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MATHeMATICS

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DIGITAL TEXT BOOK

PLEDGE

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index content Page

number

1. Introduction 42. Addition and

subtraction 4

3. algebra 74. multiplication

and division 9

5. solutions of 11

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equations

EQUATIONSTeacher 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 mulitiplied’’Subtract5’’Ok,we did’, Subtract the original number,

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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?

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

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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

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?

The first section Līlāvatī (also known as pāṭīgaṇita or aṅkagaṇita) consists of 277 verses.[6] It covers calculations, progressions, mensuration, permutations, and other topicsThe second section Bījagaṇita has 213 verses.[6] It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using a kuṭṭaka method.[6]

In particular, he also solved the

case that was to elude Fermat and his European contemporaries centuries later.[6]In the third section Grahagaṇita, while treating the motion of planets, he considered their instantaneous speeds.[6] He arrived at the approximation:[10]

for close to , or in modern notation:[10]

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First multiplication by 3, then addition of 2.It became 50,when the last 2 was added. So 50-2=48This means the original number multiplied by 3 gives 48.The number =48/3=16Thus 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 314 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

ALGEBRAWe 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?

The first section Līlāvatī (also known as pāṭīgaṇita or aṅkagaṇita) consists of 277 verses.[6] It covers calculations, progressions, mensuration, permutations, and other topicsThe second section Bījagaṇita has 213 verses.[6] It discusses zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell's equation, solving it using a kuṭṭaka method.[6]

In particular, he also solved the

case that was to elude Fermat and his European contemporaries centuries later.[6]In the third section Grahagaṇita, while treating the motion of planets, he considered their instantaneous speeds.[6] He arrived at the approximation:[10]

for close to , or in modern notation:[10]

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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=15x/3=15-8=7x=7*3=21=21Thus the original number =21

3. From the point on a line another Line is to be drawn such in wayThat, the angle on one side should Be 500 more than the angle on theOther. What should be the smallsAngle?

4. A hundred rupees note was changedInto 100 rupees notes. There were7 notes in all .How many of eachDemonization were their?

ACTIVITY

To any number if another number isadded and then be added number

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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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 forFinding a number if the some of the

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?

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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If there a number unchanged by multiplication?

Yes! ZeroThat 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 = 1Then 2x1 +1=3If x=1 then 3x1 =3 Thus if the x= 1Then 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?

Lets write x for the number and translate the problem to algebra.

If 2x +10= 4x, then what is x?2x – 4x = -10-2x =-10X=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 xThen ajayans age = x+10After 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

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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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

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?

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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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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