Vedic Mathematics

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What is Vedic mathematics?

Books on Vedic Maths

Vedic Maths Tutorial

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Vedic Mathematics

What is Vedic Mathematics?
Vedic Mathematics is the name given to the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to his research all of mathematics is based on sixteen Sutras or word-formulae. For example, 'Vertically and Crosswise` is one of these Sutras. These formulae describe the way the mind naturally works and are therefore a great help in directing the student to the appropriate method of solution.

Perhaps the most striking feature of the Vedic system is its coherence. Instead of a hotch-potch of unrelated techniques the whole system is beautifully interrelated and unified: the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. And these are all easily understood. This unifying quality is very satisfying, it makes mathematics easy and enjoyable and encourages innovation.

In the Vedic system 'difficult' problems or huge sums can often be solved immediately by the Vedic method. These striking and beautiful methods are just a part of a complete system of mathematics which is far more systematic than the modern 'system'. Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, direct and easy.

The simplicity of Vedic Mathematics means that calculations can be carried out mentally (though the methods can also be written down). There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one 'correct' method. This leads to more creative, interested and intelligent pupils.

Interest in the Vedic system is growing in education where mathematics teachers are looking for something better and finding the Vedic system is the answer. Research is being carried out in many areas including the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc.

But the real beauty and effectiveness of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most refined and efficient mathematical system possible.

The Vedic Mathematics Sutras

This list of sutras is taken from the book Vedic Mathematics, which includes a full list of the sixteen Sutras in Sanskrit, but in some cases a translation of the Sanskrit is not given in the text and comes from elsewhere.

This formula 'On the Flag' is not in the list given in Vedic Mathematics, but is referred to in the text.

The Main Sutras

By one more than the one before. |

All from 9 and the last from 10. |

Vertically and Cross-wise |

Transpose and Apply |

If the Samuccaya is the Same it is Zero |

If One is in Ratio the Other is Zero |

By Addition and by Subtraction |

By the Completion or Non-Completion |

Differential Calculus |

By the Deficiency |

Specific and General |

The Remainders by the Last Digit |

The Ultimate and Twice the Penultimate |

By One Less than the One Before |

The Product of the Sum |

All the Multipliers |

The Sub Sutras

Proportionately |

The Remainder Remains Constant |

The First by the First and the Last by the Last |

For 7 the Multiplicand is 143 |

By Osculation |

Lessen by the Deficiency |

Whatever the Deficiency lessen by that amount and
set up the Square of the Deficiency |

Last Totalling 10 |

Only the Last Terms |

The Sum of the Products |

By Alternative Elimination and Retention |

By Mere Observation |

The Product of the Sum is the Sum of the Products |

On the Flag |

**Try a Sutra**

Mark Gaskell introduces an alternative

system of calculation based on Vedic philosophy

At the Maharishi School in Lancashire we have developed a course on Vedic mathematics for key stage 3 that covers the national curriculum. The results have been impressive: maths lessons are much livelier and more fun, the children enjoy their work more and expectations of what is possible are very much higher. Academic performance has also greatly improved: the first class to complete the course managed to pass their GCSE a year early and all obtained an A grade.

Vedic maths comes from the Vedic tradition of India. The Vedas are the most ancient record of human experience and knowledge, passed down orally for generations and written down about 5,000 years ago. Medicine, architecture, astronomy and many other branches of knowledge, including maths, are dealt with in the texts. Perhaps it is not surprising that the country credited with introducing our current number system and the invention of perhaps the most important mathematical symbol, 0, may have more to offer in the field of maths.

The remarkable system of Vedic maths was rediscovered from ancient Sanskrit texts early last century. The system is based on 16 sutras or aphorisms, such as: "by one more than the one before" and "all from nine and the last from 10". These describe natural processes in the mind and ways of solving a whole range of mathematical problems. For example, if we wished to subtract 564 from 1,000 we simply apply the sutra "all from nine and the last from 10". Each figure in 564 is subtracted from nine and the last figure is subtracted from 10, yielding 436.

This can easily be extended to solve problems such as 3,000 minus 467. We simply reduce the first figure in 3,000 by one and then apply the sutra, to get the answer 2,533. We have had a lot of fun with this type of sum, particularly when dealing with money examples, such as £10 take away £2. 36. Many of the children have described how they have challenged their parents to races at home using many of the Vedic techniques - and won. This particular method can also be expanded into a general method, dealing with any subtraction sum.

The sutra "vertically and crosswise" has many uses. One very useful application is helping children who are having trouble with their tables above 5x5. For example 7x8. 7 is 3 below the base of 10, and 8 is 2 below the base of 10.

The whole approach of Vedic maths is suitable for slow learners, as it is so simple and easy to use.

The sutra "vertically and crosswise" is often used in long multiplication. Suppose we wish to multiply

32 by 44. We multiply vertically 2x4=8.

Then we multiply crosswise and add the two results: 3x4+4x2=20, so put down 0 and carry 2.

Finally we multiply vertically 3x4=12 and add the carried 2 =14. Result: 1,408.

We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. By the end of Year 8, I would expect all students to be able to do a "3 by 2" long multiplication in their heads. This gives enormous confidence to the pupils who lose their fear of numbers and go on to tackle harder maths in a more open manner.

All the techniques produce one-line answers and most can be dealt with mentally, so calculators are not used until Year 10. The methods are either "special", in that they only apply under certain conditions, or general. This encourages flexibility and innovation on the part of the students.

Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8 below.

We can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of the answer and multiplying the "differences" vertically 4x8=32 gives the second part of the answer.

This works equally well for numbers above the base: 105x111=11,655. Here we add the differences. For 205x211=43,255, we double the first part of the answer, because 200 is 2x100.

We regularly practise the methods by having a mental test at the beginning of each lesson. With the introduction of a non-calculator paper at GCSE, Vedic maths offers methods that are simpler, more efficient and more readily acquired than conventional methods.

There is a unity and coherence in the system which is not found in conventional maths. It brings out the beauty and patterns in numbers and the world around us. The techniques are so simple they can be used when conventional methods would be cumbersome.

When the children learn about Pythagoras's theorem in Year 9 we do not use a calculator; squaring numbers and finding square roots (to several significant figures) is all performed with relative ease and reinforces the methods that they would have recently learned.

For many more examples, try elsewhere on this page, the Vedic Maths Tutorial

Mark Gaskell is head of maths at the Maharishi School in Lancashire

'The Cosmic Computer'

by K Williams and M Gaskell,

(also in an bridged edition),

Inspiration Books, 2 Oak Tree Court,

Skelmersdale, Lancs WN8 6SP. Tel: 01695 727 986.

Saturday school for primary teachers at

Manchester Metropolitan University on

October 7. See website. www.vedicmaths.org

19th May 2000 Times Educational Supplement (Curriculum Special)

http://www.tes.co.uk/

http://www.vedicmaths.org/resources/articles

**Articles ****on Vedic Maths** from journals, magazines and periodicals (a small selection with url links (from a very large list).

copyright to the ACADEMY OF VEDIC MATHEMATICS

"Vertically and Crosswise", Mathematics in School, Sept 1999 (published by the Mathematical Association). K R Williams

Summing Up "Vedic Math" by Iraja Sivadas, Hawaii, USA in "Hinduisn Today" journal Jan/Feb 2001.

Vedic Mathematics Today by James Glover, partly printed in India in "Education Times" titled “Only a matter of 16 sutras”, 7th January 2002

"Practical Application of Vedic Mathematics" by Chetan Dalal published in December 10-23, 2001 issue of Business India, column Accountancy, page 95.

- "The Sutras of Vedic Mathematics", by Kenneth Williams, in the Journal of the Oriental Institute, Vol. L, Nos 1-4, Sept 2000 - June 2001, pages 145 to 156; and in the book "Glimpses of Vedic Mathematics" by Dr S K Kapoor, 2003, published by Arya Book Depot, New Delhi.
- "The System of Vedic Mathematics - a Comparison", by Kenneth Williams, in the book "Glimpses of Vedic Mathematics" by Dr S K Kapoor, 2003, published by Arya Book Depot, New Delhi.

- "Multiplication with the Vedic Method", 2010, by Syed Azman bin Syed Ismai and Pumadevi a/p Sivasubramniama. International Conference on Mathematics Education Research 2010 (ICMER 2010)
- "High Speed Reconfigurable FFT Design by Vedic Mathematics ", 2010. By Ashish Raman, Anvesh Kumar and R.K.Sarin
- "Instructor develops formula for success", 2011, by Audrey Parente. Article on a course by Rick Blum
- "The Vedic Inventive Principles", by Karthikeyan Iyer, 2011 (original Link).
- "Vedische Mathematik - ihre Geschichte", 2011. Summary of 5 page article from the German magazine P.M.
- "Singapore students learn Vedic Maths", The New Indian Express, 20th October 2011.
- "Poor numeracy 'blights the economy and ruins lives'", 2012, by Judith Burns Education reporter, BBC News.
- "Numeristics'", a paper on Numeristics (an alternative, number-based foundational theory for mathematics, using principles of Maharishi Vedic Mathematics) by Kevin Carmody. See also Equinfinitesimal Analysis: A Numeristic Approach to Calculus. And: Divergent Series: A Numeristic Approach
- "Nothing Vedic in ‘Vedic Maths’", The Hindu, 3rd September 2014.
- "Everything Vedic in ‘Vedic Maths’", 2014, by James Glover. From The Hindu, October 15, 2014.

- "Fun with Numbers’", by James Glover. From The Speaking Tree, Nov 29, 2014.
- "PTU ready for running Vedic maths course", Hindustan Times, 30th December 2014.
- "It's no rocket science: Here's a Vedic add-on for speed maths", Hindustan Times, 4th January 2015.
- "Harsh Vardhan is right: Vedic maths needs to be taught", dailyO, 5th January 2015
- "India’s next gift to the world could be Vedic mathematics", The Telegraph, 7th January 2015.
- "Vedic Mathematics in Microcontrollers", by Jai Sachith Paul. From Electronics For You, Feb 2015.

- "Punjab to hold Vedic maths test in schools", The Indian Express, January 18, 2016.
- "Magic Maths & Advaita", by James Glover. From The Speaking Tree, Apr 23 2016

- "Teaching Calculus", by Kenneth Williams. From Mathematics Today, June 2016
- "Applications of Triples", by Kenneth Williams. From Mathematics Today, April 2017

Books on Vedic Maths

VEDIC MATHEMATICS

Or Sixteen Simple Mathematical Formulae from the Vedas The original introduction to Vedic Mathematics.

Author: Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja,

1965 (various reprints).

Paperback, 367 pages, A5 in size.

ISBN 81 208 0163 6 (cloth)

ISBN 82 208 0163 4 (paper)/p

MATHS OR MAGIC?

This is a popular book giving a brief outline of some of the Vedic Mathematics methods.

Author: Joseph Howse. 1976

ISBN 0722401434

Vedic Mathematics

Master Multiplication tables, division and lots more!

We recommed you check out this ebook, it's packed with tips,

tricks and tutorials that will boost your math ability, guaranteed!

www.vedic-maths-ebook.com

A PEEP INTO VEDIC MATHEMATICS

Mainly on recurring decimals.

Author: B R Baliga, 1979.

Pamphlet./p

INTRODUCTORY LECTURES ON VEDIC MATHEMATICS

Following various lecture courses in London an interest arose for printed material containing the course material. This book of 12 chapters was the result covering a range topics from elementary arithmetic to cubic equations.

Authors: A. P. Nicholas, J. Pickles, K. Williams, 1982.

Paperback, 166 pages, A4 size./p

DISCOVER VEDIC MATHEMATICS

This has sixteen chapters each of which focuses on one of the Vedic Sutras or sub-Sutras and shows many applications of each. Also contains Vedic Maths solutions to GCSE and 'A' level examination questions.

Author: K. Williams, 1984, Comb bound, 180 pages, A4.

ISBN 1 869932 01 3./p

VERTICALLY AND CROSSWISE

This is an advanced book of sixteen chapters on one Sutra ranging from elementary multiplication etc. to the solution of non-linear partial differential equations. It deals with (i) calculation of common functions and their series expansions, and (ii) the solution of equations, starting with simultaneous equations and moving on to algebraic, transcendental and differential equations.

Authors: A. P. Nicholas, K. Williams, J. Pickles

first published 1984), new edition 1999. Comb bound, 200 pages, A4.

ISBN 1 902517 03 2./p

TRIPLES

This book shows applications of Pythagorean Triples (like 3,4,5). A simple, elegant system for combining these triples gives unexpected and powerful general methods for solving a wide range of mathematical problems, with far less effort than conventional methods use. The easy text fully explains this method which has applications in trigonometry (you do not need any of those complicated formulae), coordinate geometry (2 and 3 dimensions) transformations (2 and 3 dimensions), simple harmonic motion, astronomy etc., etc.

Author: K. Williams (first published 1984), new edition 1999. Comb bound.,168 pages, A4.

ISBN 1 902517 00 8/p

VEDIC MATHEMATICAL CONCEPTS OF SRI VISHNU SAHASTRANAMA STOTRAM

Author: S.K. Kapoor, 1988. Hardback, 78 pages, A4 size./p

ISSUES IN VEDIC MATHEMATICS

Proceedings of the National workshop on Vedic Mathematics

25-28 March 1988 at the University of Rajasthan, Jaipur.

Paperback, 139 pages, A5 in size.

ISBN 81 208 0944 0/p

THE NATURAL CALCULATOR

This is an elementary book on mental mathematics.

It has a detailed introduction and each of the nine chapters covers one of the Vedic formulae. The main theme is mental multiplication but addition, subtraction and division are also covered.

Author: K. Williams, 1991. Comb bound ,102 pages, A4 size.

ISBN 1 869932 04 8./p.

VEDIC MATHEMATICS FOR SCHOOLS BOOK 1

Is a first text designed for the young mathematics student of about eight years of age, who have mastered the four basic rules including times tables. The main Vedic methods used in his book are for multiplication, division and subtraction. Introductions to vulgar and decimal fractions, elementary algebra and vinculums are also given.

Author: J.T,Glover, 1995. Paperback, 100 pages + 31 pages of answers, A5 in size.

ISBN 81-208-1318-9./p

JAGATGURU SHANKARACHARYA SHRI BHARATI KRISHNA TEERTHA

An excellent book giving details of the life of the man

who reconstructed the Vedic system.

Dr T. G. Pande, 1997

B. R. Publishing Corporation, Delhi-110052

INTRODUCTION TO VEDIC MATHEMATICS

Authors T. G. Unkalkar, S. Seshachala Rao, 1997

Pub: Dandeli Education Socety, Karnataka-581325

THE COSMIC COMPUTER COURSE

This covers Key Stage 3 (age 11-14 years) of the

National Curriculum for England and Wales. It consists of three books each of which has a Teacher's Guide and an Answer Book. Much of the material in Book 1 is suitable for children as young as eight and this is developed from here to topics such as Pythagoras' Theorem and Quadratic Equations in Book 3. The Teacher's Guide contains a Summary of the Book, a Unified Field Chart (showing the whole subject of mathematics and how each of the parts are related), hundreds of Mental Tests (these revise previous work, introduce new ideas and are carefully correlated with the rest of the course), Extension Sheets (about 16 per book) for fast pupils or for extra classwork, Revision Tests, Games, Worksheets etc.

Authors: K. Williams and M. Gaskell, 1998.

All Textbooks and Guides are A4 in size, Answer Books are A5.

GEOMETRY FOR AN ORAL TRADITION

This book demonstrates the kind of system that could have existed before literacy was widespread and takes us from first principles to theorems on elementary properties of circles. It presents direct, immediate and easily understood proofs. These are based on only one assumption (that magnitudes are unchanged by motion) and three additional provisions (a means of drawing figures, the language used and the ability to recognise valid reasoning). It includes discussion on the relevant philosophy of mathematics and is written both for mathematicians and for a wider audience.

Author: A. P. Nicholas, 1999. Paperback.,132 pages, A4 size.

ISBN 1 902517 05 9

THE CIRCLE REVELATION

This is a simplified, popularised version of "Geometry for an Oral Tradition" described above. These two books make the methods accessible to all interested in exploring geometry. The approach is ideally suited to the twenty-first century, when audio-visual forms of communication are likely to be dominant.

Author: A. P. Nicholas, 1999. Paperback, 100 pages, A4 size.

ISBN 1902517067

VEDIC MATHEMATICS FOR SCHOOLS BOOK 2

The second book in this series.

Author J.T. Glover , 1999.

ISBN 81 208 1670-6

Astronomica; Applications of Vedic Mathematics

To include prediction of eclipses and planetary positions,

spherical trigonometry etc.

Author Kenneth Williams, 2000.

ISBN 1 902517 08 3

Vedic Mathematics, Part 1

We found this book to be well-written, thorough and easy to read.

It covers a lot of the basic work in the original book by B. K. Tirthaji

and has plenty of examples and exercises.

Author S. Haridas

Published by Bharatiya Vidya Bhavan, Kulapati K.M. Munshi Marg, Mumbai - 400 007, India.

INTRODUCTION TO VEDIC MATHEMATICS – Part II

Authors T. G. Unkalkar, 2001

Pub: Dandeli Education Socety, Karnataka-581325

VEDIC MATHEMATICS FOR SCHOOLS BOOK 3

The third book in this series.

Author J.T. Glover , 2002.

Published by Motilal Banarsidass.

THE COSMIC CALCULATOR

Three textbooks plus Teacher's Guide plus Answer Book.

Authors Kenneth Williams and Mark Gaskell, 2002.

Published by Motilal Banarsidass.

TEACHER’S MANUALS – ELEMENTARY & INTERMEDIATE

Designed for teachers (of children aged 7 to 11 years,

9 to 14 years respectively)who wish to teach the Vedic system.

Author: Kenneth Williams, 2002.

Published by Inspiration Books.

TEACHER’S MANUAL – ADVANCED

Designed for teachers (of children aged 13 to 18 years)

who wish to teach the Vedic system.

Author: Kenneth Williams, 2003.

Published by Inspiration Books.

FUN WITH FIGURES

(subtitled: Is it Maths or Magic?)

This is a small popular book with many illustrations, inspiring quotes and amusing anecdotes. Each double page shows a neat and quick way of solving some simple problem. Suitable for any age from eight upwards.

Author: K. Williams, 1998. Paperback, 52 pages, size A6.

ISBN 1 902517 01 6.

Please note the Tutorial below is based on material from this book 'Fun with Figures'

Book review of 'Fun with Figures'

From 'inTouch', Jan/Feb 2000, the Irish National Teachers Organisation (INTO) magazine.

"Entertaining, engaging and eminently 'doable', Williams' pocket volume reveals many fascinating and useful applications of the ancient Eastern system of Vedic Maths. Tackling many number operations encountered between First and Sixth class, Fun with Figures offers several speedy and simple means of solving or double-checking class activities. Focusing throughout on skills associated with mental mathematics, the author wisely places them within practical life-related contexts." "Compact, cheerful and liberally interspersed with amusing anecdotes and aphorisms from the world of maths, Williams' book will help neutralise the 'menace' sometimes associated with maths.

It's practicality, clear methodology, examples, supplementary exercises and answers may particularly benefit and empower the weaker student." "Certainly a valuable investment for parents and teachers of children aged 7 to 12." Reviewed by Gerard Lennon, Principal, Ardpatrick NS, Co Limerick. The Tutorial below is based on material from this book 'Fun with Figures'

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Index Alphabetical [Index to Pages]

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Vedic Maths Tutorial

Vedic Maths is based on sixteen Sutras or principles. These principles are general in nature and can be applied in many ways. In practice many applications of the sutras may be learned and combined to solve actual problems. These tutorials will give examples of simple applications of the sutras, to give a feel for how the Vedic Maths system works.

**These tutorials do not attempt to teach the systematic use of the sutras.**

For more advanced applications and a more complete coverage of the basic uses of the sutras, we recommend you study one of the texts available at www.vedicmaths.org

N.B. The following tutorials are based on examples and exercises given in the book 'Fun with figures' by Kenneth Williams, which is a fun introduction to some of the applications of the sutras for children.

Tutorial 1

Use the formula ALL FROM 9 AND THE LAST FROM 10 to

perform instant subtractions.

For example 1000 - 357 = 643

We simply take each figure in 357 from 9 and the last figure from 10.

So the answer is **1000 - 357 = 643**

And thats all there is to it!

This always works for subtractions from numbers consisting of a 1 followed by noughts: 100; 1000; 10,000 etc.

Similarly **10,000 - 1049 = 8951**

For 1000 - 83, in which we have more zeros than figures in the numbers being subtracted, we simply suppose 83 is 083.

So **1000 - 83** becomes **1000 - 083 = **__917__

Exercise 1 Tutorial 1

Try some yourself:

**1)** 1000 - 777 =

**2)** 1000 - 283 =

**3)** 1000 - 505 =

**4)** 10,000 - 2345 =

**5)** 10,000 - 9876 =

**6)** 10,000 - 1011 =

**7)** 100 - 57 =

**8)** 1000 - 57 =

**9)** 10,000 - 321 =

**10)** 10,000 - 38 =

Answers to excercise 1 Tutorial 1 < Click

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Index Alphabetical [Index to pages]

Tutorial 2

Using VERTICALLY AND CROSSWISE you do not need the multiplication tables beyond 5 X 5

.

Suppose you need **8 x 7**

8 is 2 below 10 and 7 is 3 below 10.

Think of it like this:

The answer is 56.

The diagram below shows how you get it.

You subtract crosswise 8-3 or 7 - 2 to get 5,

the first figure of the answer.

And you multiply vertically: 2 x 3 to get 6,

the last figure of the answer.

That's all you do:

See how far the numbers are below 10, subtract one number's deficiency from the other number, and multiply the deficiencies together.

**7 x 6 = **__42__

Here there is a carry: the 1 in the 12 goes over to make 3 into 4.

Exercise 1 Tutorial 2

Multply These:

**1)** 8 x 8 =

**2)** 9 x 7 =

**3)** 8 x 9 =

**4)** 7 x 7 =

**5)** 9 x 9 =

**6)** 6 x 6 =

Answers to excercise 1 Tutorial 2

Here's how to use VERTICALLY AND CROSSWISE for multiplying numbers close to 100.

Suppose you want to multiply 88 by 98.

Not easy,you might think. But with VERTICALLY AND CROSSWISE you can give the answer immediately, using the same method as above

Both 88 and 98 are close to 100.

88 is 12 below 100 and 98 is 2 below 100.

You can imagine the sum set out like this:

As before the 86 comes from subtracting crosswise: 88 - 2 = 86 (or 98 - 12 = 86: you can subtract either way, you will always get the same answer).

And the 24 in the answer is just 12 x 2: you multiply vertically.

So 88 x 98 = 8624

**Exercise 2 Tutorial 2**

This is so easy it is just mental arithmetic

.

Try some:

**1)** 87 x 98 =

**2)** 88 x 97 =

**3)** 77 x 98 =

**4)** 93 x 96 =

**5)** 94 x 92 =

**6)** 64 x 99 =

**7)** 98 x 97 =

Answers to Excercise 2 Tutorial 2 < Click

Multiplying numbers just over 100.

103 x 104 = 10712

The answer is in two parts: 107 and 12,

107 is just 103 + 4 (or 104 + 3),

and 12 is just 3 x 4.

Similarly 107 x 106 = 11342

107 + 6 = 113 and 7 x 6 = 42

Exercise 3 Tutorial 2

Again, just for mental arithmetic

Try a few:

**1)** 102 x 107 =

**2)** 106 x 103 =

**3)** 104 x 104 =

**4)** 109 x 108 =

**5)** 101 x123 =

**6)** 103 x102 =

Answers to excercise 3 Tutorial 2 <Click

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Index Alphabetical [Index to pages]

Tutorial 3

The easy way to add and subtract fractions.

Use VERTICALLY AND CROSSWISE to write the answer straight down!

Multiply crosswise and add to get the top of the answer:

2 x 5 = 10 and 1 x 3 = 3. Then 10 + 3 = 13.

The bottom of the fraction is just 3 x 5 = 15.

You multiply the bottom number together.

So:

Subtracting is just as easy: multiply crosswise as before, but the subtract:

Exercise 1 Tutorial 3

Try a few:

Answers to excercise 1Tutorial 3 <Click

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Index Alphabetical [Index to pages]

Tutorial 4

A quick way to square numbers that end in 5 using the formula BY ONE MORE THAN THE ONE BEFORE.

**75 = **__5625__

75² =**5625.**

75² means 75 x 75.

The answer is in two parts: 56 and 25.

The last part is always **25.**

The first part is the first number, 7, multiplied by the number "one more", which is 8:

so 7 x 8 = **56**

Similarly **852 = **__7225__ because 8 x 9 = 72.

Exercise 1 Tutorial 4

Try these:

**1)** 452 =

**2)** 652 =

**3)** 952 =

**4)** 352 =

**5)** 152 =

Answers to excercise 1Tutorial 4 <Click

Method for multiplying numbers where the first figures are the same and the last figures add up to 10.

**32 x 38 = **__1216__

Both numbers here start with 3 and the last figures (2 and 8) add up to 10.

So we just multiply 3 by 4 (the next number up) to get 12 for the first part of the answer.

And we multiply the last figures: 2 x 8 = 16 to get the last part of the answer.

Diagrammatically:

And **81 x 89 = **__7209__

We put 09 since we need two figures as in all the other examples.

Exercise 2 Tutorial 4

Practise some:

**1)** 43 x 47 =

**2)** 24 x 26 =

**3)** 62 x 68 =

**4)** 17 x 13 =

**5)** 59 x 51 =

**6)** 77 x 73 =

Answers to excercise 2 Tutorial 4 <Click

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Index Alphabetical [Index to pages]

Tutorial 5

An elegant way of multiplying numbers using a simple pattern

**21 x 23 = **__483__

This is normally called long multiplication butactually the answer can be written straight downusing the VERTICALLY AND CROSSWISEformula.

We first put, or imagine, 23 below 21:

There are 3 steps:

a) Multiply vertically on the left: 2 x 2 = 4.

This gives the first figure of the answer.

b) Multiply crosswise and add: 2 x 3 + 1 x 2 = 8

This gives the middle figure.

c) Multiply vertically on the right: 1 x 3 = 3

This gives the last figure of the answer.

And thats all there is to it.

Similarly 61 x 31 = 1891

6 x 3 = 18; 6 x 1 + 1 x 3 = 9; 1 x 1 = 1

Exercise 1 Tutorial 5

Try these, just write down the answer:

**1)** 14 x 21

**2)** 22 x 31

**3)** 21 x 31

**4)** 21 x 22

**5)** 32 x 21

Answers to excercise 1 Tutorial 5

Exercise 2a Tutorial 5

Multiply any 2-figure numbers together by mere mental arithmetic!

If you want 21 stamps at 26 pence each you can

easily find the total price in your head.

There were no carries in the method given above.,/p>

However, there only involve one small extra step.

21 x 26 = 546

The method is the same as above

except that we get a 2-figure number, 14, in the

middle step, so the 1 is carried over to the left

(4 becomes 5).

So 21 stamps cost £5.46.

Practise a few:

**1)** 21 x 47

**2)** 23 x 43

**3)** 32 x 53

**4)** 42 x 32

**5)** 71 x 72

Answers to excercise 2a Tutorial 5

Exercise 2b Tutorial 5

33 x 44 = __1452__

There may be more than one carry in a sum:

Vertically on the left we get 12.

Crosswise gives us 24, so we carry 2 to the left

and mentally get 144.

Then vertically on the right we get 12 and the 1

here is carried over to the 144 to make 1452.

**6)** 32 x 56

**7)** 32 x 54

**8)** 31 x 72

**9)** 44 x 53

**10)** 54 x 64

Answers to excercise 2b Tutorial 5

Any two numbers, no matter how big, can be

multiplied in one line by this method.

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Index Alphabetical [Index to pages]

Tutorial 6

Multiplying a number by 11.

To multiply any 2-figure number by 11 we just put

the total of the two figures between the 2 figures.

26 x 11 = 286

Notice that the outer figures in 286 are the 26

being multiplied.

And the middle figure is just 2 and 6 added up.

So 72 x 11 = 792

Exercise 1 Tutorial 6

Multiply by 11:

**1)** 43 =

**2)** 81 =

**3)** 15 =

**4)** 44 =

**5)** 11 =

Answers to excercise 1 Tutorial 6

77 x 11 = 847

This involves a carry figure because 7 + 7 = 14

we get 77 x 11 = 7147 = 847.

Exercise 2 Tutorial 6

Multiply by 11:

**1)** 11 x 88 =

**2)** 11 x 84 =

**3)** 11 x 48 =

**4)** 11 x 73 =

**5)** 11 x 56 =

Answers to excercise 2 Tutorial 6

234 x 11 = 2574

We put the 2 and the 4 at the ends

.

We add the first pair 2 + 3 = 5.

and we add the last pair: 3 + 4 = 7.

Exercise 3 Tutorial 6

Multiply by 11:

**1)** 151 =

**2)** 527 =

**3)** 333 =

**4)** 714 =

**5)** 909 =

Answers to excercise 3 Tutorial 6

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Index Alphabetical [Index to pages]

Tutorial 7

Method for dividing by 9.

23 / 9 = 2 remainder 5

The first figure of 23 is 2, and this is the answer.

The remainder is just 2 and 3 added up!

43 / 9 = 4 remainder 7

The first figure 4 is the answer

and 4 + 3 = 7 is the remainder - could it be easier?

Exercise 1a Tutorial 7

Divide by 9:

**1)** 61 / 9 = remainder

**2)** 33 / 9 = remainder

**3)** 44 / 9 = remainder

**4)** 53 / 9 = remainder

**5)** 80 / 9 = remainder

Answers to excercise 1a Tutorial 7

134 / 9 = 14 remainder 8

The answer consists of 1,4 and 8.

1 is just the first figure of 134.

4 is the total of the first two figures 1+ 3 = 4,

and 8 is the total of all three figures 1+ 3 + 4 = 8.

Exercise 1b Tutorial 7

Divide by 9

:

**6)** 232 = remainder

**7)** 151 = remainder

**8)** 303 = remainder

**9)** 212 = remainder

**10)** 2121 = remainder

Answers to excercise 1b Tutorial 7

**842 / 9 = 812 remainder 14 = **__92 remainder 14__

Actually a remainder of 9 or more is not usually

permitted because we are trying to find how

many 9's there are in 842.

Since the remainder, 14 has one more 9 with 5

left over the final answer will be 93 remainder 5

Exercise 2 Tutorial 7

Divide these by 9:

**1)** 771 / 9 = remainder

**2)** 942 / 9 = remainder

**3)** 565 / 9 = remainder

**4)** 555 / 9 = remainder

**5)** 2382 / 9 = remainder

**6)** 7070 / 9 = remainder

Answers to excercise 2 Tutorial 7

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Index Alphabetical [Index to pages]

Answers

Answers to exercise 1 Tutorial 1

1) 223

2) 717

3) 495

4) 7655

5) 0124

6) 8989

7) 43

8) 943

9) 9679

10) 9962

Return to Exercise1 Tutorial 1

Answers to exercise 1 tutorial 2

1) 64

2) 63

3) 72

4) 49

5) 81

6)216= 36

Return to Exercise1 Tutorial 2

Answers to Exercise 2 Tutorial 2

1) 8526

2) 8536

3) 7546

4) 8928

5) 8648

6) 6336

7) 9506 (we put 06 because, like all the others,

we need two figures in each part)

Return to Exercise2 Tutorial 2

Answers to exercise 3 Tutorial 2

1) 10914

2) 10918

3) 10816

4) 11772

5) 12423

6) 10506 (we put 06, not 6)

Return to Exercise3 Tutorial 2

Answers to Exercise 1 Tutorial 3

1) 29/30

2) 7/12

3) 20/21

4) 19/30

5) 1/20

6) 13/15

Return to Exercise 1 Tutorial 3

Answers to Exercise 1 Tutorial 4

1) 2025

2) 4225

3) 9025

4) 1225

5) 225

Return to Exercise 1 Tutorial 4

Answers to Exercise 2 Tutorial 4

1) 2021

2) 624

3) 4216

4) 221

5) 3009

6) 5621

Return to Exercise 2 Tutorial 4

Answers to Exercise 1 Tutorial 5

1) 294

2) 682

3) 651

4) 462

5) 672

Return to Exercise 1 Tutorial 5

Answers to Exercise 2a Tutorial 5

1) 987

2) 989

3) 1696

4) 1344

5) 5112

Return to Exercise 2a Tutorial 5

Answers to Exercise 2b Tutorial 5

6) 1792

7) 1728

8) 2232

9) 2332

10) 3456

Return to Exercise 2b Tutorial 5

Answers to Exercise 1 Tutorial 6

1) 473

2) 891

3) 165

4) 484

5) 121

Return to Exercise 1 Tutorial6

Answers to Exercise 2 Tutorial 6

1) 968

2) 924

3) 528

4) 803

5) 616

Return to Exercise 2 Tutorial 6

Answers to Exercise 3 Tutorial 6

1) 1661

2) 5797

3) 3663

4) 7854

5) 9999

Return to Exercise 3 Tutorial 6

Answers to Exercise 1a Tutorial 7

1) 6 r 7

2) 3 r 6

3) 4 r 8

4) 5 r 8

5) 8 r 8

Return to Exercise 1a Tutorial 7

Answers to Exercise 1b Tutorial 7

1) 25 r 7

2) 16 r 7

3) 33 r 6

4) 23 r 5

5) 235 r 6 (we have 2, 2 + 1, 2 + 1 + 2, 2 + 1 + 2 + 1)

Return to Exercise 1a Tutorial 7

Answers to Exercise 2 Tutorial 7

1) 714 r15 = 84 r15 = 85 r6

2) 913 r 15 = 103 r15 = 104 r6

3) 516 r16 = 61 r16 = 62 r7

4) 510 r15 = 60 r15 = 61 r6

5) 714 r21 = 84 r21 = 86 r3

6) 2513 r15 = 263 r15 = 264 r6

7) 7714 r14 = 784 r14 = 785 r5

Return to Exercise 2 Tutorial 7

copyright to the

**ACADEMY OF VEDIC MATHEMATICS**

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Index Alphabetical [Index to pages]

**Tutorial 8**

Vedic Maths - Tips & Tricks

Courtesy **www.vedic-maths-ebook.com**

By Kevin O'Connor

__Is it divisible by four?__ ** **

This little math trick will show you whether a number is divisible by four or not.

So, this is how it works.

Let's look at 1234

Does 4 divide evenly into 1234?

**For 4 to divide into any number we have**

to make sure that the last number is even

If it is an odd number, there is no way it will go in evenly.

So, for example, 4 will not go evenly into 1233 or 1235

Now we know that for 4 to divide evenly into any

number the number has to end with an even number.

Back to the question...

4 into 1234, the solution:

**Take the last number and add it to 2 times the second last number**

If 4 goes evenly into this number then you know that 4 will go evenly into the whole number.

So

4 + (2 X 3) = 10

4 goes into 10 two times with a remainder of 2 so it does not go in evenly.

Therefore 4 into 1234 does not go in completely.

Let’s try 4 into 3436546

So, from our example, take the last number, 6 and add it to

two times the penultimate number, 4

6 + (2 X 4) = 14

4 goes into 14 three times with two remainder.

So it doesn't go in evenly.

Let's try one more.

4 into 212334436

6 + (2 X 3) = 12

4 goes into 12 three times with 0 remainder.

Therefore 4 goes into 234436 evenly.

So what use is this trick to you?

Well if you have learnt the tutorial at Memorymentor.com about telling the day in any year, then you can use it in working out whether the year you are calculating is a leap year or not.

__Multiplying by 12 - shortcut__ ** **

So how does the 12's shortcut work?

Let's take a look.

12 X 7

The first thing is to always multiply the 1 of the twelve by the

number we are multiplying by, in this case 7. So 1 X 7 = 7.

Multiply this 7 by 10 giving 70. (Why? We are working with BASES here.

Bases are the fundamentals to easy calculations for all multiplication tables.

To find out more check out our Vedic Maths ebook at **www.vedic-maths-ebook.com**

Now multiply the 7 by the 2 of twelve giving 14. Add this to 70 giving 84.

Therefore 7 X 12 = 84

Let's try another:

17 X 12

Remember, multiply the 17 by the 1 in 12 and multiply by 10

(**Just add a zero to the end**)

1 X 17 = 17, multiplied by 10 giving 170.

Multiply 17 by 2 giving 34.

Add 34 to 170 giving 204.

So 17 X 12 = 204

lets go one more

24 X 12

Multiply 24 X 1 = 24. Multiply by 10 giving 240.

Multiply 24 by 2 = 48. Add to 240 giving us 288

24 X 12 = 288 (these are Seriously Simple Sums to do aren’t they?!)

__Converting Kilos to pounds__ ** **