173=4913
83=512	183=5832
93=729	193=6859
103=1000	203=8000

Example 1:

³√3869893 = ?
Answer :
Step 1: Last digit of cube number from right side is 3 that we consider 343 = 7³ we put down 7. Then
Step 2: Take the number whose cube is nearest to 3869.That is 3869 is nearest to 15³ and 16³ we take small one cube digit that is 15.
³√3869893
So the answer is 157.

Example 2:

³√1728000 = ?
Answer :
Step 1: Last digit of cube number from right side is 0 that we consider 1000 = 10³ we put down 0. Then
Step 2: Take the number whose cube is nearest to 1728.That is 1728 is nearest to 12³ and 13³ we take small one cube digit that is 12.
³√3869893
So the answer is 120.

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Shortcut to find cuberoot of any 5 or 6 digit number

Shortcut to find cube root is especially helpful in competitive exams where every second counts.
By using this method you can save a lot of time and also get accurate results.

Shortcut to mentally find the cube root of any 5 or 6 digit number

Step 1:Find the cube root of the last digit.
Points to be remembered while using this method.

(1)If the last digit is 8 then cube root will be 2.
(2)If the last digit is 2 then cube root will be 8.
(3)If the last digit is 7 then cube root will be 3.
(4)If the last digit is 3 then cube root will be 7.
(5)If the last digit is any other digit other than 2,8,3,7 then put the same number.
From this step you will get the unit's or one's place digit.

To find the tenth place digit you need to follow the below steps.

Step 2:Strike out the last 3 digits of the given number.

Step 3: find the nearest cube of the remaining number.

Step 4: find the cube root of the nearest cube which will give you ten's place digit.

List of cubes to be memorized
1³=1               4³=64              7³=343
2³=8               5³=125            8³=512
3³=27             6³=216            9³=729


NOTE: Shortcut to find cuberoot of any 5 or 6 digit number is applicable only if the given number is a perfect cube.
You can verify whether the given number is a perfect cube or not by using the Prime Factorization Method. If you are not familiar with this method check it here.


Let me explain this with an example to make things more clear and easy to understand.
So lets say we want to find the cube root of a 6 digit number 15746

Example 1:Find the cube root of 157464 in 5 seconds.
                ∛157464=?
            Step 1:First we need to find the cube root of the last digit of the given number.
                       Here the last digit is 4 . 4 is a number other than 2, 8, 3, 7
                      hence we put the number as it is.
                     We get our one's place digit as 4.
          
            Now to get tenth place digit 
            Step 2: We need to strike the last 3 digits of the given number.
                       In this example the last 3 digits are 464 which we will strike off as shown below
                       157464

           

           Step 3:We need to find the nearest cube to the remaining number(157)

                      We find that 125 is the nearest cube to 157.

           

           Step 4: We need to find the cube root of the nearest cube(125)

                     ∛125=5

                      From this step we get our ten's place digit as 5.

          From step 1 and step 4 we get the 

          ∛157464=54

Ans:54

Square and Square Root Shortcut Tricks - Basic Math Tricks Part 5

Basic Tricks - PART- 5

Square & Square Root Shortcut Tricks

SQUARE up to 50

12 = 1	112= 121	212=441
22 = 4	122 = 144	222=484
32 = 9	132 = 169	232=529
42 = 16	142 = 196	242=576
52 =25	152 =225	252=625
62 = 36	162 = 256	262=676
72 = 49	172 = 289	272=729
82 = 64	182 = 324	282=784
92 = 81	192 = 361	292=841
102= 100	202 = 400	302=900

312= 961	412=1681
322= 1024	422=1764
332= 1089	432=1849
342= 1156	442=1936
352= 1225	452=2025
362=1296	462=2116
372=1369	472=2209
382= 1444	482=2304
392= 1521	492=2401
402= 1600	502=2500

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Square and Square Root of two digit get using formula1

Square and Square Root get using formula
Formula: (a+b)² = a²+2ab+b² i.e, (a / b)²= a² / 2ab / b²
we applied this formula to obtain the square of a number

Example 1:

( 57 )²
= ( 5 / 7 )²

Answer :
Apply formula of a²+2ab+b²
Consider,
57² = ?
A as 5
B as 7 (we break the number in two parts i.e, A as 5 and B as 7 and applied formula )
= 5² / 2 x 5 x 7 / 7²
= 25 / 2 x 5 x 7 / 49
a²= 25
b²= 49
2ab = 2 x 5 x 7 = 70
= 25 / 70 / 49
Step 1: Put down 9 carry 4
Step 2: add carry 4 to 70 = 74 put down 4 carry 7
Step 3: add carry 7 to 25 = 32 put down 32
and answer is 3249,
= 3249
All this do on your mind which will help in fast calculation to obtain the answer of Square and Square Root of a number.
we applied this formula to obtain the square of a number
This is similar to the above Example.

Example 2:

(69)²
= (6/9)²

Answer :
Consider A as 6, and B as 9.
= 6² / 2 x 6 x 9 / 9²
= 36 / 2 x 6 x 9 / 81 (we break the number in two parts i.e, A as 6 and B as 9 and applied formula )
a² = 36
b²= 81
2ab = 2 x 6 x 9 = 108
=36 / 108 / 81
Step1: put down 1 carry 8
Step2 : add 8 to 108 =116 then put down 6 carry 11
Step3 : and add 11 to 36 = 47 and put down 47
= so answer is 4761,
= 4761

Note: All this do on your mind which will help in fast calculation to obtain the answer of Square and Square Root of a number.

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Square and Square Root of three digit get using formula1

we applied this formula to obtain the square of three digit number

Example 1:

Square and Square Root of 114²
Answer :
Firstly we separate the 114 like this (11/4)²
then applied previous formula on it
=11² / 2x11x4 / 4²
=112 / 2x11x4 / 16
=121 / 88 / 16
=12996
we apply the formula a² + 2.a.b + b²
Step 1: note down 6 carry 1
Step 2: add carry 1 to 88 =  89, note down 9 carry 8.
Step 3: add carry 8 to 121 and note down 129
= 12996

Note: we can also separate 114 to find square like (1/14)2

Example 2:

Square and Square Root of 223²
Answer :
Firstly we separate the 223 like this(22/3)²
then applied previous formula on it
we apply the formula a² + 2.a.b + b²
= 22² + 2 x 22 x 3 + 3²
=484 / 132 / 9
= 49729
Step 1: note down 9
Step 2:note down 2 carry 13.
Step 3:add carry 13 to 4 = 17, note down 7
Step 4: add carry 1to 48 = 49 put down
= 49729
Note: All this do on your mind which will help in fast calculation to obtain the answer of Square and Square Root of a number.

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Square and Square Root of 100 base method

Example 1: 98²= ?

Answer :
Step 1: First we know that 98² is double of 98 that is = 98 x 98 = ?,
At first we count the number of less from 100. that is the above 98 is 2 less from 100.
Step2: Now we are going to multiply 2 x 2 = 4 and note down this 4 ( that are come from both less 98 x 98 from 100).
Step 2: put one Zero left from 4 and now subtract the less number is 2 from 98 that is = 96and the answer is 9604.

Example 2 : 96² = ?

Answer :
Step 1: First we know that 96² is double of 96 that is = 96 x 96 = ?,
At first we count the number of less from 100. that is the above 96 is 4 less from 100.
Step 2: Now we are going to multiply 4 x 4= 16 and note down this 16 ( that are come from both less 96 x 96 from 100).
Step 3: now subtract the less number is 4 from 96 that is = 92 and put down it
that is 9261
and the answer is 9216.

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Square and Square root a number ending in 6

Example 1:

76²
Step 1:put down 6
Step 2:Multiply 2 with (7 + 1) = 16 and add 16 +1 = 17.put down 7 and carry 1 .
Step 3:Multiply 7 with (7 + 1) = 56 + carry 1 = 57 put down 57
Answer is 5776

Example 2:

96²
Step 1:put down 6
Step 2:Multiply 2 with (9 + 1) = 20 and add 20 + 1 = 21.put down 1 and carry 2 .
Step 3:Multiply 9 with (9 + 1) = 90 + carry 2 = 92 put down 92
Answer is 9216

Example 3:

36²
Step 1:put down
Step 2:Multiply 2 with (3 + 1) = 8 and add 8 +1 = 9.put down 9.
Step 3:Multiply 3 with (3 + 1) = 12 put down 12
Answer is 1296

Example 4:

56²
Step 1:put down 6
Step 2:Multiply 2 with (5 + 1) = 12 and add 12 +1 = 13.put down 3 and carry 1 .
Step 3:Multiply 5 with (5 + 1) = 30 + carry 1 = 31 put down 31
Answer is 3136

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How to square a number ending with 5 within seconds

Squaring a number ending in 5 is the easiest if you know the trick.
Number given to you can be a two digit number or three digit number or five or six digit number, 

it does'n t matter.
The same trick can be applied to find the square of any number ending in 5.

Let us now discuss the steps to follow

Step 1:Multiply the ten's digit of the given number with its immediate next number.This will be the 1st digit of the answer.

Step 2:Put 25 next to the result of step 1.
Logic behind putting 25 is that the number will always end with 5 and 5² is 25. 

You get the answer

Let us find squares of numbers ending in 5 using this trick

Example 1:(35)²=?

Step 1: 3 is the ten's digit in the given number 35.
Immediate next number to 3 is 4.
Hence,by multiplying 3 & 4 we get 12(3x4=12).which is the 1st digit of the answer.

Step 2: Now putting 25 next to 12 we get the answer as 1225.

Answer: (35)²= 1225

Now you try to square 25,45,15,55,65,75,85,95 & see how quickly you can square.  

Example 2:(125)²=?

Step 1: 12 is the ten's digit in the given number 125.
Immediate next number to 12 is 13.
Hence,by multiplying 12 & 13 we get 156(12x13=156).which is the 1st digit of the answer.

[To multiply a 2digit by 2digit number when ten's digit of both numbers is 1 you can apply a trick to get your answer quickly.
1st digit =1 
middle digit =add unit's place digit of both numbers(2+3=5)
unit's place digit=Multiply unit place digit of both numbers(2x3=6)]

Step 2: Now putting 25 next to 156 we get the answer as 15625.

Ans (125)²= 15625

Now you try to square 105,115,135,145,155,165,175,185,195 & see how quickly you can square.  

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square number from 10 to 19 math trick

Easiest method to find the square of any number between 10 and 19 

Step 1: First add the given number and the units digit .

Step 2: Then Square the unit's digit and put the number next to the result obtained in STEP 1.

Ex-1:Find the square of 13 in 5 seconds

(13)² = ?

Step 1: 13 + 3 = 16_

Step 2: 3² = 9

put the result obtained in step 2 in step 1 we get

Ans=169

Ex-2:Find the square of 16 in 5 seconds

(16)² = ?

        Step 1: 16 + 6 = 22_

Step 2: 6² = 36

put the result obtained in step 2 in step 1 as shown below                                                                          

22_
 36

Ans=256

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Shortcut to square any number from(20-29)

Easy way to mentally square any number from 20 to 29

Step 1:First add the given number and the unit's digit.
Step 2:Double the result obtained in step 1.
Step 3:Square the unit's digit of the number given in question and place it next to the result obtained in step 2.

Example 1:Find the square of 22 in 5 seconds

            (22)²=?

             Step 1:22 + 2=24

             Step 2:24x2=48_

             Step 3:2²=4

Now put the result obtained in step 3 in step 2,we get

Ans:484 

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Shortcut to square any number from (30-79)

It is always better to have some maths tricks handy when you are planning to take any arithmetic aptitude test either in the competitive exams such as Bank exams,CAT,MAT.
Another easy trick to find square of the numbers between 30 to 79 is to take a common base as 50 and see how far the number is from 50.This method works well when the number is very close to 50.

Easy way to find the square of  number from 30 to 79

Step 1:Find how many more or less the given number is from 50.

Step 2:Add the number to 25 if more than 50 or subtract the number from 25 if less than 50.

Step 3:Then find the square of the number added or subtracted and put next to the result arrived at in step 2.

Let us now apply the trick that we learnt in the example below

Example 1:Find the square of 52 in 5 seconds

(52)²=?

Step 1: 52-50=2

           We find 52 is 2 more than 50

 Step 2: Here we notice that the given number is more than 50 so we add 25 as follows

            25+2=27

                                                                 
Step 3: Now we find the square  of 2
           2²=4

           putting the result obtained in step 3 next to the result obtained in step 2 after adding a 0 before it as it is a single digit,we get

           2704

Ans: (52)²=2704

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Shortcut to square any number from 90 to 99

This shortcut method is one of my favorite that I like to apply when I see a number such as 98 or 99 as all I need to know is how to subtract and square of numbers from 1 to 9.

Now lets go through the steps

Step 1:Assume 100 as base and find the difference between the number to be squared and 100.

Step 2:Subtract the difference you found in step 1 from the number to be squared to find tenth place digit.

Step 3:Square the difference and place it next to step 2.

Once you know the technique you can square any number from 90 to 99 mentally.

Let us see few examples to understand better

Example 1: 98²=?

Step 1:Assuming 100 as base, we shall find the difference
100-98=2
We get the difference as 2

Step 2:Subtracting the difference(2) from the number to be squared(98) we get,
98-2=96
96 is the tenth place digit

Step 3:Squaring the difference we found in step 1 we get
2²=4
4 is the Unit place digit.
98²=9604

Example 2: 99²=?

Step 1:Assuming 100 as base, we shall find the difference between the number to be squared(99) and 100.
100-99=1
We get the difference as 1

Step 2:Subtracting the difference(2) from the number to be squared(98) we get,
99-1=98
98 is the tenth place digit

Step 3:Squaring the difference we found in step 1 we get
1²=1
1 is the Unit place digit.
99²=9801

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Shortcut to square any number from 100-120

Base method is popularly used to find square of numbers when the numbers are close to a base.By choosing the right base you will arrive at your answer quickly.
So whenever you see 102,101,112,106 it should strike to you immediately .

Hey! this is quite close to 100 so let me take the base as 100 and see if I can get the answer soon.

Here we shall discuss the shortcut method to square any number from 100 to 120 using the base method.

Lets go through the steps now

Step 1:Take 100 as base and see how far the number is from 100.
Add the given number and its deviation.

Lets say you need to square 102.

Here you can notice that the number is 2 more than 100.
Adding the given number(102) and its deviation(2)
we get,

102+2=104

Step 2:Square the deviation(2)and place it next to the result obtained in step 1.
Squaring we get,
2²=4

Ans 10404

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Shortcut to find squareroot of any number

In every bank exam you are asked either to find the square root or cube root of a number. By knowing the shortcut to find the square root of a number, you will be able to find out the square root of any number within seconds.  

Now lets go through the steps...

Step 1: First of all group the number in pairs of 2 starting from the right.

Step 2: To get the ten’s place digit, Find the nearest square (equivalent or greater than or less than) to the first grouped pair from left and put the square root of the square.

Step 3: To get the unit’s place digit of the square root

Remember the following

If number ends in	Unit’s place digit of the square root
1	1 or 9(10-1)
4	2 or 8(10-2)
9	3 or 7(10-3)
6	4or 6(10-4)
5	5
0	0

Lets see the logic behind this for a better understanding

We know,

1²=1

2²=4

3²=9

4²=16

5²=25

6²=36

7²=49

8²=64

9²=81

10²=100

Now, observe the unit’s place digit of all the squares.

Do you find anything common?

We notice that,

Unit’s place digit of both 1² and 9² is 1.

Unit’s place digit of both 2² and 8² is 4

Unit’s place digit of both 3² and 7² is 9

Unit’s place digit of both 4² and 6² is 6.

Step 4: Multiply the ten’s place digit (found in step 1) with its consecutive number and compare the result obtained with the first pair of the original number from left.

Remember,

 If first pair of the original number > Result obtained on multiplication then  select the greater number  out of the two numbers as the unit’s place digit of the square root.

If firstpair of the original number < the result obtained on multiplication,then select the lesser number out of the two numbers as the unit’s place digit of the square root.

Let us consider an example to get a better understanding of the method

Example 1: √784=?

Step 1: We start by grouping the numbers in pairs of two from right as follows

7 84

Step 2: To get the ten’s place digit,

We find that nearest square to first group (7) is 4 and √4=2

Therefore ten’s place digit=2

Step 3: To get the unit’s place digit,

We notice that the number ends with 4, So the unit’s place digit of the square root should be either 2 or 8(Refer table).

Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(2) and its consecutive number(3) we get,

2x3=6

ten’s place digit of original number > Multiplication result
7>6

So we need to select the greater number (8) as the unit’s place digit of the square root.

Unit's place digit =8

Ans:√784=28