Short tricks for Squares, Cubes and Multiplication

This series is based on Simplification. As simplification is an important part of Quantitative Aptitude, We will learn how to know squares, cubes, square root and cube root. In this series we will also cover addition, subtraction, multiplication and division.

How to calculate Squares ?

Type I: (80-100): Assume base 100. There are two zeroes in the base so up to 2 digits will be added.
Examole1: 97²
The base is 100.
Step 1: Calculate the difference between base and number.
100 -97 = 3
Step 2: Subtract difference from the number
97-3 = 94
Step 3: Calculate Square of the difference.
3² = 09
Answer: 94_09
Example2: 87² In order to calculate this, we will follow the similar approach. First, take the difference of 87 from 100. 
100 – 87 = 13
Subtracting 13 from 87 = 87 – 13 = 74 (This will be the first 2 numbers of square)
Now taking the square of 13, it is 169. 
Now, you need to pay attention here. This square is a 3 digit number, however, we are calculating from 100, for which we will assume base 2 (2 zeroes in 100).
Now, here we will keep the digits at tens and units place intact (i.e. 69). The digit at hundreds place (i.e.1) will be transferred to the difference of 87 and 13 (i.e. 74) and added to it. It becomes 74 + 1 = 75
So, our square will be 7569.

Type II (100-120): Assume base 100.
Example3: 107²
Step 1. difference between 107 and 100 = 7
Step 2. Add this to 107 i.e. 107+7 = 114
Step 3. Calculate Square of 7 = 49
Answer 11449
Consider the square of 112. In order to calculate this, we will follow a similar approach. First, take the difference of 112 from 100. 
112-100 = 12
Adding 12 to 112 = 112+12 = 124
Now taking square of 12, it is 144 
Now, you need to pay attention here. This square is a 3 digit number, however, we are calculating from 100, for which we will assume base 2 (2 zeroes in 100).
Now, here we will keep the digits at tens and units place intact (i.e. 44). The digit at hundreds places (i.e.1) will be transferred to the addition of 112 and 12 (i.e. 124) and added to it. It becomes 124 + 1 = 125
So, our square will be 12544.

Type III (50 -70).Here base will be 50.
25+extra from_ square of extra value.
51² = 25+1 _ (1²) = 26_01 = 2601
59² = 25+9 _ (9²) = 34_81 = 3481
62² = 25+12_ (12²) = 37_144 (It is wrong)
We have to transfer 1 from 144 to 37 so it will become 38
So, 62² = (37+1)_44 = 3844
68² = 25+18_(18²) = 43_324 = (43+3)_24 = 4624
Similarly transfer 3 from 324 to 43, so it will become 46 and answer will be 4624

Type IV (30-50): Here base will be 50.
25 – less from 50 _ square of less value
46² = 25 – 4 _ 16 = 2116
49² = 25 – 1 _ (01) = 2401
43² = 25 – 7 _ (49) = 1849
34² = 25 – 16 _(256) = 9256 (It is wrong)
We have to transfer 2 from 256 to 9 so it will become 11
So, 34² = 1156
36² = 25-14_(196) = 1296

Type V (71-79)
71²
By 50 method: (25+21)_ 21² = 46_441 = 5041
By 100 method: (71-29)_(29)² = 42_841 = 5041
73² = (25+23)_23² = 48_529 = 5329
79² = (79-21)_21² = 58_441 = 6241

Multiplication of Numbers having 5 at their unit places

Now, we will learn how to multiply two numbers which have 5 in their unit place.

Type I: When numbers are same.
65×65 = (6×7)_25 = 4225 (Fix 25 in last, multiply 6 from 7 i.e. 42)
85×85 = (8×9)_25 = 7225 (Fix 25 in last, multiply 8 from 9 i.e. 72)
115×115 = (11×12)_25 = 13225 (Fix 25 in last, multiply 11 from 12 i.e. 132)

Type II: When numbers have difference of 10
65×75 = (6×8)_75 = 4875 (Fix 75 in last, multiply 6 from 8 i.e. 48)
85×95 = (8× 10)_75 = 8075 (Fix 75 in last, multiply 8 from 10 i.e. 80)
115×125 = (11×13)_75 = 14375 (Fix 75 in last, multiply 11 from 13 i.e. 143) 

Type III: When numbers have difference of 20
65×85 = (6×9)_125 = 54_125 (Fix 125 in the last and multiply 6 from 9 i.e. 54)
Note: In this 1 from 125 has to be transferred to 55. So, answer will be 5525.
85×105 = (8×11)_125 = 88_125 = 8925
115×135 = (11×14)_125 = 154_125 = 15525

Type IV: When numbers have difference of 30
65×95 = (6×10)_175 = (Fix 175 in the last and multiply 6 from 10 i.e. 60)
In this 1 from 175 has to be transferred to 60. So answer will be 6175
85×115 = (8×12)_175 = 96_175 = 9775

Multiplication of different numbers

Type 1: When the difference of two numbers is even.
Multiplication = (Middle number)² – (difference/2)²
19×21 = 20² – (2/2)² = 400-1 = 399
47×53 = 50² – (6/2)² = 2500-9 = 2491
73×77 = 75² – (4/2)² = 5625-4 = 5621

Type 2: Consecutive Number Multiplication:
Square of Small number + small number
12×13 = 12²+12 = 144+12 = 156
48×49 = 48²+48 = 2304+48 = 2352
how this formula has been derived
12×13 = 12×(12+1)= 12× 12+12 = 12²+12

Type 3: Different numbers (>100)
103×108
 +8     +3 and (8×3) = 24
(103+8)_ (+3)×(+8) = 11124 or (108+3)_8×3 = 11124
109×117
 +17   +9
(109+17)_(+9)×(+17) = 126_153 = 12753

Type 4: Different Numbers (<100)
96×91
-9   -4
(96-9)_(-9)×(-4) = 8736 or (91-4)_9×4 = 8736
92×87 
-13  -8
(92-13)_(-13)×(-8) = 79_104 = 8004

Type 5: Different Numbers (<100>)
103×96
-4     +3
(103-4)_(-4×3) = 99_ (-12) = 9900-12= 9888   
or (96+3)_(-4×3) = 99_-12 = 9900-12 = 9888

How to calculate CUBE ?

Follow the undermentioned steps while calculating cubes of numbers –

Step 1: Put down the cube of tens place digit in a row of 4 figures. The other three numbers in the row of the answer should be written in a geometrical ratio in the exact proportion which is there between the digits of a number.
Step 2: Note down the two times of 2^nd and 3^rd number just below the 2^nd and 3^rd number in the next row.
Step 3: Then add up the two rows.
Example: 13³
Solution:
Step 1:    Note down the cube of 1 (i.e. at tens place). And also the ratio between 1 and 3 is 1:3, So the first row is   1   3     9   27
Step 2: Note down the 2 times of 2^nd and 3^rd number. (i.e. 6 and 18).
Step 3: Add the rows
      1        3         9        27
               6       18
      2        ₁1      ₂9        ₂7
Answer : 2197
Example: 24³
Step 1:     8       16       32         64
Step 2:              32       64
Step 3:    13      ₅8       ₁₀2         ₆4
Answer: 13824
Example: 19³
Step 1:  1      9       81         729
Step 2:        18     162
Step 3:  6     ₅8      ₃₁5          ₇₂9
Answer: 6859
Example: 92³
Step 1:  729    162     36        8
Step 2:           324     72
Step 3:  778    ₄₉6     ₁₀8        8
Answer: 778688