digital text book

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MATHeMATICS
Viii
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DIGITAL TEXT BOOK

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PLEDGE

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index
content
Page number
1. Introduction 4
2. Addition and subtraction 4
3. algebra 7
4. multiplication and division 9
5. solutions of equations 11

4. 4
EQUATIONS
Teacher who has to handle the 5th period is on leave.
So the students of std VIIIA invites their maths teacher.
‘Our teacher is on leave would you please, come?’
‘Why not?’, the teacher agreed. Students are happy. For they knew that maths
teacher would discuss even problems outside the course book. Teacher would present
puzzles, games etc,. maths club very interestingly.
Teacher: Today, why don’t we start with a puzzle?
Student: Yes, teacher
‘Please take a piece of paper and pen’ said the teacher.
The students did the same ‘Write a number you like on the
paper and keep it’ Don’t show it to anybody’.’We have written said the class.’
Add 2 to the number’.’Yes.teacher’’Ok.now multiply it by 3’Yes,we
multipiled’’Yes,we mulitip lied’’Subtract5’’Ok,we did’, Subtract the original number,
multiply by 2 and then subtract1’’It ‘s OK, teacher ‘Now it’s my turn. You say the
final number and I’II say the original number.
Well begin with kripa.’61’Kripa said. “your number is 15.Is that
correct,Kripa?”Yes,teacher,’Now haritha say the number’ “65’’Is it
16,Haritha?’Yes,HarithaTeacher ‘You’re absolutely correct teacher. How do you
make it?’Well learn the trick in our new lesson ‘EQUATIONS’
ACTIVITY
ADDITION AND SUBTRACTION:
Appu came back from the market with a bag of vegetables and other things.
Mother asked him to keep the change. It keep the change. It was 5 rupees.’Now my
saving have reached 50’.Appu said how much did he have before getting this 5 rupees?

5. 5
His savings became 50,when he got 5 rupees more. So he must have 50-5=45rupees.
 Ammu bought a pen for 10 rupees from her ‘vishukaineettam’Now she
has 40 rupees remaining .It became 40 rupees,when it was reduced by 10
rupees.So it must have been 10 more than 40.
 That is 40+10=50
 Can’t you similarly fine the answer to the questions below?
1. Gopalan bought a bunch of bananas for his shop.7 of them had slightly turned
bad. After removing them, he had 46 left. How many were there in the bunch at
first?
The number of bananas is the bunch at first=46+7=53
2. A number subtracted from 500gave 234.what is the number subtracted?
Say the number=x
Subtracted from 500=500-x
That is 500-x=234
X=500-234=266
The number=266

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ACTIVITY
 In a certain savings scheme money invested doubles in 5 years. To get
10 thousand rupees after 5 years ,how much should be invested now?
 Joseph got 1500 as his share a profit from a sale. This is one-third the
total profit. What is the total profit?
 The perimeter of a pentagon with equal sides is 65cms.What is the
length of each sides?
 A number divided 12 gives 25.What is the number?
LOOK AT THE PROBLEM
 Thrice a number and 2 together make 50.what is
the number?
Here whet were the operations done to the
number to get 50?
First multiplication by 3, then addition of 2.It
became 50,when the last 2 was added. So 50-
2=48
This means the original number multiplied by 3
gives 48.
The number =48/3=16
Thus 16 multiplied by 3 gives 48 and 2 added to
this makes 50.
 What if we change the question like this. From
thrice a number ,2 is subtracted and this gives
40.What is the number?
Here what was the number before 2 was
subtracted?
40+2=42 And this is got on multiplication by 3
The first sect ion Līlāvatī (also known as
pāṭīgaṇita or aṅkagaṇita) consists of 277
verses.[6] It covers calculations, progressions,
mensurat ion, permutations, and other
topicsThe second sect ion Bījagaṇita has 213
verses.[6] It discusses zero, infinity, posit ive
and negat ive numbers, and indeterminate
equat ions including (the now called) Pell's
equat ion, solving it using a kuṭṭaka method.[6]
In part icular, he also solved the
case that was to
elude Fermat and his European
contemporaries centuries later.[6]In the third
sect ion Grahagaṇita, while t reating the
mot ion of planets, he considered their
instantaneous speeds.[6] He arrived at the
approximation:[10]
fo r clo se to , or in modern notation:[10]
.

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14 multiplied by3 gives 42and 2 subtracted from
this gives 40
ACTIVITY
1. Anitha and her friends bought some pens. For a packet of 5 pens, they got 2
rupees reduction in price. They had to pay only 18 rupees. Had they bought the
pens separately, how much would have been the price for each pen?
2. Three added to half a number gives 23. What is the number?
3. 2 Subtracted from one-third of a number gives 40. What is the number?
ACTIVITY
ALGEBRA
We are used given a number got by doing some operations on another number.
We must find the number we started with. What was the general method used?
LOOK AT THE PROBLEM ALGEBRA
1. 8 added to one-third of a number gives 15.what is
2. the number?
Let’s first write the problem in algebra
. If x/3+8=15. What is x?
Next method let’s look at the method
of solution
x/3+8=15

8. 8
x/3=15-8=7
x=7*3=21=21
Thus the original number =21
3. From the point on a line another
Line is to be drawn such in way
That, the angle on one side should
Be 500 more than the angle on the
Other. What should be the smalls
Angle?
4. A hundred rupees note was changed
Into 100 rupees notes. There were
7 notes in all .How many of each
Demonization were their?
ACTIVITY
To any number if another number is
added and then be added number
subtracted we get the original back.
This can be written using algebra like
this
(x+a)-a=x
This same fact can be put in the
different form.
If x+a=b then x= b-a
This is the algebra form of the rule for
Finding a number if the some of the
Written evidence of the use of mathematics dates
back to at least 3000 BC with the ivory labels
found in Tomb U-j at Abydos. These labels
appear to have been used as tags for grave goods
and some are inscribed with numbers.[1] Further
evidence of the use of the base 10 number system
can be found on the Narmer Macehead which
depicts offerings of 400,000 oxen, 1,422,000
goats and 120,000 prisoners.[2]
The evidence of the use of mathematics in the Old
Kingdom (ca 2690–2180 BC) is scarce, but can be
deduced from inscriptions on a wall near a
mastaba in Meidum which gives guidelines for
the slope of the mastaba.[3] The lines in the
diagram are spaced at a distance of one cubit and
show the use of that unit of measurement.[1]
The earliest true mathematical documents date to
the 12th dynasty (ca 1990–1800 BC). The
Moscow Mathematical Papyrus, the Egyptian
Mathematical Leather Roll, the Lahun
Mathematical Papyri which are a part of the much
larger collection of Kahun Papyri and the Berlin
Papyrus 6619 all date to this period. The Rhind
Mathematical Papyrus which dates to the Second
Intermediate Period (ca 1650 BC) is said to be
based on an older mathematical text from the 12th
dynasty.[4]

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number with another number and the
number added known .
Similarly, we have the following rules also
If x-a=b then x= b+a
This is the algebra form of the rule for Finding a number when the result of subtracting
another number form. This number and the number subtracted are known.
MULTIPLICATION AND DIVISION
To get a number from its product with another number, we must divide the product
by the number with which the original number was multiplied. Similarly to get a
number its quotient by another number, we must multiply the quotient by the number
by which the original was divided. Using algebra we can write this as :
If ax=b and (a±0) then x=b/a and if x/a = b then x= ab
LOOK AT THE PROBLEM ALGEBRA
1. If there a number for which its double and triple are equal?
If there a number unchanged by multiplication?
Yes! Zero
That is, if x=0 then 2x = 3x
2. Is there a number such that one added to its double gives its triple?
In the language of algebra, the question becomes, is there a number x, such that
2x+1 = 3x(there x is not equal to 0)
It can be one only.
If x = 1
Then 2x1 +1=3
If x=1 then 3x1 =3
Thus if the x= 1
Then the number 2+1 and 3x are both equal to 3
3. When we added to 10 to 2 times a number, we get four times that number.
What’s number its it?

10. 10
Lets write x for the number and translate the problem to algebra.
If 2x +10= 4x, then what is x?
2x – 4x = -10
-2x =-10
X=5
4. Ajayan is 10 years olde than
Vijayan. Next year, ajayans age
Would be twice vijayans age.
How old are they now?
Lets vijayan be x
Then ajayans age = x+10
After 1 year, vijayan’s age would
Be(x+1)and ajayans age would we
(X+10)+1 = x+11 algebra form of
the problem being
x+11= x(x+1)=2x+2
how do we find x from this.
If from the sum x+11
These subtract x then we get 11
At this statement x+11= 2x+2
Tells that the numbers x+11 and
2x+2 are the same
So the as in the first example
(2x+2)- x= (x+11)-x=11
This means x+2=11
X= 11-2=9
So vijayan’s age is 9 and ajayan’s
age is 19
Algebra (from Arabic al-jebr meaning "reunion of
broken parts"[1]) is one of the broad parts of
mathematics, together with number theory, geometry
and analysis. In its most general form algebra is the
study of symbols and the rules for manipulating
symbols[2] and is a unifying thread of all of
mathematics.[3] As such, it includes everything from
elementary equation solving to the study of abstractions
such as groups, rings, and fields. The more basic parts of
algebra are called elementary algebra, the more abstract
parts are called abstract algebra or modern algebra.
Elementary algebra is essential for any study of
mathematics, science, or engineering, as well as such
applications as medicine and economics. Abstract
algebra is a major area in advanced mathematics, studied
primarily by professional mathematicians. Much early
work in algebra, as the Arabic origin of its name
suggests, was done in the Near East, by such
mathematicians as Omar Khayyam (1050-1123).
Elementary algebra differs from arithmetic in the use of
abstractions, such as using letters to stand for numbers
that are either unknown or allowed to take on many
values.[4] For example, in the letter is
unknown, but the law of inverses can be used to discover
its value: . In , the letters and
are variables, and the letter is a constant. Algebra gives
methods for solving equations and expressing formulas
that are much easier (for those who know how to use
them) than the older method of writing everything out in
words
.

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SOLUTIONS OF EQUATIONS:
We have seen many examples in the lesson of how we can translate mathematical
problems to algebraic equations and the numbers for which these are true, are the
answers to the problem.
The numbers for which an algebraic equation is true are called solutions of the
equation and the process of finding the solutions is called solving the equation.
Example : solving the equation 2x=10 means finding the number whose double is 10
and the solution is x=5
FORMATIVE EVALUATION:
1. Cash prize is distributed among the first three places in a science exhibition. The
second place is 5/6 part of the money of the first place. The third prize is 4/5 part
of the second. If the cash distributed is 1500 rupees. How much is each prize?
2. One angle of triangle is 1/3 of another angle. The third angle is 260 more than
that angle. Find the three angles?
3. Find the three consecutive negative numbers whose sum is -54?
4. The perimeter of a triangle is 49cm.One side is 7cm more than the second side
and 5cm less than the third side. Find the length of the three sides?
5. Ramesan framed the equation 4(2x-3)+5(3x-4)=14 to find the number in a verbal
problem. What is the number?

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