The First Sutra: Ekādhikena Pūrvena
The relevant Sutra reads Ekādhikena Pūrvena which rendered
into English simply says “By one more than the previous one”.
Its application and “modus operandi” are as follows.
(1) The last digit of the denominator in this case being 1 and the
previous one being 1 “one more than the previous one”
evidently means 2. Further the proposition ‘by’ (in the sutra)
indicates that the arithmetical operation prescribed is either
multiplication or division. We illustrate this example from pp. 1
to 3. 
Let us first deal with the case of a fraction say 1/19. 1/19
where denominator ends in 9.
By the Vedic one - line mental method.
A. First method
.0 5 2 6 315 7 89 4 7 368 4 2 i
1 1 111 1 1 11
B. Second Method
.0 5 2 6 3 1 5 7 8 / 9 4 7 3 6 8 4 2 i
1 1 11 1 1 1 1 1
This is the whole working. And the modus operandi is
A. First Method
Modus operandi chart is as follows:
(i) We put down 1 as the right-hand most digit 1
(ii) We multiply that last digit 1 by 2 and put the 2
down as the immediately preceding digit. 2 1
(iii) We multiply that 2 by 2 and put 4 down as the
next previous digit. 4 2 1
(iv) We multiply that 4 by 2 and put it down thus 8 4 2 1
(v) We multiply that 8 by 2 and get 16 as the
product. But this has two digits. We therefore
put the product. But this has two digits we
therefore put the 6 down immediately to the
left of the 8 and keep the 1 on hand to be
carried over to the left at the next step (as we
always do in all multiplication e.g. of 69 × 2 =
138 and so on). 6 8 4 2 1
(vi) We now multiply 6 by 2 get 12 as product, add
thereto the 1 (kept to be carried over from the
right at the last step), get 13 as the
consolidated product, put the 3 down and keep
the 1 on hand for carrying over to the left at
the next step. 3 6 8 4 2 1
(vii) We then multiply 3 by 2 add the one carried
over from the right one, get 7 as the
consolidated product. But as this is a single
digit number with nothing to carry over to
the left, we put it down as our next
multiplicand. 7 3 6 8 4 2 1
((viii) and xviii) we follow this procedure
continually until we reach the 18th digit
counting leftwards from the right, when we
find that the whole decimal has begun to
repeat itself. We therefore put up the usual
recurring marks (dots) on the first and the last
digit of the answer (from betokening that the
whole of it is a Recurring Decimal) and stop
the multiplication there.
Our chart now reads as follows:
= . 0 5 2 6 3 1 5 7 8 / 9 4 7 3 6 8 4 2 i .
1 1 1 1 1 1 / 1 1 1
B. Second Method
The second method is the method of division (instead of
multiplication) by the self-same “Ekādhikena Pūrvena” namely
2. And as division is the exact opposite of multiplication it
stands to reason that the operation of division should proceed
not from right to left (as in the case of multiplication as
expounded here in before) but in the exactly opposite direction;
i.e. from left to right. And such is actually found to be the case.
Its application and modus operandi are as follows:
(i) Dividing 1 (The first digit of the dividend) by
2, we see the quotient is zero and the
remainder is 1. We therefore set 0 down as the
first digit of the quotient and prefix the
remainder 1 to that very digit of the quotient
(as a sort of reverse-procedure to the carrying
to the left process used in multiplication) and
thus obtain 10 as our next dividend. 0
(ii) Dividing this 10 by 2, we get 5 as the second
digit of the quotient, and as there is no
remainder to be prefixed thereto we take up
that digit 5 itself as our next dividend. . 0 5
(iii) So, the next quotient – digit is 2, and the
remainder is 1. We therefore put 2 down as the
third digit of the quotient and prefix the
remainder 1 to that quotient digit 2 and thus
have 12 as our next dividend. . 0 5 2
(iv) This gives us 6 as quotient digit and zero as
remainder. So we set 6 down as the fourth
digit of the quotient, and as there is no
remainder to be prefixed thereto we take 6
itself as our next digit for division which gives
the next quotient digit as 3. . 0 5 2 6 3 1
1 1 1
(v) That gives us 1 and 1 as quotient and
remainder respectively. We therefore put 1
down as the 6th quotient digit prefix the 1
thereto and have 11 as our next dividend. . 0 5 2 6 3 1 5
1 1 1 1
(vi to xvii) Carrying this process of straight continuous
division by 2 we get 2 as the 17th quotient digit and 0 as
(xviii) Dividing this 2 by 2 are get 1 as 18th
quotient digit and 0 as remainder. But this is
exactly what we began with. This means that
the decimal begins to repeat itself from here.
So we stop the mental division process and
put down the usual recurring symbols (dots)
on the 1st and 18th digit to show that the
whole of it is a circulating decimal.
Now if we are interested to find 1/29 the student should
note down that the last digit of the denominator is 9, but the
penultimate one is 2 and one more than that means 3. Likewise
for 1/49 the last digit of the denominator is 9 but penultimate is
4 and one more than that is 5 so for each number the
observation must be memorized by the student and remembered.
The following are to be noted
1. Student should find out the procedure to be followed.
The technique must be memorized. They feel it is
difficult and cumbersome and wastes their time and
repels them from mathematics.
2. “This problem can be solved by a calculator in a time
less than a second. Who in this modernized world take
so much strain to work and waste time over such simple
calculation?” asked several of the students.
3. According to many students the long division method
was itself more interesting.
Posted by GMAT books at 11:46 AM
When you think of a square, you probably think of a box-shaped figure with four equal sides
that’s a good way to think about squares and square
A square of a number is just the number multiplied by itself. So the square of 4 is 4 × 4 = 16. How does
this relate to a square-shaped figure? The area of a square is the amount of space a square takes up. To calculate the area of a square, you multiply the length of one side by itself. That is why the area of a square is sometimes written as s squared, or s2. Any time a number is written with a 2 raised after it, it means to multiply the number by itself, or to square the number.
Posted by GMAT books at 11:27 AM
The strategy for adding 3-digit numbers is the same as for adding 2-digit numbers
you add left to right. After each step, you arrive at a new (and smaller) addition problem
All mental addition problems can be worked using this method. The goal is to
keep simplifying the problem until you are left adding a 1-digit number. It is important to
reduce the number of digits you are manipulating because human short-term memory is
limited to about 7 digits. Notice that 538 + 327 requires you to hold on to 6 digits in your
head, whereas 838 + 27 and 858 + 7 require only 5 and 4 digits, respectively. As you
simplify the problems, the problems get easier
Did you reduce and simplify the problem by adding left to right? After adding the
hundreds digit (623 + 100 = 723), you were left with 723 + 59. Next you should have
added the tens digit (723 + 50 = 773), simplifying the problem to 773 + 9, which you
easily summed to 782
Your mind-talk may not sound exactly like mine, but whatever it is you say to
yourself, the point is to reinforce the numbers along the way so that you don't forget
where you are and have to start the addition problem over again.
Posted by GMAT books at 11:24 AM
I remember the day in third grade when I discovered that it was easier to add and
subtract from left to right than from right to left, which was the way we had all been
taught. Suddenly I was able to blurt out the answers to math problems in class well before
my classmates put down their pencils. And I didn't even need a pencil! The method was
so simple that I performed most calculations in my head. Looking back, I admit I did so
as much tc show off as for any mathematical reason. Most kids outgrow such behavior.
Those who don't probably become either teachers or magicians.
These mental skills are not only
important for doing the tricks in this book but are also indispensable in school or at work,
or any time you use numbers. Soon you will be able to retire your calculator and use the
full capacity of your mind as you add, subtract, multiply, and divide 2-digit, 3-digit, and
even 4-digit numbers.
There are many good reasons why adding left to right is a superior method for
mental calculation. For one thing, you do not have to reverse the numbers (as you do
when adding right to left). And if you want to estimate your answer, then adding only the
leading digits will get you pretty close. If you are used to working from right to left on
paper, it may seem unnatural to add and multiply from left to right. But with practice you
will find that it is the most natural and efficient way to do mental calculations.
With the first set of problems—2 digit addition—the left to right method may not
seem so advantageous. But be patient. If you stick with me, you will see that the only
easy way to solve 3-digit and larger addition problems, all subtraction problems, and
most definitely all multiplication and division problems, is from left to right. The sooner you get accustomed to computing this way, the better
My assumption is that you know how to add and subtract 1-digit
numbers. We will begin with 2-digit addition, something I suspect you can already do
fairly well in your head. The following exercises are good practice, however, because you
will use the 2-digit addition skills you polish here for larger addition problems, as well as
virtually all multiplication problems in later chapters. It also illustrates a fundamental
principle of mental arithmetic—namely, to simplify your problem by breaking it into
smaller, more manageable components to success—simplify, simplify, simplify.
The easiest 2-digit addition problems, of course, are those that do not require you
to cany any numbers.
To add 32 to 47, you can simplify by treating 32 as 30 + 2, add 30 to 47 and then
add 2. In this way the problem becomes 77 + 2, which equals 79.
Adding from left to right, you can simplify the problem by adding 67 + 20 = 87;
then 87 + 8 = 95
Posted by GMAT books at 11:10 AM
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Author: Barnett Rich Philip A. Schmidt
Publication Date: 1989-02
Number Of Pages: 272
Schaum's Outline of Geometry is a revised edition of this very successful solved-problem outline first published in 1963. "Measure of an angle" terminology and metric as well as customary units are used throughout. Dates and amounts, sections on quantifiers, and laws of reasoning have been updated. Analytic geometry and transformational geometry has been added and the section on solid geometry shortened in line with the recent national curriculum changes. The book integrates plane geometry with arithmetic, algebra, numerical trigonometry, analytic geometry, and simple logic. For plane geometry courses in high schools and colleges and for those who want independent self study. 712 Solved Problems are included. Additionally, this book includes hundreds of supplementary problems.
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a consequence of
enough X that Y
a debate over
a responsibility to
as X as to Y
in contrast to
ask X to Y
in danger of
believed X to be Y
better served by X than by Y
just as X, so Y
compare to (similarities)
compare with (differences)
mistake X for Y
consider X Y (without 'to be')
not so much X as Y
not X but rather Y
contrast X with Y
declare X Y
rates for (not 'of')
declare Y X
requiring that X Y
requiring X to Y
depends on whether
so as not to be hindered by
so X s to (be) Y
so X as to constitute Y
the more X the greater Y
distinguish between X and Y
X enough to Y
distinguish X from Y
X out of Y (numbers)
X regarded as Y
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Principle: something fundamental that we do not question. This would be somewhat stronger than a fact because it is not specific to a limited number of cases but instead, apply to a broader range of scenarios(and often deeper in meaning). For instance, you will not talk about the principle that crime is increasing in large cities. Instead, it is a fact which applies to large cities. However, you will talk about the principles of Physics or the fundamental principles of Human Rights. I believe principles convey a stronger connotation than mere facts.
Fact: something taken as true at face value (stats, historical events)
Evidence: what is used to support a conclusion (examples, stats, historical events). Although these may include facts, it is usually stronger than facts because they are direct elements needed for the conclusion to stand whereas facts are not necessary for the latter to stand
Pre-evidence: This is a bit of a stretch. It will not often be on the test but it seems very similar to "background" information as described below.
Background: Elements needed to put the evidence into context but which, as stand alone pieces of information, might not constitute what is called an evidence necessary to arrive at a conclusion. For instance, blood tests performed on one thousand persons may reveal that 35% of those persons were HIV infected. However, the background information could be that the test was performed in more underinformed regions of the world where AIDS knowledge is at a minimum. As you can see, the fact that the test was performed in more underinformed regions is not in and of itself an evidence because it does not allow us to come to a conclusion. Instead, the 35% stats, as a stand-alone piece of info, is what will lead us to the conclusion we want. However, the background info is also crucial and cannot be omitted; it is required background info.
Consideration: Something which was taken into account or given some thought before arriving to the conclusion.
Premise: This is usually a required statement to arrive at a conclusion. Evidence and facts want to prove something to you whereas premises are there to logically lead you to a conclusion. The best example of premises is the ones included in syllogisms. For instance, you can say that(premise1) when it rains, you go outside. Then, it rains(premise2). You have to be outside(conclusion).
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