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1

5

12

Square – Numbers Consists All Digit of 1 17





Square – Numbers Consists Unit Digit 1 & Rest of All Digits 9 22








Multiplication by 5 30







Multiplication – Teen Numbers 52

Multiplication of Two Digit Numbers – Line Method 54

Vertically & Crosswise 56

Multiplying 2 Numbers With The Same Ten's Digit 61

Multiplying Two Numbers That End In 5 64

Square of Numbers Ending in 1 66





Adding Consecutive Numbers 78

Adding Consecutive Numbers Starting From 1 80

Finding The Sum of All Odd Numbers Starting From 1 82

Finding The Sum of All Even Numbers Starting From 2 84

Complements Using Nikhilam Navataha 86

Subtraction by Ekadhikena Purvena 88

Subtraction by Nikhilam Navataha 91

Starting Complements from the Middle of the Sum 92

Leaving Complements in the Middle of the Sum 93

Complements More than Once in the Same Sum 94

Problems for You 95

Chapter-1

Indian Mathematics

Indian mathematics emerged in the Indian subcontinent from 1200 BC until the end of the 18th century. In the

classical period of Indian mathematics (400 AD to 1200 AD), important contributions were made by scholars like

Aryabhata, Brahma gupta, and Bhaskara II. The decimal number system in use today were first recorded in Indian

mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number,

negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in

particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were

transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations

of many areas of mathematics.

Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras

in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a

student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries

by different scholars) that explained the problem in more detail and provided justification for the solution. In the

prose section, the form (and therefore its memorization) was not considered as important as the ideas involved. All

mathematical works were orally transmitted until approximately 500 BC; thereafter, they were transmitted both

orally and in manuscript form.

Many references of knowledge of mathematics are found in Vedic and consequent literature. Some of such sacred

scripts are listed below:

Vedas Rugveda, Yajurveda, Atharvaveda

Samhitas Yajurveda, Taitiriya, Vashista

Brhamanas Panchavisha, Shatpatha

Puranas Agni, Vayu, Narad

Shulva Sutras Baudhayana, Katyana, Manava, Apasthamb

Baudha and Jaina Scripts Lalitavistara, Suryaprajnyapti, Anuyogadvara sutra

Prosody Chanda sutra

Sidhantas Vashistha, Surya, Paulisha

Manuscript Bakshali and others

VASISTA Eduventures vasistagroup.com 1

The chronology of Indian mathematics spans from the Indus valley civilization and the Vedas to Modern times.

Indian mathematicians have made a number of significant contributions to mathematics including place-value

arithmetical notation and the concept of zero.

Sl.No. Name Period Script

1 Baudhāyana 800 BC Baudhāyana Śulbasûtra

2 Aryabhata (I) 499 AD Aryabhtiyam

3 Varahmihira 505 AD Panchasidhantika, Bruhatsamhita

4 Jinabhadra Gani 575 AD Vaisheshikabhashya

5 Brahmagupta 628 AD Brahmasphutasidhanta

6 Bhaskara (I) 629 AD Mahabhaskariya, Aryabhatiyabhashya

7 Shridhara 750 AD Patiganita, Trishatika

8 Virasena 816 AD Dhavalatika

9 Mahaviracharya 850 AD Ganitasarasamgraha

10 Govindaswamin 850 AD Mahabhaskariya Bhashya

11 Pruthudakswami 864 AD Brahmasiddanta vacanabhasya

12 Vateshvara 904 AD Vateshvara siddhanta

13 Munjala 932 AD Laghumansa

14 Aryabhata (II) 978 AD Mahasidhanta

15 Nemichandra 980 AD Gomatasara

16 Shripati 1039 AD Ganita tilaka

17 Bhaskara (II) 1150 AD Sidhantashriomani, Lilavati Bijaganitam

18 Madhava 1340-1425 AD Madhyamanayanaprakara, Venvaroha

19 Narayana Pandita 1356 AD Ganita Kaumudi

20 Parameshvara 1430 AD Lilavati vyakhya

21 Nilakantha 1444-1545 AD Tantrasamgraha, Aryabhatiya Bhashya

22 Jyeshthadeo 1475 AD Yuktibhasha

23 Shankara Variyavara 1500-1560 AD Kriyakramakari

24 Ganesh Daivadnya 1545 AD Buddhivalasani

25 Putumana Somayaji 1660-1740 AD Karanapaddhati

26 Shankar Varman 1823 AD Sadratnamala


BEAUTY OF MATHEMATICS:

Here are some funny math interesting facts. All the below tricks are based around the

sequential manipulation of the numbers being used for input and output.

SEQUENTIAL INPUTS OF NUMBERS WITH 8 SEQUENTIAL 1'S WITH 9

1 X 8 + 1 = 9 1 X 9 + 2 = 11

12 X 8 + 2 = 98 12 X 9 + 3 = 111

123 X 8 + 3 = 987 123 X 9 + 4 = 1111

1234 X 8 + 4 = 9876 1234 X 9 + 5 = 11111

12345 X 8 + 5 = 98765 12345 X 9 + 6 = 111111

123456 X 8 + 6 = 987654 123456 X 9 + 7 = 1111111

1234567 X 8 + 7 = 9876543 1234567 X 9 + 8 = 11111111

12345678 X 8 + 8 = 98765432 12345678 X 9 + 9 = 111111111

123456789 X 8 + 9 = 987654321 123456789 X 9 + 10 = 1111111111

SEQUENTIAL INPUTS OF 9 NUMERIC PALINDROME WITH 1'S

9 X 9 = 81 1 X 1 = 1

99 X 99 = 9801 11 X 11 = 121

999 X 999 = 998001 111 X 111 = 12321

9999 X 9999 = 99980001 1111 X 1111 = 1234321

99999 X 99999 = 9999800001 11111 X 11111 = 123454321

999999 X 999999 = 999998000001 111111 X 111111 = 12345654321

9999999 X 9999999 = 99999980000001 1111111 X 1111111 = 1234567654321

99999999 X 99999999 = 9999999800000001 11111111 X 11111111 = 123456787654321

999999999 X 999999999 = 999999998000000001 111111111 X 111111111 = 12345678987654321

WITHOUT 8 SEQUENTIAL INPUTS OF 6

12345679 X 9 = 111111111 6 X 7 = 42

12345679 X 18 = 222222222 66 X 67 = 4422

12345679 X 27 = 333333333 666 X 667 = 444222

12345679 X 36 = 444444444 6666 X 6667 = 44442222


12345679 X 45 = 555555555 66666 X 66667 = 4444422222

12345679 X 54 = 666666666 666666 X 666667 = 444444222222

12345679 X 63 = 777777777 6666666 X 6666667 = 444444422222

12345679 X 72 = 888888888 66666666 X 66666667 = 444444442222

12345679 X 81 = 999999999 666666666 X 666666667 = 444444444222

22

2222

222222

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

SHAKUNTALA DEVI: That’s Incalculable!

In 1976 the New York Times reported that an Indian woman named Shakuntala Devi (b. 1939)

added 25,842 + 111,201,721 + 370,247,830 + 55,511,315, and then multiplied that sum by 9,878,

for a correct total of 5,559,369,456,432, all in less than twenty seconds. Hard to believe, though

this uneducated daughter of impoverished parents has made a name for herself in the United

States and Europe as a lightning calculator.

Unfortunately, most of Devi’s truly amazing feats not done by obvious “tricks of the trade” are

poorly documented. Her greatest claimed accomplishment—the timed multiplication of two

thirteen digit numbers on paper—has appeared in the Guinness Book of World Records as an

example of a “Human Computer.” The time of the calculation, however, is questionable at best.

Devi, a master of the criss-cross method, reportedly multiplied 7,686,369,774,870 x

2,465,099,745,799, numbers randomly generated at the computer department of Imperial

College, London, on June 18, 1980. Her correct answer of 18,947,668,177,995,426,773,730 was

allegedly generated in an incredible twenty seconds.


Chapter-2

Vedic Mathematics

The astonishing system of calculation, which was originally born in the Vedic Age and was deciphered during the

start of the 20th century, is what we know as Vedic Mathematics. It is the ancient system of Indian Mathematics

which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krishna Tirthaji (1884-1960).

According to his research all of mathematics is based on sixteen sutras, or word-formulae. 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.

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.

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 practicing

the system. One can then see that it is perhaps the most refined and efficient mathematical system possible.

ADVANTAGES OF VEDIC MATHS ARE:

It reduces the burden of remembering large amount of stuff because it requires you to learn tables up to 9

only.


It enables faster calculations when compared to the conventional method. Thus, the time that gets saved in

the process can be used to answer more questions

It acts as a tool for reducing finger counting and scratch work.

It plays an important role in increasing concentration as well as improving confidence.

It is very simple, direct, totally unconventional, original and straight forward.

It encourages mental calculations.

It enriches our understanding of maths and enables us to see links and continuity between different branches

of maths.

Vedic Maths system also gives us a set of checking procedures for independent crosschecking of whatever

we do.

It keeps the mind alert and lively because of the element of choice and flexibility at each.

Holistic development of the human brain takes place through Vedic Mathematics along with

multidimensional thinking.

Vedic Mathematics system to quite an extent also helps us in developing our spiritual part of personality.

It can introduce creativity in intelligent and smart students, while helping the slow-learners grasp the basic

concepts of mathematics. More and more use of Vedic math can without any doubts generate interest in a

subject that is generally dreaded by children.

Thus, Vedic math’s is considered to be more than the blessing of the Veda for the entire humanity.

HOW IT WORKS:

It is all based mainly on 16 sutras and 14 sub sutras

There are many features of the Vedic system which contrast significantly with conventional mathematics.

Coherence - Perhaps the most striking feature of the Vedic system is its coherence (consistency and logic).

Flexibility - In modern teaching you usually have one way of doing a calculation. This is rigid and boring, and

intelligent and creative students rebel against it. Once you allow variations you get all sorts of benefits.


Children become more creative. The teacher is encouraging innovation and the children respond. In the

Vedic system there are general methods, that always work,

Having only one method of, say, multiplying is like a carpenter who uses a screwdriver for every job. The

skilled craftsman selects the tool most appropriate for the job and gets it done quicker, better and with more

satisfaction. Calculations can often be carried out from right to left or from left to right.

You can represent numbers in more than one way; we can work 2 or more figures at a time if we wish.

In this rapidly changing world adaptability and flexibility are absolutely essential for success. For the future

we can expect more change and perhaps at a more rapid pace.

Mental, improves memory - The ease and 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.

Promotes creativity - All these features of Vedic math encourage students to be creative in doing their math.

Being naturally creative students like to devise their own methods of solution.

Appeals to everyone - The Vedic system appears to be effective over all ability ranges: the able child loves

the choice and freedom to experiment and the less able may prefer to stick to the general methods but loves

the simple patterns they can use.

Increases mental Agility - Because the Vedic system uses these ultra-easy methods mental calculation is

preferred and leads naturally to develop mental agility. And this in turn leads to growth in other subjects.

Efficient and fast - In the Vedic system 'difficult' problems or huge sums can often be solved immediately.

These striking and beautiful methods are just a part of a complete system of mathematics which is far more

systematic than the modern 'system'

Easy, fun - The experience of the joy of mathematics is an immediate and natural consequence of practicing

Vedic Mathematics. And this is the true nature of maths - not the rigid and boring 'system' that is currently

widespread.

Methods apply in algebra - Another important feature of the Vedic system is that once an arithmetic method

has been mastered the same method can be applied to algebraic cases of that type - the beautiful coherence

between arithmetic and algebra is clearly manifest in the Vedic system.


FATHER OF VEDIC MATHEMATICS

His Holiness Jagadguru Shankaracharya Shri Bharati Krishna Tirthaji Maharaja is the

founder father of Vedic Mathematics and was the ecclesiastical head of most of

Hindu India and was the apostolic successor of the first Shankaracharya ( 9th

Century; India's greatest philosopher).

EARLY LIFE

Born on 14th March, 1884 at Tinnievelly, Tamil Nadu, India of pious parents, Shri

Bharti Krishna Tirtha was known as Venkatraman before he became a saint.

He was blessed with a scholarly father, P. Narasimha Shastri and surrounded by well educated, pious and religious

family members. He was exceptionally brilliant since childhood. This became evident when he passed his

matriculation examination in 1899 from Madras University by securing top most position.

In July 1899 he was awarded the title of “Saraswati” for all round proficiency and gifted oratory in Sanskrit by the

Sanskrit Association of Madras. He had a brilliant academic record with a Masters (MA) degree in six subjects –

Sanskrit, English, History, Philosophy, Mathematics and Science from the Bombay Centre of the American College of

Science, Rochester, New York.

After a brilliant University career he became a lecturer in Mathematics and

Science in the Baroda College. Thereafter he became Principal of National

College in Rajamundary, Andhra Pradesh, India.

In 1905 when the Freedom movement started in Bengal, Bharti Krishnaji participated in the freedom movement

along with Shri Aurobindo Ghosh and Gopal Krishna Gokhale - an ardent nationalist. Bharti Krishnaji wrote to

numerous newspapers propagating the freedom movement. He was also appointed “Warden of the Sons of India”

by Dr.Annie Besant in 1908.

There seemed a devout urge in Him to devote his life in the service of humanity, and he held that man could render

such service only after attaining Self Realization . Therefore he proceeded in 1909 to Shringeri Matha to realize it at

the feet of Shri Shankaracharya Shri Sachchidananda Shiva Abhinava Narasimha Saraswati.


From 1911-1918 Bharati Krishnaji practiced deep meditation and studied metaphysics and Vedas which led him to

practice an arduous life of a Sadhu (saint). He was leading a purely saintly life living on roots and fruits. His life was

continuous “Sadhana” – (meditation) and he devoted himself to the study of Vedanta and resorted to the forest for

deep meditation and spiritual attainments.

In his solitude he discerned the “Ganita-Sutras” or easy Mathematical Formulas on which he compiled the

monumental work “Vedic Mathematics” an original contribution in the field of Mathematics and Research. Bharati

Krishnaji got the key to Ganita Sutra coded in the Atharva Veda and rediscovered Vedic Mathematics with the help

of lexicographs . He found “Sixte en S utras” or word formulas which cover all the branches of Mathematics

- Arithmetic, Algebra, Geometry, Trigonometry, Physics, plan and spherical geometry, conics, calculus- both

differential and integral, applied mathematics of all various kinds, dynamics, hydrostatics and all.

LATER LIFE

Thereafter in July 1919 Sri Venkatraman Saraswati was initiated into Sanyas by Shri Trivikram Teerthaji at Varanasi

and since then he was called with the new name “Shri Bharati Krishna Tirtha”.

Shri Bharati Krishna Tirthaji Maharaja was a staunch follower of the Vedic Principles and he never went astray from

its rules. He was appointed as the Head of Dwarkapeeth by Shri Trivikram Tirthaji in 1921. Since then, He started the

life of Shankaracharya and delivered discourses wherever he went.

In 1925 he became the Head of the Govardhan Matha Monastery in Puri,Orissa and was the pontiff till 1960 the

year of his “Maha Samadhi” (departure of a Self-realized saint from his mortal coil.) Because of His Holiness's spiritual

authority over millions of Hindus, the Government of India consults the Jagadguru on policies relating to spiritual

rules and Hindu Religious Matters.

1958 USA TOUR

His Holiness Sri Jagadguru Shankaracharya of Puri, Head of the thousand year old monastic Shankaracharya Order

and the first of its leaders ever to visit the west, went for a three month tour of the United States and the United

Kingdom on an invitation by the Self Realization Fellowship (SRF) founded by Paramahansa Yogananda. The tour was

hosted by Sister Daya Mata the President of the SRF.


He spoke at various lectures attended by thousands of students of various universities and organizations such as

Stanford University, University of California , San Diego College, Vedanta Society – Hollywood, Church Of Religious

Science- Los Angeles, Los Angeles City College, and American Academy of Asian Studies etc.

He addressed a select group of Caltech graduate students in mathematics at California Institute of Technology. He

was introduced by Mr. Wesley L. Hershey executive secretary of Caltech YMCA and thereafter Bharati Krishnaji

discoursed and gave black board demonstrations. This meeting was the first in the United States in which His Holiness

presented his mathematical discoveries. The talk aroused such interest that His Holiness was invited to return to the

institute for further demonstrations.

A turn of events brought His Holiness back to Caltech. In his second discourse he took up Algebra and Quadratics;

and a third discourse at Caltech Sri Sankaracharya demonstrated the application of his theories in the field of Calculus

.

DEVELOPMENT OF VEDIC MATH

Vedic math was immediately hailed as a new alternative system of mathematics, when a copy of the book reached

London in the late 1960s. Some British mathematicians, including Kenneth Williams, Andrew Nicholas and Jeremy

Pickles took interest in this new system. They extended the introductory material of Bharati Krishna's book, and

delivered lectures on it in London. In 1981, this was collated into a book entitled Introductory Lectures on Vedic

Mathematics. A few successive trips to India by Andrew Nicholas between 1981 and 1987, renewed the interest on

Vedic math, and scholars and teachers in India started taking it seriously.

The Sutras (aphorisms) apply to and cover each and every part of each and every chapter of each and every branch

of mathematics (including arithmetic, algebra, geometry - plane and solid, trigonometry - plane and spherical, conics-

geometrical and analytical, astronomy, calculus - differential and integral etc., etc. In fact, there is no part of

mathematics, pure or applied, which is beyond their jurisdiction; The Sutras are easy to understand, easy to apply

and easy to remember; and the whole work can be truthfully summarized in one word "mental"!

Dr L M Singhvi, the former High Commissioner of India in the UK, also an avid endorser of the system says: "A single

sutra would generally encompass a varied and wide range of particular applications and may be likened to a

programmed chip of our computer age".

Another Vedic maths enthusiast, Clive Middleton feels, "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."


THE SUB SUTRAS

Together with the sixteen Sutras, Sri Bharati Krishna Tirthaji lists thirteen sub-Sutras. For example Proportionately,

By Alternate Elimination and Retention and By Mere Observation are three of them. Two of the sixteen Sutras (By

One More than the One Before and By Addition and By Subtraction) are indicated to be sub-Sutras also. However it

is believed that although there are exactly sixteen Sutras, the sub-Sutras are not fixed in number.

THE GROWING INTEREST IN VEDIC MATH

Quite a few years ago, a few schools in Europe started teaching Vedic Maths on experimental basis. Today this

remarkable system is taught in many schools and institutes in India and abroad, and even to MBA and economics

students. Interest in Vedic maths is growing in the field of education where mathematics students are looking for a

new and better approach to the subject. Even students at the Indian Institute of Technology (IIT) are said to be using

this ancient technique for quick calculations. No wonder, in a Convocation speech addressed to the students of IIT,

Delhi, by Dr. Murli Manohar Joshi, former Union Minister for Science & Technology, stressed the significance of Vedic

maths, while pointing out the important contributions of India in the realm of Mathematics, such as Aryabhatta, who

laid the foundations of algebra, Baudhayan, the great geometer, and Medhatithi and Madhyatithi, the saint duo,

who formulated the basic framework for numerals.

Researches are being undertaken in many areas, including the effects of learning Vedic maths on children. A great

deal of research is also being done on how to develop more powerful and easy applications of the Vedic sutras in

geometry, calculus, and computing.


Chapter-3

Sixteen Sutras & Upa Sutras

The lexical meaning of the word Sutra is thread, string, a key or a formula. Here of course, the last two meanings are

applicable. The tradition of expressing all types of knowledge in Sutra form is as old as Vedas.

Panini codified Sanskrit grammar in form of nearly 4000 sutras. Veda Vyasa formulated Vedanta philosophy in

Brahma Sutra.

Vedic Mathematics is a technique, which is derived from 16 sutras and 16 sub sutras in the Vedas, which are

thousands of years old and among the earliest literature of ancient Hindus in India. They are an endless source of

knowledge and wisdom, providing practical knowledge of wisdom, providing practical knowledge in all spheres of

life.

SUTRAS

SUTRA 1 Ekadhikena Purvena

Meaning By one more than the one before.

Use This sutra is used to get a next number to given number.

SUTRA 2 Nikhilam Navatas Caramam Dasatah

Meaning All from nine and the last from ten.

Use This sutra is used to get a complement number from its functional base.

SUTRA 3 Urdhva-Triyagbhyam

Meaning Vertically and crosswise.

Use This sutra is used to multiply numbers vertically and crosswise.

SUTRA 4 Paravartya Yojayet

Meaning Transpose and apply.

Use This sutra is used to get adhikaya and complement from last functional base.


SUTRA 5 Shunyam Samyasamuchchaye

Meaning If the samuchchaya is the same, it is zero.

Use This sutra is used to determine a such number which is common to all, that must

be equal to zero.

SUTRA 6 Aanyarupaye Sunyamanyat

Meaning If one is the ratio, the other is zero.

Use This sutra is used to find the value of one variable, if in a system of linear

equations, the ratio of co-efficient of other variable is equal to the ratio of

constant terms.

SUTRA 7 Sankalana Vyavakalanabhyam

Meaning By addition and subtraction.

Use This sutra is used to find the value of variables in a system of linear equations, if

the coefficients of variables are found interchanged.

SUTRA 8 Purna Puranabhyam

Meaning By the completion or non-completion.

Use This sutra is used to find the value of variable by making equation perfect square

or perfect cube.

SUTRA 9 Chalana-kalanabhyam

Meaning Differential Calculus.

Use This sutra is used in the following two cases:

To find the roots of a quadratics

To factorizing expression of 3rd, 4th and 5th degree.

SUTRA 10 Yavadunam

Meaning By the deficiency.


To find the complement to the nearest functional base.

Used to find the direct cube of two digits numbers. VASISTA Eduventures vasistagroup.com 13

SUTRA 11 Vyasti Samastih

Meaning Specific and General.


To find the average.

HCF=Product of common factors.

SUTRA 12 Sesanyankena Caramena

Meaning The remainders by the last digit.

Use This sutra is used to express a fraction as a decimal, to all its decimal places.

SUTRA 13 Sopantyadvayamantyam

Meaning The ultimate and twice the penultimate.

Use This sutra is used to find the value of variable which is such that in equation

and A,B,C,D are in AP, then D+2C=0 is one solution.

SUTRA 14 Ekanyunena Purvena

Meaning By one less than the one before.

Use This sutra is used in the following three cases:

To find the number one less than the given number

To multiply the two numbers in which a number is in repeating 9’s.

In expressing a fraction into decimal.

SUTRA 15 Gunitasamuchchaya

Meaning The product of the sum.

Use This sutra is used to verify the factors of an expression with it’s a corollary.

SUTRA 16 Gunakasamuchchaya

Meaning All the multipliers


To find the number of zeroes in the product.


To find the place of decimal in the product of numbers.

UPA-SUTRAS

UPA SUTRA 1 Anurupyena

Meaning Proportionately

UPA SUTRA 2 Sisyate Sesamjnah

Meaning The remainder remains constant.

UPA SUTRA 3 Adyamadye Nantyamantyena

Meaning The first by the first and the last by the last.

UPA SUTRA 4 Kevalaih Saptakam Gunyat

Meaning For seven, the multiplier is 143

UPA SUTRA 5 Vestanam

Meaning By osculation

UPA SUTRA 6 Yavadunam Tavadunam

Meaning Less on by the deficiency

UPA SUTRA 7 Yavadunam Tavadunikritya Vargancha Yogayet

Meaning Whatever the deficiency, less on by that amount and set up the square of the

deficiency.

UPA SUTRA 8 Antyayordashake ‘pi’

Meaning Last totalling ten

UPA SUTRA 9 Antyayoreva VASISTA Eduventures vasistagroup.com 15

Meaning Only the last terms

UPA SUTRA 10 Samuchchaya Gunitah

Meaning The sum of the products.

UPA SUTRA 11 Lopanasthapanabhayam

Meaning By alternate elimination and retention

UPA SUTRA 12 Vilokanam

Meaning Mere observation

UPA SUTRA 13 Gunita Samuchchaya Samuchchaya Gunitah

Meaning The product of the sum is the sum of the products.

UPA SUTRA 14 Dhvajanka

Meaning On the flag

UPA SUTRA 15 Dwandwa Yoga

Meaning Duplex combination process

UPA SUTRA 16 Suddha

Meaning Purification

Sūtra is an aphorism (or line, rule, formula) or a collection of such aphorisms in the form of a

manual. Literally it means a thread or line that holds things together and is derived from the

verbal root.


Chapter-4

Square – Numbers Consists All Digit of 1

EXERCISE 1: MENTAL MATHS

Sl.No. Problem Write your answer here Time Taken in Secs

1 112

2 1112

3 11112

4 111112

5 1111112

6 11111112

7 111111112

8 1111111112

9 11111111112

10 111111111112

11 1111111111112

12 11111111111112

13 111111111111112

14 11111111111111112


Chapter-5




1 222

2 2222

3 22222

4 222222

5 2222222

6 22222222

7 222222222

8 2222222222

9 22222222222

10 222222222222

11 2222222222222

12 22222222222222

13 222222222222222

14 2222222222222222


Chapter-6




1 332

2 3332

3 33332

4 333332

5 3333332

6 33333332

7 333333332

8 3333333332

9 33333333332

10 333333333332

11 3333333333332

12 33333333333332

13 333333333333332

14 3333333333333332


Chapter-7




1 662

2 6662

3 66662

4 666662

5 6666662

6 66666662

7 666666662

8 6666666662

9 66666666662

10 666666666662

11 6666666666662

12 66666666666662

13 666666666666662

3 6666666666666662


Chapter-8




1 992

2 9992

3 99992

4 999992

5 9999992

6 99999992

7 999999992

8 9999999992

9 99999999992

10 999999999992

11 9999999999992

12 99999999999992

13 999999999999992

14 9999999999999992


Chapter-9

Square – Numbers Consists Unit Digit 1 & Rest of All Digits 9



1 912

2 9912

3 99912

4 999912

5 9999912

6 99999912

7 999999912

8 9999999912

9 99999999912

10 999999999912

11 9999999999912

12 99999999999912

13 999999999999912

14 9999999999999912


Chapter-10




1 922

2 9922

3 99922

4 999922

5 9999922

6 99999922

7 999999922

8 9999999922

9 99999999922

10 999999999922

11 9999999999922

12 99999999999922

13 9999999999999922

14 99999999999999922


Chapter-11




1 932

2 9932

3 99932

4 999932

5 9999932

6 99999932

7 999999932

8 9999999932

9 99999999932

10 999999999932

11 9999999999932

12 99999999999932

13 999999999999932

14 9999999999999932


Chapter-12




1 942

2 9942

3 99942

4 999942

5 9999942

6 99999942

7 999999942

8 9999999942

9 99999999942

10 999999999942

11 9999999999942

12 99999999999942

13 999999999999942

14 9999999999999942


Chapter-13

Squares – Numbers Consists Unit Digit 5 & Rest of All Digits 9



1 952

2 9952

3 99952

4 999952

5 9999952

6 99999952

7 999999952

8 9999999952

9 99999999952

10 999999999952

11 9999999999952

12 99999999999952

13 999999999999952

14 9999999999999952


Chapter-14



Sl.No.

Problem

Write your answer here

Time Taken in Secs

1

962

2

9962

3

99962

4

999962

5

9999962

6

99999962

7

999999962

8

9999999962

9

99999999962

10

999999999962

11

9999999999962

12

99999999999962

13

999999999999962

14

9999999999999962


Chapter-15




1 972

2 9972

3 99972

4 999972

5 9999972

6 99999972

7 999999972

8 9999999972

9 99999999972

10 999999999972

11 9999999999972

12 99999999999972

13 999999999999972

14 9999999999999972


Chapter-16




1 982

2 9982

3 99982

4 999982

5 9999982

6 99999982

7 999999982

8 9999999982

9 99999999982

10 999999999982

11 9999999999982

12 99999999999982

13 999999999999982

14 9999999999999982


Chapter-17

Multiplication by 5

EXERCISE 1: STEP WISE

Problem 1: 28 x 5

Steps

Problem 2: 88 x 5

Steps

Problem 3: 129 x 5

Steps

Problem 4: 988 x 5

Steps

Problem 5: 5269 x 5

Steps


Problem 6: 68954 x 5

Steps


Steps

Problem 8: 9658647 x 5

Steps

Problem 9: 235648799 x 5

Steps

EXERCISE 2: QUICK STEPS

Sl.No. Problem Answer with Quick Steps

1 36 x 5

2 68 x 5

3 156 x 5

4 358 x 5


5 655 x 5

6 1234 x 5

7 5678 x 5

8 98765 x 5

9 65465 x 5

10 363538 x 5

11 320665 x 5

12 553260 x 5

13 6969087 x 5

14 789230 x 5

15 1230026 x 5

16 1236587 x 5



1 48 x 5

2 165 x 5

3 238 x 5

4 654 x 5


5 712 x 5

6 821 x 5

7 933 x 5

8 1036 x 5

9 1951 x 5

10 2339 x 5

11 56812 x 5

12 12345 x 5

13 36365 x 5

14 659812 x 5

15 1203650 x 5

16 36523002 x 5

17 365897455 x 5

18 234567810 x 5

19 659879871 x 5

20 36598740025 x 5

21 36985748547 x 5

22 965965963265 x 5


Chapter-18

Multiplication by 8


Problem 1: 246 x 8

Steps

Problem 2: 968 x 8

Steps

Problem 3: 4652 x 8

Steps

Problem 4: 9532 x 8

Steps

Problem 5: 3532267 x 8

Steps


Problem 6: 49982351 x 8

Steps

Problem 7: 6324127 x 8

Steps

Problem 8: 21457841 x 8

Steps



1 2345 x 8

2 4652 x 8

3 9994 x 8

4 32654 x 8

5 42517 x 8

6 63254 x 8

7 236541 x 8

8 632547 x 8

9 985742 x 8


10 9658745 x 8

11 3265451 x 8

12 6532654 x 8

13 3659784 x 8

14 61987487 x 8

15 48751523 x 8



1 3261 x 8

2 4455 x 8

3 5678 x 8

4 7895 x 8

5 32654 x 8

6 652487 x 8

7 215478 x 8

8 975487 x 8

9 875471 x 8

10 875487 x 8

11 654654 x 8

12 3254141 x 8

13 63256487 x 8

14 565874511 x 8


Chapter-19

Multiplication by 9


Problem 1: 132 x 9

Steps

Problem 2: 965 x 9

Steps

Problem 3: 5681 x 9

Steps


Steps

Problem 5: 3526781 x 9

Steps


Problem 6: 2999772 x 9

Steps



1 1234 x 9

2 4325 x 9

3 6555 x 9

4 45489 x 9

5 6985 x 9

6 144747 x 9

7 123234 x 9

8 567878 x 9

9 68974588 x 9

10 799999999 x 9

11 387683281 x 9

12 75468743 x 9

13 1564836581 x 9

14 697689232 x 9

15 47123561 x 9




1 1568 x 9

2 2424 x 9

3 4253 x 9

4 6578 x 9

5 212354 x 9

6 869852 x 9

7 777122 x 9

8 452414 x 9

9 253636 x 9

10 758966 x 9

11 741245 x 9

12 4742456 x 9

13 23265412 x 9

14 36598744 x 9

15 654877712 x 9

16 63258741254 x 9


Chapter-20

Multiplication by 11


Problem 1: 13231 x 11

Steps

Problem 2: 96587 x 11

Steps

Problem 3: 568999 x 11

Steps

Problem 4: 458932 x 11

Steps

Problem 5: 356781 x 11

Steps


Problem 6: 9997723 x 11

Steps



1 1234 x 11

2 4325 x 11

3 6565 x 11

4 45689 x 11

5 69889 x 11

6 74747 x 11

7 12341234 x 11

8 56785678 x 11

9 68974588 X 11

10 799999999 X 11

11 387683281 x 11

12 975468743 X 11

13 1564836581 X 11


14 3897689232 x 11

15 45987123561 X 11

16 65326598123 x 11



1 2341 x 11

2 1365 x 11

3 7878 x 11

4 6868 x 11

5 59966 x 11

6 868686 x 11

7 696989 x 11

8 3654265 x 11

9 85687452 x 11

10 326512425 x 11

11 5684213451 x 11

12 56897487231 x 11

13 42685812132 x 11

14 986572365124 x 11

15 123456789251 x 11

16 6365658732145 x 11


Chapter-21



Problem 1: 124 x 12

Steps

Problem 2: 568 x 12

Steps

Problem 3: 985 x 12

Steps

Problem 4: 926125 x 12

Steps

Problem 5: 178926 x 12

Steps


Problem 6: 99998 x 12

Steps



1 123 x 12

2 623 x 12

3 836 x 12

4 986 x 12

5 3654 x 12

6 5555 x 12

7 8632 x 12

8 9865 x 12

9 32321 x 12

10 69867 x 12

11 87654 x 12

12 256478 x 12

13 953684 x 12

14 4564564 x 12


15 8697451 x 12

16 65326540 x 12

17 230005620 x 12

18 653248700 x 12



1 26 x 12

2 72 x 12

3 98 x 12

4 365 x 12

5 487 x 12

6 689 x 12

7 6548 x 12

8 8888 x 12

9 9123 x 12

10 36526 x 12

11 46821 x 12

12 84354 x 12


Chapter-22



Problem 1: 14 x 25

Steps

Problem 2: 38 x 25

Steps

Problem 3: 198 x 25

Steps

Problem 4: 546 x 25

Steps


Steps

Problem 6: 98562 x 25

Steps


Problem 7: 98562 x 25

Steps

Problem 8: 98562 x 25

Steps

Problem 9: 98562 x 25

Steps

Problem 10: 98562 x 25

Steps


Sl.No.

Problem

Answer with Quick Steps

1 52 x 25

2 73 x 25

3 89 x 25

4 198 x 25

5 561 x 25

6 733 x 25


7 1234 x 25

8 5678 x 25

9 36584 x 25

10 65847 x 25


Sl.No.

Problem


Time Taken in Secs

1 5 x 25

2 16 x 25

3 36 x 25

4 125 x 25

5 268 x 25

6 641 x 25

7 3214 x 25

8 7172 x 25

9 34341 x 25

10 93265 x 25

11 2658411 x 25

12 23234577 x 25

13 98988 x 25

14 2356842 x 25

15 657842897 x 25


Chapter-23



Problem 1: 24 x 75

Steps

Problem 2: 46 x 75

Steps

Problem 3: 146 x 75

Steps

Problem 4: 263 x 75

Steps

Problem 5: 328 x 75

Steps


Problem 6: 555 x 75

Steps


Steps

Problem 8: 23654 x 75

Steps


Sl.No.

Problem

Answer with Quick Steps

1 456 x 75

2 698 x 75

3 777 x 75

4 1962 x 75

5 8754 x 75

6 32654 x 75

7 42536 x 75

8 74523 x 75


9 123456 x 75

10 3265324 x 75


Sl.No.

Problem


Time Taken in Secs

1 128 x 75

2 276 x 75

3 369 x 75

4 988 x 75

5 1963 x 75

6 2982 x 75

7 6568 x 75

8 7172 x 75

9 12121 x 75

10 33445 x 75

11 234563 x 75

12 27675

13 54784 x 75

14 321652 x 75


Chapter-24

Multiplication – Teen Numbers


Problem 1: 12 x 14

Steps

Problem 2: 15 x 16

Steps

Problem 3: 18 x 19

Steps




1 16 x 12

2 18 x 16

3 17 x 18

4 18 x 13

5 16 x 14



1 19 x 13

2 19 x 16

3 18 x 12

4 12 x 17

5 16 x 13

6 12 x 15

7 19 x 17

8 18 x 14

9 13 x 18

10 19 x 16


Chapter-25

Multiplication of Two Digit Numbers – Line Method

EXERCISE 1: STEPWISE

Problem 1: 12 x 16

Problem 2: 22 x 33

Problem 3: 15 x 14


Problem 4: 25 x 32

Problem 5: 24 x 43

Problem 6: 41 x 72


Chapter-26

Vertically & Crosswise


Problem 1: 12 x 23

Problem 2: 22 x 28

Problem 3: 48 x 74

Problem 4: 56 x 77


Problem 5: 98 x 97

Problem 6: 16 x 69

Problem 7: 65 x 92

Problem 8: 45 x 86

Problem 9: 56 x36


Problem 10: 22 x 98

Problem 11: 78 x 65

Problem 10: 72 x 86



1 19 x 91

2 28 x 69

3 33 x 63

4 44 x 19

5 58 x 88

6 64 x 25


7 26 x 66

8 56 x 77

9 23 x 55

10 44 x 99

11 64 x 68

12 72 x 85

13 27 x 72

14 34 x 61

15 15 x 68

16 35 x 48

17 99 x 98

18 74 x 76



1 99 x 91

2 82 x 96

3 36 x 63

4 55 x 19


5 85 x 48

6 46 x 25

7 46 x 67

8 65 x 88

9 32 x 77

10 64 x 81

11 74 x 78

12 75 x 15

13 29 x 73

14 53 x 63

15 98 x 53

16 99 x 99

17 12 x 32

18 26 x 65

18 32 x 23

19 86 x 92

20 98 x 44

221 96 x 55

22 44 x 26


Chapter-27

Multiplying 2 Numbers With The Same Ten's Digit


Problem 1: 43 x 48

Steps

Problem 2: 54 x 52

Steps

Problem 3: 92 x 98

Steps



Steps


Steps



1 25 x 28

2 31 x 39

3 46 x 49

4 57 x 52

5 86 x 88

6 126 x 122

7 156 x 157


8 184 x 187

9 238 x 237

10 546 x 541



1 26 x 23

2 38 x 33

3 46 x 46

4 57 x 58

5 65 x 63

6 77 x 78

7 89 x 87

8 122 x 124

9 148 x 143

10 155 x 153

11 163 x 162

12 197 x 191


Chapter-28

Multiplying Two Numbers That End In 5


Problem 1: 25 x 105

Steps

Problem 2: 65 x 85

Steps


Steps

Problem 4: 35 x 65

Steps


Steps


Steps




1 45 x 195

2 55 x 225

3 245 x 265

4 315 x 15

5 475 x 455

6 655 x 955



1 95 x 145

2 35 x 135

3 25 x 125

4 45 x 135

5 555 x 525

6 325 x 315

7 415 x 155

8 225 x 255

9 995 x 15

10 335 x 625


Chapter-29

Square of Numbers Ending in 1


Problem 1: 312

Steps

Problem 2: 612

Steps

Problem 3: 912

Steps

Problem 4: 1212

Steps

Problem 5: 1812

Steps

Problem 6: 2212

Steps




1 912

2 1312

3 1712

4 2612

5 3912

6 4612

7 5212

8 5512

9 6612

10 7312



1 712

2 512

3 812

4 1112

5 1512

6 1912

7 2612

8 3312

9 4412

10 9912


Chapter-30



Problem 1: 352

Steps

Problem 2: 452

Steps

Problem 3: 652

Steps

Problem 4: 1252

Steps

Problem 5: 2552

Steps

Problem 6: 3652

Steps


Problem 7: 4552

Steps

Problem 8: 7552

Steps



1 552

2 1752

3 2952

4 3152

5 3952

6 4452

7 4852

8 5752

9 6352

10 6952

11 7152


12 7752



1 152

2 352

3 852

4 1052

5 1552

6 3552

7 4152

8 4752

9 5152

10 5352

11 6652

12 7552

13 8452

14 9552

15 9952

16 10152

17 11252

18 20052


Chapter-31



Problem 1: 262

Steps

Problem 2: 462

Steps

Problem 3: 662

Steps

Problem 4: 862

Steps

Problem 5: 1162

Steps




1 762

2 1262

3 1562

4 1962

5 2362

6 3362

7 3862

8 4662

9 4862

10 5662

11 6762

12 7862

13 8262

14 9762

15 9962

16 10262




1 362

2 562

3 962

4 1062

5 2162

6 2762

7 3662

8 4562

9 4962

10 5162

11 4462

12 5162

13 6662

14 7662

15 8562

16 9662


Chapter-32



Problem 1: 242

Steps

Problem 2: 442

Steps

Problem 3: 642

Steps

Problem 4: 1242

Steps

Problem 5: 8542

Steps




1 342

2 542

3 942

4 1342

5 1642

6 3342

7 3642

8 4342

9 5642

10 5842



1 142

2 742

3 842

4 1042

5 2142

6 2342

7 3142

8 4142


Chapter-33



Problem 1: 292

Steps

Problem 2: 492

Steps

Problem 3: 1092

Steps

Problem 4: 7292

Steps

Problem 5: 8592

Steps

Problem 6: 9292

Steps




1 792

2 1292

3 1592

4 2392

5 2892

6 3192

7 4492

8 5392

9 5592

10 6192



1 892

2 1192

3 1792

4 1892

5 2192

6 2492


Chapter-34

Adding Consecutive Numbers


Problem 1: 12 to 18

Problem 2: 15 to 45

Problem 3: 99 to 124





1 7 to 44

2 5 to 45

3 6 to 99

4 9 to 66

5 16 to 44

6 22 to 58

7 33 to 65

8 128 to 988

9 256 to 369

10 567 to 984

11 874 to 1025

12 986 to 2568

13 999 to 12387

14 1026 to 6557

15 5687 to 5689

16 2500 to 3698

17 3265 to 3365

18 1232 to 1238

19 3164 to 3264

20 10001 to 12568


Chapter-35

Adding Consecutive Numbers Starting From 1


Problem 1: 1 to 14

Problem 2: 1 to 444

Problem 3: 1 to 987






1 1 to 24

2 1 to 33

3 1 to 41

4 1 to 89

5 1 to 99

6 1 to 999

7 1 to 1212

8 1 to 4567

9 1 to 7890

10 1 to 7676



1 1 to 11

2 1 to 30

3 1 to 66

4 1 to 88

5 1 to 155

6 1 to 200

7 1 to 781

8 1 to 2997

9 1 to 5748

10 1 to 9999


Chapter-36

Finding The Sum of All Odd Numbers Starting From 1


Problem 1: 1 to 11

Problem 2: 1 to 46

Problem 3: 1 to 110

Problem 4: 1 to 333

Problem 5: 1 to 455

Problem 6: 1 to 500




1 1 to 14

2 1 to 58

3 1 to 65

4 1 to 99

5 1 to 125

6 1 to 268

7 1 to 398

8 1 to 857

9 1 to 999

10 1 to 5687



1 1 to 22

2 1 to 69

3 1 to 87

4 1 to 198

5 1 to 254

6 1 to 457

7 1 to 666

8 1 to 988

9 1 to 1265

10 1 to 3256


Chapter-37

Finding The Sum of All Even Numbers Starting From 2


Problem 1: 2 to 28

Problem 2: 2 to 120

Problem 3: 2 to 325

Problem 4: 2 to 998






1 2 to 16

2 2 to 54

3 2 to 86

4 2 to 128

5 2 to 256

6 2 to 512

7 2 to 778

8 2 to 996

9 2 to 1636

10 2 to 8659



1 2 to 28

2 2 to 46

3 2 to 78

4 2 to 120

5 2 to 198

6 2 to 296

7 2 to 299

8 2 to 345

9 2 to 431

10 2 to 557

11 2 to 687

12 2 to 889

13 2 to 985

14 2 to 1234


Chapter-38

Complements Using Nikhilam Navataha


Problem 1: Find the complement of 78









1 59

2 98

3 196

4 368

5 569

6 800

7 8001

8 96587

9 123456

10 5678901



1 99

2 126

3 365

4 9998

5 56897

6 98988

7 321654

8 9879879

9 32654812

10 265659812


Chapter-39

Subtraction by Ekadhikena Purvena

EXERCISE 1: TWO DIGITS SUBTRACTION

Problem No. 1 2 3 4 5

9 3 3 5 8 1 5 4 6 7

4 5 2 9 6 8 3 5 4 8

Answer


8 5 7 8 4 5 7 4 5 2

3 8 3 9 3 6 3 7 3 8

Answer

EXERCISE 2: THREE DIGITS SUBTRACTION


4 5 6 3 2 3 5 6 7 1 2 3 6 8 7

2 8 9 1 8 5 4 1 9 9 9 2 8 8

Answer


9 9 8 8 7 4 4 2 3 5 6 3 8 7 6

9 8 9 4 9 9 2 8 6 3 9 8 2 9 8

Answer


EXERCISE 3: FOUR DIGITS SUBTRACTION

Problem No. 1 2 3 4

5 5 5 5 4 6 8 1 2 3 4 5 6 5 4 3

1 6 7 8 2 3 4 5 9 9 9 4 2 4 8

Answer

Problem No. 5 6 7 8

9 8 7 6 6 4 5 2 3 5 6 7 6 8 7 1

1 9 9 9 5 4 9 8 2 8 6 8 3 9 8 4

Answer

Problem No. 9 10 11 12

5 2 5 4 9 2 6 6 3 8 8 1 2 6 8 1

3 9 9 7 8 1 9 7 1 5 9 7 1 9 8

Answer

Problem No. 13 14 15 16

2 6 3 1 9 1 9 4 3 2 1 0 3 6 0 0

1 9 9 9 2 9 8 7 2 8 7 9 3 5 8 9

Answer

Problem No. 17 18 19 20

9 6 4 5 6 8 2 0 1 2 6 0 2 0 1 0

2 9 7 8 5 9 8 2 1 1 9 9 1 9 9 9

Answer


EXERCISE 4: FIVE DIGITS SUBTRACTION

Problem No. 1 2 3

9 8 7 6 5 8 7 6 2 3 6 5 4 6 5

2 8 8 7 6 8 7 6 1 9 2 6 5 8 9

Answer

Problem No. 4 5 6

6 5 5 4 5 2 3 2 3 4 5 8 9 7 5

2 9 9 8 7 1 3 2 4 7 3 2 5 8 9

Answer

Problem No. 7 8 9

4 2 3 5 8 4 3 6 2 7 6 1 1 1 1

3 1 1 8 9 3 8 8 9 9 5 1 1 1 9

Answer

Problem No. 10 11 12

3 5 6 8 1 6 8 7 2 3 9 8 7 4 5

2 2 2 9 9 5 7 9 4 5 9 8 7 3 9

Answer


7 2 6 4 2 5 4 3 2 8 7 1 2 0 4

4 8 8 7 5 2 7 9 8 9 1 2 3 4 5

Answer


Chapter-40

Subtraction by Nikhilam Navataha

EXERCISE 1:

Problem No. 1 2 3

6 3 4 2 2 9 7 5 3 2 8 4 6 7 4

4 7 4 4 5 5 2 4 7 9 6 3 9 9 8

Answer

Problem No. 4 5 6

8 9 7 6 6 1 9 7 4 5 6 4 5 3 2

1 7 8 8 9 0 7 8 9 5 4 9 9 8 9

Answer

Problem No. 7 8 9

9 6 9 7 7 8 5 3 2 1 8 4 5 3 7

5 4 7 9 9 5 9 7 7 9 4 9 9 8 8

Answer


2 2 3 4 5 9 3 2 1 0 8 4 5 3 2

1 3 5 9 8 5 6 7 8 8 6 5 9 7 4

Answer


Chapter-41

Starting Complements from the Middle of the Sum

EXERCISE 1:

Problem No. 1 2 3

5 6 7 8 9 6 5 3 2 8 3 8 2 6 8

2 9 8 9 2 5 6 9 1 6 1 9 4 9 3

Answer

Problem No. 4 5 6

9 5 6 8 1 3 2 6 5 7 6 7 5 8 2

5 8 8 9 0 2 9 7 4 3 5 9 6 2 1

Answer

Problem No. 7 8 9

5 6 7 2 9 9 6 5 4 8 3 2 1 6 9

2 8 9 1 3 8 8 9 7 3 2 8 9 4 6

Answer


8 7 3 2 8 2 3 6 2 4 3 6 6 8 2

6 9 7 1 2 1 7 9 8 2 1 9 8 2 1

Answer


Chapter-41

Leaving Complements in the Middle of the Sum

EXERCISE 1:

Problem No. 1 2 3

6 5 3 7 8 3 2 1 2 2 8 5 6 3 5

3 2 9 9 1 1 1 9 8 6 2 2 3 9 6

Answer

Problem No. 4 5 6

9 8 7 5 4 3 2 6 5 3 6 7 8 5 7

3 5 6 9 8 2 1 5 8 7 4 5 9 8 8

Answer

Problem No. 7 8 9

7 8 2 6 5 3 6 7 4 1 2 6 5 4 8

4 6 3 7 8 2 2 9 7 6 1 3 6 9 9

Answer


7 5 2 1 4 3 2 6 5 7 6 8 4 2 1

6 2 6 7 6 2 1 9 9 8 5 6 7 8 3

Answer


Chapter-42

Complements More than Once in the Same Sum

EXERCISE:

Problem No. 1 2 3

9 6 3 2 1 4 6 7 1 2 4 5 5 6 9 1 5 2

2 9 1 8 7 3 3 8 0 6 7 4 3 8 9 4 3 9

Answer

Problem No. 4 5 6

7 2 4 3 5 7 6 9 2 8 7 4 5 1 5 4 6

2 8 1 1 3 5 4 0 9 8 9 0 2 3 7 2 8

Answer

Problem No. 7 8 9

3 3 1 2 8 6 9 7 3 6 8 3 6 3 4 5 3 4

2 8 0 6 9 7 4 6 6 2 9 2 2 9 9 9 1 8

Answer


5 9 6 3 2 4 5 0 1 2 3 4 4 2 1 3 5 4

1 6 8 8 0 5 3 2 8 4 1 8 3 3 3 3 3 8

Answer


PROBLEMS FOR YOU

Use exactly four 4's to form every integer from 0 to 50, using only the operators +, -, x, /, () (brackets) x2

(square), and ! (factorial).

Answer Steps

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Documents

VASISTA maths workbook.pdfVASISTA Head Office: #41, Bhanavi Complex, Vishwamanava Double Road, Kuvempu Nagara, Mysore-570023 vedic-maths.com 8748885500,8748886600