Math e Magician by Math e Magician

By Uma Shrikant Rale Ria-Niki Publishers

Book Name

Be a Math-e-magician

Publication

Ria-Niki Publishers

Sushila Nagari(phase1), 301, Jai,

Patwardhan Baug, Kothrud, Pune 411052

Email : b.a.mathemagician@gmail.com Copy Right

Uma Shrikant Rale

Publishing Date : 1/01/2015 Book Design, Type Setting & Printing By

Infini Artographics

info@infini-arts.com

+ 91 20 65004647, + 91 98225 68882

Price :

Rs. 400

This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisherâ&#x20AC;&#x2122;s prior written consent in any form of binding or cover other than condition being imposed on the subsequent purchaser and without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the above mentioned publisher of this book.

I dedicate this book to my children Anagha and Ashutosh, To my wonderful grandchildren Ria, Nikita, Anush and Anay And to all my dear students - With a wish that they never stop learning!

Why did I write this book ??

Why did I write this book ?? Being an educator for 30+ years has been an amazing and extremely fulfilling journey! As a teacher of Math, sometimes I would see some children struggling with big multiplications and divisions and would want to do something to help them and boost their confidence. While working to enhance my teaching-learning skills, I came across a lot of new stuff in various books. Self-learning from Trachtenburgâ&#x20AC;&#x2122;s methods and Vedic Mathematics was a journey in itself. These amazing method of calculations have been lost in the fast pace of changes and technology. These are patterns in Math converted to simple rules, a gift to students by mathematicians of long ago. These rules were used when there were no calculators and believe me once you learn the rules and practice hard you can beat a calculator in speed and accuracy, isnâ&#x20AC;&#x2122;t it amazing? It then became my hobby to learn, simplify the rules and teach them to my students as a reward for doing good work in class. For more than 10 years I conducted a program called Math++ in the Indian High School, Dubai. I was thrilled with the feedback from the students who attended the program, some of these are included in this book. The program was so satisfying, that the idea to compile all those amazing rules in one place, in this book, took shape in my mind. A dream of a book that will make Math enjoyable. And now any student from anywhere in the world can learn all the rules conveniently, using this book and at their own pace. These rules can be applied to any syllabus as it helps to make your Math base strong. Learn the rules, apply them in your day to day calculations and experience how much you can enjoy your math classes in school. The rules can be mastered by anyone who is willing to learn them, even if you have not always been very comfortable with Math. And if you learn well, you will do better at any Math test. You will also have more time to re-check your exam papers as you are bound to finish earlier and you will be able to calculate faster than you ever did. If you really give it a sincere shot, I promise you will soon Be a MATH-E-MAGICIAN! One requires a lot of support when such a project is taken up. My husband, my mom in law, and the entire extended family has been a strong support system for me. Appreciation comes in many forms and I have been lucky to get loads of it from students and friends too. The Indian High school has always been a great place for learning and growing as a teacher, where good work is always is appreciated. Thank you everyone for helping me do this. I miss my parents at this important time in my life, but I can visualize their delight and feel the pat on my back. This book has fulfilled my dream of reaching out to children, to hand over to them a tool, that will make day-to-day Math exciting and easy. My heartfelt thanks to my nephew, Santosh Subhedar. It was his perseverance and trust that was instrumental in getting this book printed!

Math E Magician

Who will benefit from the use of this book?

Who will benefit from the use of this book? Any student age 8 to 80 years can learn to do fast calculations from these simple rules, if they are committed to learning well. Don’t stop till your good becomes better and your better, best !!

How to use this book? It is very simple to follow : • The book has 47 rules with solved examples. • Read and understand one rule at a time, and try the sample questions. • Try using the rule with more examples till you are confident. • Then start solving the worksheet of that topic given at the back of the book. • Do 5 sums regularly at a time and check your answers always. • Time yourself for every 5 sums from the sets. Speed will come with practice. • Accuracy is more important than speed. So try to get the answers right. • Proceed with the worksheet doing one section at a time till you are 100% confident. • Go to the next rule and repeat the process. • It will be great if you can get a partner in this venture as it is fun to work together. Your partner could be your mom or dad too as they will enjoy the process with you. • You can make your own worksheets too. • Check your answers by the two methods in the book. (Chapters 5,6) • Compete with yourself and analyze your progress. • Remember there is no substitute for hard and smart work

Math E Magician

What Will You Find In The Book?

What will you find in the book? Topic

Page

11 Magic square and its use. 05 2 Addition facts and speed addition 09 3 Subtraction facts and speed subtraction. 11 4 Speed Multiplication 14 • Multiplication by 11-19 in a flash • Multiplication by multiples of 11 - 19 5 Checking by the 9s rule 29 6 Checking by the 11s rule 30 7 Zip-Zap Multiplication in one step using star patterns 32 • One step multiplication of 2D by 2D • One step multiplication of 3D by 2D • One step multiplication of 3D by 3D • One step multiplication of 4D by 2D • One step multiplication of 4D by 3D • One step multiplication of 4D by 4D 8 Zip-Zap multiplication by 5, 25, 50, 125 42 9 Zip-Zap division by 5, 25, 50, 125 43 10 Zip-zap converting fractions to decimals 44 11 Multiplication by 9, 99, 999 and so on 46 12 Multiplication using base 10 and its multiples 49 13 Squares and square roots. 55 • Squaring any two digit number • Squaring a three digit number • Square roots of perfect squares 14 Cubes and Cube roots 62 15 Generation of multiplication tables up to 3 digit numbers 65 16 Divisibility tests for prime numbers till 41 68 17 Division 72 • Short division • Division with a 2 digit divisor • Division with a 3 digit divisor 18 Zip-zap calculation of percentage 85 19 Addition of many numbers 87 20 Some quick useful conversions. 91 21 Some more practice - topic-wise worksheets. 97

Math E Magician

The Magic Square

THE MAGIC SQUARE This simple square of numbers can be used magically to train your mind to add or subtract quickly. This is how you can proceed.

Remember that • To add in ones you must go down the line as 1, 2, 3, 4 .... - 10, 11 ... • To subtract ones you must go up as 30, 29, 28, ....– 21, 20,19 ... • To add tens you must go to the right as 33, 43, 53 ... • To subtract tens you must go to the left as 86, 76, 66 ... +

Jump 1 square for every +10. Jump 1 square for every -10.

+ One step down for every +1. - One step up for every -1.

101 111 121

102

100 110 120

132

114 135 106 117

109 140

2 is the only even prime number. 3, 5... are prime numbers, others are composite. Math E Magician

Using The Magic Square

Using The Magic Square The magic square can be used for addition and subtraction. You will not need to count on your fingers once you are able to visualize the Magic Square and the method of using it. Let us begin with the operation of addition.

Example 01

23 +10

Example 02

23 +30

Example 03

23 +50

• Put your finger on 23 • Jump with the finger, one place to the right 23 to 33 as you are adding 1 ten. • Stop. • The answer is 33

• Put your finger on 23 • Jump with the finger, three places to the right as you are adding 3 tens. • So 23 to 33, 43, 53 and Stop • The answer is 53

• Put your finger on 23 • Jump with the finger, five places to the right as you are adding 5 tens. • So 23 to 33, 43, 53, 63, 73 and Stop • The answer is 73

Repeat this with different sets of numbers. Now let’s try subtraction with the help of the same Magic Square!

Example 01

23 -10

• Put your finger on 23 • Jump with the finger, one place to the left as you are subtracting 1 ten. • 23 to 13, Stop • The answer is 13

Math E Magician

Using The Magic Square

Example 02

23 -20

Example 03

53-40

• Put your finger on 23 • Jump with the finger, two places to the left as you are subtracting 2 tens. • 23 to 13, 3 and stop • The answer is 3

• Put your finger on 53 • Jump with the finger, four places to the left as you are subtracting 4 tens. So we go from 53 to 43, 33, 23, 13 and stop. • The answer is 13

Repeat this with different sets of numbers. Now let’s go further with more additions and subtractions.

Example 01

34+25

Example 02

34-25

Math E Magician

• Put your finger on 34 • You have to add 2 tens and 5 ones to 34, so there are 2 jumps for tens and 5 steps for ones • Jump with the finger, two places to the right as you are adding 2 tens to reach 54 • Step down 5 places so we go from 54 to 55, 56, 57, 58, 59 and stop. • The answer is 59

• Put your finger on 34 • You have to subtract 2 tens and 5 ones, so there are 2 jumps for tens and 5 steps for ones but to the left as this is subtraction. • Jump with the finger, two places to the left, 34 to 24, 14 as you are subtracting 2 tens, to reach 14 • Step up 5 placesfrom 14 to 13, 12, 11, 10, 9 and stop. • The answer is 9

Using The Magic Square

Observe the path of addition of 9 to any number

Observe the path of subtraction of 9 from any number

Observe the path of addition andsubtraction of 11 from any number Adding 11

subtracting 11

Try out more additions and subtractions and soon you will not need the Magic square in front of you. You will still be able to see it when you close your eyes as it will be retained by your brain. Do quick calculations while travelling and use your time productively. And the side effect of this will be a sharp brain with good thinking skills. 10

Math E Magician

Additions Facts

ADDITION FACTS Knowing the addition facts well, will speed up your addition skill to a great extent. You can then avoid using your fingers to calculate . 1+1=2

1+2=3

1+3=4

1+4=5

1+5=6

2+1=3

2+2=4

2+3=5

2+4=6

2+5=7

3+1=4

3+2=5

3+3=6

3+4=7

3+5=8

4+1=5

4+2=6

4+3=7

4+4=8

4+5=9

5+1=6

5+2=7

5+3=8

5+4=9

5 + 5 = 10

6+1=7

6+2=8

6+3=9

6 + 4 = 10

6 + 5 = 11

7+1=8

7+2=9

7 + 3 = 10

7 + 4 = 11

7 + 5 = 12

8+1=9

8 + 2 = 10

8 + 3 = 11

8 + 4 = 12

8 + 5 = 13

9 + 1 = 10

9 + 2 = 11

9 + 3 = 12

9 + 4 = 13

9 + 5 = 14

10 + 1 = 11

10 + 2 = 12

10 + 3 = 13

10 + 4 = 14

10 + 5 = 15

1+6=7

1+7=8

1+8=9

1 + 9 = 10

1 + 10 = 11

2+6=8

2+7=9

2 + 8 = 10

2 + 9 = 11

2 + 10 = 12

3+6=9

3 + 7 = 10

3 + 8 = 11

3 + 9 = 12

3 + 10 = 13

4 + 6 = 10

4 + 7 = 11

4 + 8 = 12

4 + 9 = 13

4 + 10 = 14

5 + 6 = 11

5 + 7 = 12

5 + 8 = 13

5 + 9 = 14

5 + 10 = 15

6 + 6 = 12

6 + 7 = 13

6 + 8 = 14

6 + 9 = 15

6 + 10 = 16

7 + 6 = 13

7 + 7 = 14

7 + 8 = 15

7 + 9 = 16

7 + 10 = 17

8 + 6 = 14

8 + 7 = 15

8 + 8 = 16

8 + 9 = 17

8 + 10 = 18

9 + 6 = 15

9 + 7 = 16

9 + 8 = 17

9 + 9 = 18

9 + 10 = 19

10 + 6 = 16

10 + 7 = 17

10 + 8 = 18

10 + 9 = 19

10 + 10 = 20

Memorizing these tables (especially the coloured part) will help you add fast. There are also ways in which you can train your brain to think a bit differently. Math E Magician

Additions Facts

EXAMPLES 1 34 + 8

34 + 10 = 44 – 2 = 42 (add 10 instead of 8 and then subtract 2)

2 65 + 18

65 + 20 = 85 – 2 = 83 (add 20 instead of 18, then subtract 2)

3 47 + 36

47 + 30 = 77 + 6 = 83 (add 3 tens the 6 ones)

236 + 613 236 + 600 = 836 + 10 = 846 + 3 = 849. This is done step-wise adding numbers from the left-hand side instead of the right-hand side. Do these steps mentally and speed will follow. Practice is the key word here!

658 + 284 658 + 200 = 858 + 80 = 938 + 4 = 942. Here you must be able to recognize and use the addition facts table to add the carry over figure quickly to the previous number. In the sum above, 5 + 8 = 13 so the 800 becomes 900.

826 + 649 826 + 600 = 1426 + 40 = 1466 + 9 = 1475 6 (66 + 9 = 75 using magic square technique). Try the worksheet below using the magic square and addition facts.

PRACTISE makes you PERFECT – WS/1

QUESTION

11 + 24

61 + 15

45 + 34

333 +112

25 + 55

125 + 77

71 + 33

712 + 334

19 + 14

10.

215 + 134

Math E Magician

Subtraction Facts

SUBTRACTION FACTS 18 - 9 = 9

17 - 8 = 9

16 - 7 = 9

15 - 6 = 9

14 - 5 = 9

13 - 4 = 9

17 - 9 = 8

16 - 8 = 8

15 - 7 = 8

14 - 6 = 8

13 - 5 = 8

12 - 4 = 8

16 - 9 = 7

15 - 8 = 7

14 - 7 = 7

13 - 6 = 7

12 - 5 = 7

11 - 4 = 7

15 - 9 = 6

14 - 8 = 6

13 - 7 = 6

12 - 6 = 6

11 - 5 = 6

10 - 4 = 6

14 - 9 = 5

13 - 8 = 5

12 - 7 = 5

11 - 6 = 5

10 - 5 = 5

13 - 9 = 4

12 - 8 = 4

11 - 7 = 4

10 - 6 = 4

12 - 9 = 3

11 - 8 = 3

10 - 7 = 3

11 - 9 = 2

10 - 8 = 2

12 - 3 = 9

11 - 2 = 9

11 - 3 = 8

10 - 2 = 8

10 - 3 = 7

10 - 9 = 1 9-9=0

8-8=0

7-7=0

6-6=0

5-5=0

4-4=0

9-8=1

8-7=1

7-6=1

6-5=1

5-4=1

4-3=1

9-7=2

8-6=2

7-5=2

6-4=2

5-3=2

4-2=2

9-6=3

8-5=3

7-4=3

6-3=3

5-2=3

4-1=3

9-5=4

8-4=4

7-3=4

6-2=4

5-1=4

9-4=5

8-3=5

7-2=5

6-1=5

9-3=6

8-2=6

7-1=6

9-2=7

8-1=7

3-3=0

2-2=0

3-2=1

2-1=1

3-1=2

9-1=8

Both additions and subtractions can be done faster if you know the facts given above very well. EXAMPLES 1 56 - 9 = 47 (use the magic square)

746 - 534 2

746 - 500 = 246 - 30 = 216 - 4 = 212 (gostep-wise from left to right)

852 - 684 852 - 600 = 252 - 80 = 172 3 (to subtract 80, first subtract 100 and then add 20), lastly, 172 - 4 = 168 which is our answer! Math E Magician

Make Your Own Memory Game!

Make Your Own Memory Game! These facts can be made into a card game where each block is written as a card like this, 4 + 2 on one side and 6 on the other side as given below. 4+2

Side 1

Side 2

7+5

Side 1

Side 2

When all the cards are made, arrange the cards as per the columns in the add/sub facts.

Take the cards of the addition fact of one or two numbers at a time, say 2 and 3 only.

Shuffle the cards with side 1 on top like a normal card game.

Select one card.

Suppose the card you select has 2 + 6 on it. Then you must call out the answer as 8.

Check at the back of the card, as the answer is written there.

If you get the right answer give yourself one point, and keep the card by your side.

If your answer is wrong then put the card back in the pack.

Shuffle again and take another card and continue till you know all the facts of the pack.

Add more and more cards till all the facts are done.

You can play this game with your friends too.

Once you know the addition and subtraction facts, you will save time while doing these operations. While you are at it, learn your multiplication tables upto 20. Our brain is amazing but not used fully. So committing certain facts to memory is good for quick recall, when there is limited time.

Math E Magician

Practise Makes You Perfect

PRACTISE makes you PERFECT – WS/2 SN

QUESTION

654 - 531

645 – 143

942 - 419

593 – 421

789 - 145

8174 – 5342

555 - 234

6473 – 5161

715 - 427

10.

416 + 637 – 349

Amazing Facts If you add up all of the consecutive numbers from 1 to 100, (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + ...) the total will be 5, 050. 111 111 111 × 111 111 111 = 12345678 9 87654321

PRACTISE makes you PERFECT – WS/3 37 + 46 49 + 24 45 + 61 55 + 12 67 + 34 49 + 61 69 + 23 55 + 67 26 + 39 74 + 61 54 + 69 28 + 46 17 + 71 56 + 79 84 + 29 Math E Magician

43 – 19 24 - 16 55 – 39 71 – 46 78 – 62 88 – 59 62 – 58 43 – 28 94 – 67 82 – 64 96 - 39 74 - 61 154 - 69 88 - 46 117 - 71

637 + 285 214 + 523 852 + 123 741 + 642 951 + 159 456 +753 124 + 624 864 + 382 715 + 198 972 + 279 541 + 698 258 + 465 197 + 271 564 + 719 841 + 729

746 – 324 88 – 259 162 – 58 343 – 128 894 – 657 782 – 364 67 – 391 745 – 641 854 – 696 188 - 146 117 – 71 674 –261 154 – 69 588 - 346 417 – 371 15

Speed Multiplication

SPEED MULTIPLICATION To memorize multiplication tables is a good trait indeed. Multiplication can also be done very fast using some other rules based on patterns in Math. In the next few pages I am going to share with you some methods that cut down your calculation time to a half and at the same time, accuracy is not compromised. Let’s check out the rules of simple multiplication by 11 to 19

RULE 1 ZIP-ZAP Multiplication By 11 EXAMPLE 1: 23456 X 11. Steps for Multiplication: As usual, we will begin our multiplication starting from ones place proceeding to the tens, hundreds and so on. Step 1 First put a 0 in front of the leftmost number – 023456 X 11 Step 2 The digit that we will multiply will be denoted as “N” Step 3 The number to its right will be its neighbor “n”. Step 4 In the example above, if I am multiplying 5 by 11, 5 will be called the number “N” while 6, the number to the right, will be its neighbor “n” Step 5 4 will be the neighbor of 3 and 6 has no neighbor at all! Step 6 The Rule is: To the number, add its neighbor on the right hand side. Step 7 FORMULA: (N+n) Step 8 Now instead of multiplying by 11 we will multiply by 1, which is the digit in ones place of the multiplier 11 Step 9 So number (6 x 1) + neighbor (o) = 6 Step 10 Repeat this step for each digit of the multiplicand (number you are multiplying). Understand the steps done below and start enjoying speed while calculating. Check out the steps done in example given below

Math E Magician

Speed Multiplication

EXAMPLE 1 0 2 3 4 5 6 x 1 1 .......... Start from 6 .......... 6 (N) x 1 + 0 (n), so write it down as 6 6 .......... (5 x 1) + 6 = 11, write down 1, carry over 1 16 .......... (4 x 1) + 5 + 1 = 10, write down 0, carry over 1 016 .......... (3 x 1) + 4 + 1 = 8, write 8, no carry over 8016 .......... (2 x 1) + 3 = 5, write 5, no carry over 58016 .......... (0 x 1) + 2 = 2 2 58016 The answer is 23456 X 11 = 258016 EXAMPLE 2 0 4 2 8 3 X 11 3 13 113 7 113 4 7113

.......... .......... .......... .......... .......... ..........

Start from 3 3 (N) x 1 + 0 (n), so write it down as 3 (8 x 1) + 3 = 11, write down 1, carry over 1 (2 x 1) + 8 + 1 = 11, write down 1, carry over 1 (4 x 1) + 2 + 1 = 7, write 7, no carry over (0 x 1) + 4 = 4

The answer is 4283 X 11 = 47113

Amazing Facts Sequential 1â&#x20AC;&#x2122;s with 9 1 x 9 + 2 = 11 12 x 9 + 3 = 111 123 x 9 + 4 = 1111 1234 x 9 + 5 = 11111 12345 x 9 + 6 = 111111 123456 x 9 + 7 = 1111111 1234567 x 9 + 8 = 11111111 12345678 x 9 + 9 = 111111111 123456789 x 9 + 10 = 1111111111

Math E Magician

Speed Multiplication

RULE 2 ZIP-ZAP Multiplication By 12 Steps for Multiplication: Step 1 Put a 0 in front of the leftmost number. Step 2 Begin as usual from ones place and proceed to the tens, hundreds etc. Step 3 Multiply each digit by 2 and then add the neighbor on the right. Step 4 FORMULA: (2N + n)

Letâ&#x20AC;&#x2122;s check out the steps. EXAMPLE 1 Multiply 75321 by 12 0 7 5 3 2 1 X 1 2 .......... start from 1 2 .......... (1 x 2) = 2, write it below 1 52 .......... (2 x 2) + 1 = 5 852 .......... (3 x 2) + 2 = 8 3852 .......... (5 x 2) + 3 = 13, write 3, carry over 1 03852 .......... (7 x 2) + 5 + 1 = 20, write 0, carry over 2 9 0 3852 .......... (0 x 2) + 7 + 2 = 9 The answer is 75321 x 12 = 903852

EXAMPLE 2 Multiply 68945 by 12 068945X12 .......... (5 x 2) = 10, write 0, carry over 1 0 .......... (4 x 2) + 5 +1 = 14, write 4, carry over 1 40 .......... (9 x 2) + 4 + 1 = 23, write 3, carry over 2 340 .......... (8 x 2) + 9 + 2 = 27, write 7, carry over 2 7340 .......... (6 x 2) + 8 + 2 = 22, write 2, carry over 2 27340 .......... (0 x 2) + 6 + 2 = 8 8 27340 The answer is 68945 X 12 = 827340

Math E Magician

Speed Multiplication

PRACTISE makes you PERFECT – WS/4 SN

QUESTION

X 12

11.

8421103

6102356 X 11

12.

67883566 X 12

X 11

9938874 X 12

13.

23557

56421

X 11

3211342 X 11

14.

10283746 X 12

78963

X 11

10.

799445

15.

289514

QUESTION

21546

X 11

78334

65423

X 11

85231

4. 5.

X 12

X 11

Amazing Facts Sequential 8’s with 9 9 x 9 + 7 = 88 98 x 9 + 6 = 888 987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888 98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888 9876543 x 9 + 1 = 88888888 98765432 x 9 + 0 = 888888888

Math E Magician

Speed Multiplication

RULE 3 ZIP-ZAP Multiplication By 13 Steps for multiplication Step 1 Put a 0 in front of the leftmost number. Step 2 Begin from the digit in ones place. Step 3 Multiply each digit by 3 and then add the neighbor on the right. Step 4 FORMULA: (3N + n)

EXAMPLE 1 Multiply 25021 by 13 025021X13 .......... (1 x 3) = 3, write it below 1 3 .......... (2 x 3) + 1 = 7 73 .......... (0 x 3) + 2 = 2. 273 .......... (5 x 3) + 0 = 15, write 5, carry over 1 5273 .......... (2 x 3) + 5 + 1 = 12, write 2, carry over 1 25273 .......... (0 x 3) + 2 + 1 = 3 3 25273 The answer is 25021 x 13 = 325273 EXAMPLE 2 Multiply 68742 by 13 068742X13 .......... (2 x 3) = 6, write 6 6 .......... (4 x 3) + 2 = 14, write 4, carry over 1 46 .......... (7 x 3) + 4 + 1 = 26, write 6, carry over 2 646 .......... (8 x 3) + 7 + 2 = 33, write 3, carry over 3 3646 .......... (6 x 3) + 8 + 3 = 29, write 9, carry over 2 93646 .......... (0 x 3) + 6 + 2 = 8 8 93646 The answer is 68742 x 13 = 893646 Now try this in one step 541256 x 13

Math E Magician

Speed Multiplication

RULE 4 ZIP-ZAP Multiplication By 14 Steps for Multiplication Step 1 Put a 0 in front of the leftmost number. Step 2 Begin from the digit in ones place. Step 3 Multiply each digit by 4 and then add the neighbor on the right. Step 4 FORMULA (4N + n)

EXAMPLE 1 Multiply 75321 by 14 075321x14 .......... (1 x 4) = 4, write it below 1 4 .......... (2 x 4) + 1 = 9 94 .......... (3 x 4) + 2 = 14, write 4, carry over 1 494 .......... (5 x 4) + 3 + 1= 24, write 4, carry over 2 4494 .......... (7 x 4) + 5 + 2 = 35, write 5, carry over 3 54494 .......... (0 x 4) + 7 + 3 = 10 1054494 The answer is 75321 x 14 = 1054494 EXAMPLE 2 Multiply 68945 by 14 068945x14 .......... (5 x 4) = 20, write 0, carry over 2 0 .......... (4 x 4) + 5 + 2 = 23, write 3, carry over 2 30 .......... (9 x 4) + 2 + 4 = 42, write 2, carry over 4 230 .......... (8 x 4) + 9 + 4 = 45, write 5, carry over 4 5230 .......... (6 x 4) + 8 + 4 = 36, write 6, carry over 3 65230 .......... (0 x 4) + 6 + 3 = 9 965230 The answer is 68945 X 14 = 965230 Now try this in one step 124563 x 14

Math E Magician

Speed Multiplication

PRACTISE makes you PERFECT – WS/5 SN

QUESTION

64837264 X 13

976667 X 14

11.

246987 X 12

978686

X 14

535326 X 12

12.

56745

136288

X 12

65435

X 11

13.

649238 X 13

548364

X 11

689607 X 13

14.

918367 X 14

1378237 X 13

10.

137879 X 14

15.

10372

QUESTION

X 11

X 12

LOOKING BACK TO REVIEW

To Multiply by 14

To Multiply by 11

• Multiply the number in ones place by 4

• Multiply the number in ones place by 1

Multiplication rules of 11-14 in a mind map

To Multiply by 13

To Multiply by 12

• Multiply the number in ones place by 3

• Multiply the number in ones place by 2

Math E Magician

Speed Multiplication

RULE 5 ZIP-ZAP Multiplication By 15 Steps for multiplication Step 1 Put a 0 in front of the leftmost number. Step 2 Begin from the digit in ones place. Step 3 Multiply each digit by 5 and then add the neighbor on the right. Step 4 FORMULA: (5N + n)

EXAMPLE 1 Multiply 75421 by 15 075421X15 .......... (1 x 5) = 5, write 5 below 1 5 .......... (2 x 5) + 1 = 11, write 1, carry over 1 15 .......... (4 x 5) + 2 + 1 = 23, write 3, carry over 2 315 .......... (5 x 5) + 4 + 2 = 31, write 1, carry over 3 1315 .......... (7 x 5) + 5 + 3 = 43, write 3, carry over 4 31315 .......... (0 x 5) + 4 + 7 = 11 1131315 The answer is 75421 x 15 = 1131315 EXAMPLE 2 Multiply 68945 by 15 068945X15 .......... (5 x 5) = 25, write 5, carry over 2 5 .......... (4 x 5) + 5 + 2 = 27, write 7, carry over 2 75 .......... (9 x 5) + 4 + 2 = 51, write 1, carry over 5 175 .......... (8 x 5) + 9 + 5 = 54, write 4, carry over 5 4175 .......... (6 x 5) + 8 + 5 = 43, write 3, carry over 4 34175 .......... (0 x 5) + 6 + 4 = 10 1034175 The answer is 68945 X 15 = 1034175

Math E Magician

Speed Multiplication

RULE 6 ZIP-ZAP Multiplication By 16 Steps for multiplication Step 1 Put a 0 in front of the leftmost number. Step 2 Begin from the digit in ones place. Step 3 Multiply each digit by 6 and then add the neighbor on the right. Step 4 FORMULA: (6N + n)

EXAMPLE 1 Multiply 75321 by 16 075321X16 .......... (1 x 6) = 6, write it below 1 6 .......... (2 x 6) + 1 = 13, write 3, carry over 1 36 .......... (3 x 6) + 2 + 1 = 21, write 1, carry over 2 136 .......... (5 x 6) + 2 + 3 = 35, write 5, carry over 3 5136 .......... (7 x 6) + 5 + 3 = 50, write 0, carry over 5 05136 .......... (0 x 6) + 7 + 5 = 12 1205136 The answer is 75321 x 16 = 1205136 EXAMPLE 2 Multiply 68945 by 16 068945x16 .......... (5 x 6) = 30, write 0, carry over 3 0 .......... (4 x 6) + 5 + 3 = 32, write 2, carry over 3 20 .......... (9 x 6) + 4 + 3 = 61, write 1, carry over 6 120 .......... (8 x 6) + 9 + 6 = 63, write 3, carry over 6 3120 .......... (6 x 6) + 8 + 6 = 50, write 0, carry over 5 03120 .......... (0 x 6) + 5 + 6 = 11 1103120 The answer is 68945 X 16 = 1103120

Math E Magician

Speed Multiplication

RULE 7 ZIP-ZAP Multiplication By 17 Steps for multiplication Step 1 Put a 0 in front of the leftmost number. Step 2 Begin from the digit in ones place. Step 3 Multiply each digit by 7 and then add the neighbor on the right. Step 4 FORMULA: (7N + n)

EXAMPLE 1 Multiply 75321 by 17 075321X17 .......... (1 x 7) = 7, write it below 1 7 .......... (2 x 7) + 1 = 15, write 5, carry over 1 57 .......... (3 x 7) + 2 + 1 = 24 write 4, carry over 2 457 .......... (5 x 7) + 3 + 2= 40, write 0, carry over 4 0457 .......... (7 x 7) + 5 + 4 = 58, write 8, carry over 5 80457 .......... (0 x 7) + 7 + 5 = 12 1280457 The answer is 75321 x 17 = 1280457 EXAMPLE 2 Multiply 68945 by 17 068945X17 .......... (5 x 7) = 35, write 5, carry over 3 5 .......... (4 x 7) + 5 + 3 = 36, write 4, carry over 3 65 .......... (9 x 7) + 4 + 3 = 70, write 0, carry over 7 065 .......... (8 x 7) + 9 + 7 = 72, write 2, carry over 7 2065 .......... (6 x 7) + 8 + 7 = 57, write 7, carry over 5 72065 .......... (0 x 7) + 6 + 5 = 11 1172065 The answer is 68945 X 17 = 1172065

Math E Magician

Speed Multiplication

RULE 8 ZIP-ZAP Multiplication By 18 Steps for multiplication Step 1 Put a 0 in front of the leftmost number. Step 2 Begin from the digit in ones place. Step 3 Multiply each digit by 8 and then add the neighbor on the right . Step 4 FORMULA: (8N + n)

EXAMPLE 1 Multiply 75321 by 18 075321X18 .......... (1 x 8) = 8, write it below 1 8 .......... (2 x 8) + 1 = 17, write 7, carry over 1 78 .......... (3 x 8) + 2 + 1= 27, write 7, carry over 2 778 .......... (5 x 8) + 3 + 2 = 45, write 5, carry over 4 5778 .......... (7 x 8) + 5 + 4 = 65, write 5, carry over 6 55778 .......... (0 x 8) + 7 + 6 = 13 1355778 The answer is 75321 x 18 = 1355778 EXAMPLE 2 Multiply 68945 by 18 068945X18 .......... (5 x 8) = 40, write 0, carry over 4 0 .......... (4 x 8) + 5 + 4 = 41, write 1, carry over 4 10 .......... (9 x 8) + 4 + 4 = 80, write 0, carry over 8 010 .......... (8 x 8) + 9 + 8 = 81, write 1, carry over 8 1010 .......... (6 x 8) + 8 + 8 = 64, write 4, carry over 6 41010 .......... (0 x 8) + 6 + 6 = 12 1241010 The answer is 68945 x 18 = 1241010

Math E Magician

Speed Multiplication

RULE 9 ZIP-ZAP Multiplication By 19 Steps for multiplication Step 1 Put a 0 in front of the leftmost number. Step 2 Begin from the digit in ones place. Step 3 Multiply each digit by 9 and then add the neighbor on the right. Step 4 FORMULA: (9N + n)

EXAMPLE 1 Multiply 75321 by 19 075321X19 .......... (1 x 9) = 9, write it below 1 9 .......... (2 x 9) + 1 = 19, write 9, carry over 1 99 .......... (3 x 9) + 2 + 1 = 30, write 0, carry over 3 099 .......... (5 x 9) + 3 + 3 = 51, write 1, carry over 5 1099 .......... (7 x 9) + 5 + 5 = 73, write 3, carry over 7 31099 .......... (0 x 9) + 7 + 7= 14 1431099 The answer is 75321 x 19 = 1431099 EXAMPLE 2 Multiply 68945 by 19 068945X19 .......... (5 x 9) = 45, write 5, carry over 4 5 .......... (4 x 9) + 5 + 4 = 45, write 5, carry over 4 55 .......... (9 x 9) + 4 + 4 = 89, write 9, carry over 8 955 .......... (8 x 9) + 9 + 8 = 89, write 9, carry over 8 9955 .......... (6 x 9) + 8 + 8 = 70, write 0, carry over 7 09955 .......... (0 x 9) + 6 + 7 = 13 1309955 The answer is 68945 x 19 = 1309955 * You may apply the same rules to multiplication of decimal numbers, taking care to put the decimal points in the correct place in the answer.

Math E Magician

Speed Multiplication

PRACTISE makes you PERFECT – WS/6 SN

QUESTION

65783

X 16

5676211 X 17

11.

57659

43649

X 13

5473264 X 18

12.

657812 X 16

5478564 X 15

669843 X 12

13.

987987 X 15

123687

X 12

364873 X 15

14.

34565

876987

X 14

10.

123273 X 19

15.

123623 X 19

QUESTION

QUESTION X 11

X 16

LOOKING BACK TO REVIEW

To Multiply by 18

To Multiply by 15

• Multiply the number in ones place by 8

• Multiply the number in ones place by 5

Multiplication rules of 15-18 in a mind map

To Multiply by 17

To Multiply by 16

• Multiply the number in ones place by 7

• Multiply the number in ones

Math E Magician

Multiplication by Multiples

RULE 10 ZIP-ZAP Multiplication By Multiples Of 11-19 Now that you know how to speed multiply by 11 to 19 let us look at some applications of the same rules to multiply by bigger numbers. *Understand that multiplication of any number by 2 means just doubling each digit starting from the ones place *Dividing by 2 is simply halving the number starting from the left most digits. EXAMPLE 1 238485769x2 =476971538 Now look at this example: 54123 X 22 Since the factors of 22 are 11, 2 (22 = 11 x 2), we can multiply 54123 by 11 first and then double the answer obtained. 0 5 4 1 2 3 X 22 (11 x 2) 54123 x 11 = 595353 (Rule N + n) for factor 11 595353 x 2 = 1190706. Doubling for factor 2 So 54123 X 2 = 1190706 Any big multiplier can be resolved into factors first and then using rules of multiplication, multiply the given number by the factors, one at a time. We used the (N + n) Rule for x 11 first and then doubled each digit of the answer thus obtained, in example 1. EXAMPLE 2 0 3 4 1 2 3 x 36 - Factors of 36 are 18 and 2 (18 x 2 = 36). So use the Rule (8N + n) and then double up the answer obtained, so that you get the final answer. Check this out 0 3 4 1 2 3 x 36 (18 x 2). Multiply first by factor 18 then by factor 2 34123 x 18 = 614214 ......... (8N+n) multiplication by 18 as it is the bigger factor 614214 x 2 = 1228428 ......... (doubling) or multiplying by factor 2 So, 34123 x 36 = 1228428 28 ......... Factors of 28 are 14 and 2 (14 x 2 = 28), 1 so use the Rule (4N + n) and then double up the number thay you abtain.

Math E Magician

Multiplication by Multiples

144 ......... Factors are 12 and 12 (12 x 12 = 144), so use the Rule (2N + n) twice! 2 225 ......... Factors are 15 and 15 (15 x 15 = 225), 3 so use the Rule (4N + n) and then double up the answer that you get. 48 ......... Factors are 12, 2 and 2 (12 x 2 x 2 = 48), 4 so use the Rule (2N + n) and then double up the number you get two times for 2 x 2 *Factorize the multiplier and multiply in steps.

PRACTISE makes you PERFECT â&#x20AC;&#x201C; WS/7 SN

QUESTION

54160 x 32

71562 x 36

11.

7482 x 144

64132 x 22

54123 x 121

12.

29472 x 48

46521 x 24

6231 x 132

13.

56412 x 38

58026 x 26

5236 x 225

14.

75321 x 22

78123 x 28

10.

4563 x 169

15.

5546 x 1331

Answers

1733120

2576232

1077408

1410904

6548883

1414656

1116504

8225492

2143656

1508676

1178100

1657062

2187444

771147

7381726 Math E Magician

Checking the Answers

CHECKING THE ANSWERS RULE 11 Method 1 : Digit-Sum and Casting Out 9s What is the digit sum of the number 435673? Adding the digits of the number to obtain a single digit is called its digit sum. This can be done as a simple addition, till you get a single digit as shown below. Digit sum of 435673 will be 4 + 3 + 5 + 6 + 7 + 3 = 28. We go on to obtain a single digit for 28 as 2 + 8 = 10, and further as 1 + 0 = 1. So the digit sum (DS) of 435673 is 1 Or knock off the 9s in the number and then add the digits. You can even knock off digits giving a sum of 9. 4 3 5 6 7 3 ......... 5 + 4 = 9, 6 + 3 = 9 So knock it out and add only 7 + 3 = 10, 1 + 0 = 1 (DS is the same as above) Let us find the digit sum of 6 7 5 3 9 2 4 4 ......... 7 + 2, 6 + 3, 5 + 4, 9 these combinations of 9 can be knocked off to get the digit sum as 4 or you may add all the digits to a total of 40, then 4 + 0 = 4 so the digit sum is 4. Casting out nines makes it easier to find the digit sum.

To Check an Answer : Method 1 In order to check if the answer is correct we can use the Digit sum rule. The digit sum of the left hand side (LHS) must equal the digit sum of the right hand side. (RHS) EXAMPLE 1 561 x 121 = 67881 Digit sum of LHS = (5+6+1) x (1+2+1) add digits of each of the two numbers = 12 x 4 = 3 x 4 = 12 (adding digits to get a single digit) = 3 (1+2) Digit sum of RHS = 6 + 7 + 8 + 8 + 1 = 30 = 3 (3+0) So LHS = RHS, and our answer is correct

Math E Magician

Checking the Answers

EXAMPLE 2 6734 x 821 Digit sum of LHS = (6+7+3+4) x (8+2+1) = 20 x 11 = 2 x 2 Digit sum of RHS = 5 + 5 + 2 + 8 + 6 + 1 + 4 = 31 = 4 (3+1) So LHS = RHS, and our answer is correct

5528614

This check method can be used to check any answer for multiplication, addition and subtraction. For division you must use the formula: Dividend = (divisor x quotient) + remainder

RULE 12 Method 2 : Elevens Rule Checking by this rule for the accuracy of your answer will leave no doubt for error. The rule is simple for a two digit number. Subtract the digit in the tens place FROM the digit in the ones place to get the elevens check number. EXAMPLE: The elevens check number for 79 will be 9 – 7 = 2 But if the number is 97 and the tens digit is greater than the ones digit, you must add 11 to the ones digit and then subtract. So for 97, the working will be : Step 1 : 7 + 11 = 18; Step 2 : 18 – 9 = 9 For a bigger number the method resembles the divisibility test for 11 Let us take the number 284769. We need to add the alternate digits starting from ones place. Number - 2 8 4 7 6 9 Now 9 + 7 + 8 = 24 (odd places) and 6 + 4 + 2 = 12 (even places) Subtract sum of evenly placed digits from sum of oddly placed digits, that is 24 – 12 = 12 This can be still simplified as 2 – 1 = 1 to get a single digit answer So 1 is the elevens check number. Sometimes you may get the sum of the evenly placed digits to be greater than the oddly placed ones. You have to then add 11 to the oddly placed digits and then subtract to get the elevens check number.

Math E Magician

Checking the Answers

EXAMPLE 64532 x 12 = 774384 Left hand side (LHS) is 6 4 5 3 2 x 1 2. Start from ones place of the multiplicand 2, 5, 6 are in odd places while 3, 4 are in even places. 2 + 5 + 6 = 13 and 3 + 4 = 7; then 13 – 7 = 6. The elevens check number of the multiplier is 2 – 1 = 1. So 6 X 1 = 6 is the check number. Right hand side (RHS) is 774384. To find the elevens check number we will add 4 + 3 + 7 = 14 and 8 + 4 + 7 = 19. Now since we cannot do 14 – 19 we must add 11 to 14. So (11 + 14) – 19, 25 – 19 = 6 So the elevens check numbers of the two sides are the same and the answer is correct.

PRACTISE makes you PERFECT – WS/8 Check the answers by both the methods. 546321 x 12 = 6555852

75321 x 14 = 1054494

15963 x 18 = 287334

85214 x 13 = 1107782

46318 x 15 = 694770

71395 x 16 = 1142320

15987 x 17 = 271779

5689 x 1237 =7037293

96193 x 19 = 1827667

CHECKING DIFFERENT MATHEMATICAL OPERATIONS Multiplication

Addition

Square of a number

Find the digit sum of each number. DS* of multiplicand x DS* of multiplier should be equal to DS* of the product. To check use the formula Dividend = (divisor x quotient) + remainder DS* of the total or the sum should be equal to the DS* of all the numbers added The DS* of the bigger number – the DS* of the smaller number = DS* of the difference DS* of the number multiplied by itself is equal to the DS* of the square number. DS* of the number multiplied itself thrice is equal to the DS* of the cube number.

Division

Subtraction

Cube of a number

Digit Sum

Math E Magician

Zip - Zap Multiplication

RULE 13 ZIP-ZAP Multiplication of A 2-Digit Nymber By A 2-Digit Number STAR PATTERNS FOR MULTIPLICATION 1

ADD Follow the star pattern given above to do the multiplication EXAMPLE 1 2 4 Step 1 X 3 5 0

Multiply 4 x 5 = 20, write 0, carry over 2

2 4 Step 2 X 3 5 4 0

Multiply (5 x 2) + (4 x 3) + 2 = 24, write 4, carry over 2

2 4 Step 3 X 3 5 8 4 0

Multiply (2 x 3) + 2 = 8, write the answer.

The answer is 24 X 35 = 840 EXAMPLE 2 Step 1 6 8 Step 2 X 4 6 Step 3 3 1 2 8

Multiply 8 x 6 = 48, write 8, carry over 4 Multiply (6x6) + (4x8) + 4 =72, write 2, carry over 7 Multiply (4x6) + 7 = 31, write the answer.

The answer is 68 X 46 = 3128 You may even attempt to do the sum horizontally like this: 51 x 63 = 3213, keep the steps in mind. As 3 goes ahead multiplying the number starting from ones place in the first step, tens place in the second step, multiplication by 6 follows close behind.

Math E Magician

Zip - Zap Multiplication

PRACTISE makes you PERFECT – WS/9 QUESTION

QUESTION ANSWER

QUESTION

ANSWER

65 x 45

2945

68 x 54

3672

82 x 15

1230

41 x 63

2583

92 x 76

6992

72 x 93

6696

84 x 71

5964

64 x 36

2304

78 x 36

2808

81 x 67

5427

97 x 96

9312

93 x 47

4371

Try and write your birth date in Roman numerals. 2015 is written as MMXV

ROMAN NUMERALS

Math E Magician

5000

10000

100

50000

500

100000

1000

500000

1000000

Zip - Zap Multiplication

RULE 14 ZIP-ZAP Multiplication of A 3-Digit Nymber By A 2-Digit Number STAR PATTERNS FOR MULTIPLICATION 1

ADD

Follow the star pattern given above to do the multiplication EXAMPLE 1 Step 1 5 6 4 (7 x 4 = 28), write 8, carry 2 X 3 7 8

5 6 4 Step 2 X 3 7 (7 x 6) + (3 x 4) + 2 = 56, write 6, carry over 5 6 8 5 6 4 Step 3 X 3 7 (7 x 5) + (3 x 6) + 5 = 58, write 8, carry 5 8 6 8 5 6 4 Step 4 X 3 7 (3 x 5) + 5 = 20 2 0 8 6 8 The answer is 564 x 37 = 20868 In one step : EXAMPLE 2 Step 1 3 7 2 (6 x 2=12), write 2 and carry-over 1 X 4 6 (6 x 7) + (4 x 2) + 1 = 51. Write 1 and carry-over 5 Step 2 1 7 1 1 2 (6 x 3) + (4 x 7) + 5 = 51. Write 1 carry-over 5. Step 3 (4 x 3) + 5 = 17, write it to get the answer Step 4

Math E Magician

Zip - Zap Multiplication

PRACTISE makes you PERFECT – WS/10 QUESTION

QUESTION ANSWER

QUESTION

ANSWER

165 x 45

7425

618 x 54

33372

282 x 15

4230

421 x 63

26523

392 x 76

29792

732 x 93

68076

484 x 71

34364

644 x 36

23184

578 x 36

20808

851 x 67

57017

697 x 96

66912

963 x 47

45261

612 x 45

27540

571 x 46

26266

854 x 32

27328

627 x 16

10032

541 x 78

42198

312 x 43

13416

*Check each answer by both the rules.

∏ (Pi) written to 50 decimal places: 3.14159265358979323846264338327950288419716939937510

RAMANUJAN'S MAGIC SQUARE 22

What is the total of each row and each column? Math E Magician

Zip - Zap Multiplication

RULE 15 ZIP-ZAP Multiplication of A 3-Digit Nymber By A 3-Digit Number STAR PATTERNS FOR MULTIPLICATION 1

ADD

EXAMPLE 1 Step 1 7 6 2 (4 x 2 = 8), write 8 X 1 3 4 8

7 6 2 Step 2 X 1 3 4 (4 x 6) + (3 x 2) = 30, write 0, carry over 3 0 8 7 6 2 Step 3 X 1 3 4 (4 x 7) + (3 x 6) + (1 x 2) + 3 = 51, write 1, carry over 5 1 0 8 7 6 2 X 1 3 4 (3 x 7) + (1 x 6) + 5 = 32, write 2, carry over 3 Step 4 2 1 0 8 7 6 2 Step 5 X 1 3 4 (1 x 7) + 3 = 10, 1 0 2 1 0 8 The answer is 762 x 134 = 102108 EXAMPLE 2 Step 1 3 7 2 (6 x 2=12), write 2 and carry-over 1 Step 2 X 2 4 6 (6 x 7) + (4 x 2) + 1 = 51. Write 1 and carry-over 5 Step 3 9 1 5 1 2 (6 x 3) + (4 x 7) + (2 x 2) + 5 =55. Write 5 and carry-over 5. Step 4 (4 x 3) + (2 x 7) + 5 = 31, Write 1 and carry-over 3 Step 5 (2 x 3) + 3 = 9, write 9 to get the answer.

Math E Magician

Zip - Zap Multiplication

PRACTISE makes you PERFECT – WS/11 QUESTION

QUESTION ANSWER

QUESTION

ANSWER

a) 546 X 123

67158

e) 294 X 449

132006

b) 852 X 417

355284

f) 748 X 602

450296

c) 996 X 421

419316

g) 202 X 115

23230

d) 661 X 902

596222

h) 258 X 147

37926

The word ‘mathematics’ comes from the Greek máthēma, which means learning, study, science. Do you know a word known as Dyscalculia? Dyscalculia means difficulty in learning arithmetic, such as difficulty in understanding numbers, and learning maths fact Notches (cuts or indentation) on animal bones prove that humans have been doing mathematics since around 30,000 BC. What comes after a million, billion and trillion? A quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion and undecillion. The number 5 is pronounced as ‘Ha’ in Thai language. 555 is also used by some as slang for ‘HaHaHa’. Different names for the number 0 include zero, nought, naught, nil, zilch and zip. Zero ( 0 ) is the only number which cannot be represented by Roman numerals. Courtesy Google

Math E Magician

Zip - Zap Multiplication

RULE 16 ZIP-ZAP Multiplication of A 4-Digit Nymber By A 2-Digit Number STAR PATTERNS FOR MULTIPLICATION 1

ADD

EXAMPLE 1 Step 1 5 7 3 2 (5 x 2 = 10), write 0, carry over 1 X 4 5 0

5 7 3 2 Step 2 X 4 5 (5 x 3) + (4 x 2) +1 = 24, write 4, carry over 2 4 0 5 7 3 2 Step 3 X 4 5 (5 x 7) + (4 x 3) + 2 = 49, write 9, carry over 4 9 4 0 5 7 3 2 Step 4 X 4 5 (5 x 5) + (4 x 7) + 4= 47, write 7, carry over 4 7 9 4 0 5 7 3 2 Step 5 X 4 5 (4 x 5) + 4 = 24, write 24 2 5 7 9 4 0 The answer is 5732 x 45 = 257940 Speed Multiplication 6 3 0 7 (6 x 7 = 42), write 2, carry over 4 Step 1 X 3 6 (6 x 0) + (7 x 3) +4 = 25, write 5, carry over 2 Step 2 2 2 7 0 5 2 (6 x 3) + (3 x 0) + 2 = 20, write 0, carry over 2 Step 3 (6 x 6) + (3 x 3) + 2= 47, write 7, carry over 4 Step 4 (3 x 6) + 4 = 22, write 22 Step 5

Math E Magician

Zip - Zap Multiplication

RULE 17 ZIP-ZAP Multiplication of A 4-Digit Nymber By A 3-Digit Number STAR PATTERNS FOR MULTIPLICATION 1

ADD

EXAMPLE 1 5 2 4 7 Step 1 X 4 3 5 (5 x 7 = 35), write 5, carry over 3 5 5 2 4 7 X 4 3 5 (5 x 4) + (3 x 7) + 3 = 44, write 4, carry over 4 Step 2 4 5 5 2 4 7 Step 3 X 4 3 5 (5 x 2) + (3 x 4) + (4 x 7) + 4 = 54, write 4, carry over 5 4 4 5 5 2 4 7 Step 4 X 4 3 5 (5 x 5) + (3 x 2) + (4 x 4) + 5 = 52, write 2, carry over 5 2 4 4 5 5 2 4 7 Step 5 X 4 3 5 (3 x 5) + (4 x 2) + 5 = 28, write 8, carry over 2 8 2 4 4 5 5 2 4 7 Step 6 X 4 3 5 (4 x 5) + 2 = 22, write 22. 2 2 8 2 4 4 5 The answer is 5247 x 435 = 2282445 Now we can try a sum and get the answer in one step. You may revise the star pattern before going ahead.

Math E Magician

Zip - Zap Multiplication

In one Step Step 1 6 3 2 9 Step 2 X 7 4 1 Step 3 4 6 8 9 7 8 9 Step 4 Step 5 Step 6

(1 x 9 = 9), write 9 (1 x 2) + (4 x 9) =38, write 8, carry over 3 (1 x 3) + (4 x 2) + (7 x 9) + 3 = 77, write 7, co 7 (1 x 6) + (4 x 3) + (7 x 2) + 7 = 39, write 9, co 3 (4 x 6) + (7 x 3) + 3 = 48, write 8, carry over 4 (7 x 6) + 4 = 46, write 46

The answer is 6329 x 741 = 4689789

PRACTISE makes you PERFECT â&#x20AC;&#x201C; WS/12 QUESTION

QUESTION ANSWER

QUESTION

ANSWER

5487 x 654

3588498

6541 x 236

1543676

7452 x 213

1587276

9631 x 523

5037013

8521 x 521

4439441

6543 x 753

4926879

9512 x 247

2349464

1236 x 456

563616

7412 x 258

1912296

7592 x 515

3909880

The biggest prime number yet discovered is 2 raised to 32,582,857 minus 1. The number has 9,808,358 digits, enough to fill more than 10 books with 350 pages.

Math E Magician

Zip - Zap Multiplication

RULE 18 ZIP-ZAP Multiplication of A 4-Digit Nymber By A 4-Digit Number STAR PATTERNS FOR MULTIPLICATION 1

ADD

EXAMPLE 1 Step 1 (3 x 1 = 3), write 3 7 2 3 1 X 6 2 4 3 3

7 2 3 1 6 2 4 3 Step 2 (3 x 3) + (4 x 1) = 13, write 3, carry over 1 3 3 7 2 3 1 Step 3 (3 x 2) + (4 x 3) + (2 x 1) + 1 = 21, write 1, carry over 2 X 6 2 4 3 1 3 3 7 2 3 1 X 6 2 4 3 Step 4 (3 x 7) + (4 x 2) + (2 x 3) + (6 x 1) + 2 = 43, write 3, carry over 4 3 1 3 3 7 2 3 1 Step 5 (4 x 7) + (2 x 2) + (6 x 3) + 4 = 54, write 4, carry over 5 X 6 2 4 3 4 3 1 3 3 7 2 3 1 Step 6 (2 x 7) + (6 x 2) + 5 = 31, write 1, carry over 3 X 6 2 4 3 1 4 3 1 3 3 7 2 3 1 Step 7 (6 x 7) + 3 = 45, write 45 X 6 2 4 3 4 5 1 4 3 1 3 3 The answer is 7231 x 6243 = 45143133 Math E Magician

Speed Multiplication and Division

ZIP-ZAP Multiplication By 5, 25, 50, 125 Basic knowledge 5 = 10/2

25 = 100/4

50 = 100/2

125 = 1000/8

RULE 19 ZIP-ZAP Multiplication By 5 (10/2) EXAMPLE 1 65214 x 5 = 65214 x 10/2 = 652140 / 2 = 326070 – Answer EXAMPLE 2 47086 x 5 = 47086 x 10/2 = 470860 / 2 = 235430 – Answer

multiply by 10 and divide by 2

RULE 20 ZIP-ZAP Multiplication By 25 (100/4) EXAMPLE 3 25416 x 25 = 25416 x 100/4 = 2541600 / 4 = 635400 – Answer

multiply by 100 and divide by 4

EXAMPLE 4 62584 x 25 = 62584 x 100/4 = 6258400 / 4 multiply by 100 and divide by 4 = 1564600 – Answer

RULE 21 ZIP-ZAP Multiplication By 50 (100/2) EXAMPLE 5 321456 x 50 = 321456 x 100 /2 = 32145600 / 2 multiply by 100 and divide by 2 = 16072800 – Answer EXAMPLE 6 951246 x 50 = 951246 x 100 / 2 = 95124600 / 2 multiply by 100 and divide by 2 = 47562300 – Answer

Math E Magician

Speed Multiplication and Division

RULE 22 ZIP-ZAP Multiplication By 125 (1000/8) EXAMPLE 7 17935 x 125 = 17935 x 1000 / 8 = 17935000 / 8 = 2241875 – Answer EXAMPLE 8 654123 x 125 = 654123 x 1000 / 8 = 654123000 / 8 = 81765375 – Answer

multiply by 1000 and divide by 8

ZIP-ZAP Division By 5, 25, 50, 125 Basic knowledge 5 = 10/2

25 = 100/4

50 = 100/2

125 = 1000/8

RULE 23 ZIP-ZAP Division By 5 (10/2) EXAMPLE 1 254136 / 5 = 254136 ÷ 10 / 2 = 254136 x 2 / 10 = 508272 / 10 = 50827.2 – Answer

multiply by 2 and divide by 10

RULE 24 ZIP-ZAP Division By 25 (100/4) EXAMPLE 2 741258 / 25 = 741258 ÷ 100 / 4 = 741258 x 4 / 100 = 2965032 / 100 = 29650.32 – Answer

multiply by 4 and divide by 100

RULE 25 ZIP-ZAP Division By 50 (100/2) EXAMPLE 3 456321 / 50 = 456321 ÷ 100 / 2 = 456321 x 2 / 100 = 912642 / 100 = 9126.42 – Answer

Math E Magician

multiply by 2 and divide by 100

Speed Multiplication and Division

RULE 26 ZIP-ZAP Division By 125 (1000/8) EXAMPLE 4 15463 / 125 = 15463 ÷ 1000/8 = 15463 x 8 / 1000 multiply by 8 divide by 1000 = 123704 / 1000 = 123.704 - Answer

PRACTISE makes you PERFECT – WS/13 QUESTION

QUESTION ANSWER

QUESTION

ANSWER

2456 x 50

122800

251364 / 50

5027.28

84526 x 25

2113150

145236 / 5

29047.2

7531 x 125

941375

54682 / 25

2187.28

85214 x 5

426070

685214 / 125

5481.712

RULE 27 ZIP-ZAP Conversion of Fractions to Decimals Using your knowledge of division on the previous page, you can apply the same to certain conversions of fractions to decimals. EXAMPLES Convert 7/2 to a decimal Convert 3/4 to a decimal Convert 3/ 25 to a decimal Convert 7/50 to a decimal Convert 3/5 to a decimal

: : : : :

7 = 7 X 5 = 2 2 X 5 3 = 3 X 25 = 4 4 X 25 3 = 3 X 4 = 25 25 X 4 7 = 7 X 2 = 50 50 X 2 3 = 3 X 2 = 5 5 X 2

35 10 75 100 12 100 14 100 6 10

= 3.5 - answer = 0.75 - answer = 0.12 - answer = 0.14 - answer = 0.6 - answer

Math E Magician

Review

PRACTISE makes you PERFECT – WS/14 - Conversion QUESTION

QUESTION ANSWER

QUESTION

ANSWER

2/5

0.4

84 / 50

1.68

7 / 25

0.28

9/2

4.5

LOOKING BACK TO REVIEW

To Divide by 125

To Multiply by 5

• Write 1000 instead of 125 8 • Multiply by 8 and divide by 1000

• Write 10 instead of 5

2 • Multiply by 10 and divide by 2.

To Multiply by 25

To Divide by 50

• Write 100 instead of 50 2 • Multiply by 2 and divide by 100

• Write 100 instead of 25 4 • Multiply by 4 and divide by 100

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-- 50

Review in a nutshell

-- 25

To Divide by 25

--125

• Write 100 instead of 25

4 • Multiply by 100 and divide by 4.

X 25

X 50

-- 5

To Multiply by 50 • Write 100 instead of 50

X 125

2 • Multiply by 100 and divide by 2.

To Divide by 5

To Multiply by 125

• Write 10 instead of 5 2 • Multiply by 2 and divide by 10

• Write 1000 instead of 125

8 • Multiply by 1000 and divide by 8.

Speed Multiplication by 9s

Multiplication By 9, 99, 999, 99999 and So On RULE 28 ZIP-ZAP Multiplication By 9 Basic knowledge : Multiplication by 9 is the same as multiplication by (10– 1) EXAMPLE 1 541236 x 9 = 541236 x (10 – 1) = (541236 x 10) – (541236 x 1) = 5412360 – 541236 (you may go to this step directly) = 4871124 - Answer EXAMPLE 2 172839 x 9 = 1728390 – 172839 (Put 0 after 9 and subtract original number) = 1555551 - Answer EXAMPLE 3 458231 x 9 = 4582310 – 458231 = 4124079 - Answer

RULE 29 ZIP-ZAP Multiplication By 99 Basic knowledge : multiplication by 99 is the same as multiplication by (100 –1) EXAMPLE 1 125463 x 99 = 125463 x (100 – 1) = 12546300 – 125463 (Put 00 after 3 and subtract original number) = 12420837 – Answer EXAMPLE 2 62145 x 99 = 6214500 – 62145 (99 = 100 – 1) = 6152355 - Answer

RULE 30 ZIP-ZAP Multiplication By 999 Basic knowledge: Multiplication by 999 is the same as multiplication by (1000 – 1) EXAMPLE 1 475123 x 999 = 475123000 – 475123 (Put 000 after 3 and subtract original number) = 474647877 Observe that the multipliers 9, 99, 999 have lesser number of digits than the number they multiply (multiplicand).

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Speed Multiplication by 9s

RULE 31 ZIP-ZAP Multiplication By 9999 If the number of 9s is equal to the number of digits in the multiplicand, then the multiplication becomes easier! EXAMPLE 1 8721 x 9999 (4-digit number X 4 nines) 8721 is written as .........8720. That is the last digit is reduced by 1. Step 1 Step 2 Then every digit of the new number is subtracted from 9 to get the last 4 digits of the answer. 8721 x 9999 = 8720 / 1(9-8)2(9-7)7(9-2)9(9-0) = 87201729 EXAMPLE 2 4563 x 9999 = 4562 5437 (underlined digits are obtained by subtracting 4562 from 9999) EXAMPLE 3 4712639 x 9999999 = 4712638 5287361 EXAMPLE 4 89320 x 99999 = 89319 10680

RULE 32 If the number of 9s is more than the number of digits in the multiplicand, then the multiplication is slightly different. The steps are: 1 Reduce the number by 1 and write it down.

2 Insert the EXTRA nines. 3 Subtract every digit the number in step 1 from 9

EXAMPLE 1 76943 x 999999 = 76942 9 23057 (the number has 5 digits and it is multiplied by 6 nines. Hence the extra 9 is inserted in between the two sections of the answer.)

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Speed Multiplication by 9s

EXAMPLE 2 546213 x 99999999 = 546212 99 453787. EXAMPLE 3 6725 X 9999999 = 6724 999 3275 Quite easy, isn’t it?

PRACTISE makes you PERFECT – WS/2 QUESTION

ANSWER

245698 x 999999

245697754302

2468912 x 99999999

246891197531088

147852 x 99999999

14785199852148

3214569 x 9999999

3214568 6785431

Magical Nines 99998 00001 9 98 0 0 1

9 99

001 0 8

1 00000 0 8 9 9 9999

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Speed Multiplication Using Base 10, 100, 1000

RULE 33 Multiplication Using Base 10 and Its Multiples This method is a gift from Vedic Mathematics. The base numbers used are 10, 100, 1000 which are multiples of 10. The base taken depends on the number of digits involved in multiplication. Basic knowledge: 9 x 8 ( 1D x 1D - base 10), 97 x 99 ( 2D x 2D - base 100), 9945 x 9995 ( 4D x 4D - base 10000 )

RULE 33a EXAMPLE 1 97 x 94 (2D x 2D - base 100) Base: 100 LHS RHS 97 - 3

97 is 3 less than base 100 (-3)

94 - 6 94 is 6 less than base 100 (-6) 91 / 18 RHS : multiply: the right hand numbers (-3 x -6 = 18) LHS : Simplify across : (97 â&#x20AC;&#x201C; 6 = 91) or (94 â&#x20AC;&#x201C; 3 = 91) The Answer is 9118

The right hand side must have two digit in the answer corresponding to the base 100. Let us work out an example so that the steps are clear.

996 X 998 Base: 1000 LHS RHS 996 - 4 998 - 2 / xxx LHS RHS 996 - 4 998 - 2 / 008 LHS RHS 996 - 4

996 is 4 less than the base 1000 998 is 2 less than the base 1000 RHS answer will be 3 digits (no. of 0 in the base) (- 4 x - 2 = 8) RHS answer = 008 LHS answer : 996 - 2 or 998 - 4 = 994

998 -2 994 / 008

The answer is 994008 Math E Magician

Speed Multiplication Using Base 10, 100, 1000

EXAMPLE 2 99992 X 99997 Base : 100000 LHS RHS 99992 - 8 99992 is 8 less than the base 100000 99997 - 3 99997 is 3 less than the base 100000 / xxxxx denotes the 5 digit answer of the RHS Corresponding to the number of 0 in the base. LHS RHS 99992 - 8 - 8 x - 3 = 24 99997 - 3 this is the answer to be written for the RHS / 00024 LHS RHS 99992 - 8 LHS answer: simplify the digits across

99997 - 3 99989 / 00024

(99992 – 3 = 99989; or 99997 - 8 = 99989)

The answer is 9998900024 EXAMPLE 3 999987 x 999990 Base1000000 LHS RHS 999987 - 13 999990 - 10 999977 / 000130

999987 is 13 lesser than base 1000000 (- 13) 999990 is 10 lesser than base 1000000 (- 10) (- 13 x – 10 = 130) – RHS answer LHS answer : (999987 – 10 or 9999990 – 13 = 999977)

The asnwer is 999977000130 Now, if the numbers are GREATER than the base, then the RHS will be positive (+) but the steps remain the same as you can see in the next example.

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Speed Multiplication Using Base 10, 100, 1000

RULE 33b EXAMPLE 4 1005 x 1006 Base1000 LHS RHS 1005 + 005 1006 + 006 1011 030 The answer is 1011030

1005 is 5 more than the base 1000 (+ 5) 1006 is 6 more than the base 1000 (+ 6) RHS answer (5 x 6 = 030), LHS answer (1005 + 6 or 1006 + 5 = 1011)

EXAMPLE 5 1095 x 1008 Base1000 LHS RHS 1095 + 095 1008 + 008 1103 760

1095 is 95 more than the base 1000 (+ 95) 1008 is 8 more than the base 1000 (+ 8) RHS answer if obtained by (+95 x +8 = 760), LHS answer (1095 + 8) or (1008 + 95 = 1103)

The answer is 1103770 EXAMPLE 6 100011 x 100012 Base100000 LHS RHS 100011 + 00011 100012 + 00012 100023 00132

100011 is 11 more than the base 100000 (+ 11) 100012 is 12 more than the base 100000 (+ 12) RHS answer obtained by (+11 x +12= 00132), LHS answer (100011 + 00012) or (100012 + 00011 = 100023)

The answer is 10002300132

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Speed Multiplication Using Base 10, 100, 1000

RULE 33c Now suppose you have to multiply 9995 x 10005 EXAMPLE 5 9995 x 10005 Base 10000 LHS RHS 9995 - 0005 10005 + 0005 10000 / - 0025 Base 10000 LHS RHS 9995 - 0005 10005 + 0005 9999 / - 9975

Step 1

9995 is 5 less than the base 10000 (- 5) 10005 is 5 more than the base 10000 (+ 5) RHS, obtained by (-0005 x +0005 = - 0025) LHS answer (9995 + 0005) or (10005 - 0005) = 10000 Step 2

Now borrow a 10000 from the LHS and subtract 0025 RHS = 10000 - 25 = 9975. Also the LHS reduces by 1 Also the LHS reduces by 1 to 9999

The answer is 99999975 EXAMPLE 6 9985 x 10010 Base 10000 LHS RHS 9985 - 0015 10010 + 0010 9995 / 0150 9994 9850

9985 is 15 less than the base 10000 (- 15) 10010 is 10 more than the base 10000 (+ 10) RHS: (-0015 x + 0010 = - 0150), a negative number LHS answer (9985 + 0010) or (10010 - 0015) = 9995 Borrow 10000 from LHS. 10000 - 150 = 9850 = RHS LHS is reduced by 1

The answer is 99949850

PRACTISE makes you PERFECT â&#x20AC;&#x201C; WS/16

QUESTION

QUESTION ANSWER

QUESTION

ANSWER

1005 X 1006

1011030

983 X 1001

983983

9993 X 9996

99890028

996 X 1007

1002972

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Speed Multiplication Using Base 10, 100, 1000

RULE 33d Application of base multiplication to other situations. The base that you can take for a multiplication could be a multiple too like (2 x 10 = 20), (3 x 100 = 300), (5 x 1000 = 5000) , (100 Ăˇ 2 = 50) In Step 1 follow the rule and get the answer for the RHS. Step 2 is the product of the two RHS numbers. In examples 1 â&#x20AC;&#x201C; 4 given below, the actual bases are 20, 500, 3000, 50 respectively. Hence you need to follow step 3. In step 3 multiply the LHS answer that you obtained, by the factor that balances the product as per the actual base. In example 1 the answer of step 1 is multiplied by 2, as the base is 20, while for example 4 it is divided by 2 as the base is 50.

RULE 33e EXAMPLE 1 23 x 24 Working Base 10 x 2=20 LHS RHS 23 +3 Step 1 24 +4 27 / 12 Step 2 x2 / 12 Step 3 54(+1) / 2 Step 4 55 / 2

a) 23 is 3 more than base 20 (+3) b) 24 is 4 more than base 20 (+4) c) LHS has a carry-over of 1 d) RHS = 23 + 4 or 24 + 3 = 27 e) Multiply LHS first with 2 as the base is 20: 27 X 2 = 54 f ) Add the carry over to get the final answer: 54 + 1 = 55

The answer is 552

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Speed Multiplication Using Base 10, 100, 1000

EXAMPLE 2 495 x 490 Working Base 100 x 5 = 500 LHS RHS 495 -5 Step 1 490 -10 485 / 50 Step 2 x 5 / 50 Step 3 2425 / 50

a) 495 is 5 less than base 500 (- 5) b) 490 is 10 less than base 500 (- 10) c) RHS:(- 5 x –10 = 50 ), LHS :(495 – 10 = 485) d) LHS multiplied by 5(as working base is 500)

The answer is 242550 EXAMPLE

and

3005 X 3004 W Base 1000 x 3 =3000 LHS RHS Step 1 3005 +5 3004 +4 Step 2 3009 / 020 Step 3 x 3 / 020 9027 / 020

58 X 56 100 ÷ 2 = 50 LHS RHS 58 +8 56 +6 64 / 48 x1/2 / 48 32 / 48

The answer is 3248

The answer is 9003020

EXAMPLE 5 5006 X 4992 W Base 1000 X 5 = 5000 LHS RHS 5006 +6 4992 -8 4998 /-048 X 5 24990 /-048 24989 / 952

RHS is negative Borrow from LHS : (1000 - 48 = 952) Reduce LHS by 1

The answer is 24989 952

PRACTISE makes you PERFECT – WS/17

QUESTION

ANSWER

QUESTION

ANSWER

a)1005 X 1006

a) 1011030

c) 2001 X 1998

c) 3997998

b) 997 X 992

b) 989024

d) 506 X 502

d) 254012 Math E Magician

Squares and square Roots

SQUARES AND SQUARE ROOTS When a number is multiplied by itself the product is called a square number or a perfect square. EXAMPLES 2 x 2 = 4 ------- 4 is the square of 2 while 2 is called the square root of 4 Memorizing squares of numbers 1 â&#x20AC;&#x201C; 30 will make your calculations go faster and you will save a lot of time too.

SQUARES AND CUBES OF NUMBERS

Number

Square

Cube

Number

Square

Cube

Number

Square

Cube

121

1331

441

9261

144

1728

484

10648

169

2197

529

12167

196

2744

576

13824

125

225

3375

625

15625

216

256

4096

676

17576

343

289

4913

729

19683

512

324

5832

784

21952

729

361

6859

841

24389

100

1000

400

8000

900

27000

NOTE : Square numbers never have 2, 3, 7 ,8 as their digit in ones place.

Math E Magician

Squares and square Roots

RULE 34 Squaring a two digit number with â&#x20AC;&#x153;5â&#x20AC;? in ones place When a number has a 5 in the ones place, squaring it is very easy if you apply this rule! Let us say we want to find the square of 15. You can do this by multiplying 15 by 15 = 225 to get the answer. OR 1 / 52 Zip-zap Rule: let us write 152 as (1 / 5)2 1 x 2 / 25 The last two digits of the answer will be 25 as 52 = 25 2 / 25 The previous digit 1 must be multiplied by its successor 2 The answer is 225 Let us take some more examples Find the square of 25 2 / 5 = 2 x 3 / 52 = 6 / 25 252 = 625 Find the square of 65 6 / 5 = 6 x 7 / 52 = 42 / 25 652 = 4225 Now that is easy, right?

RULE 35 Squaring any two digit number Using this method you will be able to square numbers up to 99 in a jiffy! EXAMPLES 1 What is the square of 34? 3 4 1) To calculate this, start from the RHS number 4 x 3 4 2) 42 = 4 x 4 = 16. Write 6 and carry over 1. 1 1 5 6 3) cross multiply (4 x 3) x 2 = 24 + 1 = 25. Write 5 and carry over 2 4) 32=3 x 3 = 9 + 2 = 11. Write 11 to get the answer The answer is 1156 58

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Squares and square Roots

EXAMPLES 2 What is the square of 78? 7 8 2 STEPS 1) 82 = 64. Write 4, carry-over 6 2) (8 x 7) x 2 = 56 x 2 = 112 + 6 = 118. Write 8, carry-over 11. 3) 72 = 49 + 11 (carry-over) = 60 6011864 The answer is 7 8 2 = 6084 EXAMPLE 3 What is the square of 93? 1) 32 = 9. Write 9 9 3 2 2) (3 x 9) x 2 = 27 x 2 = 54. Write 4, carry-over 5 3) 92 = 81 + 5 (carry-over) = 86 86549 The answer is 9 3 2 = 8649 Observe example 3, you can do this even by the base method. So next time you have to calculate, think of the best method and the quickest too and get your answer in a jiffy

PRACTISE makes you PERFECT â&#x20AC;&#x201C; WS/18 QUESTION

QUESTION ANSWER

QUESTION

ANSWER

562

3136

942

8836

952

9025

632

3969

672

4489

442

1936

732

5329

752

5625

562

3136

492

2401

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Squares and square Roots

RULE 36 Squaring a Three digit number To square a three digit number, we start-off just as we do for a two digit number but from step 3, the method differs slightly. Check it out. EXAMPLE 1 What is the square of 543? 543 9 1) 32 = 3 x 3 = 9. Write 9 49 2) (4 x 3) x 2 = 12 x 2 = 24, write 4 and carry-over 2 49 3) 42= 4 x 4 = 16 + 2 = 18, carry-over full18 849 4) 2(5 x 3) = 15 x 2 = 30 + 18 = 48. Write 8, carry-over 4 4849 5) (5 x 4) x 2 = 20 x 2 = 40 + 4 = 44. Write 4, carry-over 4 294849 6) 5 x 5 = 25 + 4 (carry-over) = 2 9 The answer is 294849 EXAMPLE 2 What is the square of 721? 7 2 1 1) 1 x 1 = 1. Write 1 1 2) Twice (2 x 1) = 2 x 2 = 4, write 4. 41 3) 2 x 2 = 4, carry-over full 4 41 4) 2 (7 x 1) = 7 x 2 = 14 + 4 = 18. Write 8, carry-over 1 841 5) 2 (7 x 2) = 14 x 2 = 28 + 1 = 29. Write 9, carry-over 2 9841 6) 7 x 7 = 49 + 2 (carry-over) = 51 519841 The answer is 514841

* WD - Working Dividend * 2D - Two Digit Number * DS - Digit Sum

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Squares and square Roots

EXAMPLE 3 What is the square of 596? 5 9 6 6 1) 6 x 6 = 36. Write 6, carry-over 3 16 2) 2(9 x 6) = 2 x 54 = 108 + 3 = 111, write 1, carry-over 11 16 3) 9 x 9 = 81 + 11 = 92, carry-over full 92 216 4) 2(5 x 6) = 30 x 2 = 60 + 92 = 152. Write 2, carry-over 15. 5216 5) 2 (5 x 9) = 45 x 2 = 90 + 15 = 105. Write 5, carry-over 10 355216 6) 5 x 5 = 25 + 10 = 35 The answer is 355216 EXAMPLE 4 What is the square of 996? Observe that for this one we could use base 1000 and work the answer easily. Base 1000 996 - 4 x 996 - 4 992 / 016 The answer is 992016 You could also calculate using the 3D x 3D method to get the answer. So you see, practice will tell you the best method to be used in a particular situation and any of these will give you the correct answer. And donâ&#x20AC;&#x2122;t forget to check your answer by the 9s or 11s method every time.

Amazing Pattern Squaring Nines!!! Sequential Inputs of 9 9 x 9 = 81 99 x 99 = 9801 999 x 999 = 998001 9999 x 9999 = 99980001 99999 x 99999 = 9999800001 999999 x 999999 = 999998000001 9999999 x 9999999 = 99999980000001 99999999 x 99999999 = 9999999800000001 999999999 x 999999999 = 999999998000000001 Math E Magician

Squares and square Roots

SQUARE ROOTS OF PERFECT SQUARES Number

Square

100

In the table above, observe the digit in ones places for all the square numbers. The digits 2, 3, 7, 8 are not there in any of the ones places! So a perfect square will never have these four digits in its ones place. Numbers 1 and 9 both have 1 in ones place. Numbers 2 and 8 both have 4 in ones place. Numbers 3 and 7 both have 9 in ones place. Numbers 4 and 6 both have 6 in ones place. Number 5 has a 5 in ones place while 10 has 0 in ones place.

Number

100

Square

100

400

900

1600

2500

1360

4900

6400

8100

10000

It is necessary to be familiar with the above tables to be able to estimate/calculate the square root of a given number which is a perfect square number.

RULE 37 Square Roots by Estimation EXAMPLE 1 Calculate the square root of 2809 * Split the number in groups of 2 digits --------- 28 09 * 28 is between square numbers 25 and 36, so estimated root of 28 = 5 * Square root could have a 3 or 7 in ones place as the square number ends in 9. So square root could be 53 or 57 * Now the square of 50 = 2500 and square of 60 = 3600. * 2809 is nearer 2500 hence the estimated square root is 53 * Check the answer by squaring 53 in the method learned already.

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Squares and square Roots

EXAMPLE 2 Calculate the square root of 3844 * Split the number in groups of 2 digits ---------- 38 44 * 38 is between square numbers 36 and 49, so estimated root of 38 = 6 * Square root could have a 2 or 8 in ones place as the square number ends in 4 So square root could be 62 or 68 * Square of 60 = 3600 and square of 70 = 4900. * 3844 is nearer 3600 hence the estimated square root is 62 * Check the answer by squaring 62 in the method learned already. EXAMPLE 3 Calculate the square root of 7056 * Split the number in groups of 2 digits --------- 70 56 * 70 is between square numbers 64 and 81, so estimated root of 70 = 8 * Square root could have a 4 or 6 in ones place as the square number ends in 6 so it could be 84 or 86 * Square of 80 = 6400 and square of 90 = 8100. * 7056 is nearer 6400 hence the estimated square root is 84 * Check the answer by squaring 84 in the method learned already.

The biggest prime number yet discovered is 2 raised to 32, 582, 857 minus 1. The number has 9,808,358 digits, enough to fill more than 10 books with 350 pages.

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Cubes and Cube Roots

CUBES AND CUBE ROOTS RULE 38 Cube of a Number A number multiplying itself thrice is called a cube number. We will take the help of the algebraic formula: (a + b) 3 = a3 + 3a2b + 3ab2 + b3 We will further simplify this formula for our calculation of cubes of numbers. (a + b) 3 = a3 + a2b + ab2 + b3 + 2a2b + 2ab2 (a + b)3 = a3 + 3a2b + 3ab2 + b3

The middle two terms a2b + ab2 are doubled

EXAMPLE 1 Using the above, let us calculate the cube of 53. So a = 5, b = 3. 533 = 53 + (52 x 3) + (5 x 32) + 33 taking a = 5, b = 3 27 1350 = 125 + 75 + 45 + 27 a3 + a2b + ab2 + b3 2 2 22500 + 150 + 90 + 2a b + 2ab 3 3 2 2 2 + 1 2 5 0 0 0 53 = 125 + 225 + 135 + 27 a + 3a b + 3ab + b 148877 3 The answer is 53 = 148877 EXAMPLE 2 Let us calculate the cube of 76. So a = 7, b = 6. 763 = 73 + (72 x 6) + (7 x 62) + 63 a = 7 , b = 6 216 7560 = 343 + 294 + 252 + 216 a3 + a2b + ab2 + b3 2 2 88200 + 588 + 504 + 2a b + 2ab 3 2 2 3 + 3 4 3 0 0 0 = 343 + 882 + 756 + 216 a + 3a b + 3ab + b 438976 The answer is 438976 EXAMPLE 3 calculate the cube of 84 6 4 843 = 512 + 256 + 128 + 64 + 512 + 256 3840 76800 = 512 + 768 + 384 + 64 + 512000 The answer is 592704 592704

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Cubes and Cube Roots

RULE 39 Cube Roots of Perfect Cubes Number

Cube

125

216

343

512

729

1000

In the table above, observe the digit in ones places for all the cube numbers. Cubes of numbers 1, 4, 5, 6, 9 and 10 all have 1, 4, 5, 6, 9, 0 respectively in one’s place. Number with 2 in its ones place will have a 8 in ones place of its cube. Number with 8 in its ones place will have a 2 in ones place of its cube. Number with 3 in its ones place will have a 7 in ones place of its cube. Number with 7 in its ones place will have a 3 in ones place of its cube. You need to know the rules above and the table of cubes 1 – 10 very well to do your mental calculations of cube roots. Let us take a look at some examples. EXAMPLE 1 What is the cube root of 262144? • Split the number in groups of 3 digits 262 • Digit in one’s place in the rightmost group 144 is 4 cube root .4 • 262 is between cubes of 6 and 7 6 • So digit for cuberoot of 262 will be 6 • Hence the cube root of 262114 will be 64

144 4 4

EXAMPLE 2 What is the cube root of 50653? • Split the number in groups of 3 digits starting from the right. 50 • Here the left group has 2 digits which is acceptable as the number has 5 digits. 3 • Digit in one’s place in the rightmost group 653 is 3, so cube root is 7• • 27 < 50 < 64 ( between cubes of 3 and 4) So the digit for 50 will be 3 • Hence the cube root of 50653 will be 37

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653 7

Cubes and Cube Roots

EXAMPLE 3 What is the cube root of 373248? • Split the number in groups of 3 digits starting from ones place • Check the digit in one’s place in the rightmost group • Check the digit in one’s place in the rightmost group 7 • Hence the cube root of 373248 will be 72

373 7

248 2

EXAMPLE 4 What is the cube root of 110592? Let us do this in one step! 110 592 4

The answer is - Cube root of 110592 is 48

PRACTISE makes you PERFECT – WS/19 Calculate the cube of

1) 72

2) 49

3) 61

PRACTISE makes you PERFECT – WS/20 FIND THE CUBE ROOT OF 1) 166375

3) 262144

2) 373248

4) 79507

Giving you the answers would spoil the fun of getting them yourself !!

Math E Magician

Generation of Multiplication Tables

RULE 40 Generation of Multiplication Tables

MULTIPLICATION TABLE OF 2

BASIC STEPS

Write the table of 2 times • In the table, the digit in the Tens place changes at 10 and 20

• The star shows the change in the tens place

• So we can generate tables of 32, 42, 82 and so on with the help of the basic 2 times table.

• Let us see how that is done

• The star is like a carry-over of 1 • This basic multiplication table of 2 can help us generate any other number with 2 in ones place

(+4) 8

(+4) 12

126

(+4) 16

168

(+5)21

210

(+4) 25

252

• Wherever there is a star, add 5 as a star is a carry-over of 1

(+4) 29

294

(+4) 33

336

• So in the places that are highlighted, 5 has been added instead of 4

(+4) 37

378

(+5)42

420

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MULTIPLICATION TABLE OF 42 • In column B, write only the digits in ones place from the table of 2 times with the stars (as shown above) • Generate column A with a 4 on top. Add 4 as you come down the column A.

• Confirming that the table is accurate is easy. The last number is 10 times the first. In this case, 42 x 10 = 420

Generation of Multiplication Tables

Multiplication Table of 4

1 1 2 2 2 3 3 4

Multiplication Table of 64

B 4 8 2* 6 0* 4 8 2* 6 0*

A1 6 (+6) 1 2 (+7) 1 9 (+6) 2 5 (+7) 3 2 (+6) 3 8 (+6) 4 4 (+7) 5 1 (+6) 5 7 (+7) 6 4

B 4 8 2* 6 0* 4 8 2* 6 0*

64 128 192 256 320 384 448 572 576 640

For a table of 64 we must take the help of a 4 times table. • Write the multiplication table of 4 • Put a * where the tens place value changes • Observe that the place value of tens place changes 4 times for the table of 4 • Write column B for the table of 64 • Add 6 for every step down but 7 for the step that has a star. • Your last number ( here it is 640 ) must be 10 times the first number: 64 x 10 = 640

You can generate any table in this manner. The method can be extended to 3 or 4 digit multiplication tables too. Below, the multiplication table of 236 is generated from the table of 6. Remember wherever there is a * it denotes a carry-over of 1 for the next column. Steps 1

Write the table of 6 putting * for places where the number of the tens place changes.

Rewrite column B in column E

In column D go down adding 3 for every step without *

Add 4 if there is an * in coloum E ( denoted in red )

In column D write a single digit only with an * for the carry-over. That is if your sum is 11, write it as 1* and carry on the addition with 1 with the * as the carry-over for the next column.

The final column C is obtained by adding 2 as you go down. Add 3 when you come across an * in column D

Observe that the top line is the clue to the number of carry-overs. The column with 6 on top has 6 carry-overs, while the column with 3 has 3 carry-overs. A good way of knowing you are on the right track!!!

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Generation of Multiplication Tables

6 Times

Working

1 Times

236

2 Times

(+2)4

(+4)7

472

3 Times

(+3)7

(+3)0*

708

4 Times

(+2) 9

(+4)4

944

5 Times

(+2)11

(+4)8

1180

6 Times

(+3)14

(+3)1*

1416

7 Times

(+2)16

(+4)5

1652

8 Times

(+2)18

(+3)8

1888

9 Times

(+3)21

(+4)2*

2124

10 Times

(+2)23

(+4)6

2360

You can now be adventurous and try writing the table of 1253 too based on the same principle !!!

RAMANUJAN'S MAGIC SQUARE 22

The sum of the identically coloured boxes is 139 Math E Magician

Ramanujan's date of birth 22 - 12 - 1887 69

Divisibility Tests for Prime Numbers

RULE 41 Divisibility tests for Prime Numbers A divisibility test is time saving for factorization for LCM, HCF or also for reducing numbers by common terms. Common tests for numbers 2 – 10 are known by most students. Divisibility tests for prime numbers like 7, 13, 17 and so on follow a special pattern. For these numbers we need a check multiplier to begin with. To find the check multiplier we must find a multiple of the given prime number, with 1 or 9 in ones place. A check multiplier with 1 in ones place is negative. A check multiplier with 9 in ones place is positive. EXAMPLE 1 Find whether 2135 is divisible by 7. • First we must find the check multiplier for 7. • The multiples of 7 with 1 or 9 in ones place are 21 and 49 respectively. • With 21 the check multiplier will be (– 2). • With 49 the check multiplier will be (+ 5) that is 1 more than the digit in tens place. • To use the check multiplier (- 2) 2 1 3 5 the number to be tested 2 1 3 /5 x (- 2) the last digit x check multiplier, 5 x (-2) = -10 - 1 0 subtracting 10 from 213 we get, 213 – 10 = 203 2 0 / 3 x (- 2) the last digit x check multiplier, 3 x (-2) = - 6 -6 subtracting 6 from 20 we get, 20 – 6 = 14 1 4 14 is divisible by 7 so 2135 is also divisible by 7 Using the check multiplier +5 2 1 3 5 the number to be tested 2 1 3 /5 x (+5) the last digit x check multiplier, 5 x (+5) = 25 +2 5 adding 25 to 213 we get, 213 + 25 = 238 2 3 /8 x (+5) the last digit x check multiplier, 8 x (+5) = 40 4 0 adding 40 to 23 we get, 20 – 6 = 63 6 3 6 3 is divisible by 7 so 2135 is also divisible by 7

Math E Magician

Divisibility Tests for Prime Numbers

The remainder must be a multiple of the divisor or it could be a zero to conclude that the given number is divisible by the divisor. EXAMPLE 2 Is 496321 divisible by 7 Using the check multiplier +5 4 9 6 3 2 1 the number to be tested 4 9 6 3 2 / 1 x 5 the last digit x check multiplier, 1 x 5 = 5 + 5 add 5 to the number 4 9 6 3 / 7 x 5 the last digit x check multiplier, 7 x 5 = 35 + 3 5 add 35 to the number 4 9 9 / 8 x 5 the last digit x check multiplier, 8 x 5 = 40 + 4 0 add 40 to the number 5 3 / 9 x 5 the last digit x check multiplier, 9 x 5 = 45 + 4 5 add 45 to the number 9 / 8 x 5 the last digit x check multiplier, 8 x 5 = 40 + 4 0 add 40 to the number 4 9 49 is divisible by 7, so 496321 will be divisible by 7 EXAMPLE 2 Is 3913 divisible by 13 The multiples of 13 with 1 or 9 in ones place are 91 and 39 respectively. So the check multipliers of 13 are (- 9) or (+ 4) 3 9 1 / 3 x 4 check multiplier +4 + 1 2 4 0 / 3 x 4 + 1 2 5 2 52 is divisible by 13 so 3913 is also divisible by 1 The same number can be checked for divisibility using the other check multiplier 3 9 1 / 3 x - 9 check multiplier - 9 - 2 7 3 6 / 4 x - 9 - 3 6 0 remainder = 0, so 13 is divisible by 13

Math E Magician

Ready Recknor for Divisibility Tests

READY RECKNOR FOR DIVISIBILITY TESTS Divisibility to be tested for

Rule for positive divisibility check

Conclusion

Number to be tested should be an even number

Divisible by 2

Sum of the digits of the number should be divisible by 3

Divisible by 3

Last two digits of the number should be divisible by 4

Divisible by 4

Digit in ones place should be 5 or 0

Divisible by 5

Number should be divisible by both 2 and 3

Divisible by 6

Last three digits of the number should be divisible by 8

Divisible by 8

Sum of digits should be divisible by 9

Divisible by 9

The digit in ones place should be 0

Divisible by 10

Sum of alternate digits should be 0 or a multiple of 11

Divisible by 11

Factors of 12 are 3, 4. So the number should be divisible by both 3 and 4

Do both the test to confirm divisibility by 12

Factors of 36 are 4, 9. So the number should be divisible by both 4 and 9.

Do both the test to confirm divisibility by 36

RAMANUJANâ&#x20AC;&#x2122;S NUMBER 1729 is the natural number following 1728 and preceding 1730. 1729 is known as the Ramanujan number after the famous Indian mathematician Srinivasa Ramanujan. It is a very interesting number. It is the smallest number that can be expressed as the sum of two cubes in two different ways 1729 = 13 + 123 1729 = 93 + 103

Math E Magician

Ready Recknor for Divisibility Tests

READY RECKNOR FOR DIVISIBILITY BY PRIMES Divisibility to be tested for

Multiplier with 1in one’s place

Multiplier with 9 in ten’s place

Negative check multiplier

Positive check multiplier

Easy to use check multiplier

(-2) or 5

119

(-5)

171

161

261

279

(-3)

111

259

369

(-4)

PRACTISE makes you PERFECT – WS/21 1. Is 24516 divisible by 23?

2. Is 654782 divisible by 37?

Math E Magician

Speed Division

DIVISION RULE 42 Short Division When doing prime factorizing for HCF or LCM, you divide the number by its factor. This oral division or short division should be mastered by practice and knowledge of divisibility tests. By short division the quotient and remainder can be obtained. It is done without any working or orally. EXAMPLE 1 Find the quotient and remainder when 65421 is divided by 6 6 6 5 54 2 21 1 0 9 0 3 – remainder 3 Quotient 10903 The quotient is written down as you go from left to right. The first digit 6 is divided by 6 and quotient 1 is written below 6. The next digit is 5 so dividing by 6 we get a 0 and 5 is carried forward to get the working number as 54. The next digit of the quotient 9 is obtained when 54 is divided by 6. Again 2 divided by 6 gives a quotient of 0, carrying it forward. Lastly 21 divided by 6, quotient is 3 and remainder is 3. As there is a remainder the number is not divisible by 6. EXAMPLE 2 Divide 65421 by 8 8 6 5 14 62 61 8 1 7 7 – remainder 5 Quotient 8177

PRACTISE makes you PERFECT – WS/22 Short Division

QUESTION

QUESTION ANSWER

QUESTION

ANSWER

654782 / 7

93540 rem 2

874521 / 9

97169

2415879 / 8

301984 rem 7

547812 / 6

91302

Math E Magician

Speed Division

SPEED DIVISION Dividing two numbers without any working saves a lot of time. Of course in the middle school, Grades 4 â&#x20AC;&#x201C; 6, where a question is mainly testing your division skill, you must do the traditional long division to get your marks. The method shown below is taught in Vedic Math books and requires oral work which gives you the quotient quickly. Let us see how that is done.

RULE 43 ZIP-ZAP Division by 9 EXAMPLE 1 Divide 1221 by 9 9 1 2 2 1 * Bring down the first digit as the quotient. +1 +3 +5 * Add it to the second digit, 1 + 2 = 3 to get the next digit Q 1 3 5 6 * Continue in the same pattern : 2 + 3 = 5 , 5 + 1 = 6 The answer is 135.6 NOTE THIS IMORTANT RULE * If the last digit of the quotient is less than seven, write the answer as it is. * If the last digit of the quotient is 7 or greater than 7, we must follow one more Rule which is as follows : * If the last digit of the quotient is between 7 to 16, add 1 to it to get the final answer. * Now follow the pattern as given below If the last digit is from 17 to 26 ----- to the last digit add 2 27 to 36 ----- to the last digit add 3 37 to 46 ----- to the last digit add 4 47 to 56 ----- to the last digit add 5 57 to 66 ----- to the last digit add 6 and so on. Observe the pattern! The above rule is only for division by 9

Math E Magician

Speed Division

EXAMPLE 2 9 6 4 1 2 3 +6 +10 +11 +13 The last digit of the Q is 16, 6 (10) (11) (13) (16)+1 so we have to add 1 to it (refer to the rule) 6 10 11 13 17 carry forward the digit in the tens place, (6+1) (10+1) (11+1) (13+1) for a 2D quotient 7 1 2 4 / 7 the quotient The answer is 7124.7

(Observe the position of the decimal point)

RULE 44 ZIP-ZAP Division by 11 - 19 EXAMPLE 1 6 1 8 0 . 7 11 6 7 9 8 / 7 -6 -1 - 8 -0 for division by 11 we subtract the previous digit. 6 (1) (8) (0) (7) 6 1 8 0 7 the quotient The answer is 6180.6 EXAMPLE 2 6 1 3 9 . 3 11 6 7 5 13 / 12 -6 -1 - 3 - 9 The Q digit 4 and 10 can not be subtracted so 6 (1) (4) (10) (3) Reduce the quotient by 1 6 1 3 9 1 Carry over the reduced number. 4 is reduced to 3 so 1 is carried over. Now subtraction is possible The answer is 6139.3 * WD - Working Dividend * 2D - Two Digit Number * DS - Digit Sum

Math E Magician

Speed Division

EXAMPLE 3 1 2 8 5 . 2 Steps 12 1 5 14 22 / 13 * bring down 1 as the 1st digit of the quotient -2 -4 -16 -10 * Subtract Q x 2 from the next digit of the dividend 1x2 3x2 10 6 3 * Reduce the quotient by 1 till subtraction is possible 2x2 9x2 5x2 * Reduced quotients 8x2 1 2 8 5 / 2 Quotient The answer is 1285.2 EXAMPLE 4 3 5 1 7 . 9

Steps

13 4 15 17 13 / 33 * Subtract Q x 3 from the next digit of the dividend -9 -15 - 3 -21 * Reduce the quotient by 1 till subtraction is possible 4x3 6x3 2x3 10x3 12 3x3 5x3 1x3 7x3 9 * Reduced quotients The answer is 3517.9 EXAMPLE 5 1 1 7

6 . 7

Steps

14 1 6 14 37 / 35 * when we are unable to subtract we will -4 -4 -28 -24 reduce the quotient by 1 till subtraction is possible 1x4 2x4 10x4 9x4 11x4 1x4 7x4 6x4 7x4 *Reduced quotients The answer is 1176.7 EXAMPLE 6 1 6 7

7 . 9

Steps

15 2 15 41 46 / 49 ** every time we are unable to subtract -5 -30 -35 -35 we will reduce the quotient by 1 2x5 10x5 11x5 11x5 14 till subtraction is possible 1x5 6x5 7x5 7x5 9 *Reduced quotients The answer is 1677.9

Math E Magician

Speed Division

EXAMPLE 7 0 9 0 4 . 5 16 1 14 54 -0 -54 1x6 14x6 0x6 0x6 9x6

7 / 33 -0 -24 7x6 9x6 4x6 5x6

*every time we are unable to subtract, we will reduce the quotient by 1 till subtraction is possible *Reduced quotients

The answer is 904.5 EXAMPLE 8 3 17

9 . 7

5 62 37 / 75 -21 -56 -2 -63 6x7 14x7 6x7 16x7 12x7 3x7 8x7 3x7 9x7 7x7 6

* every time we are unable to subtract, we will reduce the quotient by 1 till subtraction is possible *Reduced quotients

The answer is 3839.7 EXAMPLE 9 1 0 2 7 . 3 18 1 8 4 29 / 62 ** Remember every time we are unable to -8 -0 -16 -56 subtract, we will reduce the quotient 1x8 0x8 4x8 13x8 6x8 by 1 till subtraction is possible 2x8 7x8 3x8 *Reduced quotients The answer is 1027.3 EXAMPLE 10 3 4 1 8 . 7 19 6 34 39 25 / 87 -27 -36 -9 -72 6x9 7x9 3x9 16x9 15x9 3x9 4x9 1x9 8x9 7x9

** every time we are unable to subtract, we will reduce the quotient by 1 till subtraction is possible *Reduced quotients

The answer is 3418.7

Math E Magician

Speed Division

PRACTISE makes you PERFECT â&#x20AC;&#x201C; WS/23 QUESTION

QUESTION ANSWER

5648214 / 19

297274.42

6547821 / 17

385165.94

2743527 / 11

249411.54

4523715 / 9

502635

864512 / 13

66500.92

9132456 / 12

761038

7158719 / 16

447419.93

4685543 / 14

334681.64

RULE 45 ZIP-ZAP Division by a Two Digit Divisors When dividing any number by a divisor of 2 digits, we have to remember certain rules and patterns that must be followed. For a divisor of 2 digits, the position of the decimal point has to be determined in the beginning. The position is between the ones and tens place. Example: If 54782 is divided by 62 the decimal point is between 8 and 2, that is 5478/2 and denoted by a /. The digit in the tens place is taken for active division, while the digit in ones place is adjusted to get a working dividend. The pattern of working goes as shown in the working of the following sums. When we divide by 62, we really divide by 6 and make adjustments for the second digit of the divisor. Follow the steps closely.

Math E Magician

Speed Division

EXAMPLE 1 Divide 7456321 by 62 Step 1

1 62 7 4 5 6 3 2 / 1 - 6 14 1 -2 -12 1

62 7 4 5 6 3 2 / 1 14 05 16 -2 -4 12 1 -12 0 1 2 0 2 62 7 4 5 6 3 2 / 1 16 43 - 0 -4 16 39 -12 4 1 2 0 2 6 62 7 4 5 6 3 2 / 1 43 32 21 -4 -12 39 20 -36 3 1 2 0 2 6 3 . 2 62 7 4 5 6 3 2 / 1 0 32 21 -12 -6 20 15 -18 -12 2 3 The answer is 120263.2

* 7 ÷ 6 = Q1, carry forward 1. bring down 4 * Multiply the quotient 1 by the ones digit in the divisor, 2 * Subtract the product from 14 to get the working dividend 12 Step 2

* 12 ÷ 6 = Q2, carry forward 0 * Bring down 5 Quotient 2 x 2 = 4, subtract 5 – 4 = 1 The working dividend is 16 Step 3

* 1 ÷ 6 = Q 0, * Carry forward 1 to the next digit of the divisor * Q 0 x 2 = 0, the working dividend becomes 16 * 16 ÷ 6 = Q2 r4. carry forward 4 * Q2 x 2 = 4, 43 – 4 = 39 Step 4

* 39 ÷ 6 = Q6, carry forward 3 * Q6 x 2 = 12, 32 – 12 = 20 * 20 ÷ 6 = Q3, carry forward 2 * Q3 x 2 = 6, 21 – 6 = 15

Step 5

* 20 ÷ 6 = Q3, carry forward 2 * Q3 x 2 = 6, 21 – 6 = 15 * 15 ÷ 6 = Q2, carry forward 3 * Q 2 is beyond the decimal point

Math E Magician

Speed Division

The division has now a quotient up to one decimal point. When you learn to do division by this method, you will be able to do it in one step as all the small subtractions can be done orally saving a lot of time. Till you master the method you may require write down the subtractions, but later on writing just the working dividend (WD) will suffice. You will do a 1 step division as soon as you are confident with the steps.

EXAMPLE 1 Divide 7456321 by 62 1 2 0 2 6 3 . 2 62 7 4 - 6 14 (Q X 2) 1 -2 WD 12 -12 0

5 05 -4 1 - 0 1

6 16 0 16 -12 4

3 43 - 4 39 -36 3

2 . 1 32 21 12 6 20 15 18 12 2 3

0 30 * Follow the arrows while working - 4 * carry forward number goes up 26 * Subtract (Q x digit in ones place) of the division to get the working dividend (WD)

The answer is 120263.2 Observe the arrows above so that you will understand the method easily. WD stands for working dividend EXAMPLE 2 Divide 849376 by 64 1 3 2 7 1 . 5 64 8 4 9 -6 24 29 2 -4 -12 WD 20 17 -18 -12 2 5

3 7 . 6 0 53 37 36 20 -8 -28 -4 -20 45 9 32 0 - 42 -6 -30 3 3 2

* Follow the arrows while working * carry forward number goes up * Subtract (Q x digit in ones place) of the division to get the working dividend (WD)

The answer is 13271.5

Math E Magician

Speed Division

*if this product (Q x digit in ones place) canâ&#x20AC;&#x2122;t be subtracted, reduce the quotient by 1 EXAMPLE 3 Divide 5482467 by 73 2 * Quotient 3 is reduced to 2 for 7 5 1 0 2 . 3 8 subtraction to be possible 73 5 4 58 22 04 16 / 27 70 -21 -15 -3 -0 -6 -6 WD 54 37 7 1 16 21 64

* The digits in red are the carry-overs quotient x digit in ones place of the divisor gives the working dividend for actual division

The answer is 75102.28 EXAMPLE 3 Divide 5147896 by 82 7 6 2 7 8 9 . 2 1 82 5 1 34 67 78 89 / 36 20 -12 -4 -14 -14 -18 -4 WD 51 22 63 64 75 18 16 The answer is 62779.21

* Digits in red are the carry overs Q x digit in ones place of the divisor gives the working dividend for actual division

Go over the sums again if you need to. You have to be thorough with the division steps to go to the next topic.

PRACTISE makes you PERFECT â&#x20AC;&#x201C; WS/24 Division by 2D

Divide 521756 by 54 QUESTION

Quotient: 9662.14

Divide 217756 by 47

Quotient: 4633.10

Divide 4523179 by 32

Quotient: 141349.34

Math E Magician

Speed Division

RULE 46 3D Divisor When dividing any number by a divisor of 3 digits, rules and patterns that must be followed are slightly different from the previous rule. Let us check out the stepwise method. For a divisor of 3 digits, the position of the decimal point has to be determined in the beginning. The position is between the hundreds and tens digit. Example: If 54782 is divided by 362 the decimal point is between 7 and 8, that is 547/82 and denoted by a /. The digit in the hundreds place of the divisor is taken for active division. The digit in tens and ones place is adjusted to get a working dividend. The pattern of working is similar for all the digit of the quotient. Active divisor is the hundreds digit of the divisor Divide 1 or 2 digits of the dividend by the active divisor and write the quotient, Q

Q x digit in tens place of divisor has to be subtracted from the dividend digit, to give the working dividend for the next digit of the quotient. When you have 2 digits in the quotient the working is : (New Q x digit in tens place) + (Previous Q x digit in ones place of divisor) Subtract the above from working dividend

EXAMPLE 1 Divide 7465321 by 362 2 362 7 4 5 6 3 / 2 1 *7 รท 3 = Q2, carry forward 1. - 6 14 *multiply the quotient 2 by the 1 -12 tens digit in the divisor, 2 x 6 = 12 2 *subtract the product 12 from 14 to get the working dividend 2

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Speed Division

5 2 0 7 362 7 4 5 6 3 / 2 1 *Again 2 ÷ 3 = Q0, carry forward 2 14 25 66 Bring down 5. Now (0 x 6) + (2 x 2) = 4 -12 -4 -30 *Subtract 4 from 25 to get the working WD 2 21 36 dividend 21 -15 21 ÷ 3 = 7. But (7 x 6) + (0 x 2) = 42 > 6 6 So reduce the quotient to 5 for subtraction to be possible (5 x 6) + (0 x 2) = 30, ( 66 – 30) = 36 (WD) 2 0 5 9 362 7 4 5 6 3 / 2 1 36 ÷ 3 = Q9, 66 93 carry forward 9 to the next digit of divisor -30 -64 (9 x 6) + (5 x 2) = 64, 93 – 64 = 29 (WD) 36 29 -27 9 7 2 0 5 9 9 . 5 7 362 7 4 5 6 3 / 2 1 38 ÷ 3 = Q9 which must be reduced to 7 93 82 (7 x 6) + (9 x 2) = 42 + 18 = 60, -64 - 22 ÷ 3 = Q7, 29 -21 8 2 0 5 9 7 . 5 7 362 7 4 5 6 3 / 2 1 22 ÷ 3 = Q5, carry forward 7 82 71 (5 x 6) + (7 x 2) = 30 + 14 = 44, -60 -44 71 – 44 = 27 22 27 -15 7 2 0 5 9 7 . 5

362 7 4 5 6 3 / 2 1 71 Quotient 5 is beyond the decimal point -44 27 ÷ 3 = 7, carry forward 6 27 - 21 6 The answer is 20597.57

Math E Magician

Speed Division

EXAMPLE 2 Divide 8532174 by 253 3 3 7 2 4 . 0 0 253

8 25 43 52 41 / 27 14 -6 -15 -24 -44 -31 -26 -12 2 10 19 8 10 1 2 -6 -14 -4 -8 -0 -0 4 5 4 2 1 2

1) 25 - q 3 x 5 = 10 2) 43 – (q3 x 5 + q3 x 3) = 19 3) 52 – (q7 x 5 + q3 x 3) = 44 and so on

The answer is 33724.00 EXAMPLE 3 Divide 8532174 by 462 1 8 4 6 7 . 9 0 462 8 45 73 72 81 / 97 74 -4 - 6 -50 -40 -44 -54 -68 4 39 23 32 37 43 6 -32 -16 -24 -28 -36 -0 7 7 8 9 7 6

1) 45 - (q 1 x 6 ) = 39 2) 73 – (q8 x 6 + q 1 x 2) = 50 3) 72 – (q 4 x 6 + q 8 x 2) = 40 and so on

The answer is 18467.9 EXAMPLE 4 Divide 8532174 by 215 3 9 6 8 4 . 5 3 215 8 25 43 72 51 / 57 34 -6 -3 -24 -51 -38 -44 -25 2 22 19 21 13 13 9 -18 -12 -16 -8 -10 -6 4 7 5 5 3 3

1) Active divisor 2 2) New Q X 1 3) Previous Q X 5

The answer is 39684.53 Later you may even avoid writing the last two lines in the Zip-Zap division as you can do the subtractions mentally once you know the steps. * WD - Working Dividend * 2D - Two Digit Number * DS - Digit Sum Math E Magician

Speed Division

PRACTISE makes you PERFECT – WS/25 Division by 3D Divide 543678 by 234 QUESTION

Divide 876543 by 365 QUESTION

Divide 948576 by 412 QUESTION

Divide 5084956 by 311 QUESTION

RAMANUJAN'S MAGIC SQUARE 22

The sum of any diagonal is also 139

Sum of the squares with same colour is 139

Math E Magician

Percentage

PERCENTAGE Per cent as you know means out of 100. It is denoted by the symbol %. Though you have to use paper and pencil to calculate complex problems, you can use speedy methods to calculate percent of a number especially when you are in a shopping mall or you need to calculate your marks as a percent. Basic Knowledge 100%

75%

50%

25%

10%

12 ½ %

6 1/4 %

1/4

1/10

1/8

1/16

0.75

0.5

0.25

0.1

0.125

0.0625

Follow the patterns given below Percent

100%

75%

50%

25%

10%

400

300

200

100

Percent

100%

75%

50%

25%

10%

500

375

250

125

Percent

100%

75%

50%

25%

10%

600

450

300

150

Percent

100%

75%

50%

25%

10%

750

562.5

375

187.5

37.5

7.5

Math E Magician

Percentage

Percent

Rs.1200

Rs.8000

Rs.1500

100%

1200

8000

1500

50%

600

4000

750

25%

300

2000

375

75%

900 (50%+25%)

6000 (50%+25%)

1075 (50%+25%)

10%

120

800

150

35%

420 (10%+25%)

2800 (10%+25%)

525 (10%+25%)

60 (1/2 x 10%)

400 (1/2 x 10%)

75 (1/2 x 10%)

12 (1200 ÷ 100)

80 (8000 ÷ 100)

15 (1500 ÷ 100)

55%

660 (50%+5%)

4400 (50%+5%)

825 (50%+5%)

24 (1% x 2)

160(1% x 2)

30 (1% x 2)

60%

720 (50%+ 10%)

4800 (50%+ 10%)

900 (50%+ 10%)

Did you understand the rule? Go over it well to master it and do practice a lot. Don’t forget the worksheets at the back of the book. EXAMPLE 1 Find 75%, 12% of 200 75% = 50% + 25% 12% = 10% +1% + 1% = 100 + 50 = 20 + 2 + 2 = 150 = 24

PRACTISE makes you PERFECT – WS/26

QUESTION

QUESTION ANSWER

QUESTION

ANSWER

52% of 1500

780

150% of 400

600

65% of 300

195

62% of 650

403

90% of 8400

7560

80% of 700

560

Math E Magician

Addition of Many Numbers

ADDITION OF MANY NUMBERS RULE 47 The Tens Rule When you have to add together a lot of numbers, it becomes cumbersome for some to keeping adding and as a result the sum is incorrect. The method given below simplifies addition. Let us learn it with the help of an example. EXAMPLE 1 Add the following 4* 2* 3* 4* 2 5 6* 7 Start adding from ones column. 8* 4* 5 6*

7 + 6 = 13, put a dot or star as you cross 10, carry over 3

7 5 1 3 3 + 3 + 8 = 14, put a star for 10, carry over 4 2* 0 1 8*

4 + 7 = 11, put a star for 10 carry over 1

9* 5* 6* 7*

1 + 3 + 9 = 13, put a star for 10

5 2 6 3*

write 3 as your sum of the column

+ 6* 0 4* 9 Count the stars and carry them to the next (tens) column (4*) 4 1 4 3 3

4 + 6= 10, Put a star for the 10

5 +1 + 1 + 6 = 13, Put a star for the 10 and carry ahead 3

3 + 6 + 4 = 13, Put a star for the 10 write 3 as your column sum. Count the stars and carry them to the next (hundreds)column. (3*)

3 + 5 = 8, 8 + 4 = 12, Put a star for the 10 and carry ahead 2.

2 + 5 = 7, 7 + 0 + 5 = 12, Put a star for the 10 and carry ahead 2. 2 + 2 = 4. Write 4 as your column sum. Count the stars and carry them to the next (thousands) column (2*)

2 + 2 = 4, 4 + 8 = 12, Put a star for the 10 and carry ahead 2 2 + 7 + 2 = 11. Put a star for the 10 and carry ahead 1.

1 + 9 = 10, Put a star for the 10. 6 + 5 = 11 Put a star for the 10

and write 1 as your column sum.

Count the stars and put the number as the last digit. (4*) The answer is 41433 Math E Magician

Addition of Many Numbers

EXAMPLE 2 Add the following numbers and check your answer 3* 1* 2* 1* 2* 6 5 5 3 4 5 Use the Digit-sum rule for checking the answer: 5* 0 2 4 1 3 Find the digit sum of each row. (6+5+5+3+4) = 5 7 5* 7* 1* 9* 2 (7+5+7+1+9) = 29 = 2+9 = 11= 2 5* 3 1 0 7* 7 (5+3+!+0+7) = 16 = 1+6 = 7 2 3 6* 5 4 2 (2+3+6+5+4) = 20 = 2+0 = 2 4* 1 2 4 0 2 (4+1+2+4+0) = 11 = 1+! = 2 3 0 9 4 9 5 3 the digit sum of the above (5+3+2+7+2+2) = 12= 3 Now find the digit sum of the answer ---- 3 + 0 + 9 + 4 + 9 + 5 = 3 Both the digit sum answers are 3, so 309495 is the correct answer! The answer is 309495 EXAMPLE 3 Add the following numbers and check your answer by digit sum 3* 1* 2* 2* 2* 6 4 5 2 4 3 9* 0 4* 4 1 9 Find the digit sum of each row horizontally. 7* 2 1 5* 9* 6 Find the digit sum of the vertical column. 5 4* 1 7* 7* 6 (3+9+6+6+7+1) = 32 = 3 + 2 = 5. 3* 1 6 2 4 7 2 1 4* 3 0 1 3 3 4 3 5 5 5 Now find the digit sum of the answer (3 + 3 + 4 + 3 + 5 + 5) = 5 Hence our answer is correct! The answer is 334355

Math E Magician

Addition of Many Numbers

RULE 48 Addition by the Elevens Rule In this rule you must add till you go beyond 11. Put a star for 11 and carry forward the excess. For example if the total is 14 , put 1 star for 11 and carry forward 14 - 11 = 3. EXAMPLE 1 Add the following 2 5 6 7 • Start adding from ones column. 8 4 5* 6* • 7 + 6 = 13, put a star as you cross 11 and carry 13 - 11 = 2 7* 5* 1 3 • 2 + 3 + 8 = 13, put a star for 11, go ahead with 13 - 11 = 2 2 0 1 8* • 2 + 7 + 3 = 12, put a star for 11. Carry forward 1. 1 + 9 = 10 9* 5 6 7 • write 10 as your sum of the column. Do not carry over 5* 2 6* 3* • write the number of stars below the sum + 6 0 4 9 • Add the next column in the same way. 1 0 16 110 17 10 working total. *0 3 1 2 3 No. of stars 4 1 4 3 3 To get the answer, add the units column as 10+3=13 Write 3, carry over 1. Now add the numbers in a L-shape as shown by arrows 7+2+3 +1=13, write 3 carry over 1 Then 10+1+2+1= 14. Write 4 ans carry over 1 6+3+1+1= 11. Write 1, carry over 1. Lastly 0+0+3+1=4 The answer is 41433 For checking: Find the digit sum of each of the 4 columns above 3 3 2 7 For the next part of the check, 6 10 7 10 Working total from the example + *3 1 2 3 No. of stars + *3 1 2 3 Repeat no. of stars (12) (12) (11) (16) (1+2)(1+2)(1+1) (1+6) 3 3 2 7 digit sum of each bracket. The Two digit sums are identical. So the answer is correct You may even check the answer by the nines method by comparing the total of the DS of the rows and the DS of the answer. If they are the same, your answer is correct.

Math E Magician

Addition of Many Numbers

EXAMPLE 2 Add the following numbers. A B C D E 6 4 5 2 4 9* 0 4 4 1 Find the digit sum of each column. 7* 2 1 5* 9* A B C D E 5 4 1* 7 7 5 3 3 5 7 3 1* 6 2 4* 2 1 4 3* 0 0 10 1 10 1 3 Working Total 0 2 1 1 2 2 stars in each column 3 3 4 3 5 5 The answer is 334355 To check the answer - Working total: 10 Stars: 2 Stars : 2 (14) Digit sum: 5 Answer checked!

1 10 1 1 1 2 1 1 2 3 (12) 5 3 3 5

3 2 Add 2 7 7 *

PRACTISE makes you PERFECT â&#x20AC;&#x201C; WS/27 Addition of many big numbers

ADD

5426, 4785, 8054, 6521, 9025, 7312

8214, 6541, 6012, 8012, 3421, 6521

Math E Magician

Some Conversions

SOME CONVERSIONS Multiply by 10 per box from left to right for conversion of bigger to smaller Kilometers to millimeter Kilometer Hectometer Decameter Km Hm Dac

Meter m

Decimeter Centimeter Millimeter dm cm mm

Kilogram Kg

Hectogram Decagram Hg Dag

Gram g

Decigram dg

Centigram cg

Milligram mg

Kiloliter Kl

Hectoliter Hl

Liter l

Deciliter dl

Centiliter cl

Milliliter ml

Decaliter Dal

Divide by 10 per box from right to left for conversion of small to big Millimeters to kilometers

Weight: 1 Kg = 2.2 pounds Capacity: 1 Liter = 1000 milliliter

Distance: 60 miles = 100 Km; 6 miles = 10 Km; 0.6 miles = 1 Km

Temperature: â&#x20AC;˘ C x 9 / 5 + 32 = F zip-zap: F = 2C + 32 â&#x20AC;˘ (F - 32) x 5/9= C zip-zap: C = F - 32 (approximately) 2 Know the currency of different countries and their coversions to the currency of your country.

RAMANUJAN'S MAGIC SQUARE 22

The sum of the identically coloured boxes is 139 Math E Magician

Names of Big Numbers

NAMES OF

BIG

NUMBERS

Names

10n

NAMES

10n

NAMES

10n

Million

106

Octillion

1027

Sexdecillion (Sedecillion)

1051

Nonillion

1030

Septendecillion

1054

Milliard Billion

109

Decillion

1033

Octodecillion

1057

Trillion

1012

Undecillion

1036

Novemdecillion (Novendecillion)

1060

Quadrillion

1015

Duodecillion

1039

Vigintillion

1063

Quintillion

1018

Tredecillion

1042

Centillion

10303

Sextillion

1021

Quattuordecillion

1048

Septillion

1024

Quindecillion (Quinquadecillion)

1048

RAMANUJAN'S MAGIC SQUARE 22

Sum of the squares with same colour is 139

Math E Magician

Worksheets

WORKSHEETS The worksheet should be done after you are confident about the topic. Check your answer by both methods so that you master the techniques given. Make your own worksheet for extra practice. After you have finished with the rules attempt the mixed bags. Remember accuracy is more important than speed but speed has to be developed Answers are given, but see them after you have finished checking. Note the time you take for 5 sums / 10 sums and try to beat that score gradually.

Math E Magician