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: 97^{2}

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.

3^{2} = 09 **Answer: 94_09**

Example2: 87^{2}** ^{ }**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.

Type II (100-120):

**Example3: 107**

^{2}

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.

Type III (50 -70).

**25+extra from_ square of extra value.**

51

^{2}= 25+1 _ (1

^{2}) = 26_01 = 2601

59

^{2 }= 25+9 _ (9

^{2}) = 34_81 = 3481

62

^{2}= 25+12_ (12

^{2}) =

**37_144 (It is wrong)**

We have to transfer 1 from 144 to 37 so it will become 38

So, 62

6825+18_(18

We have to transfer 1 from 144 to 37 so it will become 38

So, 62

^{2}= (37+1)_44 = 384468

^{2}=^{2}) = 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

Type IV

**(30-50)**: Here base will be 50.

**25 – less from 50 _ square of less value**

46

^{2}= 25 – 4 _ 16 = 2116

49

^{2}= 25 – 1 _ (01) = 2401

43

^{2}= 25 – 7 _ (49) = 1849

34

^{2}= 25 – 16 _(256) =

**9256 (It is wrong)**

We have to transfer 2 from 256 to 9 so it will become 11

So, 34

36

Type V (71-79)

We have to transfer 2 from 256 to 9 so it will become 11

So, 34

^{2}= 115636

^{2 }= 25-14_(196) = 1296Type V (71-79)

71

^{2}

By 50 method: (25+21)_ 21

^{2 }= 46_441 = 5041

By 100 method: (71-29)_(29)

^{2}= 42_841 = 5041

73

^{2}= (25+23)_23

^{2}= 48_529 = 5329

79

^{2}= (79-21)_21

^{2}= 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 = (6x7)_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 = 20^{2} – (2/2)^{2} = 400-1 = 399

47×53 = 50^{2} – (6/2)^{2} = 2500-9 = 2491

73×77 = 75^{2} – (4/2)^{2} = 5625-4 = 5621

Type 2: Consecutive Number Multiplication:

Square of Small number + small number

12×13 = 12^{2}+12 = 144+12 = 156

48×49 = 48^{2}+48 = 2304+48 = 2352

how this formula has been derived

12×13 = 12×(12+1)= 12× 12+12 = 12^{2}+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)

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>)

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: 13^{3}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**

_{1}1

_{2}9

_{2}7

**Answer : 2197**

Example: 24

^{3}

Step 1: 8 16 32 64

Step 2: 32 64

Step 3: 13

_{5}8

_{10}2

_{6}4

**Answer: 13824**

Example: 19

^{3}

Step 1: 1 9 81 729

Step 2: 18 162

Step 3: 6

_{5}8

_{31}5

_{72}9

**Answer: 6859**

Example: 92

^{3}

Step 1: 729 162 36 8

Step 2: 324 72

Step 3: 778

_{49}6

_{10}8 8

Answer: 778688

Answer: 778688

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