*ALGEBRA Made Easy as Arithmetic*

# How to use Arithmetic to

Simplify Algebra!

Discover the art of 'ALGEBRA Made Easy as Arithmetic,' Ike Prudente's book which demystifies a seemingly difficult subject. This book teaches students creative Math thinking, empowers teachers and parent with teaching tools and tricks and entertains math enthusiasts.

Be a Math Wizard Now!

Can you solve these problems in 60 seconds?

Many schools teach students to perform their Arithmetic operations from right to left. Most Algebraic operations however, are done from left to right, more naturally in the same way as we read and write.

A false impression that Algebra is unrelated to Arithmetic is thus created in the minds of students.

Watch 9-year old Franz, after learning Math shortcuts, solve arithmetic problems left to right, in 30 seconds

Algebra is a just a generalization of Arithmetic. Franz’s use of left to right techniques in addition and multiplication prepares him for Algebra. He used applications of Algebraic generalizations to specific Arithmetic problems

'ALGEBRA made easy as Arithmetic' enables the student to **solve algebraic problems as easy as Franz solves arithmetic problems**.
'ALGEBRA made easy as Arithmetic' enables the student to **solve algebraic problems as easy as Franz solves arithmetic problems**.

Using the multiplication by 11 arithmetic short cut enables us to quickly compute 143 x 11.

First we write 1, then the sum of 1+4=5, then the sum of 4+3=7 then 3, to get 1573.

Using the same multiplication technique enables us to quickly compute (x^{2}+4x+3) * (x+1).

First we write 1x^{3} (or just x^{3}), then the sum of 1+4=5x^{2}, then the sum of 4+3=7x then 3, to get 1x^{3}+5x^{2}+7x+3.

That is....

# ALGEBRA Made Easy as Arithmetic!

**From The Desk Of**

** Ike Prudente**

** Math Advocate**

**Dear Math Enthusiast,**

UP Diliman Chancellor Michael Tan, wrote “What I see in this book’s collection of ‘tricks’ is practical algebra,” referring to my first printed book, “25 Math Short Cuts,” which contains algebraic proofs to most of the arithmetic techniques presented.

'ALGEBRA Made Easy as Arithmetic' is somewhat the reverse: it is an initial attempt to demonstrate how application of some basic arithmetic procedures can greatly simplify many algebraic operations. It also tries to show that there are other approaches to algebraic solutions which some students may find simpler to use.

This is not an introductory course to Algebra. For those students who are about to take their first course in Algebra, we recommend that they first read Kenneth William’s e-book “Algebra for the Terrified” which is available at www.amazon.com

This is also not intended to be an Algebra textbook. Instead, this is a handy reference book that aims to show that Mathematics as exemplified by Vedic Math is a unified field of study rather than segregated into Arithmetic, Algebra, Geometry and other subjects as taught in conventional Mathematics. It is a perfect supplement to Algebra textbooks.

The vertically-and-crosswise general multiplication method used extensively in Vedic Mathematics, for example, can be applied to polynomial multiplication resulting into one-line solutions. The result of this polynomial multiplication can then be quickly verified using the Vedic Sutra or word formula: “The product of the sums is the sum of the product”. This method using digit sums used to be taught in the intermediate grades during the early nineteen sixties and is referred to as “Cuenta Viejo” by old folks.

Math skills is required for most high paying jobs in today’s technology-driven economy. As Algebra is considered the gateway to higher Mathematics, understanding and appreciating Algebra is vital to any student’s life.

In Part I, we included some simple arithmetic techniques not commonly taught in schools today but have valuable applications in algebra.

In Part II, Arithmetic thinking is applied in solving linear equations. Some later chapters may seem difficult at first but will become easy with a little practice.

In Parts III and IV, some specific techniques discussed in Part I are applied in polynomial multiplication, division and factoring.

Some chapters not directly based on the Arithmetic techniques are added for completeness.

This book will prove useful to many sectors:

- For Elementary and High school Math teachers, Part I presents the additional arithmetic techniques they need to teach their pupils to greatly improve their confidence in solving Algebraic problems. The digit sum can be taught as early as grade 1 while its application can be discussed in later grades.
- For Algebra instructors, this will enable them to apply the lessons in Part I to simplify some of the conventional “long” methods in Algebra as shown in Parts II-IV.
- For students of from all ages, this book provides a handy reference from elementary up to high school and even college math courses.
- For parents, this is an invaluable tool for parents, especially of the home schooler, who struggle in Algebra.

We hope that this book will inspire many teachers and students to study and create more applications of the basics of Arithmetic in other Math course so that we can create a series of “Made Easy as Arithmetic” books.

Ike Prudente

# What do People Say About

'ALGEBRA Made Easy as Arithmetic'

*
In the age of digital technology, daily activities are anchored on computations by men and machines. Mathematics isn’t just knowing your multiplication table. It is also about creative and analytical thinking. This book unlocks the ideas behind Algebra as applied to Arithmetic. To teachers, students and enthusiasts of Mathematics, go forth and multiply (no pun intended) using the techniques in 'ALGEBRA Made Easy As Arithmetic.'
*

**Joselito Bautista**

Corporate Planner and Math Enthusiast

Finalist, Inter-University Search for the Math Wizard 1978, 1979

1980 Topnotcher, Electrical Engineering Board Exam

*
'Algebra Made Easy As Arithmetic' is a terrific book that offers something for everyone. Math lovers will be challenged to understand why the tips and techniques actually work. On the other hand, students who need some help with math can try out all the techniques, and pick out the ones that they like best.*

*I can recommend this book to all algebra students, and to their teachers as well!*

**Annette Lagman, PhD**

System IT Consultant, University of the Philippines

Former Math and Computer Science Professor, James Madison University, Effat University

*
In general, it is a good resource for teachers especially for coaches of learners who joined different math competitions. But the book can also help regular teachers to show to their learners that Math is not hard to understand and learn. The book can really help improve the performance of learners especially in algebra.
*

**Dr. Severa Salamat**

EPS Mathematics, Division of Sta Rosa City

*
I like it! I'm so glad that you included the casting out nines.
I used to teach that several decades ago and I seldom find it in new books. Your new book 'ALGEBRA Made Easy as Arithmetic,' only shows how brilliant the author is. I salute you. It will be one of a kind and I'm sure this will click for both teachers and students. Students find it so difficult to study Algebra so this book will make them learn it the easy way.
Congratulations Ike, this will be another milestone in your career and I'm happy to know about the invitation you received for you to present your work at Vedic's Math Conference.
*

**Encarnacion Reyes**

President. Math Teacher's Association of the Philippines, San Pablo Chapter

*
In all my more than 3 decades of computer programming experience, never have I written code that would include concepts that I have learned in Calculus.
Always in Algebra.
For me, Algebra is the root of all problem-solving techniques.
Thanks to my good old friend, Virgilio 'Ike' Ylagan Prudente, Algebra can now be as simple as Arithmetic. I just wish this book was written earlier. Had it been so, I would have gotten straight 1's in all my post Math 11 subjects. Good job, Ike! Very good job, indeed!
*

**Danny Lopez**

IT Consultant

Co-Founder, MSC Institure of Technology

San Pablo City

*
I have been teaching mathematics for many years. It is always a joy to see students who love the subject. Sad to say, majority of the students nowadays, especially the non-engineering ones, hate the subject. Algebra is out of this world for them. With all the x’s and the y’s plus all the other letters that sometimes accompany them, most students do not know what to do.
With Ike Prudente’s Algebra-Made-Easy-As-Arithmetic, the students will have a better grasp of algebra. The approach is very much different from the algebra we used to know. The student will discover that algebra is something to enjoy, not something to be afraid of.
*

**Isaac Pitcairn Nemenzo Yap**

Former Dean, College of engineering, Architecture and Technology, Palawan State University.

*
Math was never easy back in high school. With this book, however, Math has never been easier. Mr. Ike Prudente has done us a great service. If Ike can, you can. If you can, I can. If Ike can, I can!
*

**Vincent Ragay**

Pisay_Dos

# Foreword to

'ALGEBRA Made Easy as Arithmetic'

It is good to see the appearance of this book as it shows how the simple processes we all learn in arithmetic have a natural counterpart in algebra. In addition, the easy and unified Vedic methods make the transition seem almost obvious.

From there, i.e. from the algebraic formulations, the process can be reversed again to provide and suggest further arithmetical applications. This is the power of algebra: to represent a whole class within a single algebraic statement. Then careful study of that statement can reveal further unknown and unexpected relationships.

Algebra is generalised arithmetic and the mental process of generalisation is one we all unconsciously perform all the time. So there is nothing really new here, it is a matter of applying our normal thinking in the realm of number. Spotting patterns is doing algebra: it is as simple as that.

This book therefore offers a much needed path from simple number work to the next level up, which involves the use of letters as symbols that have a meaning. This in turn expands our ability to handle mental processes that involve generalisation and abstraction. That is to say it develops our normal thinking processes and abilities.

This excellent book from Virgilio “Ike” Prudente is a welcome and much needed addition to the Vedic Maths literature and will undoubtedly be a great help to all those who wish to expand and enrich their knowledge of mathematics.

**Kenneth Williams
Vedic Mathematics Academy**

# Table of Contents

Part I: Some Important Math Short Cuts

1: Digit Sum

2: Using Digits Sums to Check Arithmetic Computations

3: The First by the First and the Last by the Last

4: Multiplication by 11, 12 & 13

5: Multiplication by 9, 8 & 7

6: Division by 9, 8 & 7

7: Division by 11, 12 & 13

8: Vertically and Crosswise -- 2 x 2 left-to-right multiplication

9: Vertically and Crosswise -- Longer Multipliers

10: Vertically and Crosswise -- Sliding Multiplier

11: Vertically and Crosswise – Fractions

Part II – Solving Linear equations

Warm-Up: Arithmetic Review

Work Out: Introduction to Algebra

12: “Think of a Number” – One-step Solution

13: “Think of a Number” – Two-step Solution

14: “Think of a Number” – Three-step Solution

15: Solution of Equations with Two Variable Terms: Type I – ax + b = cx + d

16: Solution of Equations with Two Variable Terms: Type II – x/a + x/b = c

17: Solution of Equations with Two Variable Terms: Type III – (ax + b)/(cx + d) = p/q

18: Solution of Simultaneous Equations: By Substitution

19: Solution of Simultaneous Equations: By Addition and By Subtraction

20: Solution of Simultaneous Equations: Special Case – x and y coefficients interchanged

Part III – Multiplication Techniques:

21: Algebraic Multiplication – Binomials

22: Algebraic Multiplication – Trinomials

23: Multiplication of Longer Polynomials

24: Multiplying Polynomials of Unequal Length: Sliding Multiplier

25: The First by the First and the Last by the Last

26: The Product of the Sum is the Sum of the Products

27: Special Multiplication: Multiplying by (x + 1)

28: Special Multiplication - Multiplying by (x - 1)

29: General Multiplication - Multiplying by (ax±b)

Part IV: Solving quadratic equations

30: Reversing DPMA

31: Determining the Signs

32: Finding the Constants in the Factors

33: The AC Method of Factoring

34: Tirthaji’s Method

35: The "root" method

36: Using the Sum of Coefficients

37: Special Case: One factor is (x + 1)

38: Special Case: One Factor is (x - 1)

39: The differential is equal to the square root of the discriminant

40: Special Division by (x - 1)

41: Special Division by (x + 1)

42: Special Division by (x ± a)

### p.s. Who loves Math?

Author Ike Prudente did a demo to the **Kariton Teacher Volunteers, Dynamic Teen Company** of Cavite City

Unang tanong ni Ike sa teacher-volunteers, *"How many of you like Math?"* Very few raised their hands.

After a two-hour session, demonstrating some of the math techniques, tinanong ulit sila, *"Who loves Math?"*

Halos lahat sila nag-taas ng kamay!

Teacher-volunteers, spread the word: Math can be made **Fun, Fast and Easy!**

**From The Desk Of**

** Ike Prudente**

** MATH Advocate**

**RE:** The Only Way To Teach Your Child Math from Home!

**Dear Parent,**

I
am writing you today because I know you love your kids and you want them to ace through every Math subject in school. I also know that deep down in your heart you wish that Math could be more fun, for you and for your kids.

Maybe, if There is a MATH Teacher in the House!

You could be that Math Teacher, and we will help you be exactly that!

The parents are in a better position to understand their children’s learning style and better help them with their lessons than their teachers provided they have the confidence and the knowhow.

In fact, Math education should start in the home as early as possible, much earlier that school age. And the parent being the most influential person in a child’s life, is in the best position to help cultivate the child’s positive attitude towards Math.

Our books seek to provide resources for parents to increase their knowledge in Math so that they will develop the confidence for their children. Specifically, we aim to show the parents

- That Math is Fun, Fast, and Easy
- How to “Mathematize Your Home”
- How to help children get good grades in math

# What’s Algebra But a Lost Number?

**By Rheea Hermoso-Prudente**

I remember my high school Math days as bleak days of terror. Geometry was bearable, but Algebra, Trigonometry, and Calculus were the stuff my nightmares were made of. I remember in one Calculus class, our teacher asked me to stand and recite and I couldn't answer the question. I was totally humiliated (The Hubby loved that particular Math teacher, though, and was most likely loved back; in fact, that teacher was a guest at our wedding).

I do not want my kids to have the same mathematical distaste. Although I have learned to -- if somewhat begrudgingly -- like Math, I want them to be more enthusiastic about it, the way their grandfather is (I always tell the story of Lolo Ike's bewilderment when no one quite shared the thrill of discovery over ways to divide with some number like 26 or such). So far, I think we've built some decent foundational skills with minimal pain and some fun, thanks to jellybeans and board games.

Raine will be moving up to college in a few months, and obviously, so will the complexity of the Math concepts she'll be learning. I'm glad that she's had some rudimentary exposure to formal Algebra in her old schools, so she's fairly confident that "Algebra's easy. I like Algebra!" Good for her. I think our brainwashing is working.