Class VI I.I.T. Foundation, N.T.S.E.& Mathematics Olympiad Curriculum & Chapter Notes

8/13/2019 Class VI I.I.T. Foundation, N.T.S.E.& Mathematics Olympiad Curriculum & Chapter Notes

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2012

Jai Kumar Gupta

Brilliant Public School, Sitamarhi

20/04/2012

VI I.I.T. Foundation, N.T.S.E.&Mathematics Olympiad

Curriculum & Chapter Notes


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VI I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Curriculum Page 1

VI I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Curriculum

Chapter as per NCERT Text Book Topics

1. Knowing Our Numbers

Comparing Numbers Estimation of the Numbers

Roman Numerals Importance of Brackets

2. Whole NumbersWhole Numbers Properties of Whole Numbers

3. Playing with NumbersPrime and Composite Numbers Divisibility of Numbers Prime Factorization, HCF and LCM

4. Basic Geometrical IdeasPoints, Lines and Curves Angles, Polygons and Circles

5. Understanding Elementary Shapes

Line and Angles

Two Dimensional Figures

Three Dimensional Shapes 6. Integers Integers

7. FractionsTypes of Fractions Comparing Fractions Addition and Subtraction of Fractions

8. DecimalsComparing Decimals Addition and Subtraction of Decimals

9. Data HandlingData handling Pictograph Bar Graph

10. MensurationPerimeter Area

11. AlgebraVariables Use of Variables Equations

12. Ratios and ProportionsRatios Proportions

13. SymmetryLine SymmetryMirror Symmetry

14. Practical Geometry

Basic Constructions

Construction of Lines Constructing of Angles
http://www.learnnext.com/class6/maths/Comparing-Numbers.htmhttp://www.learnnext.com/class6/maths/Comparing-Numbers.htmhttp://www.learnnext.com/class6/maths/Estimation-of-the-Numbers.htmhttp://www.learnnext.com/class6/maths/Estimation-of-the-Numbers.htmhttp://www.learnnext.com/class6/maths/Roman-Numerals.htmhttp://www.learnnext.com/class6/maths/Roman-Numerals.htmhttp://www.learnnext.com/class6/maths/Importance-of-Brackets.htmhttp://www.learnnext.com/class6/maths/Importance-of-Brackets.htmhttp://www.learnnext.com/class6/maths/Whole-Numbers.htmhttp://www.learnnext.com/class6/maths/Whole-Numbers.htmhttp://www.learnnext.com/class6/maths/Properties-of-Whole-Numbers.htmhttp://www.learnnext.com/class6/maths/Properties-of-Whole-Numbers.htmhttp://www.learnnext.com/class6/maths/Prime-and-Composite-Numbers.htmhttp://www.learnnext.com/class6/maths/Prime-and-Composite-Numbers.htmhttp://www.learnnext.com/class6/maths/Divisibility-of-Numbers.htmhttp://www.learnnext.com/class6/maths/Divisibility-of-Numbers.htmhttp://www.learnnext.com/class6/maths/HCF-and-LCM.htmhttp://www.learnnext.com/class6/maths/HCF-and-LCM.htmhttp://www.learnnext.com/class6/maths/Lines-and-Curves.htmhttp://www.learnnext.com/class6/maths/Lines-and-Curves.htmhttp://www.learnnext.com/class6/maths/Polygons-and-Circles.htmhttp://www.learnnext.com/class6/maths/Polygons-and-Circles.htmhttp://www.learnnext.com/class6/maths/Line-and-Angles.htmhttp://www.learnnext.com/class6/maths/Line-and-Angles.htmhttp://www.learnnext.com/class6/maths/Two-Dimensional-Figures.htmhttp://www.learnnext.com/class6/maths/Two-Dimensional-Figures.htmhttp://www.learnnext.com/class6/maths/Three-Dimensional-Shapes.htmhttp://www.learnnext.com/class6/maths/Three-Dimensional-Shapes.htmhttp://www.learnnext.com/class6/maths/Integers.htmhttp://www.learnnext.com/class6/maths/Integers.htmhttp://www.learnnext.com/class6/maths/Types-of-Fractions.htmhttp://www.learnnext.com/class6/maths/Types-of-Fractions.htmhttp://www.learnnext.com/class6/maths/Comparing-Fractions.htmhttp://www.learnnext.com/class6/maths/Comparing-Fractions.htmhttp://www.learnnext.com/class6/maths/Addition-and-Subtraction-of-Fractions.htmhttp://www.learnnext.com/class6/maths/Addition-and-Subtraction-of-Fractions.htmhttp://www.learnnext.com/class6/maths/Comparing-Decimals.htmhttp://www.learnnext.com/class6/maths/Comparing-Decimals.htmhttp://www.learnnext.com/class6/maths/Addition-Subtraction-Decimals.htmhttp://www.learnnext.com/class6/maths/Addition-Subtraction-Decimals.htmhttp://www.learnnext.com/class6/maths/Data-handling-class6.htmhttp://www.learnnext.com/class6/maths/Data-handling-class6.htmhttp://www.learnnext.com/class6/maths/Pictograph.htmhttp://www.learnnext.com/class6/maths/Pictograph.htmhttp://www.learnnext.com/class6/maths/Bar-Graph.htmhttp://www.learnnext.com/class6/maths/Bar-Graph.htmhttp://www.learnnext.com/class6/maths/Perimeter.htmhttp://www.learnnext.com/class6/maths/Perimeter.htmhttp://www.learnnext.com/class6/maths/Area.htmhttp://www.learnnext.com/class6/maths/Area.htmhttp://www.learnnext.com/class6/maths/Variables.htmhttp://www.learnnext.com/class6/maths/Variables.htmhttp://www.learnnext.com/class6/maths/Use-Variables.htmhttp://www.learnnext.com/class6/maths/Use-Variables.htmhttp://www.learnnext.com/class6/maths/Equations.htmhttp://www.learnnext.com/class6/maths/Equations.htmhttp://www.learnnext.com/class6/maths/Ratios.htmhttp://www.learnnext.com/class6/maths/Ratios.htmhttp://www.learnnext.com/class6/maths/Proportions.htmhttp://www.learnnext.com/class6/maths/Proportions.htmhttp://www.learnnext.com/class6/maths/Line-Symmetry-class6.htmhttp://www.learnnext.com/class6/maths/Mirror-Symmetry.htmhttp://www.learnnext.com/class6/maths/Basic-Constructions6.htmhttp://www.learnnext.com/class6/maths/Basic-Constructions6.htmhttp://www.learnnext.com/class6/maths/Construction-Lines.htmhttp://www.learnnext.com/class6/maths/Construction-Lines.htmhttp://www.learnnext.com/class6/maths/Construction-of-Angles.htmhttp://www.learnnext.com/class6/maths/Construction-of-Angles.htmhttp://www.learnnext.com/class6/maths/Construction-of-Angles.htmhttp://www.learnnext.com/class6/maths/Construction-Lines.htmhttp://www.learnnext.com/class6/maths/Basic-Constructions6.htmhttp://www.learnnext.com/class6/maths/Mirror-Symmetry.htmhttp://www.learnnext.com/class6/maths/Line-Symmetry-class6.htmhttp://www.learnnext.com/class6/maths/Proportions.htmhttp://www.learnnext.com/class6/maths/Ratios.htmhttp://www.learnnext.com/class6/maths/Equations.htmhttp://www.learnnext.com/class6/maths/Use-Variables.htmhttp://www.learnnext.com/class6/maths/Variables.htmhttp://www.learnnext.com/class6/maths/Area.htmhttp://www.learnnext.com/class6/maths/Perimeter.htmhttp://www.learnnext.com/class6/maths/Bar-Graph.htmhttp://www.learnnext.com/class6/maths/Pictograph.htmhttp://www.learnnext.com/class6/maths/Data-handling-class6.htmhttp://www.learnnext.com/class6/maths/Addition-Subtraction-Decimals.htmhttp://www.learnnext.com/class6/maths/Comparing-Decimals.htmhttp://www.learnnext.com/class6/maths/Addition-and-Subtraction-of-Fractions.htmhttp://www.learnnext.com/class6/maths/Comparing-Fractions.htmhttp://www.learnnext.com/class6/maths/Types-of-Fractions.htmhttp://www.learnnext.com/class6/maths/Integers.htmhttp://www.learnnext.com/class6/maths/Three-Dimensional-Shapes.htmhttp://www.learnnext.com/class6/maths/Two-Dimensional-Figures.htmhttp://www.learnnext.com/class6/maths/Line-and-Angles.htmhttp://www.learnnext.com/class6/maths/Polygons-and-Circles.htmhttp://www.learnnext.com/class6/maths/Lines-and-Curves.htmhttp://www.learnnext.com/class6/maths/HCF-and-LCM.htmhttp://www.learnnext.com/class6/maths/Divisibility-of-Numbers.htmhttp://www.learnnext.com/class6/maths/Prime-and-Composite-Numbers.htmhttp://www.learnnext.com/class6/maths/Properties-of-Whole-Numbers.htmhttp://www.learnnext.com/class6/maths/Whole-Numbers.htmhttp://www.learnnext.com/class6/maths/Importance-of-Brackets.htmhttp://www.learnnext.com/class6/maths/Roman-Numerals.htmhttp://www.learnnext.com/class6/maths/Estimation-of-the-Numbers.htmhttp://www.learnnext.com/class6/maths/Comparing-Numbers.htm


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VI I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 1

1. Knowing Our Numbers

Comparing Numbers

The arrangement of numbers from the smallest to the

greatest is called ascending order .

The arrangement of numbers from the smallest to the greatest is called ascending order .

Ex: 2789, 3560, 4567, 7662, 7665

The arrangement of numbers from the greatest to the smallest is called descending

order . Ex: 7665, 7662, 4567, 3560, 2789

If two numbers have an unequal number of digits, then the number with the greater

number of digits is greater.

If two numbers have an equal number of digits, then the number with the greater digit isgreater.

The greatest single-digit number is 9. When we add 1 to this single-digit number, we get

10, which is the smallest two-digit number. Therefore, the greatest single-digit number

+1=the smallest two-digit number.

The greatest two digit-number is 99. When we add 1 to this two-digit number, we get

100, which is the smallest three-digit number. Therefore, the greatest two-digit number+1=the smallest three-digit number.

The greatest three-digit number is 999. When we add 1 to this three-digit number, we get

1000, which is the smallest four-digit number. Therefore, the greatest three-digit number

+1=the smallest four-digit number.


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The greatest four-digit number is 9999. When we add 1 to this four-digit number, we get

10,000, which is the smallest five-digit number. Therefore, the greatest four-digit

number +1=the smallest five-digit number.

The greatest five-digit number is 99999. When we add 1 to this five-digit number, we get

1,00,000, which is the smallest six digit number. Therefore, the greatest five-digitnumber +1=the smallest six-digit number.

The number, that is, one with five zeroes (100000), is called one lakh.

Crores Lakhs Thousands Ones

Tens Ones Tens Ones Tens Ones Hundreds Tens Ones

Commas in international system:

As per international numeration, the first comma is placed after the hundreds place . Commas

are then placed after every three digits .

Ex: (i) 8,876,547

The number can be read as eight million eight hundred seventy-six thousand five hundred and

forty-seven.

(ii) 56,789, 056

The number can be read as fifty-six million seven hundred eighty-nine thousand and fifty-six.

Billions Millions Thousands Ones

Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones

Use the following place value chart to identify the digit in any place in the international system.

Comparison of the Indian and the international numeration systems:

Indian Numeration Crore Ten Lakh Lakh Ten Thousand Thousand Hundred Tens Ones

Numbers 10000000 1000000 100000 10000 1000 100 10 0

International Numeration Ten Million Million Hundred

Thousand

Ten Thousand Thousand Hundred Tens Ones

Units of measurement:


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1 metre=100 centimetres

1 kilogram = 1,000 grams

1 kilometre = 1000 metres

1 litre=1,000 millilitres

The greatest six-digit number is 999999. When we add 1 to this six-digit number, we get

10,00,000, which is the smallest seven-digit number. Therefore, the greatest six-digit number

+1=the smallest seven-digit number.

The number, that is, one with six zeroes (1000000), is called ten lakh.

The greatest seven-digit number is 9999999. When we add 1 to this seven-digit number, we get

10000000, which is the smallest eight-digit number. Therefore, the greatest seven-digit number

+1=the smallest eight-digit number.

The number, that is, one with seven zeroes (10000000),is called one crore.

Commas are placed to the numbers to help us read and write large numbers easily.

Commas in Indian numeration:

As per Indian numeration, the first comma is placed after the hundreds place . Commas are then placed

after every two digits .

Ex: (i) 88,76,547

The number can be read as eighty-eight lakh seventy-six thousand five hundred and forty-seven.

(ii)5 , 67, 89, 056

The number can be read as five crore sixty-seven lakh eighty-nine thousand and fifty-six.

Use the following place value chart to identify the digit in any place in the Indian system.

Crores Lakhs Thousands Ones

Tens Ones Tens Ones Tens Ones Hundreds Tens Ones

Commas in international system:

As per international numeration, the first comma is placed after the hundreds place . Commas are then

placed after every three digits .

Ex: (i) 8,876,547

The number can be read as eight million eight hundred seventy-six thousand five hundred and forty-

seven.


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(ii)56,789, 056

The number can be read as fifty-six million seven hundred eighty-nine thousand and fifty-six.

Billions Millions Thousands Ones

Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones

Use the following place value chart to identify the digit in any place in the international system.

Comparison of the Indian and the international numeration systems:

Indian Numeration Crore Ten Lakh Lakh Ten Thousand Thousand Hundred Tens Ones

Numbers 10000000 1000000 100000 10000 1000 100 10 0

International Numeration Ten Million Million Hundred

Thousand

Ten Thousand Thousand Hundred Tens Ones

Units of measurement:

1 metre=100 centimetres

1 kilogram = 1,000 grams

1 kilometre = 1000 metres

1 litre=1,000 millilitres

The greatest six-digit number is 999999. When we add 1 to this six-digit number, we get

10,00,000, which is the smallest seven-digit number. Therefore, the greatest six-digit number

+1=the smallest seven-digit number.

The number, that is, one with six zeroes (1000000), is called ten lakh.

The greatest seven-digit number is 9999999. When we add 1 to this seven-digit number, we get

10000000, which is the smallest eight-digit number. Therefore, the greatest seven-digit number

+1=the smallest eight-digit number.

The number, that is, one with seven zeroes (10000000),is called one crore.

Commas are placed to the numbers to help us read and write large numbers easily.

Commas in Indian numeration:

As per Indian numeration, the first comma is placed after the hundreds place . Commas are then placed

after every two digits .

Ex: (i) 88,76,547


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The number can be read as eighty-eight lakh seventy-six thousand five hundred and forty-seven.

(ii)5 , 67, 89, 056

The number can be read as five crore sixty-seven lakh eighty-nine thousand and fifty-six.

Use the following place value chart to identify the digit in any place in the Indian system.


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Estimation of the Numbers

The estimation of a number is a reasonable guess of the

actual value.

The estimation of a number is a reasonable guess of the actual value. Estimation means approximating

a quantity to the accuracy required. This is done by rounding off the numbers involved and getting a

quick, rough answer.

The numbers 1, 2, 3 and 4 are nearer to 0. So, these numbers are rounded off to the lower ten. The

numbers 6, 7, 8 and 9 are nearer to 10. So, these numbers are rounded off to the higher ten. The number

5 is equidistant from both 0 and 10, so it is rounded off to the higher ten.

Eg:

(i) We round off 31 to the nearest ten as 30

(ii) We round off 57 to the nearest ten as 60

(iii) We round off 45 to the nearest ten as 50

The numbers 1 to 49 are closer to 0. So, these numbers are rounded off to the nearest hundred. The

numbers 51 to 99 are closer to the lower hundred. So, these numbers are rounded off to the higher

hundred. The number 50 is rounded off to the higher hundred.

Eg:

(i) We round off 578 to the nearest 100 as 600.

(ii) We round off 310 to the nearest 100 as 300.

Similarly, 1 to 499 are rounded off to the lower thousand, and 501 to 999 to the higher thousand. The

number 500 is equidistant from both 0 and 1000, and so it is rounded off to the higher thousand.

Eg:

(i)We round off 2574 to the nearest thousand as 3000.

(ii)We round off 7105 to the nearest thousand as 7000.


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Estimation of sum or difference:

When we estimate sum or difference, we should have an idea of the place to which the rounding

is needed.

Eg: (i) Estimate 4689 + 19316

We can say that 19316 > 4689We shall round off the numbers to the nearest thousands.

is rounded off to 19000

4689 is rounded off to 5000

Estimated sum:

19000 + 5000=24000

(ii) Estimate 1398-526

We shall round off these numbers to the nearest hundreds.

1398 is rounded off to 1400526 is rounded off to 500

Estimated difference:

1400-500=900

Estimation of the product:

To estimate the product , round off each factor to its greatest place, then multiply the rounded

off factors .

Eg: Estimate 92 x 578

The first number, 92, can be rounded off to the nearest ten as 90.

The second number, 578, can be rounded off to the nearest hundred as 600.

Hence, the estimated product =90 x 600 = 54,000

Estimating the outcome of number operations is useful in checking the answer.

Roman Numerals


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Many years ago, Hindus and Arabs developed a

number system called the Hindu-Arabic number

system.

Hindu-Arabic number system:

Many years ago, Hindus and Arabs developed a number system called the Hindu-Arabic number system.

It is the name given to the number system that we use today.

Roman numerals:

It is the numeral system that originated in ancient Rome . This numeral system is based on certain

letters, which are given values and are used as numerals. The following are the seven number symbols

used in the Roman numeral system, and their values:

I V X L C D M

1 5 10 50 100 500 1000

Seven letters of English alphabet, i.e. I, V, X, L, C, D and M , are used to represent Roman numerals.

Roman numerals do not have a symbol for zero . Roman numerals are read from left to right , and are

arranged from the largest to the smallest . Multiplication, division and other complex operations were

difficult to perform on Roman numerals. So Arabic numerals were used. The Roman numerals for the

numbers 1 - 15 are shown below:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

I II III IV V VI VII VIII IX X XI XII XIII XIV XV

We can find these roman numerals in some clocks.

Rules for Roman numerals:

1. In Roman numerals, a symbol is not repeated more than thrice. If a symbol is repeated, its value

is added as many times as it occurs.


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For example, if the letter I is repeated thrice, then its value is three.

1. The symbols V, L and D are never repeated.

2. If a symbol of smaller value is written to the right of a symbol of greater value, then its value gets

added to the symbol of greater value.

3. For example, in case of VI, I is written to the right of V. It means that 1 should be added to 5.

Hence, its value is 6.If a symbol of smaller value is written to the left of a symbol of greater

value, then its value is subtracted from the symbol of greater value.

For example, in case of IV, I is written to the left of V. It means that 1 should be subtracted from 5.

Hence, its value is 4.

4. The symbols V, L and D are never written to the left of a symbol of greater value, so V, L and D

are never subtracted.

For example, we write 15 as XV and not VX.

The symbol I can be subtracted from V and X only. For example, the value of IV is four and the

value of VI is six.

The symbol X can be subtracted from L, M and C only. For example, X is subtracted from L to

arrive at 40, which is represented by XL

Importance of Brackets

Brackets help in simplifying an expression that has

more than one mathematical operation .

Usin g brackets:

Brackets help in simplifying an expression that has more than one mathematical operation .

If an expression that includes brackets is given, then turn everything inside the bracket into a single


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number , and then carry out the operation that lies outside.

Eg:

1. (6 + 8) x 10 = 14 x 10 = 140

2. (8 + 3) (9 - 4) = 11 x 5 = 55

Expandin g br ackets:

The use of brackets allows us to follow a certain procedure to expand the brackets systematically.

For example:

1. 8 x 109 = 8 x (100 + 9) = 8 x 100 + 8 x 9 = 800 + 72 = 872

2. 105 x 108 = (100 + 5) x (100 + 8)

= (100+5)x100+(100+5)x8

=100 x 100 + 5 x 100 + 100 x 8 + 5 x 8

=10000 + 500+ 800 + 40

=1134


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2. Whole NumbersWhole Numbers

The numbers used for counting are called natural

numbers.

The numbers used for counting are called natural numbers . The number that comes immediately before

another number in counting is called its predecessor . The number that comes immediately after another

number in counting is called its successor . To find the successor of any given natural number, just add 1

to the given number. The value of nothing is represented by the number zero .

Eg: 3 - 3 = 0

Natural numbers together with the number zero are called whole numbers . When comparing two whole

numbers, the number that lies to the right on the number line is greater. When comparing two whole

numbers, the smaller number lies to the left on the number line .

Properties of Whole Numbers

A whole number added to 0 remains unchanged.

A whole number added to 0 remains unchanged. Thus, 0 is called the additive identity in whole

numbers. The product of two whole numbers is the same, no matter in which order they are multiplied.

This is called the commutative property of multiplication . A whole number multiplied by 1 remains


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unchanged. Thus, 1 is called the multiplicative identity in whole numbers. Whole numbers are closed

under addition and multiplication. Subtraction and division are not commutative in whole numbers.

Whole numbers are not closed under subtraction and division.

While adding whole numbers, we can group the numbers in any order. This is called the associative

property of addition. While multiplying whole numbers, we can group them in any order. This is called

the associative property of multiplication. The sum of the products of a whole number with two other

whole numbers is equal to the product of the whole number with the sum of the two other whole numbers.

This is called the distributive property of multiplication over addition.


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Di visibi li ty of numbers by 6:

A number is divisible by 6 if that number is divisible by both 2 and 3 .


A number is divisible by 8 if the number formed by its last three digits is divisible by 8.


A number is divisible by 9 if the sum of its digits is divisible by 9.


A number that has 0 in its ones place is divisible by 10.


If the difference between the sum of the digits at the odd and even places in a given number is either 0

or a multiple of 11 , then the given number is divisible by 11.

Co-prime numbers:

If the only common factor of two numbers is 1, then the two numbers are called co-prime numbers.

General rules of divisibil ity for al l numbers:

If a number is divisible by another number, then it is also divisible by all the factors of the other

number. If two numbers are divisible by another number, then their sum and difference is also

divisible by the other number. If a number is divisible by two co-prime numbers, then it is also

divisible by the product of the two co-prime numbers.


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Prime Factorization, HCF and LCM

Writing a number as a product of its prime factors is

called the prime factorisation of the number.

Writing a number as a product of its prime factors is called the prime factorisation of the number.

Eg: (i) 18=2 x 3 x 3

(ii) 40=2 x 2 x 2 x 5

The greatest of the common factors of the given numbers is called their highest common factor (HCF) .

It is also known as the greatest common divisor .

Eg: Prime factorisation of 16 = 2 x 2 x 2 x 2

Prime factorisation of 40 = 2 x 2 x 2 x 5

HCF of 16 and 40 = 2 x 2 x 2 = 8

The smallest common multiple of the given numbers is called their Least Common Multiple (LCM).

Eg: The LCM of given numbers using their prime factorisation:

Prime factorisation of 4 = 2 x 2

Prime factorisation of 6 = 2 x 3

LCM of 4 and 6 = 2 x 2 x 3 =12

To find the LCM of the given numbers using the division method :

Write the given numbers in a row.

Divide the numbers by the smallest prime number that divides one or more of the given numbers.

Write the number that is not divisible, in the second row.

Write the new dividends in the second row.

Divide the new dividends by another smallest prime number.

Continue dividing till the dividends are all prime numbers or 1.

Stop the process when all the new dividends are prime numbers or 1.


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4. Basic Geometrical IdeasPoints, Lines and Curves

The most practical branch of mathematics is

geometry. The term 'geometry' is derived from the

Greek word 'geometron'.

The most practical branch of mathematics is geometry. The term 'geometry' is derived from the Greek

word 'geometron'. It means Earth's measurement. The fundamental elements of geometry are given

below:

Point:

In geometry, dots are used to represent points . A point is used to represent any specific location or

position. It neither has any size, nor dimensions such as length or breadth. A point can be denoted by a

capital letter of the English alphabet. Points can be joined in different ways.

L ine segment:

A line segment is defined as the shortest distance between two points . For example, if we mark any

two points, M and N, on a sheet of paper, then the shortest way to join M to N is a line segment. It is

denoted by Points M and N are called the end points of the line segment.


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Line:

A line is made up of an infinite number of points that extend indefinitely in either direction. For

example, if a line segment from M to N is extended beyond M in one direction and beyond N in the other,

then we get a line, MN. It is denoted by A line can also be represented by small letters of the

English alphabet.

Ray:

A ray is a portion of a line. It starts at one point and goes on endlessly in one direction . For example,if a line from M to N is extended endlessly in the direction of N, then we get a ray, MN. It is denoted by

and can be read as ray MN.

Plane:

A plane is said to be a very thin flat surface that

does not have any thickness , and is limitless. For

example, this sheet is said to plane PQR. An infinite

number of points can be contained within a plane.


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I ntersecting l in es:

If two lines pass through a point, then we say that the two lines intersect at that point. Thus, if two lines

have one point in common, then they are called intersecting lines . For example, two lines

pass though point P. These two lines are called intersecting lines.

Parall el li nes or non- intersecting l ines :

In a plane, if two lines have no point in common, then they are said to be parallel or non- intersecting

lines.

Parallel lines never meet, cut or cross each other. In the figure, it can be observed that two lines

are parallel. We write .

Curves:

Curves can be defined as figures that flow smoothly without a break . A line is also a curve, and is

called a straight curve. Curves that do not intersect themselves are called simple curves. The end points

oin to enclose an area. Such curves are called closed curves. For example, (i), (ii) and (iii) are simple

curves, whereas (iv) and (v) are closed curves.


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are called curves. Curves that do not cross themselves are called 'simple curves' . In the

diagram here, (i), (ii), (iv) and (v) are simple curves, while (iii) is not a simple curve. Also, (i),

(iii), (iv) are examples of closed curves, while (ii) and (v) are examples of open curves.

An 'angle' is made up of two rays having a common end point . The two rays forming the

angle are called 'sides' of the angle, and the common end point is called the ' vertex ' of the

angle. Simple closed curves made up of only line segments are called polygons . A polygon

made up of three line segments is called a triangle. For example,

The triangle in the diagram is called . Here, points A, B and C are the vertices.

are the sides, and are the angles of the triangle.

A four-sided polygon is said to be a quadrilateral . Points A, B, C and D are the vertices.are the sides, and are the angles of quadrilateral ABCD.

A quadrilateral has, in all, four pairs of adjacent sides. They are AB and BC, BC and CD, CD

and DA, and DA and AB.


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Also, it has two pairs of opposite sides. They are AB and DC, and BC and AD. Angles A and B,

angles B and C, angles C and D, and angles D and A are said to be adjacent angles. Angles A

and C, and angles B and D are said to be pairs of opposite angles.

A circle is formed by a point moving at the same distance from a fixed point. A circle is also a

simple closed curve; however, it does not have any sides or angles.

The line that forms the boundary of a circle is called its circumference . The part enclosed by

the circumference of a circle is called the interior of the circle. The part left outside the circle is

said to be the exterior of the circle. Some points may lie on the circumference of the circle.

An arc is a part of the circumference of a circle. A chord is a line segment joining two points

that lie on a circle. The part of a circle that is enclosed by a chord and an arc is called a segment

of the circle. A chord passing through the centre of the circle is called its diameter . A diameter

is the longest chord of a circle. A diameter of a circle divides it into two halves. Each half is

called a semi-circle . A line segment that joins the centre of the circle and a point on the

circumference is called the radius of the circle. The radius of a circle is half of the diameter.

The part of a circle enclosed by two radii and an arc is called a sector .


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5. Understanding Elementary Shapes

Line and Angles

The distance between the endpoints of a line

segment is the length of the line segment.

Length of a line segment:

The distance between the endpoints of a line segment is the length of the line segment. The

length of a line segment can be measured accurately using a ruler and a divider .

Complete angle:

An angle of measure 360 0 is called a complete angle.

One quadrant = (Complete angle)= 1/4 x 360 0900 =

Two quadrants = 1/2 (Complete angle) = 1/2 x 360 0 = 180 0

Three quadrants=3/4

(Complete angle) = 3/4 x 360 0 = 270 0

Right angle:

An angle that measures 900

is called a right angle. A right angle makes a quarter revolutions .

Straight angle:

An angle that measures 180 0 is called a straight angle. A straight angle makes a half revolution .


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Acute angle:

An angle that measures less than 90 0 is called an acute angle.

Obtuse angle:

An angle that measures more than 900 and less than 180

0 is called an obtuse angle.

Reflex angle:

An angle that measures more than 180 0 is called a reflex angle.

Intersecting lines:

Two lines that meet each other at a single point are called intersecting lines.

Perpendicular lines:

Two lines that intersect each other at right angles are said to be perpendicular to each other.

Bisector of a line segment:

A bisector of a line segment is a line that divides the line segment into two equal parts .

Perpendicular bisector of a line segment:

The perpendicular line that divides a line segment into two equal parts is called the perpendicular

bisector of the line segment.

Two Dimensional Figures

The closed figure formed by joining three line

segments end-to-end is called a triangle.


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The closed figure formed by joining three line segments end-to-end is called a triangle. Each line

segment forms a side of the triangle.

Scalene Tr iangle:

A triangle is called a scalene triangle if all the three sides are of unequal length .

I sosceles Tr iangles:

A triangle is called an isosceles triangle if two of its sides are of equal length .

Equil ateral Tr iangle:

A triangle is said to be an equilateral triangle if the lengths of all of its sides are equal .

Acute-Angled Tr iangle:

If all the angles of a triangle are less than 90, then the triangle is called an acute-angled triangle.

Right-Angled tri angle:

If one of the angles in a triangle is a right angle , then the triangle is called a right-angled triangle.

Obtuse-Angled Tri angle:

If one of the angles in a triangle is an obtuse angle, then the triangle is called an obtuse-angled triangle.

Parallelogram:

A parallelogram is a four-sided figure in which the opposite sides are parallel to each other and are of

equal length. In a parallelogram, the diagonals need not be equal in length.

Rectangle:

A rectangle is a type of parallelogram that has opposite sides equal in length and parallel to each other. Its

diagonals are equal in length. A rectangle has four right angles .

Square:

A square is a type of parallelogram in which all the four sides are equal in length. Its diagonals are equal

in length. A square has four right angles.


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Rhombus:

In a rhombus, all the sides are equal in length, and the opposite sides are parallel to each other. Its

diagonals are not equal in length. Also, the opposite angles are equal to each other.

Trapezium:

A trapezium has one pair of sides parallel to each other. The other two sides are not parallel to each other.

Polygon:

A polygon is a closed figure with three or more than three sides.

Three Dimensional Shapes

Solid figures have three dimensions - length, breadth

and height.

Solid figures have three dimensions - length, breadth and height.

Eg: A ball, a brick, an ice cream cone and a can.

Face

The flat surface of a solid shape is called a face.

Edge

An edge is a line segment two faces of a solid shape meet.

Vertex

A vertex of a solid shape is a point where three or more edges meet. A cuboid has 6 faces , 12 edges and 8

vertices .

Prisms and pyramids are named after their bases. The base of a prism can be of any polygonal shape.


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There are 5 faces , 9 edges and 6 vertices in a

Triangular prism

There are 4 faces , 6 edges and 4 vertices in a triangular pyramid.


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6. IntegersIntegers

If you move towards the right from the zero mark on

the number line , the value of the numbers

increases .

Whole numbers are represented on the number line as shown here:

If you move towards the right from the zero mark on the number line , the value of the numbers

increases . If you move towards the left from the zero mark on the number line , the value of the

numbers decreases .

The collection of the numbers, that is, -3, -2, - 1, 0, 1, 2, 3, ., is called integers . When we need to use

numbers with a negative sign, we need to go to the left of zero on the number line. These numbers are

called negative numbers.

Examples where these negative numbers are used are temperature scale, water level in a lake or river,

level of oil tank, debit account and outstanding dues.

The numbers -1, -2, -3, - 4 which are called negative numbers, are also called negative integers .

The number 1, 2, 3, 4 s, which are called positive numbers, are also called positive integers .

If we stand at the zero mark on the number line, we can either go left towards negative integers or right

towards positive integers. When we move left towards zero on the number line, the value of positive

integers decreases. When we move left further away from zero on the number line, the value of negative

integers decreases.

Additi on of i ntegers:

When two positive integers are added, then we get an integer with a positive sign.

Example: (+8) + (+6)= + 14


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When two negative integers are added, then we get an integer with a negative sign.

Example: (-3) + (-5) = -8

When a positive integer is added to a negative integer, then we subtract them and put the sign of

the greater integer. The greater integer can be decided by ignoring the signs of the integers.

Example: (+4) + (-9) = -5; (+8) + (-3) = 5

Subtr action of i ntegers:

When we subtract a larger positive integer from a smaller positive integer, the difference is a

negative integer.

Eg: (+5)-(+8) = -3

To subtract a negative integer from any given integer, we just add the additive inverse of thenegative integer to the given integer.

Eg: (-5)-(-8) = +3

Thus, the subtraction of an integer is the same as the addition of its additive inverse. Both addition and

subtraction of integers can be shown on a number line.


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7. FractionsTypes of Fractions

A fraction is a part of a whole . A whole can be a

group of objects or a single object.

A fraction is a part of a whole . A whole can be a group of objects or a single object.

For example, is a fraction. In this, 3 is called the numerator and 15 is called the denominator.

In the figure shown here, the shaded portion is represented by .

Whole numbers are represented on the number line as shown here:

A fraction can be represented on the number line.

For example,

1. Consider a fraction is greater than 0, but less than 1.

Divide the space between 0 and 1 into two equal parts. We can show one part as the fraction


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1. Consider another fraction is greater than 0, but less than 1.

Divide the space between 0 and 1 into five equal parts. We can show the first part as the second as

the third as the fourth as and the fifth part as

Proper f ractions:

A proper fraction is a number representing a part of a whole .

In a proper fraction, the number in the denominator shows the number of parts into which the whole is

divided, while the number in the numerator shows the number of parts that have been taken.

Eg:

I mproper fr actions:

A fraction in which the numerator is bigger than the denominator is called an improper fraction. Eg:

M ixed fractions:

A combination of a whole and a part is said to be a mixed fraction.

Eg:

Conversion of improper fr action i nto mixed fraction:

An improper fraction can be expressed as mixed fraction by dividing the numerator by the denominator of

the improper fraction to obtain the quotient and the remainder. Then the mixed fraction will be

.

Conversion of mixed fraction i nto improper fr action:

A mixed fraction can be written in the form an improper fraction by writing it in the following way:


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L ike fractions:

Fractions with the same denominator are said to be like fractions .

Eg:

Unl ike fractions:

Fractions with different denominators are called unlike fractions .

Eg:

Equi valent fr actions:

Fractions that represent the same part of a whole are said to be equivalent fractions.

Eg:

To find an equivalent fraction of a given fraction, multiply both the numerator and the denominator of the

given fraction by the same number.

Simplest f orm of fr action:

A fraction is said to be in its simplest form or its lowest form if its numerator and denominator have

no common factor except one . The simplest form of a given fraction can also be found by dividing its

numerator and denominator by its highest common factor (HCF).


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Comparing Fractions

Fractions with the same denominator are called like

fractions .

Fractions with the same denominator are called like fractions .

Compari ng l ike fr actions:

In like fractions, the fraction with the greater numerator is greater.

Two fractions are unlike fractions if they have different denominators .

Compari ng unl ike fr actions:

If two fractions with the same numerator but different denominators are to be compared, then the

fraction with the smaller denominator is the greater of the two.

To compare unlike fractions, we first convert them into equivalent fractions. For example, to compare the

following fractions ie.,

We find the common multiple of the denominators 6 and 8. 48 is a common multiple of 6 and 8.

24 is also a common multiple of 6 and 8. Least Common Multiple (LCM) of 6 and 8 = 24

x =

x =


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Hence, we can say that is greater than

Addition and Subtraction of Fractions

If two fractions have the same number in the

denominator , then they are said to be like fractions .

L ike fractions:

If two fractions have the same number in the denominator , then they are said to be like fractions .

To add li ke fr actions:

1. Add the numerators of the fractions to get the numerator of the resultant fraction.

2. Use the common denominator of the like fractions as the denominator of the resultant fraction.

To subtract li ke fr actions:

1. Subtract the numerators of the fractions to get the numerator of the resultant fraction.

2. Use the common denominator of the like fractions as the denominator of the resultant fraction.


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Unl ike fractions:

Fractions with different numbers in the denominators are said to be unlike fractions .

To add unli ke fr actions:

1.

Find their equivalent fractions with the same denominator.

2. Add the numerators of the fractions to get the numerator of the resultant fraction.

3. Use the common denominator of the obtained like fraction as the denominator of the resultant

fraction.

To subtract unl ik e f ractions:

1. Find their equivalent fractions with the same denominator.

2. Subtract the numerators of the fractions to get the numerator of the resultant fraction.

Use the common denominator of the obtained like fraction as the denominator of the resultant fraction.


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Additi on or subtraction of mixed fr actions:

Two mixed fractions can be added or subtracted by adding or subtracting the whole numbers of the

two fractions, and then adding or subtracting the fractional parts together . Two mixed fractions can

also be converted into improper fractions and then added or subtracted.


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8. DecimalsComparing Decimals

Every fraction whose denominator is 10 can be written

in decimal notation.

If a block of one unit is divided into 10 equal parts, then each part is (one - tenth) of the unit. It iswritten as in decimal representation. The dot denotes the decimal point .

Every fraction whose denominator is 10 can be written in decimal notation.

Eg:

If a block of one unit is divided into 100 equal parts, then each part is of the unit. It

is written as in decimal notation. Every fraction whose denominator 100 can be written in decimal

notation.

Eg:

To read decimals, we can use the following chart. The first digit to the right after the decimal point

represents the tenths parts, the second the hundredths parts, and so on.

Decimal point Tenths Hundredths Thousandths

.

All decimal numbers can be represented on the number line . Every decimal number can be represented

as a fraction . Any two decimal numbers can be compared. The comparison starts with the whole part of

the numbers. If the whole parts are equal, then the tenth parts can be compared, and so on. Decimal

numbers are used in many ways in real life. For example, in representing the units of money, length and

weight, we use decimal numbers.


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Addition and Subtraction of Decimals

To add or subtract decimal numbers , make sure that

the decimal points of the given numbers are placed

exactly one below another .

To add or subtract decimal numbers , make sure that the decimal points of the given numbers are

placed exactly one below another . While adding or subtracting two decimal numbers, the number of

digits after the decimal point should be equal . In case they are not equal, the gaps must be filled with

zeros after the last digit.

For example:

1. To add 6.82 and 5

First insert zeros in the empty places after the decimal point so that both the numbers have the same

number of digits after the decimal point. Next, write the numbers such that their decimal points are one

below another.

2. To subtract 5 from 6.82

First insert zeros in the empty places after the decimal point so that both the numbers have the same

number of digits after the decimal point. Next, write the numbers such that their decimal points are one

below another.


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Addition or subtraction should be carried out from the extreme right side . Place the decimal point

correctly after performing the addition or subtraction.


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9. Data HandlingData handling

Data is a collection of numbers gathered to get some

information.

Data is a collection of numbers gathered to get some information.

To get the required information, all observations should be recorded.

Tally marks are used to organise the observations . Record every observation by a vertical mark, but

every fifth observation should be recorded by a mark across the four earlier marks, like this: .

We depict each observation with the help of tally marks.

For example , we have a group of persons and their sizes of shoes. The tabular form representing the tallymarks is as shown here.


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Pictograph

A picture that visually helps us to understand data is

called a pictograph .

A picture that visually helps us to understand data is called a pictograph . A pictograph represents data in

the form of pictures, objects or parts of objects .

Eg:


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In real life, pictographs are used by newspapers and magazines to attract the attention of the readers. A

pictograph helps us to answer questions on the data at a glance. To draw pictographs, we use symbols to

represent a certain number of things or items.

For example, represents 100 bulbs.

The key for a pictograph tells the number that each picture or symbol represents.

Bar Graph

A picture that visually helps us to understand data is

called a pictograph .

Bar graphs or bar diagrams are helpful in representing the data visually . In bar graphs or bar

diagrams, bars of equal width are drawn horizontally or vertically with equal spacing between them.

The length of each bar represents the required information. Choosing an appropriate scale for a bar

graph is important. Scale means the number used to represent one unit length of a bar . For example,

the scale for the bar graph shown here is 1 unit length = 100 children.


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10. MensurationPerimeter

Closed figure: A figure with no open ends is a closed

figure.

Closed figure: A figure with no open ends is a closed figure.

Regular closed figures: A closed figure in which all the sides and angles equal.

Perimeter:

Perimeter is the distance covered along the boundary forming a closed figure when we go round the

figure once. The concept of perimeter is widely used in real life.

Eg: 1) For fencing land.

2) For building a compound wall around a house.

The perimeter of a regular closed figure is equal to the sum of its sides.

Per imeter of a r ectangle:


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Per imeter of a square:

Equil ateral tr iangle:

A triangle with all its sides and angles equal is called an equilateral triangle.

The perimeter of an equilateral triangle with the side 'a'


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Area

The amount of surface enclosed by a closed figure is

called its area .

The amount of surface enclosed by a closed figure is called its area .

The following conventions are to be adopted while calculating the area of a closed figure using a squared

or graph paper.

1. Count the fully-filled squares covered by the closed figure as one square unit or unit square

each.

2. Count the half-filled squares as half a square unit .

3. Count the squares that are more than half-filled as one square unit .

4. Ignore the squares filled less than half.

For example, the area of this shape can be calculated as shown:

Covered area Number Area estimate (sq. units)

Fully filled squares 6 6

Half-filled squares 7 7 x

Squares filled more than half 0 0

Squares filled less than half 0 0

Area covered by full squares = 6 x 1 = 6 sq. units

Area covered by half squares = 7 x = 7/2= 3 sq. units


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Total area of the given shape = 6 + 3 sq. units

Thus, the total area of the given shape = 9 sq. Units

Area of a rectangle can be obtained by multiplying length by breadth. Area of the square can be obtained

by multiplying side by side.


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11. AlgebraVariables

An unknown quantity can be represented by a variable

An unknown quantity can be represented by a variable . Usually, a variable is any letter from the

English alphabet that represents an unknown quantity. The relation between the unknown quantity and

other quantities can be expressed with the help of the variable. The value of the variable varies with the

given condition on the variable. A quantity whose value does not vary is called a constant . An

expression consisting of variables, constants and mathematical operators is called an algebraic

expression .

Mathematical operations such as addition, subtraction, multiplication and division can be easily

performed on variables. We can use variables to form expressions based on patterns.

The following are some branches of mathematics:

The branch of mathematics where letters are used along with numbers is called algebra.

The branch of mathematics that deals with numbers, operations on numbers and properties of

numbers is called arithmetic.

The branch of mathematics that deals with the figures and shapes is called geometry.


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Use of Variables

Variables are used to frame rules to find the perimeter

of a polygon.

Variables are used to frame rules to find the perimeter of a polygon. The perimeter of a polygon can be

obtained by adding the lengths of its sides . The following are simple rules to frame the perimeter of

geometrical figures using variables.

If the length of the side is denoted by variable's',

then the perimeter of a square is equal to

and its breadth variable

If the lengths of the sides of a triangle are denoted

by then the perimeter of the triangle is

equal to


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The following are some simple rules for the properties of numbers using variables.

Commutative property of addition:

This property states that two numbers can be added in any order. If represent any two numbers,

then

Commutative property of multiplication:

This property states that two numbers can be multiplied in any order. If represent any two

numbers, then

Associative property of addition:

This property states that three numbers can be added in any order. If represent any three

numbers, then

Associative property of multiplication:

This property states that three numbers can be multiplied in any order. If represent any three

numbers, then

Distributive property of multiplication over addition:

This property states that if represent any three numbers, then

Equations

A mathematical statement that indicates that the valueof the LHS is equal to the value of the RHS is called an

equation.


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A mathematical statement that indicates that the value of the LHS is equal to the value of the

RHS is called an equation.

An equation puts a condition on the variable .

The value for which the equation is satisfied is the solution of the equation.

The value of the variable in an equation that satisfies the equation, or makes its LHS equal to its

RHS, is the solution.

An equation can contain numbers and variables.

An equation is said to be algebraic equation if it consists of a variable.

A single variable equation will have a unique solution.

An equation that does not have any variable is called a numerical or an arithmetic equation.

Different numerical values for the variable are substituted in an algebraic equation, and the solution is

obtained by using a method called the trial and error method.

If there is no sign of equality between the LHS and the RHS, then it is not an equation.


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12. Ratios and ProportionsRatios

Usually, the comparison of quantities of the same

type can be made by the method of difference between

the quantities.

Usually, the comparison of quantities of the same type can be made by the method of difference

between the quantities. However, a more meaningful comparison between the quantities can be made by

using division, i.e. by verifying how many times one quantity is into the other quantity. This method is

known as comparison by ratio .

For example, Keertana's weight is 20 kg and her father's weight is 80 kg. So we can say that Keertana's

father's weight and Keertana's weight are in the ratio 20:80.

To calculate ratio , the two quantities have to be measured using the same unit . If not, they should be

converted to the same unit before ratio is taken. The same ratio can occur in different situations.

For example, the ratio 4:5 is different from 5:4.

Thus, the order in which the quantities are taken into consideration to express their ratio is important.

A ratio can be treated as a fraction .

For example, 5:6 can be treated as 5/6.

Two ratios are said to be equivalent if the fractions corresponding to them are equivalent.

To calculate equivalent ratio , convert the ratio into a fraction, and then multiply or divide the numerator

and the denominator by the same number.

Ex:4:5 is equivalent to 8:10 or 12:15 and so on.A ratio can be expressed in its lowest form. For example, the ratio 45:25 in its lowest form can be

written as follows:

Thus, the lowest form of 45:25 is 9:5.


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Proportions

If the ratios between Quantity A and Quantity B is

equal to the ratio between Quantity C and Quantity D,

If the ratios between Quantity A and Quantity B is equal to the ratio between Quantity C and Quantity D,

then the four quantities A, B, C and D, are said to be in proportion . Proportion is denoted by the signs

' ' or '= '. Thus, the quantities 4, 16, 5 and 20 can be written as 4:16 5:20 or 4:16=5:20

The order of the terms in a proportion carries value . The quantities 4, 16, 5 and 20 are in proportion,

whereas 4, 20, 5 and 16 are not in proportion. In the proportion a:b c:d, the quantities a and d are the

extreme terms , and b and c are the middle terms. The method of calculating the value of one unit and

using this value to calculate the value of the required number of units is called the unitary method .

For example, suppose the cost of 8 bags is Rs. 240. Now, to find the cost of 6 bags,

using the unitary method, we first find out the cost of one bag.

Cost of one bag =240/8= Rs. 30

Now, the cost of 6 bags =6 Rs.30=180

Hence, the cost of 6 bags is Rs. 180.


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13. SymmetryLine Symmetry

A figure can have line symmetry if a line can be drawn

dividing it into two equal halves . The line is called

the line of symmetry .

A figure can have line symmetry if a line can be drawn dividing it into two equal halves . The line is

called the line of symmetry . We can find examples of objects showing line symmetry in nature. For

example, a butterfly, some leaves and flowers show line symmetry.

Examples of line symmetry can also be found in many of our ancient and modern buildings.

Objects that show line symmetry appear more balanced and beautiful.

A kite shape has only one line of symmetry.

A rectangle has two lines of symmetry.

An equilateral triangle has three lines of symmetry.


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A circle has an infinite number of lines of symmetry.

A shape may have just one or more than one lines of symmetry. When completing a given figure against

a given line of symmetry, make sure that:

Each part of the constructed figure is equal in measurement to its corresponding part in the given

figure. Each point on the given figure and its corresponding point on the constructed figure are at the

same distance from the line of symmetry.

Mirror Symmetry

The line of symmetry is related to mirror reflection.


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The line of symmetry is related to mirror reflection.

An object and its mirror image are equal in shape and size . An object and its image are always at the

same distance from the surface of a mirror, which is called the mirror line .

The left and the right sides of an object appear inverted in a mirror. An object and its image show mirror

symmetry, with the mirror line being the line of symmetry.

Letters written from right to left, appear written from left to right in their mirror image.

Letters like A, M and U appear the same in their mirror image.

The letters A, H, I, M, O, T, U, V, W, X and Y appear the same in their mirror image.

All the other letters of the alphabet appear reversed in their mirror image. Symmetry has plenty of

applications in real life, as in art, architecture, textiles designing, geometrical reasoning, Kolams, Rangoli,

etc.


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14. Practical GeometryBasic Constructions

Ruler, Compass, Divider, Set squares, Protractor

The tools in our geometry box are:

Ruler

Compass

Divider

Set squares

Protractor

The description of each tool and its uses are given below:

Ruler:

A ruler is a flat and straight-edged strip , whose one side is graduated into centimetres and the other into

inches. A ruler is commonly called a scale . It is the most essential tool in geometry. It is used in all

constructions.

The basic uses of a ruler are:

Measuring lengths of line segments.

Drawing line segments.


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Compass:

A compass has two ends. One end holds a pointer, while the other end holds a pencil. It is also called a

pair of compasses.

The basic uses of a compass are:

Marking off equal lengths.

Drawing arcs.

Drawing circles.

Divider:

A divider is a tool similar in shape to a compass. It has a pair of pointer ends .


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The basic uses of a divider are:

Comparing lengths of line segments.

Helping avoid positioning errors.

Taking accurate measurements.

Set squar es:

The two triangular tools in the geometry box are called set squares. One of the set square is an isosceles

triangle with two angles measuring each . The other set square is a scalene triangle with two

angles measuring and each. The two perpendicular sides of either set square are graduated

into centimetres.

The basic uses of set squares are:

Drawing perpendicular lines.

Drawing parallel lines.

Protractor:

A semi-circular tool with degrees marked is called a protractor. The centre of the semicircle is called the

midpoint of the protractor. This point helps as a reference point for the protractor. The horizontal line is

called the base line or the straight edge of the protractor.


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The basic uses of a protractor are:

Measuring angles.

Drawing angles.

The important points to be remembered while using the tools for construction are:

Draw smooth and thin lines.

Mark points lightly.

Maintain tools or instruments with sharp pointers and fine edges.

Keep two pencils in the box. One is for drawing lines and marking points. The other is for using

in the compass.

Construction of Lines

Steps to construct a line segment of length 5 cm

Steps to construct a l ine segment of length 5 cm:

Draw line l.

Mark a point on line and name it .


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Open the compass to measure the length of the line segment

by placing the pointer on the 0 mark of the ruler and the

pencil point on the 5 cm mark.

Place the pointer of the compass on point P.

Swing an arc on the line to cut it at Q.

is the required line segment of length 5 cm.

Two lines are said to be perpendicular when they intersect

each other at an angle of 90 o.

The perpendicular bisector is a perpendicular line that bisects another line into two equal parts .

Constructing of Angles

An exact copy of a line segment can be constructed

using a ruler and a compass .

An exact copy of a line segment can be constructed using a ruler and a compass .

To constr uct a copy of an angle:

Draw a line AB.

Mark any point O on AB.


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Place the compass pointer at vertex X of the given figure and draw an arc with a convenient

radius, cutting rays XY and XZ at points E and F, respectively.

Without changing the compass settings, draw an arc on line AB from point O. It cuts line AB at

P.

Set the compass to length EF.

Without changing the compass settings, draw an arc from P cutting the previous arc at point Q.

Join points O and Q.

Hence, POQ is the required copy of YXZ

To constr uct the bisector of an angle:

Let the given angle be LMN.

Place the compass pointer at vertex M of the given angle.

Draw an arc cutting rays ML and MN at U and V, respectively.

Draw an arc with V as the centre and a radius more than half the length of UV in the interior of LMN.


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Draw another arc with U as the centre and the same radius intersecting the previous arc.

Name the point of intersection of the arcs as X.

Join points M and X.

Thus, the ray MX is the required bisector of LMN

In a similar way, we can construct:

Documents

Class VI I.I.T. Foundation, N.T.S.E.& Mathematics Olympiad Curriculum & Chapter Notes