## How to Find the Square of Number ending with 5?

We know
• 52 = 25
• 152 = 225
• 252 = 625

Now see the similarity. In all these numbers the last two digits are same. That is 25. The only thing it changes is the first digit.

How to find that?

#### a) What is 15²  ?

We know the last two digits of the square will be 25. Now take the first digit of 15; i.e. 1. Multiply 1 with it’s next number.
i.e. 2. So 1 x 2 = 2.

#### b) What is 25² ?

We know the last two digits of the square will be 25. Now take the first digit of 25; i.e. 2.

Multiply 2 with it’s next number. i.e. 3.

So 2 x 3 = 6.

#### c) What is 55² ?

(5 x 6) 25 = 3025

#### d) What is 85² ?

(8 x 9) 25 = 7225. Hardly it takes less than 5 seconds. You don’t even need to
write the numbers on the paper.

Try 1052
As usual (10 x 11) 25 = 11025 is the answer. Our usual method will take a
minimum of 30 seconds to do this simple multiplication.

Now we can easily save 25 seconds.
Practice Problems

1. 1152
2. 2052

## How to find a Square root of a number?

The first method is inter-related to this one. This method is applicable only for perfect squares.
You have to follow Three simple steps.

1. Find the range
2. Check the Unit Digit.
3. Find the square of number ending with 5 in the particular range and fix the number.

## a) What is the square root of 484?

#### Step 1

We all know 202 = 400. Also we know 302 = 900.
So our number lies between 20 and 30.

#### Step 2

The unit digit is 4. So the unit digit of square root can be either 2 or 8. The number can be either 22 or 28.

#### Step 3

Between 20 and 30, we can easily find the square of 25 using the first method.

252 = (2 x 3)25 = 625
484 is less than 25.

## b) What is the square root of 9801?

Range – 90 to 100.
Square of 95 = (9 x 10) 25 = 9025.
Answer can be either 91 or 99. (Based on Unit Digit)
9025 is less than 9801. So 99 is the Answer.

## c) What is the Square root of 112896?

Range – 300 to 350 (As it is a big number, we need to follow one more step)
Now we should reduce the range.

Square of 33 is 1089.

So 3302 will be 108900.

Square of 34 is 1156. So 3402 will be 115600.

Our number is in between 330 and 240.

As the unit digit is 6, there are two possibilities.

It can be either 334 or 336.
Square of 335 = (33 x 34) 25 = (11 x 3 x 34) 25 = (374 x 3) 25= 112225.

As our number is more than the square of 335, the Answer is 336.

Note: This method may look tough and time consuming. But it is really a simple one.

## d) What is the Square root of 390625?

Range
Square of 600 – 360000.
Square of 650 – (6 x 7)
2500 = 422500. (By Method 1).
So it is in between 60 and 65.

As the unit digit is 5, the square root should also end with 5. The options we have are 605, 615, 625, 635, and 645.

We can easily eliminate 605 and 645. (Based on Difference)

Note : These steps can be done by just seeing the number and mainly with out using pen.

Square of 615 = (61 x 62)25 = (62 (60 + 1))25 = 378225
Square of 625 = (62 x 63)25 = (63 (60 + 2))25 = 390625

## How to find a Square of any number?

Any number can be represented as the addition of two numbers. For example 32 can be written as 30 + 2. 46 can be written as 40 + 6. 116 can be written as 110 + 6.
The Steps are
• Split
• Use Formula

### a) What is the Square of 116?

Split = 110 + 6
Formula = (a + b)2 = a2 + b2 + 2ab = 12100 + 36 + 1320

### b) What is the Square of 1004?

Split = 1000 + 4
Apply Formula = 1000000 + 16 + 8000 = 1008016

(If you use traditional method, finding the square of 1004 will take a minimum of 40 seconds. By this method, you just saved 25 seconds again )

### c) What is the Square of 605?

Here you can two methods.

The unit number is 5.

Always go for the easy method.

Answer = (60 x 61)25 = (60 (60 + 1))25 = 366025

Practice Problems

Find the Square of

1. 713
2. 819
3. 1009
4. 1520

## Multiplication using Split and Merge Method

This method is the most effective method for multiplication.

We usually don’t do mistakes when we multiply a number with less than 10.

We do mistakes only when the number is big. Let’s go to the technique.

#### a) What is 63 x 15?

This is the usual method we follow. It took 20 seconds for me. For you it may take less than that.

Here is the simpler method.= 63 (10 + 5) = 630 + 315 = 945. (Hardly takes 10 seconds).

In competitive exams, every second counts and even a 0.1 mark can change your future.

#### b) What is 81 x 19?

= 81 (20 – 1). It don’t need to be only addition. It can be subtraction also. The ultimate aim is to reduce the time by using simple calculation.

= 1620 – 81 = 1539

#### c) What is 131 x 26

= 26 (100 + 30 + 1) = 2600 + 780 + 26 = 3406. (You can split the number to n number of times)

#### d) What is 147 x 150

You can split this to 150 (100 + 40 + 4 + 3) OR 147 (100 + 50) = 14700 + 7350

NOTE: Always use the simple way to find the

#### e) What is 1005 x 106

Traditional Method will surely take 30 plus seconds.

= 106 (1000 + 2 + 2 + 1)

(I have split the number to many numbers for your understanding. It can be just 106 (1000 + 5)).

#### f) What is 1256 x 516

= 1256 (500 + 10 + 3 + 3)

(It may look big. But once you practice this method, you will understand how simple and effective it is)

## Multiplication with 5 – Into 10 By 2 Method (x 10/2)

When you have to multiply any number with 5, First multiply with 10 and divide by 2.

• 2 x 5 = 20 /2 = 10

• 77 x 5 = 770/2 = 385

• 1876 x 5 = 18760/2 = 9380

• 978672 x 5 = 9786720/2 = 4893360 (You can directly write the answer)

#### g) What is 1082 x 107

= 1082 (100 + 5 + 2)

= 108200 + [10820/2] + 2164

= 108200 + 5410 + 2164 = 115774 (Note that I write this many steps only for your understanding)

#### h) What is 103 x 97

Here rather than using split and merge method, we can apply another 7th std Formula.

(a + b) (a – b) = a2 – b

(100 + 3) (100 – 3) = 1002 – 32 = 10000 – 9 = 9991.

i) What is 53 x 47

= (50 + 3) (50 – 3) = 2500 – 9 = 2491. (It just took 5 seconds).

j) What is 163 x 157

= (160 + 3) (160 – 3) = 25600 – 9 = 25591

## Multiplication with 11

a) 21 x 11 = ?

Leave the traditional method. Shortcut method is write 2 and 1 at the first and last position. (Always add from the left)

i.e. 2___1 Now add 2 and 1. That is 3. Answer is 231. (You got answer in one second).

b) 25 x 11 = 2 ___ 5 = 275

c) 32 x 11 = 3 ___2 = 352

d) 253 x 11 = 2 _______ 3 = 2783 (Just adding the adjacent numbers).

e) 531 x 11 = 5 ____1 = 5841

f) 277 x 11 = 2____7

Here 2 followed by (2 + 7) then (7 + 7) then 7

= 29 (14) 7

(Carry the 1 to the left)

= 3047

g) 9879 x 11 = 9_____9

= 9 (9 + 8) (8 + 7) (7 + 9) 9

= 9 (17) (15) (16) (9) (Add the carry over number to the left)

= (9+1) (7+1) (5+1) (6)(9)

= 108669

i) 1387 x 11 = 1 ______ 7

= 1 (1+3) (3+8) (8+7) 7

= 1 (4) (11) (15) 7

= 1(4 +1) (1+1) (5) (7)

= 15257

j) 2587 x 11 = 2______7 = 27(13)(15)7 = 28457

k) 3768912 x 11 = 3_______2 = 3(10)(13)(14)(17)(10)(3)2 = 41458032

Practice Problems

1. 98763 x 11

2. 9659213 x 11

3. 8621098 x 11

4. 55598 x 11

5. 127409 x 11

## Multiplication of Numbers near to the bases

Base is nothing but numbers like 100, 1000, 10000, 100000..etc

Case 1:

### Multiplication of Numbers Below the Base

a) 99 x 98

• 99 is one less from the base 100

• 98 is two less from the base 100

Here we ll get the answer in Two Parts.

One is LHS and another is RHS.

LHS is 97. RHS is 02. Answer is 9702.

Always ensure that the number of digits in RHS should be equal to the number of 0’s in the Base.

b) 96 x 92

• 96 is less than 100 by 4

• 92 is less than 100 by 8

c) What is 88 x 86

d) What is 981 x 991

e) What is 9987 x 9994

Winners Don’t Do Different Things They Do Things Differently

## Square of Numbers near to the Base

a) What is 97

• 97 is less than 100 by 3

So, Write like this –> (97-3) / 3

Note that the number of digits in RHS should be equal to number of Zeros in the base.

b) What is 94

• 94 is less than 100 by 6.

So, Write like this → (94-6) / 6

= 88 / 36. Answer is 8836

c) What is 88

= (88 – 12) / (122

= (76) / (144) = 7744 (Carry Over the One to LHS. Base is 100. So it should be two digits).

d) What is 998

= (998 – 2) / (22

= (996) (004). Answer = 996004

e) 9993 x 9993

= (9993 – 7) / (72

= (9986) / (49) = 99860049

f) 105 x 105

= (105 + 5) / (52) (We add as it is above the base)

= (110)/ 25. Answer = 11025.

You can answer it just by seeing the numbers. No need to Write any on paper.

g) 109 x 109

= (109 + 9) / (92

= (118)/(81) = 11881

h) 1013 x 1013

= (1013 + 13) / (132

= (1026) / (169)

= 1026169

i) 1025 x 1025

= (1025 + 25) / (252

= (1050) / (625)

= 1050625

j) 1096 x 1096

= (1096 + 96) / (962

= (1192) / [(96 – 4)/ (42)]

= (1192) / (9216) = (1201) / (216)

= 1201216 (carry over 9 to LHS. Base is 1000. Digits in RHS should be three).

k) 1113 x 1113

= (1113 + 113) / (1132

= (1226) / [(113 + 13) / (13)2] (Separate 1132 and do calculation for that alone)

= (1226) / [126 / 169]

= 1226 / 

= 1238769 (Carryover the 12 to the left) (Understand that I write this many steps only for your understanding)

Note: This is the easiest way to find the Squares. Practice more. Then only you will know how simple it is. Assume your own numbers and do the math.

## Simple Trick to remember Squares of numbers from 25 to 30

Just keep 25 as the middle number and do simple addition.

## Best Way to Calculate Percentage

In Percentage concept, there is division as well as multiplication. It is really
time consuming for many people. In fact, it is the easiest part among all the
topics. It is easier than adding two 3 digit numbers. The Split and Merge
method can also be applied for Division also.

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