# integers

## 7th Class Maths Concept

Studying maths in VII class successfully means that children take responsibility for their learning and learn to apply the concepts to solve problems.

### 1. INTEGERS

Natural numbers: All the counting numbers starting from 1 are called Natural numbers.

1, 2, 3… Etc.

Whole numbers: Whole numbers are the collection of natural numbers including zero.

0, 1, 2, 3 …

Integers: integers are the collection of whole numbers and negative numbers.

….,-3, -2, -1, 0, 1, 2, 3,…..

Integers on a number line:

Operations on integers:

3 + 4 = 7

-2 + 4 = 2

Subtraction of integers on a number line:-

6 – 3 = 3

Multiplication of integers on a number line:-

2 × 3 ( 2 times of 3) = 6

3 × (- 4 ) ( 3 times of -4) = -12

Multiplication of two negative integers:

• To multiply two negative integers, first, we multiply them as whole numbers and put plus sign before the result.
• The multiplication of two negative integers is always negative.

Ex:- -3 × -2 = 6,  -10 × -2 = 20 and so on.

Multiplication of more than two negative integers:

• If we multiply three negative integers, then the result will be a negative integer.

Ex:- -3 ×   -4 ×   -5 = -60,  -1× -7 × -4 = -28 and so on.

• If we multiply four negative integers, then the result will be a positive integer.

Ex:- -3 ×   -4 ×  -5 × -2  = 120,  -1× -7 × -4  × -2 = 56 and so on.

Note:-

1. If the no. of negative integers is even, then the result will be positive.

2. If the no. of negative integers is odd, then the result will be negative.

Division of integers:

• The division is the inverse of multiplication.
• When we divide a negative integer by a positive integer or a positive integer by a negative integer, we divide them as whole numbers then put negative signs for the quotient.

Ex:- -3 ÷ 1 = 3, 4 ÷ -2 = -2 and so on.

When we divide a negative integer by a negative integer, we get a positive number as the quotient.

Ex:- -3 ÷ -1 = 3, -4 ÷ -2 = 2 and so on.

#### Properties of integers:

1.Closure property:-

2.commutative property:-

3.associative property:-

1 + 0 = 0 + 1 = 1,   10 + 0 = 0 + 10 = 10

•For any integer ‘a’, a + 0 = 0 + a

2 + (-2) = (-2) + 2 = 0,  5 + (-5) = (-5) + 5 = 0

•For any integer ‘a’, a+ (-a) = (-a) + a = 0

•Additive inverse of a = -a and additive inverse of (-a) = a

Multiplicative identity:-

2 × 1 = 1 × 2 = 2,    5 × 1 = 1 × 5 = 5

•For any integer ‘a’, a × 1 = 1 × a = a

•1 is the multiplicative identity.

multiplicative inverse:-

For any integer ‘a’, 1/a × a = a × 1/a = 1

• multiplicative inverse of a = 1/a
• Multiplicative inverse of  1/a = a.

distributive property:-

For any three integers a, b and c,    a × (b + c) = (a × b) + (a × c).

3 × (2 + 4) = 18

(3 × 2) + (3 × 4) = 6 + 12 = 18

∴ 3 × (2 + 4) = (3 × 2) + (3 × 4).

### 2. FRACTIONS, DECIMALS AND RATIONAL NUMBERS

Fraction: A fraction is a number that represents a part of the whole. A group of objects is divided into equal parts, then each part is called a fraction.

The proper and improper fractions:

In a proper fraction, the numerator is less than the denominator.

Ex: – 1/5, 2/3, and so on.

In an improper fraction, the numerator is greater than the denominator.

Ex: – 5/2,11/5 and so on.

Comparing fractions:

Like fractions: – We have to compare the like fractions with the numerator only because the like fractions have the same denominator. The fraction with the greater numerator is greater and the fraction with the smaller numerator is smaller.

Ex: ,    and so on

Unlike fractions: –

With the same numerator: For comparing unlike fractions, we have to compare denominators when the numerator is the same. The fraction with a greater denominator is smaller and the fraction with a smaller denominator is smaller.

Ex: –     and so on.

Note: – To find the equivalent fractions of both the fractions with the same denominator, we have to take the LCM of their denominators.

∗ Like Fractions:

∗ Unlike fractions:

Subtraction of fractions:

∗ Like fractions:

Ex:

Unlike fractions: – First, we have to find the equivalent fraction of given fractions and then subtract them as like fractions

Ex:

Multiplication of fractions:

Multiplication of fraction by a whole number: –

Multiplication of numbers means adding repeatedly.

Ex: –

• To multiply a whole number with a proper or improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator the same.

2.Multiplication of fraction with a fraction: –

multiplication of two fractions =

Division of fractions:

Ex: – 2 ÷

⇒ 6 one-thirds in two wholes

Reciprocal of fraction: reciprocal of a fraction is   .

Note:

• dividing by a fraction is equal to multiplying the number by its reciprocal.
• For dividing a number by mixed fraction, first, convert the mixed fraction into an improper fraction and then solve it.

Ex:

1.Division of a whole number by a fraction: –

2.Division of a fraction by another fraction: –

Decimal number or fractional decimal:

In a decimal number, a dot(.) or a decimal point separates the whole part of the number from the fractional part.

The part right side of the decimal point is called the decimal part of the number as it represents a part of 1. The part left to the decimal point is called the integral part of the number.

Note: –

• while adding or subtracting decimal numbers, the digits in the same places must be added or subtracted.
• While writing the numbers one below the other, the decimal points must become one below the other. Decimal places are made equal by placing zeroes on the right side of the decimal numbers.
###### Comparison of decimal numbers:

while comparing decimal numbers, first we compare the integral parts. If the integral parts are the same, then compare the decimal part.

Ex: – which is bigger: 13.5 or 14.5

Ans: 14.5

Which is bigger: 13.53 or 13. 25

Ans: 13.53

Multiplication of decimal numbers:

For example, we multiply 0.1 × 0.1

Multiplication of decimal numbers by 10, 100, and 1000: –

Here, we notice that the decimal point in the product shifts to the right side by as many zeroes as in 10, 100, and 1000.

Division of decimal number:

Division of decimal number by 10,100 and 1000: –

Here, we notice that the decimal point in the product shifts to the left side by as many zeroes as in 10, 100, and 1000.

Rational numbers:

The numbers which are written in the form of p/q, where p, q are integers, and q ≠ 0, are called rational numbers.

Rational numbers are a bigger collection of integers, negative fractional numbers, positive fractional numbers.

Ex: – 1, 2, -1/2, 0 etc.

### 3. SIMPLE EQUATIONS

Equation: Equation is the condition of a variable. It says that two expressions are equal.

• An equation has two sides LHS and RHS, on both sides of the equality of sign.
• One of the expressions of the equation is must have a variable.
• If we interchange the expressions from LHS to RHS, the equation remains the same

Ex: – x + 2 = 5; 2 = x + 3

Balanced equation:

In an equation, if LHS =RHS, then that equation is balanced.

If the same number is added or subtracted on both sides of the balanced equation, the equation remains will the same.

Ex: 8 + 3 = 11

If add 2 on both sides ⇒ LHS = 8 + 3 + 2 = 13

RHS = 11 + 2 = 13

∴ LHS = RHS

8 +3 = 11 if subtract 2 on both sides

LHS = 8 + 3 – 2 = 9

RHS = 11 – 2 = 9

∴ LHS = RHS

###### Using algebraic equations in solving day to day problems:
2. Denote the unknown or quantity to be found with some letters such as x, y, z …etc.
3. Write the problem in the form of an algebraic equation by making a relation among the quantities.
4. Solve the equation.
5. Check the solution

### 4. LINES AND ANGLES

Complimentary angles: When the sum of the angles is 900, the angles are called complementary angles.

Ex: 300, 600; 200, 700 and soon.Supplementary angles: When the sum of the angles is 1800, the angles are called  Supplementary angles.

Ex: 1200, 600; 1100, 700 and soon.

Adjacent angles: The angle having a common Arm and a common vertex are called Adjacent angles.

⇒ ∠AOC and ∠BOC adjacent angles.

Vertically opposite angle: If two lines are intersecting at a point, then the angles that are formed opposite to each other at that point are called vertically opposite angles.

Transversal: A line that intersects two or more lines at distinct points is called a transversal.

#### Angles made by a transversal:

Corresponding angles:

Two angles that lie on the same side of the transversal and one interior and another one exterior are called corresponding angles.

∠1, ∠5; ∠2, ∠6; ∠3, ∠7 and ∠4, ∠8

Alternate angles:

Two angles which are the lies opposite side of the transversal and both interior or exterior are called Alternate angles.

∠1, ∠7; ∠2, ∠8 are exterior alternate angles

∠3, ∠5; ∠4, ∠6 are interior alternate angles.

∠3, ∠6; ∠4, ∠5 interior angles same side of the transversal.

Transversal on parallel lines:

If pair of parallel lines are intersected by a transversal then the angles of each pair of corresponding angles are equal

⇒ ∠1, =∠5; ∠2= ∠6; ∠3= ∠7 and ∠4= ∠8

•If pair of parallel lines are intersected by a transversal then the angles of each pair of interior alternate angles are equal.

∠3= ∠5; ∠4= ∠6

•If pair of parallel lines are intersected by a transversal then the angles of each pair of exterior alternate angles are equal.

∠1= ∠7; ∠2= ∠8

•If pair of parallel lines are intersected by a transversal then the angles of each pair of interior angles on the same side of the transversal are supplementary.

∠3+∠6= 1800; ∠4+ ∠5 = 1800

Note:

1.If a transversal intersects two lines and the pair of corresponding angles are equal, then the lines are parallel.

2.If a transversal intersects two lines and the pair of alternate angles are equal, then the lines are parallel.

3.If a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel.

### 5. TRIANGLE AND ITS PROPERTIES

Triangle:
A closed figure formed by three-line segments is called a triangle.

In ∆ABC,

• Three sides are
• Three angles are ∠ABC, ∠BCA, ∠ACB
• Three vertices are A, B, C.

Classification of triangles:

Triangles can be classified according to the properties of their sides and angles.

According to sides:

Based on sides triangles are three types:

• Scanlan triangle (ii) Isosceles Triangle (iii) equilateral triangle

According to angles:

• Acute-angled triangle (ii) Right-angled triangle (iii) Obtuse-angled triangle

Relationship between the sides of a triangle:

1. The sum of the lengths of any two sides of a triangle is greater than the third side.

1. The difference between the lengths of any two sides of a triangle is less than the third side.

The altitude of a triangle:

We can draw three altitudes in a triangle.
A perpendicular line drawn from a vertex to its opposite side of a triangle is called the Altitude of the triangle.

Median of a triangle:

In a triangle, a line drawn from the vertex to the mid-point of its opposite side is called the median of the triangle.

Medians of a triangle are concurrent. We can draw three medians in a triangle.

The point of concurrence of medians is called the centroid of the triangle. It is denoted by G

Angle-sum property of a triangle:
Some of the angles in a triangle is 1800

∠A + ∠B + ∠C = 1800

An exterior angle of a triangle:

When one side of the triangle is produced, the angle thus formed is called an exterior triangle.

Exterior angle property:- The exterior angle of a triangle is equal to the sum of two interior opposite angles.

x0+ y0 = z0

## 6.RATIO – APPLICATIONS

Ratio: Comparison of two quantities of the same kind is called ‘Ratio.

The ratio is represented by the symbol ‘:’

If the ratio of two quantities ‘a’ and ‘b’ is a : b, then we read this as ‘a is to b’

The quantities ‘a’ and ‘b’ are called terms of the ratio.

Proportion: if two ratios are equal, then they are said to be proportional.
‘a’ is called as first term or antecedent and ‘b’ is called a second term or consequent.

If a: b = c : d, then a, b, c, d are in proportion and   ⇒ ad = bc.

The product of means = the product of extremes

Unitary method:  The method in which we first find the value of one unit and then the value of the required no. of units is known as the unitary method.

Direct proportion: In two quantities, when one quantity increase(decreases) the other quantity also increases(decreases) then two quantities are in direct proportion.

Percentages:

‘per cent’ means for a hundred or per every hundred. The symbol % is used to denote the percentage.

1% means 1 out of 100, 17% means 17 out of 100.

Profit and Loss:

Selling price = SP; Cost price = CP

If SP > CP, then we get profit

Profit = SP – CP

Profit percentage =

SP = CP + profit

If SP < CP, then we get a loss

Loss = CP – SP

Loss percentage =

SP = CP – Loss

Simple interest:

Principle: – The money borrowed or lent out for a certain period is called the Principle.

Interest: – The extra money, for keeping the principle paid by the borrower is called interest.

Amount: – The amount that is paid back is equal to the sum of the borrowed principal and the interest.

Amount = principle + interest

Interest (I) =   where R is the rate of interest.

## 7.DATA HANDLING

Data: The information which is in the form of numbers or words and helps in taking decisions or drawing conclusions is called data.

Observations: The numerical entries in the data are called observations.

Arithmetic Mean: The average data is also called an Arithmetic mean.

Arithmetic Mean (A.M) =

The arithmetic mean always lies between the highest and lowest observations of the data.

When all the values of the data set are increased or decreased by a certain number, the mean also increases or decreases by the same number.

Mode: The most frequently occurring observation in data is called Mode.

If data has two modes, then it is called bimodal data.

Note: If each observation in a data is repeated an equal no. of times, then the data has no mode.

Median: The middlemost observation in data is called the Median.

Arrange given data in ascending or descending order.

If a data has an odd no. of observations, then the middle observation is the median.

If a data has even no. of observations, then the median is the average of middle observations.

Bar graph:

Bar graphs are made up of uniform width which can be drawn horizontally or vertically with equal spacing between them.

The length of each bar tells us the frequency of the particular item.

Ex:

Double bar graph:

It represents two observations side by side.

Ex:

Pie chart: A circle can be divided into sectors to represent the given data

The angle of each sector =

Ex:

 Budget Amount in rupees Food 1200 Education 800 Others 2000 Savings 5000 Total income 9000

## 8.CONGRUENCY OF TRIANGLES

Congruent figures: Two figures are said to be congruent if they have the same shape and size.

Congruency of line segments:  If two-line segments have the same length, then they are congruent. Conversely, if two-line segments are congruent, then they have the same length.

Congruency of Triangles:

Two triangles are said to be congruent if (i) their corresponding angles are equal (ii) their corresponding sides are equal.
Ex: In ∆ ABC, ∆ DEF

∠A ≅ ∠D; ∠B≅ ∠E; ∠C ≅ ∠F

AB ≅ DE; BC≅ EF; AC ≅ DF

∴∆ABC ≅ ∆DEF

The criterion for congruency of Triangles:

1.Side -Side -Side congruency (SSS): –
If three side of a triangle is equal to the corresponding three sides of another triangle, then the triangles are congruent.

∴∆ABC ≅ ∆DEF

2.Side -Angle -Side congruency (SAS): –
If two sides and the angle included between the two sides of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

∴ ∆ABC ≅ ∆DEF

3.Angle – Side -Angle congruency (ASA): –
If two angles and included side of a triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent.

∴ ∆ABC ≅ ∆DEF

4.Right angle – Hypotenuse – Side congruence (RHS): –

If the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and side of the other right-angled triangle, then the triangles are Equal.

∴∆ABC ≅ ∆DEF

## 9.CONSTRUCTION OF TRIANGLES

The no. of measurements required to construct a triangle = 3

A triangle can be drawn in any of the situations given below:

• Three sides of a triangle
• Two sides and the angle included between them.
• Two angles and the side included between them.
• The hypotenuse and one adjacent side of the right-angled triangle.

Construction of a triangle when measurements of the three sides are given:

Ex: construct a triangle ABC with sides AB = 4cm, BC = 7cm and AC = 5cm

Step of constructions:

Step -1: Draw a rough sketch of the triangle and label it with the given measurements.

Step -2: Draw a line segment of BC of length 7cm.

Step -3: with centre B, draw an arc of radius 4cm, draw another arc from C with radius 5cm such that it intersects first at A.

Step -4: join A, B and A, C. The required triangle ABC is constructed.

Construction of a triangle when two sides and the included angle given:

EX: construct a triangle ABC with sides AB = 4cm, BC = 6cm and ∠B=600

Step of constructions:

Step -1: Draw a rough sketch of the triangle and label it with the given measurements.

Step -2: Draw a line segment of AB of length 4cm.

Step -3: draw a ray BX making an angle 600 with AB.

Step -4: draw an arc of radius 5cm from B, which cuts ray BX at C.

Step -5: join C and A, we get the required ∆ABC.

Construction of a triangle when two angles and the side between the angles given:

Ex: construct a triangle PQR with sides QR = 4cm, ∠Q= 1200 and ∠R= 400

Step of constructions:

Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

Step -2: Draw a line segment QR of length 4 cm.

Step -3: Draw a ray RX, making an angle 400 with QR.

Step -4: Draw a ray QY, making an angle 1000 with QR, which intersects ray RX.

Step -5: Mark the intersecting point of the two rays as P. Required triangle PQR is constructed.

Construction of a triangle when two sides and the non-included angles are given:

Ex: construct a triangle MAN with sides MN = 4cm, AM = 3cm and ∠A= 400

Step of constructions:

Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

Step -2: Draw a line segment MA of length 0f 5cm.

Step -3: Draw a ray AX making an angle 400 with the line segment MA.

Step -4: With M as the centre and radius 3 cm draw an arc to cut ray AX. Mark the intersecting point as N.

Step -5: join M, N, then we get the required triangle MAN.

Construction of a right-angled triangle when hypotenuse and sides are given:

Ex: construct a triangle ABC, right angle at B and AB = 4cm, Ac = 5cm

Step of constructions:
Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

Step -2: Draw a line segment BC of length 0f 4cm.

Step -3: Draw a ray BX perpendicular to BC at B

Step -4: Draw an arc from C with a radius of 5cm to intersect ray BX at A.

Step -5: Join A, C, then we get the required triangle ABC.

## 10.ALGEBRAIC EXPRESSIONS

Variable: It is a dependent term. It takes different value.

Ex: m, x, a, etc.

Constant:  It is an independent term. It has a fixed value.

Ex: 1, 3,  etc.

Like terms and Unlike terms: If the terms contain the same variable with the same exponents, then they are like terms otherwise, unlike terms.

Ex: 3x, –4x, x are like terms

3x, 4y, 4 are unlike terms

Coefficient: Coefficient is a term which the multiple of another term (s)

EX: In 5x. 5 is the coefficient of x and x is the coefficient of 5

5 is a numerical coefficient

x is the literal coefficient

Expression: An expression is a single term or a combination of terms connected by the symbols ‘+’ (plus) or ‘−’ (minus).

Ex: 2x – 3. 3x, 2 +3 – 4 etc.

Numerical Expressions:  If every term of an expression is constant, then the expression is called numerical expression.

Ex: 2 + 3 + 5, 2 – 4 – 7, 1 + 5 – 4 etc.

Algebraic expression:  If an expression at least one algebraic term, then the expression is called an algebraic expression.

Ex: x + y, xy, x – 3, 4x + 2 etc.

Note: Plus (+) and Minus (−) separate the terms

Multiplication (×) and Division (÷) do not separate the terms.

Types of Algebraic expressions:

Monomial: – If an expression has only one term, then it is called a monomial.

Ex: 2x2, 3y, x, y, xyz etc.

Binomial: If an expression has two unlike terms, then it is called binomial.

Ex: 2x+ 3y, x2+ y, x +yz2 etc.

Trinomial: If an expression has three unlike terms, then it is called trinomial.  Ex: 2x+ 3y + 4xy, x2+ y + z, x2 y +yz2 + xy2 etc.

Multinomial: If an expression has more than three unlike terms, then it is called multinomial.

Ex: 2x+ 3y + 4xy +5, x2+ y + z – 4y + 6 ,

x2 y +yz2 + xy2 – 4xy + 8yz etc.

Degree of a monomial: The sum of all exponents of the variables present in a monomial is called the degree of the monomial.

Ex: Degree of 5xy3

An exponent of x is 1 and an exponent of y is 3

Sum of exponents = 1 + 3 = 4

∴ degree of 5xy3 is 4

Degree of an Algebraic Expression: The highest exponent of all the terms of an expression is called the degree of an Algebraic expression.

Ex: degree of x2 + 3x + 4x3 is 3

degree of 3xy + 6x2y + 5x2y2 is 4

The sum of two or more like terms is a like term with a numerical coefficient that is equal to the sum of the numerical coefficients of all the like terms in addition.

Ex: 3x + 2x = (3 + 2) x = 5x

4x2y + x2y = (4 + 1) x2y = 5x2y

Subtraction of like terms:

The difference of two like terms is a like term with a numerical coefficient is equal to the difference between the numerical coefficients of the two like terms.

Ex: 3x − 2x = (3 − 2) x = x

4x2y −2 x2y = (4 −2) x2y = 2x2y

Note: (i) addition and subtraction are not done for unlike terms. (ii) If no terms of an expression are alike then it is said to be in the simplified form.

The standard form of an Expression:

In an expression, if the terms are in such a way that the degree of the terms is in descending order, then the expression is said to be in standard form.

Ex: 5 – 2x2 + 4x +3x3

Standard form is 3x3 – 2x2 + 4x + 5

Finding the value of an expression:

Example: find the value of expression x3 + y + 3, when x = 1 and y = 2

Sol: given expression is x3 + y + 3

Substitute x = 1 and y = 2 in above expression

(1)3 + 2 + 3 = 1 + 2 + 3 = 6

This is in two ways: (i) Column or Vertical method (ii) Row or Horizontal method.

Column or Vertical method:

Step –1: Write the expression in standard form if necessary.

Step –2: write one expression below the other such that the like terms come in the same column.

Step –3: Add the like terms column-wise and write the result just below the concerned column.

Ex: Add x2 + 3x + 5, 3 – 2x + 3x2 and 3x – 2

Sol:

Row or Horizontal method.

Step –1: Write the expression in standard form if necessary.

Step –2:  Re-arrange them term by grouping the like terms.

Step –3: Simplify the coefficients.

Step –4: Write the resultant expression in standard form.

Ex: Add x2 + 3x + 5, 3 – 2x + 3x2 and 3x –2

Sol: (x2 + 3x + 5) + (3 – 2x + 3x2) + (3x –2)

= (x2 + 3x2) + (3x – 2x + 3x) + (5 + 3 – 2)

= (1 + 3) x2 + (3 – 2 + 3) x + 6

=4x2 + 4x + 6

For every algebraic expression there exist another algebraic expression such that their sum is zero. These two expressions are called the additive inverse of each other.

Subtraction of algebraic expressions:

This is in two ways: (i) Column or Vertical method (ii) Row or Horizontal method.

Column or Vertical method:

Step –1: Write the expression in standard form if necessary.

Step –2: write one expression below the other such that the expression to be subtracted comes in the second row and the like terms come one below the other.

Step –3: Change the sign of every term of the expression in the second row to get the additive inverse of the expression.

Step –4: Add the like terms column-wise and write the result just below the concerned column.

Ex: Subtract: x2 + 3x + 5 from 3x2 + 4x – 3

Sol:

Row or Horizontal method:

Step –1: Write the expressions in one row with the expression to be subtracted in a bracket with assigning a negative sign to it.

Step –2:  Add the additive inverse of the second expression to the first expression.

Step –3: Group the like terms and add or subtract.

Step –4: Write the resultant expression in standard form.

Ex: Subtract: x2 + 3x + 5 from 3x2 + 4x – 3

Sol:   3x2 + 4x – 3 – (x2 + 3x + 5)

= 3x2 + 4x – 3 – x2 – 3x – 5

= (3 – 1) x2 + (4 – 3) x + (– 3 – 5)

= 2x2 + x – 8

## 11.EXPONENTS

We know that,

a × a = a2 (a raised to the power of 2)

a × a × a = a3 (a raised to the power of 3)

a × a × a × a × a × a ×…. m times = am

am is in exponential form

a is called base, m is called exponent or index.

Laws of exponents:

• am × an = am + n
• (am)n = amn
• am = an ⇒ m = n
• (ab)m = am.an
• a0 = 1

Standard form:  A number that is expressed as the product of the largest integer exponent of 10 and a decimal number between 1 and 10 is said to be in standard form.

Ex: 1324 in standard form is 1.324 × 103.

Quadrilateral: A Quadrilateral is a closed figure with four sides, four angles and four vertices.

• AB, BC, CD, and AD are sides.
• A, B, C and D are the vertices.
• ∠ABC, ∠BCD, ∠CDA and ∠DAC are the angles.

The line segment joining the opposite vertices of a quadrilateral are called the diagonals of the Quadrilateral. In the above figure AC, BD is the diagonals.

The two sides of a Quadrilateral that have a common vertex are called the adjacent sides of the Quadrilateral. From the above figure, AB, BC; BC, CD; CD, DA and DA, AB are the adjacent sides.

The two angles of a Quadrilateral that have a common side are called the adjacent angles of the Quadrilateral. From the above figure, ∠A, ∠B; ∠B, ∠C; ∠C, ∠D and ∠D, ∠A is the adjacent angles.

The two sides of a quadrilateral, which do not have a common vertex are called opposite sides of a quadrilateral. From the above figure, AB, CD; BC, DA are the opposite sides.

The two angles of a quadrilateral, which do not have a common side are called opposite angles of a quadrilateral. From the above figure, ∠A, ∠C; ∠B, ∠D are the opposite angles.

Interior and exterior of a Quadrilateral:

In a Quadrilateral ABCD, S, N are interior points, M, P are exterior points and A, B, C, D and Q are lies on the Quadrilateral.

A Quadrilateral is said to be a convex Quadrilateral if all line segments joining points in the interior of the Quadrilateral also lie in the interior of the Quadrilateral.

A Quadrilateral is said to be a concave Quadrilateral if all line segments joining points in the interior of the Quadrilateral not lie in the interior of the Quadrilateral.

Angle sum property of a quadrilateral:

The Sum of the angle in a Quadrilateral is 3600

In a Quadrilateral ABCD, ∠A + ∠B + ∠C + ∠D = 3600

1.Trapezium:

In a Quadrilateral, one pair of opposite sides are parallel then it is Trapezium.

In a Trapezium ABCD, AB∥ DC; AC, BD are diagonals.

2.Kite:

In a Quadrilateral two distinct consecutive pairs of sides are equal in length then it is called a Kite.

In a Kite ABCD, AB = BC; AD = DC AC, BD are diagonals.

3.Parallelogram:

In a Quadrilateral, two pairs of opposite sides are parallel then it is Parallelogram.

In a Parallelogram ABCD, AB∥ DC, AD∥ BC; AD, BD are diagonals.

Properties of parallelogram: –

• The opposite sides of a parallelogram are equal in length.
• The opposite angles are equal in measure.
• The sum of the adjacent angles is 1800
• Diagonals are bisected to each other and not equal in length.

4.Rhombus:

In a parallelogram in which two adjacent sides are equal, then it is a Rhombus.

In a Parallelogram ABCD, AB∥ DC, AD∥ BC; AD, BD are diagonals.

Properties of Rhombus: –

• All sides of a Rhombus are equal in length.
• The opposite angles are equal in measure.
• The sum of the adjacent angles is 1800
• Diagonals are bisected to each other perpendicularly and not equal in length.

5.Rectangle:

In a parallelogram all angles are equal, then it is a Rectangle.

Properties of Rectangle: –

• The opposite sides are equal in length.
• Each angle is 900.
• The sum of the adjacent angles is 1800
• Diagonals are bisected to each other and not equal in length.
• Each diagonal divides the rectangle into two congruent triangles.

6.Square:

In a rectangle adjacent sides are equal, then it is a Square.

Properties of Square: –

• All sides of a square are equal in length.
• Each angle is 900.
• The sum of the adjacent angles is 1800
• Diagonals are bisected to each other and equal in length.
• Each diagonal divides the square into two congruent triangles.

## 13.AREA AND PERIMETER

Area of a parallelogram:

Area of parallelogram (A) = b × h square units.

The area of the parallelogram is equal to the product of its base (b) and the height(h)

Area of a Triangle:

Area of triangle = ½ b × h square units.

The area of the triangle is equal to half the product of its base (b) and height (h).

In a Right-angled triangle, two of its sides can be the height.
Area of a Rhombus:

The area of the Rhombus is equal to half the product of its diagonals

Area of rhombus = ½ d1 × d2 square units.

Circumference of the circle:

Circumference of circle = 2πr = πd

Area of the rectangular path:

Area of Rectangular path = area of the outer rectangle – are of the inner rectangle

## 14. UNDERSTANDING 2D AND 3D SHAPES

Net: Net is a short of skeleton-outline in 2d, which when folded the result in 3d shape.

Nets of 3D shapes:

1.Cube:

2.Cylinder:

3.Pyramid:

Oblique Sketches:

Oblique sketches are drawn on a grid paper to visualise 3D shapes.

Ex: Draw an oblique sketch of a 3×3×3 cube

Step-1: Draw the front face

Step-2: Draw the opposite face, which is the same as the front face. The sketch is somewhat offset from Step-1

Step-3: Join the corresponding corners.

Step-4: Redraw using dotted lines for hidden edges.

Isometric Sketches:

Isometric sketches are drawn on a dot isometric paper to visualise 3D shapes.

Ex: Draw an oblique sketch of a 2×3×4 cuboid

Step-1: Draw a rectangle to show the front face.

Step-2:  Draw four parallel line segments of length 3cm.

Step-3: Connect the corresponding corners with appropriate line segments

Step-4: This is an isometric sketch of a cuboid

## 15.SYMMETRY

Line of symmetry: The line which divides a figure into two identical parts is called the line of symmetry or axis of symmetry.

An object can have one or more than one lines of symmetry.

Regular polygon:

If a polygon has equal sides and equal angles, then the polygon is called a Regular polygon.

Lines of symmetry for Regular polygons:

 Regular polygon No. of sides No. of axes of symmetry Triangle 3 3 Square 4 4 Pentagon 5 5 Polygon n n

Rotational symmetry: If we rotate a figure, about a fixed point by a certain angle and the figure looks the same as before, then the figure has rotational symmetry.

The angle of rotational symmetry:  The minimum angle of rotation of a figure to get the same figure as the original is called the angle of rotational symmetry or angle of rotation.

The angle of rotation of the equilateral triangle is 1200

The angle of rotation of a square is 900

Order of rotational symmetry:

The no. of times a figure, rotated through its angle of rotational symmetry before it comes to the original position is called the order of rotational symmetry.

The order of rotational symmetry for an equilateral triangle is 3.

The order of rotational symmetry for a square is 4.

Note: All figures have rotational symmetry of order 1, as can be rotated completely through 3600 to come back to its original position.

An object has rotational symmetry, only when the order of symmetry is more than 1.

• Some shapes have a line of symmetry and some have rotational symmetry and some have both.

Square, Equilateral triangle and Circle have both line and rotational symmetry.

## TS 8th Class Maths Concept

Studying maths in VIII class successfully meaning that children take responsibility for their own learning and learn to apply the concepts to solve problems.

This notes is designed by the ‘Basics in Maths team’. These notes to do help students fall in love with mathematics and overcome fear.

## 1. RATIONAL NUMBERS

• Natural numbers: All the counting numbers starting from 1 are called Natural numbers.

1, 2, 3… Etc.

• Whole numbers: Whole numbers are the collection of natural numbers.

0, 1, 2, 3 …

• Integers: integers are the collection of whole numbers and negative numbers.

….., -3, -2, -1, 0, 1, 2, 3….

• Rational numbers: The numbers which are written in the form of p/q, where p, q are integers and q ≠ 0 are called rational numbers. Rational numbers are denoted by Q.

Properties of Rational numbers

• Natural numbers:

1.Closure property:-

### • Rational numbers:

#### 3. Associative  property:-

1 + 0 = 0 + 1 = 1,   3/2 + 0 = 0 + 3/2 = 3/2

• For any rational number ‘a’, a + 0 = 0 + a

• 0 is the additive identity.

2 + (-2) = (-2) + 2 = 0,  5 + (-5) = (-5) + 5 = 0

• For any rational number ‘a’, a+ (-a) = (-a) + a = 0

• Additive inverse of a = -a and additive inverse of (-a) = a

Multiplicative identity:-

2 × 1 = 1 × 2 = 2,    6 ×1/6 = 1 × 1/6 = 1/6

• For any rational number ‘a’, a × 1 = 1 × a = a

• 1 is the multiplicative identity.

Multiplicative inverse:-

2 × 1/2 = 1/2 × 2 =1

For any rational number ‘a’,

a × 1/a = 1/a × a = 1

• multiplicative inverse of a =1/a

• Multiplicative inverse of  1/a= a.

Distributive property:-

For any three rational numbers  a, b and c,

a × (b + c) = (a × b) + (a × c)

3/2×(5/3+1/5)=(3/2×5/3)+(3/2×1/5)

Representing rational numbers on a number line:

Ex: represent 29/6 on a number line

this lies between 4 and 5

Divide the number line between 4 and 5 into 6 equal parts. Mark 5th part counting from 4.

The role of zero:

• If 0 is added to any rational number, then the rational number remains the same.

For any rational number ‘a’ a + 0 = a = 0 + a

• 0 is the additive identity.

• Natural numbers does not have additive identity.

for any rational number ‘a’

a + (-a) = 0 = (-a) + a

3 + (-3) = 0,   10 + (-10) = 0

• additive inverse of ‘a’ is ‘-a’ and additive inverse of  ‘-a’ is ‘a’

The role of 1:

• If 1 is multiplied to any rational number, then the rational number remains same.

For any rational number ‘a’ a × 1 = a = 1 × a

• 1 is the multiplicative identity.

Multiplicative inverse:-

3 × 1/3 = 1 = 1/3 × 3

for any natural number ‘a’

a × 1/a = 1 = 1/a ×a

• Multiplicative inverse of ‘a’ is ‘1/a’ and multiplicative inverse of ‘1/a’ is ‘a’

Distributive property:

For any 3 rational numbers a, b and c, a (b + c) = ab + ac

Ex:-  1/3 (2/5 + 1/5) = 1/3(3/5) = 3/15

1/3× 2/5 + 1/3 × 1/5 = 2/5 + 1/5 = 3/5

Inserting rational numbers between given two numbers:

• There infinitely many rational numbers between given two numbers.

• We have two methods to find rational numbers between two numbers.

First method: – First we have to convert given rational numbers as the same denominator and write the rational numbers which come between given numbers.

Second method: – if a and b any given rational numbers then a/bis a rational number between a and b.

Decimal representation of rational numbers

The decimal expansion of rational is either terminating or non-terminating repeating decimal.

Note:-Decimal numbers with the finite no. of digits is called terminating Decimal numbers with the infinite no. of digits is called non- terminating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non- terminating repeating decimals.

Terminating decimals:

Consider a rational number $inline&space;dpi{80}&space;bg_white&space;fn_cm&space;LARGE&space;frac{3}{4}$

$dpi{50}&space;bg_white&space;fn_cm&space;LARGE&space;therefore&space;:&space;frac{3}{4}&space;:&space;=&space;0.75$

Non-terminating decimals:

Consider a rational number $dpi{50}&space;bg_white&space;fn_cm&space;LARGE&space;frac{2}{3}$

$dpi{80}&space;bg_white&space;fn_cm&space;LARGE&space;frac{2}{3}=&space;0.666...$

$dpi{80}&space;bg_white&space;fn_cm&space;LARGE&space;^{therefore&space;}&space;frac{2}{3}:&space;=&space;0.bar{6}$

## 2. LINEAR EQUATIONS IN ONE VARIABLE

Equations: An algebraic equation is the equality of algebraic expressions involving variables and constants.

• It has an equality sign.
• The expression on the left of the equality sign is called the LHS (Left Hand Side) and right of the equality is called RHS (Right Hand Side) of the equation.
• In an equation, the value of RHS and LHS are equal. This happens to be true only for certain values of the variable. This value is called the solution of the equation.

Linear equations: If the degree of the equation is 1, then it is called a linear equation.

Ex:  2x – 3 = 5, x = 3y, 5x + 3y = 3 and so on.

Simple equations or linear equations in one variable: An equation of the form ax + b = 0 or ax = b where a, b are constants and a≠0is called a linear equation in one variable or simple equation.

Ex: 2x + 3 = 7, x = 3, 2 – 3x = – 1 and so on.

Note:  if we transpose terms from LHS to RHS or RHS to LHS

‘+’ quantity becomes ‘– ‘quantity

‘–’ quantity becomes ‘+ ‘quantity

‘×’ quantity becomes ‘÷ ‘quantity

‘÷’ quantity becomes ‘× ‘quantity

Solving simple equation having the variable on one side:

Ex: solve the equation 2x + 32 = 2

Sol:   2x = 2 – 32 (transpose 32 to RHS)

2x = – 30

x = – 30/2 (transposing to RHS)

∴ solution of 2x + 32 = 2 is – 10.

Solving simple equation having the variables on both sides:

Ex:  solve the equation 9y + 5 = 15y – 1

Sol: given equation is 9y + 5 = 15y – 1

9y – 15y = – 1 – 5 (Transposing 5 to RHS and 15y to LHS)

–6y = – 6

y = –6/–6 = 1

y = 1

∴ solution of the equation 9y + 5 = 15y – 1 is 1.

Method of cross Multiplication:

Multiply the numerator of the LHS by the denominator of the RHS and multiply the numerator of the RHS by the denominator of LHS. This method is called the cross multiplication method.

Reducing Equations to simpler form – Equations Reducible to Linear form:

Ex: solve the equation

Sol: given equation is

⇒ 7(5x + 2) = 12(2x + 3) (∵ by cross multiplication method)

⇒ 35x + 14 = 24x + 36

⇒ 35x – 24x = 36 – 14 (by transposing terms)

⇒ 11x = 22

∴ x = 2  is the solution of given equation.

It has 4 sides, 4 vertices, 4 angles and two diagonals.

1.Trapezium: A quadrilateral with at least one pair of parallel sides is called a trapezium.

Opposite sides are not equal and diagonals are not equal

2.Parallelogram: A quadrilateral with two pairs of opposite sides are parallel is called a parallelogram.

Opposite sides are parallel and equal

Opposite angles are equal

Diagonals are not equal.

Diagonals bisect each other.

3.Rectangle: A parallelogram with one of the angles 900 is called a rectangle.

Opposite sides are parallel and equal

Opposite angles are equal

Diagonals are equal.

Diagonals bisects each other.

4.Rhombus: A parallelogram with adjacent sides are equal is called rhombus.

All sides are equal

Opposite angles are equal

Diagonals are not equal.

Diagonals bisect each other and angle between diagonals is 900

5.Square: A rhombus with four right angles is called a square.

All sides are equal

Opposite angles are equal

Diagonals are equal.

Diagonals bisect each other and angle between diagonals is 900

6.Kite: A quadrilateral with two pairs of adjacent sides is called a kite.

We can draw quadrilaterals when the following measurements are given.

1. When 4 sides and one angle are given
2. When 4 sides and one diagonal are given
• When three sides and two diagonals are given
1. When two adjacent sides are given and three angles are given
2. When three side and two included angles are given

Type of Quadrilateral – No, of individual measurements:

 Type of quadrilateral No. of individual measurements Quadrilateral 5 Trapezium 4 Parallelogram 3 Rectangle 3 Rhombus 2 Square 1

Example 1:

Construct the quadrilateral PQRS with the measurements: PQ = 5.5cm, QR = 3.5 cm, RS= 4 cm, PS = 5 cm and ∠P = 450.

Steps of construction:

1. Construct a line segment PQ with a radius 5.5cm.
2. With the center, P draw a ray and an arc that are equal to 450 and 5 cm.
3. These intersecting points are kept as S.
4. With centers S, Q draws two arcs equal to Radius 4 cm, 3.5 cm respectively.
1. The intersecting point of these two arcs is kept as R.
2. Join Q, R, and S, R
3. Therefore, the required quadrilateral PQRS formed.

Example 2

Construct the parallelogram PQRS with the measurements: PQ = 4.5cm, QR = 3 cm and ∠PQR = 600

In parallelogram PQRS with the measurements: PQ = 4.5cm, QR = 3 cm and ∠PQR = 600

⇒ RS = 4.5cm, PS = 3 cm (in a parallelogram opposite sides are equal)

Steps of construction

1. Construct a line segment PQ with a radius 4.5cm.
2. With the center, Q draws a ray and an arc that are equal to 450 and 3 cm.
3. These intersecting points are kept as R.
4. With centers R, P draws two arcs equal to radius 4 cm, 3.5 cm respectively.
1. The intersecting point of these two arcs is kept as S.
2. Join P, S and R, S
3. Therefore, the required parallelogram PQRS formed.

## 4. EXPONENTS  AND POWERS

We know that a2 = a × a (two times)

a3 = a × a × a (three times)

⇒ a × a × a × a × a … m times = am

Here, am is called the exponent form.

• In exponent, form am, ‘a’ is base, ‘m’ is exponent, power, or index.
• We read am as a raised to the power of m.

Laws of exponents:

Express small numbers in Standard form by using exponents:

• If a number is expressed in the form of m ×10n where 1≤m<10, n is any integer, then that number is in standard form.
• Very small numbers can be expressed in standard form using negative exponents.

Ex:    express 0.0000456 in standard form

Sol : 0.0000456 = 456/10000000 = 456/107 = 456 × 10-7.

## 5. COMPARING QUANTITIES USING PROPORTION

Ratio: comparing two quantities of same kind by using division is called ratio.

Ratio of two quantities a and b is denoted by a: b.

Per cent: per cent means ‘per hundred’ or out of hundred’. The symbol % stands for percent.

Discount:

Marked price (M.P): – The price printed on an article by manufacturer is called marked price. It is also called as list price or usual price or catalogue price.

Discount: – Discount is the reduced marked price. It is generally given as percent of the marked price. Discount is always depending on the marked price.

Net price or selling price: – The difference between the M.P and discount is called net price or selling price.

Example:  A T.V is marked at ₹ 18000 and discount allowed on it is 10%. What is the amount of discount and its sale price?

Ans:  Given marked price = ₹18000, discount percentage = 10%

Now, discount = 10% 0f 18000 = $inline&space;fn_cm&space;emph{}frac{10}{100}times&space;18000$= 1800

Selling price = marked price – discount = 18000 – 1800 = 16200.

∴ selling price = ₹ 16,200.

Profit and loss:

Cost price (C.P.): – Cost price is the price for which an article is bought or the price paid by a customer to by an article.

Selling price (S.P.): – Selling price is the price for which an article is sold.

Profit: – If the Selling price is greater than the cost price, then we get the profit.

Profit = S.P – C.P.

Loss:If the Selling price is less than the Cost price then, we get loss.

Loss = C.P – S.P.

Some formulae in profit and loss:

For-Profit:

For Loss:

Government collects taxes on every sale. This is called VAT. Shop keeper collect this from the customers and pay it to the Govt.

VAT is changed on the Selling price of an item and will be included in the bill. VAT is an increase percent of selling price.

Example:

The cost of an article is ₹ 500. The sales tax is 5%. Find the bill amount.

Ans: cost price of an article = ₹500

% of Sales tax = 5

Sales tax paid = ₹

Bill amount = cost price + sales tax paid

= 500 + 25

= ₹ 525.

Simple interest:

Principal: – The money which is borrowed is called ‘principal’.

Rate of interest: – percentage of interest per year is called rate of interest.

Time: – The period for which money is called time.

Interest: – The money which is paid for the use of the principal is called interest.

Amount: – The total money which is paid after the expiry of the time is called amount.

Compound Interest: Compound interest allows us to earn interest on interest.

• The time period after which interest is added to principal is called conversion period. When interest is compounded h yearly, there are two conversion periods in a year. In such case, alf year rate will be half of the annual rate.

## 6. SQUARE ROOTS AND CUBE ROOTS

Square:  Square number is the number raised to the power 2. The number obtained by the number multiplied by itself.

• If a natural number p can be expressed as q2, where q is also natural, then p is called a square number.

Ex: – 1) square of 9 = 92 = 9× 9 = 18, 2) square of 4 = 42 = 4× 4

Perfect Square:  A natural number is called a perfect square if it is the square of some natural number.

Ex: – 1,4,9, …etc.

Properties of perfect square:

1. The square of an even numbers is always an even number.

Ex: – 22 = 4 (4 is even), 62 = 36 (36 is even), here 2, 6 are an even number.

1. The square of an odd number is always an add number.

Ex: – 32 = 9 (9 is even), 152 = 225 (225 is even), here 3, 15 are an odd number.

• The square of a proper fraction is as proper fraction less than the given fraction.

Ex: –

1. The square of decimal fraction less than 1 is smaller than the given decimal.

Ex: – (0.3)2 = 0.09 < 0.03.

1. A number ending with 2, 3, 7 or 8 is never a perfect square.

Ex: – 72, 58, 23 are not perfect squares.

1. A number ending with an odd no. of zeros is never a perfect square

Ex: – 20, 120,1000 and so on.

Patterns in square numbers:

1. 1 + 3 = 4 = 22

1 + 3+ 5 = 9 = 32

1 + 3 + 5 +7 = 16 = 42

…………………………….

⇒ sum of n odd natural numbers = n2

1. Difference between two consecutive square numbers:

22 − 11 =4 −1 = 3 = 2 + 1

32 − 21 =9 −4 = 5 = 3 + 2

42 − 31 =16 −9 = 7 = 3 + 4

⇒ for any natural number ‘m’, (m + 1)2 – m2 = (m+1) + m

1. Pythagorean triplet:

Three natural numbers a, b and c are said to form a Pythagorean triplet if, c2 = a2 + b2

For every natural number a > 1, (2a, a2 – 1, a2 + 1).

Ex: – if we put a = 3 in (2a, a2 – 1, a2 + 1), then we get Pythagorean triplet (6, 8, 10).

1. Between two consecutive square numbers m2 and (m + 1)2, there are 2m non-perfect square numbers.

Ex: – 22, 32 are two consecutive square numbers

Non-perfect square numbers between 22 and 32 are:5, 6, 7, and 8

⇒ 2(2) = 4 Non-perfect square numbers are there in between 22 and 32

1. Using the identities (a + b)2 = a2 + 2ab + b2, (a – b)2 = a2 – 2ab + b2 to evaluate square numbers.

Ex: – 122 = (10 + 2) 2 = 102 + 2 (10) (2) + 22 = 100 + 400 + 4 = 144

92 = (10 – 1)2 = 102 – 2 (10) (1) + 12 = 100 – 20 + 1 = 81

1. Using the identity (a – b) (a + b) = a2 – b2 to find the product of two consecutive odd or two consecutive even numbers.

Ex: – 9 × 11 = (10 – 1) (10 + 1) = 102 – 1 = 99

20 × 22 = (21 – 1) (21 + 1) = 212 – 1 = 441 – 1 = 440.

Square Root: the square root of a number x is that number when multiplied by itself gives x as the product. The square root of x is denoted by $sqrt{x}$.

Ex: –

Methods of Finding Square root of given Number

Prime factorization method: –

Steps:

1. Resolve the given number into prime factors.
2. Make pairs of similar factors.
3. The product of prime factors, choosing one out of every pair gives the square root of the given number.

Ex: – 16

Prim factors of 16 = 2 ×2× 2× 2

= 2 × 2 = 4

∴ square root of 16 = 4

Division method: –

Steps:

1. Mark off the digits in pairs starting with the unit place. Each pair and remaining one digit are called a period.
2. Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient.
3. Subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. this becomes the new dividend.
4. Now, a new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.
5. Repeat steps 2, 3, and 4 till all the periods have been taken up. Thus, the obtained quotient is the required square root.

Finding the square root through subtraction of successive odd numbers:

• Subtract first odd number (1) from a given number
• Subtract the second odd number (3) from the above result.
• Continue this process until the result will be zero (0).
• Count the steps involved above the process. No. of steps is the required answer.

Ex: find square root of 16

16 – 1 = 15; 15 – 3 = 12; 12 – 5 = 7; 7 – 7 = 0

After 4 steps we got 0.

∴ square root of 16 = 4.

The square root of a number in decimal form

Make the no. of decimal places even, by affixing a zero, if necessary. Now periods and find out the square root by the long division method.

Put the decimal point in the square root as soon as the integral part is exhausted.

Ex: – To find the square root of 79.21

The square root of a decimal number which is not perfect square:

if the square root is required to correct up to two places of decimal, we shall find it up 3 places of decimal and then round it off up to two decimal places.

if the square root is required to correct up to three places of decimal, we shall find it up 4 places of decimal and then round it off up to three decimal places.

Ex: – To find the square root of 0.8 up to 2 decimal places

Cube of a number:

The cube of a number is that number raised to the power 3.

Ex: – cube of 0.3 = 0.33 = 0.027

Cube of 2 = 23 = 8

Perfect cube:

If a number is a perfect cube, then it can be written as the cube of some natural numbers.

Ex: – 1, 8, 27, and so on.

Cube root:

The cube root of a number x is that number which when multiplied by itself three times gives x as the product.

Cube root of x is denoted by

Methods of finding the cube root of given Number

Prime factorization method: –

Steps:

1. Resolve the given number into prime factors.
2. Make triplets of similar factors.
3. The product of prime factors, choosing one out of every triplet gives the cube root of the given number.

Ex: – 27

Prim factors of 27 = 3 ×3×3

= 3

∴ cube root of 27 = 3

Estimating the cube root of a number:

Ex:  estimate the cube root of 2744

Start making groups of through estimation. The first group is 744 and the second group is 2

2      744

The first group i.e., 744will give us the units digit of the cube root. As 744 ends with 4, its cube root also ends with 4. So, the unit place of cube root will be 4.

In second group number is 2

We know that 13 < 2 < 23

As the smallest number is 1, t becomes the tens place of the required cube root.

∴  cube root of 2744 = 14.

## 7. FREQUENCY DISTRIBUTION TABLES AND GRAPHS

Data: An information available in the numerical form or verbal form or graphical form that helps in taking decisions or drawing conclusions is called data.

Measures of central tendency:

The measures of central tendency are 3 types. They are: 1. Arithmetic mean 2.  Median and 3.  Mode.

1.Arithmetic Mean:

Arithmetic mean of x1, x2, x3, …. x n is   ⇒

Where ∑xi represents the sum of all xi ’s; ‘i’ takes the values from 1 to n.

Arithmetic mean by deviation Method: –

A is assumed mean.

∎Sum of the deviations of all observations from the estimated mean is zero.

∎Arithmetic mean is a representative value of the entire data.

∎Arithmetic mean depends on both no. of observations and value of each observation in a data.

∎Arithmetic mean is unique value of data.

∎When all the observations of the data are increased or decreased by a certain number, the mean also increase or decrease by the same number.

∎When all the observations of the data are multiplied or divided by a certain number, the mean also multiplied or divided by the same number.

2.Median:

Median is the middle most value of the given data.

First, we arrange given data in ascending or descending order.

If n is the no. of observation in a data, then

Median =  observation, when n is odd.

Median =   when n is even.

∎Median is the middle most observation of the data.

∎It depends on no. of observations and middle observations of the ordered data

∎ It not effected by any change in extreme values.

3.Mode:

Mode is the most frequently occurring observation of given data.

∎Mode depends neither on no. of observations nor value of all observations.

∎It is used to analyse both numerical and verbal data.

∎ There may be two or three or many modes for the same data.

Grouped data:

If we organize the data by dividing it into convenient groups, then it is called Grouped data.

Frequency distribution or Frequency table:

Representation of classified distinct observations of the data with frequencies is called frequency distribution.

Class intervals: Small groups in a data are called Class intervals

Ex: 0 – 5, 5- 10, …

Limits and boundaries:

In the class interval 5 – 10, 5 is called lower limit and 10 is called upper limit.

Boundaries: Average of upper limit of first class and lower limit of second class is becomes the upper boundary first class and lower boundary of second class.

These boundaries are also called ‘true class limits’

Length of the class: Difference between upper and lower boundaries of a class is called Length of the class.

From the above table length of the class 0.5 – 10.5 is 10.5 – 0.5 = 10

Range: The difference between highest and least value of given data is called range of the data.

Construction of grouped frequency Distribution:

Ex: 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7

Highest value = 7; least value = 1

Range = 7 – 1 = 6.

Class interval =

=  = 0.8 approximately.

Characteristics of grouped frequency distribution:

1. It divide the data onto convenient and small groups called class intervals.1.
2. Class intervals like 1 – 10, 11 – 20 … are called inclusive class intervals. Both lower and upper limits of a particular class belong to that particular class.
3. Class intervals like 0 – 10, 10 – 20 … are called exclusive class intervals. Only lower limit of particular class belongs to that class but not its upper limit.
4. In exclusive class intervals, both limits and boundaries are equal.
5. In inclusive class intervals, limits and boundaries are not equal.
6. Individual values of all observations can not be identified from the frequency table, but the value of each observation of a particular class assumed to be the average of upper and lower boundaries of that class. This value is called ‘class mark’ or ‘mid value’.

Less than and greater than Cumulative frequencies:

The distribution that represents upper boundaries of the classes and their respective less than cumulative frequencies is called ‘less than cumulative frequency distribution’

The distribution that represents lower boundaries of the classes and their respective greater than cumulative frequencies is called ‘greater than cumulative frequency distribution’

 Class intervals frequency LB Greater than cumulative frequency UB Less than cumulative frequency 0 – 5 7 0 36 + 7 = 43 5 7 5 – 10 10 5 26 + 10 = 36 10 7+10 = 17 10 – 15 15 10 11 + 15 = 26 15 17 + 15 = 32 15 – 20 8 15 3 + 8 = 11 20 32 + 8 = 40 20 – 25 3 20 3 25 40 + 3 = 43

Graphical representation of the Data:

Bar graph:

A display of information using vertical or horizontal bars of uniform width and different lengths being proportional to the respective values is called bar graph.

Ex:

Histogram:

A graphical representation of frequency distribution of exclusive class intervals is called histogram.

Ex:

Steps on Construction:

Step-1:  If class intervals are inclusive, then convert them into the exclusive form

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 10 units

Step-4: Draw rectangle with class intervals as bases and the corresponding frequencies as the corresponding heights.

Histogram with varying Base widths:

Ex:

Steps on Construction:

Step-1:  If class intervals are inclusive, then convert them into the exclusive form

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 10 units

Step-4: Draw rectangle with class intervals as bases and the corresponding frequencies as the corresponding heights.

Frequency polygon:

Ex:

Steps on Construction:

Step-1:  Calculate the mid points of every class interval given in the data.

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 10 units

Step-4: Draw the histogram for this data and mark the midpoints of the tope

Step-5: Join the mid points successfully.

Frequency curve:

Ex:

Steps on Construction:

Step-1:  find the class mark of the class intervals.

Step-2: Choose a suitable scale on the X – axis and mark the class intervals on it.

Step-3: Choose a suitable scale on the Y – axis and mark the frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 2 units

Step-4:  Plot the points (which are in the above table) on graph

Step-5: Join the consecutive points by a free hand curve.

Less than cumulative frequency curve:

Ex:

 Class Frequency UB L. C. F 0 – 10 2 10 2 10 – 20 5 20 7 20 – 30 3 30 10 30 – 40 1 40 11 40 – 50 4 50 15 50 – 60 2 60 17

Steps on Construction:

Step-1:  If class intervals are inclusive, then convert them into the exclusive form

Step-2: Construct the less than cumulative frequency table.

Step-3: Choose a suitable scale on the X – axis and mark the upper boundaries class intervals on it.

Choose a suitable scale on the Y – axis and mark the cumulative frequencies on it.

Scale: On X – axis 1cm = 10 units

On Y – axis 1 cm = 2 units

Step-4:  If ‘x’ denotes the upper boundary of class interval and ‘y’ denotes the corresponding cumulative frequency of particular class, then plot (x, y) on the graph.

## 8.EXPLORING GEOMETRICAL FIGURES

Congruent figures: The figures which have same shape and size are called congruent figures.

Flip: Flip is a transformation in which a plane figure is reflected across a line, creating a mirror image of the original figure.

Rotation: Turning round centre is called Rotation. The distance from the centre to any point on the shape stays the same. Every point makes a circular round the centre.

Similar figures: The figures which have same shape but not in size are called similar figures.

Dilation: The method of drawing enlarged or reduced similar figures is called Dilation.

Constructing Dilation:

Ex: Construct a dilation with scale factor 3, of a triangle

Steps on Construction:

Step-1:  Draw a ∆ABC and choose the centre of dilation O which is not on the triangle. Join every vertex of the triangle from O and produce

Step-2: By using compasses, mark three points A’, B’, C’ on the projection so that OA’ = 3OA; OB’ = 3OB and OC’ = 3OC.

Step-3:  Join A’B’, B’C’, C’A’. We observe that ∆ABC~∆A’B’C’

Symmetry:

In symmetry there are 3 types: 1. Line of symmetry, 2. Rotational symmetry and 3. Point symmetry.

1.Line of symmetry: The lines which cuts the figures exactly halves is called line of symmetry.

2.Rotational symmetry:

When an object is rotated about its centre, it comes same position after some rotation, then it is called rotational symmetry.

No. of rotations to get initial position of an object is called ‘order of position.

Ex: When a rectangle is rotated about its centre its shape resembles the initial position two times.

Order of rotation of rectangle is 2.

3.Point symmetry:

The figure looks the same either we see it from upside or it from down side is called ‘point of symmetry.

Ex: H, S, I have point of symmetry.

Tessellations: The patterns formed by repeating figures to fill a plane without gaps or overlaps are called ‘Tessellations’.

## 9.AREA OF PLANE FIGURES

Area of triangle:

The area of triangle with base ‘b’ and height ‘h’ is  square units

Area of Rectangle:

The area of rectangle with breadth ‘b’ and length ‘l’ is l × b square units.

Area of Square:

The area of triangle with side ‘a’ is a × a = a2 square units.
Area of parallelogram:

The area of parallelogram with base ‘b’ and height ‘h’ is b × h square units.

Area of Rhombus:

The area of Rhombus with lengths of diagonal d1, d2 is square units.

Area of Trapezium:

The area of Trapezium whose lengths of parallel sides a, b and distance between the parallel side’s ‘h’ is    square units.

The area of Quadrilateral whose lengths of perpendiculars drawn from vertices to diagonal are h1, h2 and length of the diagonal ‘a’ is     square units.

.Area of circle:

The area of circle with radius ‘r’ is π r
2 square units.

Area of circular path or area of Ring:

Area of ring = area of outer circle – area of inner circle

= πR 2 – π r2

= π(R 2 –  r2) square units.

Length of the arc:

Length of the arc of a sector (l) is

Area of sector:

area of sector
=

## 10.DIRECT AND INVERSE PROPORTIONS

Proportion: If a : b = c : d, then a, b, c and d are in proportion.

Direct proportion:

x and y are any two quantities are said to be in direct proportion, if x is increase (decrease), then y is increase (decrease).

where k is any constant

If x1 and x2 are the values of x corresponding to the values of y1 and y2 of y respectively, then

Inverse proportion:

x and y are any two quantities are said to be in inverse proportion, if x is increase (decrease), then y is decrease (increase).

xy = k where k is any constant

If x1 and x2 are the values of x corresponding to the values of y1 and y2 of y respectively, then

⇒ x1y1 = x2y2.

Compound proportion:

Change in one quantity depends upon the change in two or more quantities in some proportion, then we equate the ratio of the first quantity to the compound ratio of the other two quantities.

• One quantity may be in direct proportion with the other two quantities.
• One quantity may be in inverse proportion with the other two quantities.
• One quantity may be in direct proportion with one of the two quantities and inverse proportion to with the remaining quantity.

## 11.ALGEBRAIC EXPRESSIONS

Term:  Term is the product of constant and one or more variables.

Ex: 2x, 3xy. 5x2yz etc.

Algebraic Expression: Terms are added or subtracted to form an Algebraic Expression.

Ex; 2x + 3, 2y – 3x, 4xyz – 3x3y etc.

Monomial: If an expression contains only one term then it is called monomial.

Ex; x, 3x, – 5yz

Binomial: If an expression contains two terms, then it is called Binomial.

Ex: x + 3, x – y, 3xy – 2zx etc.

Trinomial: If an expression contains three terms, then it is called Trinomial.

Ex: x + 3 – y, x + 3xy – y, 3xy – 2z + x etc.

Like and Unlike terms: If the terms have same variable with same exponent then they are called Like terms, other wise they are called Unlike terms.

Ex: 2xy, 5yx, – 4xy are like terms

2xy, 5yz, 6zx are Unlike terms.

Ex:  Add 4x2 – 3xy + 2y2 and x2 + xy – 6y2

Subtraction of algebraic expressions:
Ex:
Subtract x2 – 2xy + 3y2 from 5x2 + 6xy – y2

Multiplication of Algebraic expressions:

For finding the product of algebraic terms we add the power of same base variables.

1.Multiplying two monomials: –

Ex: 3 × x = x + x + x = 3x.

5x × 3y = (5 × 3) × (x × y) = 15 × xy = 15xy

5x × 3x = (5 × 3) × (x × x) = 15 × x2 = 15x2(5 × 3) × (x × y) = 15 × xy = 15xy

2.Multiplying three or more monomials: –

Ex: 3 × x × y= 3xy.

5x × 3y × 4z = (5 × 3 × 4) × (x × y × z) = 60 × xyz = 60xyz

3x2 × (– 4x) × 2x3 × 2 = (3 × – 4 × 2 × 2) × (x2 × x × x3) = – 48 x6

3.Multiplying a binomial by a monomial: –

Ex: 5x (3x – 4y) = (5x × 3x) + (5x × – 4y) = 15x2 – 20xy

4.Multiplying a Trinomial by a monomial: –

Ex: 5x (3x – 4y + 4z) = (5x × 3x) + (5x × – 4y) + (5x × 4z) = 15x2 – 20xy + 20 xz

5.Multiplying a Binomial by a Binomial: –

Ex: (x + y) (2x – 3y) = x (2x – 3y) + y (2x – 3y) = 2x2 – 6 xy + 2xy – 3 y2 = 2x2 – 4xy – 3y2

6.Multiplying a Binomial by a Trinomial: –

Ex: (x + y) (2x – 3y + z) = x (2x – 3y + z) + y (2x – 3y + z)

= 2x2 – 6 xy + xz + 2xy – 3 y2 + yz

= 2x2 – 4xy – 3y2 + xz + yz

Identity:  An equation is called an identity if it is satisfied by any value that replaces its variables. An equation is true for certain values for the variable in it, where as an identity is true for all its variables. There fore it is known as universally true equation.

Symbol for identity is denoted by ‘≡’ (read as identically equal to)

Some important identities:

• (a +b)2 ≡ a2 + 2ab + b2
• (a – b)2 ≡ a2 – 2ab + b2
• (a + b) (a – b) ≡ a2 – b2
• (a + b + c)2 ≡ a2 + b2 + c2 + 2ab + 2bc + 2ca
• (x + a) (x + b) ≡ x2 + (a + b) x + ab.

Geometrical verification of (a +b)2 ≡ a2 + 2ab + b2

Consider a square with side a + b

Area of square = (a + b)2

Procedure:

•Divide the square into four regions as shown in the figure.

•It consists of two squares with side ‘a’ and side ‘b’ respectively and two rectangles with length and breadth as ‘a’ and ‘b’ respectively.

•The area of given square is equal to sum of the areas of four regions.

⇒ Area of square = area of square with side a + area of square with side b + area of rectangle with sides a and b + area of the rectangle with sides and b

⇒ (a + b)2 = a2 + b2 + ab + ba

(a + b) 2 = a2 + 2ab + b2

∴ (a +b)2 ≡ a2 + 2ab + b2

Geometrical verification of (a – b)2 ≡ a2 – 2ab + b2

Consider the square with side ‘a’

The square is divided into four regions I, II, III and IV

Area of square = area of region I + area of region II + area of region III + area of region IV

a2 = b (a – b) + b2 + b (a – b) + (a – b)2

a2 = ab – b2 + b2 + ab – b2 + (a – b)2

a2 = ab + ab – b2 + (a – b)2

⇒ (a – b)2 = a2 – ab – ab + b2

(a – b)2 = a2 – 2ab + b2

Geometrical verification of (a + b) (a – b) ≡ a2 – b2

Consider the square with side ‘a’

Remove the square from this whose side is ‘b’ units, we get

a2 – b2 = area of region I + area of region II

= a (a – b) + b (a – b)

= (a – b) (a + b)

∴ (a + b) (a – b) ≡ a2 – b2

## 12.FACFTORISATION

Factorisation:

The process of writing given expression as a product of its factors is called ‘Factorisation’.

It is helped to write the algebraic expressions in simpler form.

Irreducible factor:

A factor which can not be further expressed as product of factors is an irreducible factor.

Factorisation by Method of common factors:

Ex: Factorise 3x + 15

3x + 15 = (3 × x) + (3 ×5) (writing each term as the product of irreducible factors)

3 is the common factor of both terms

Take 3 as the common

3x + 15 = 3 × (x + 5) = 3 (x + 5)

Factorisation by grouping the terms:

Ex: Factorise ax + by + ay + bx

Firs group the like terms

ax + by + ay + bx = (ax + bx) + (ay + by)

= x (a + b) + y (a + b) (by taking out common factor from each term)

= (a + b) (x + y) (by taking out common factor from each term)

Factorisation by using identities:

• (a +b)2 ≡ a2 + 2ab + b2
• (a – b)2 ≡ a2 – 2ab + b2
• (a + b) (a – b) ≡ a2 – b2 are the algebraic identities.

Example 1:

Factorise x2 + 4x + 4

Sol: x2 + 4x + 4 = x2 + 2 (2)(x) + (2)2

It is in the form of identity (a + b)2 = a2 + 2ab + b2

x2 + 4x + 4 = (x + 2)2 = (x + 2) (x + 2).

Example 2:

Factorise x2 – 4x + 4

Sol: x2 – 4x + 4= x2 –2 (2)(x) + (2)2

It is in the form of identity (a – b)2 = a2 – 2ab + b2

x2 + 4x + 4 = = x2 –2 (2)(x) + (2)2 =(x – 2)2 = (x – 2) (x –2).

Example 3:

Factorise 4x2 – 9y2

Sol: 4x2 – 9y2 = (2x)2 – (3y)2

It is in the form of identity (a – b) (a + b) = a2 – b2

4x2 – 9y2 = (2x)2 – (3y)2 = (2x – 3y) (2x + 3y).

Factors of the form (x + a) (x + b) = x2 + (a + b) x + ab:

Ex: x2 + 12x + 35

Here we have to find out factors 35 whose sum is 12

35 = 1 × 35               1 + 35 = 36

–1 × –35           –1 –35 = –36

7 × 5                 7 + 5 = 12

–7 × –5              –7 –5 = – 12

Now x2 + 12x + 35 = x2 + (7 + 5) x + 35

= x2 + 7x + 5x + 35

= x (x + 7) + 5 (x + 7)

= (x + 7) (x + 5)

Division of algebraic Expression:

1.Division of a monomial by another monomial:

Ex: 12x5 ÷ 3x

12x5 ÷ 3x =  =

= 4x4

2.Division of an expression by a monomial:

Ex: 4x3 + 10 x2 + 8x ÷ 2x

4x3 + 10 x2 + 8x = 2 × 2 × x × x × x + 2 × 5 × x × x + 2 × 2× 2 × x

= (2x) (2x2) + (2x) (5x) + (2x) (4)

= (2x) (2x3 + 5x + 4)

4x3 + 10 x2 + 8x ÷ 2x =

=

= 2x2 + 5x + 4

3.Division of an Expression by Expression:

Ex: (5x2 + 15x) ÷ (x + 3)

5x2 + 15x = 5x (x + 3)

(5x2 + 15x) ÷ (x + 3) =

=

= 5x

## 13.VISUALIZING 3-D IN 2-D

Various Geometrical Solids:

Some solids (3 – D objects) have flat faces and some solids have curved faces.

Polyhedron: 3 – D objects which have flat surfaces are called polyhedron.

Ex: book, dice, cube etc.

Non – Polyhedron: 3 – D objects which have curved faces are called Non – polyhedron.

Ex: ball, pipe etc.

Faces, Edges, and Vertices of 3D – objects:

Regular polyhedron:

The polyhedron, which has congruent faces, equal edges and vertices are formed by equal no. of edges is called regular polyhedron.

Ex: Cube, Tetrahedron.

Prism: The soiled object with two parallel and congruent polygonal faces and lateral faces as rectangles or parallelograms is called a prism.

If the base of the prism is triangle, then it is called triangular prism.

If the base of the prism is square, then it is called square prism.

If the base of the prism is pentagon, then it is called pentagonal prism.

Euler’s Relation (Formula):

E + 2 = F + V

Where E = No. of edges;

F = No. of faces and

V = No. of vertices

Net diagrams:

A net is a short of skeleton – outline in 2 – D, which, when folded the net results in 3 – D shape.

Ex:

Tetrahedron

Cube

## 14.SURFACE AREAS AND VOLUMES

Cuboid:

Lateral surface area (L.S.A) = 2h (l + b) square units.

Total surface area (T.S.A) = 2 (lb + bh + hl) square units.

Volume = lbh cubic units.

Cube:

Lateral surface area (L.S.A) = 4 a2 square units.

Total surface area (T.S.A) = 6 a2 square units.

Volume = a3 cubic units.

We measure volume of liquids in millilitres(ml) or litres(l)

1cm3 = 1 ml.

1000 cm3 = 1l.

1m3 = 1000000cm3 = 1000 l. = 1 kl. (kilo litre).

## 15.PLAYING WITH NUMBERS

Divisibility:

If a number ‘a’ divides another number ‘b’ completely, then ‘b’ is divisible by ‘a’.

Place value of digit:

Place value of 7 is 7 000000.

Place value of 6 is 6000

Place value of 3 is 30

Divisibility Rules:

Divisibility rule by 2: –

If the unit place of a given number is 0, 2, 4, 6, 8 then that number is divisible by 2.

Ex: 10, 12, 526 etc.

Divisibility rule by 3: –

If the sum of the digits of a given number is divisible by 3, then that number is divisible by 3.

Ex: 234

Sum of the digits = 2 + 3 + 4 = 9

9 is divisible 3

∴ 234 is divisible by 3

Divisibility rule by 4: –

If the last two digits of a given number is divisible by 4, then that number is divisible by 4.

Ex: 324

24 is divisible by 4

∴ 324 is divisible by 4

∴ 324 is divisible by 4

Divisibility rule by 5: –

If the units place of given number is 0 or 5, then it is divisible by 5.

Ex: 10, 15, 235, 480 etc.

Divisibility rule by 6: –

If a number is divisible by both 3 and 2 then that number is divisible by 6.

Ex: 324

324 is divisible by both 3 and 2

∴ 324 is divisible by 6

Divisibility rule by 7: –

Fist multiply the last digit of given number by 2,

subtract this result from the number formed by remaining digits of given number.

If that result is divisible by 7, then the given number is divisible by 7.

Ex: 112

Last digit is 2 ⇒ 2 × 2 = 4

Now 11 – 4 = 7

7 is divisible by 7

∴ 112 is divisible by 7.

Divisibility by 8: –

if the last three digits of a number is divisible by 8, then that number is divisible by 8.

Ex: – 4232, last three digits 232 are divisible by 8

∴ 4232 is divisible by 8.

Divisibility by 9: –

if the sum of the digits of a number is divisible by 9, then that number is divisible by 9.

Ex: – 459, 4 + 5 + 9 = 18 → 18 is divisible by 9       ∴ 459 is divisible by 9

532, 5 + 3 + 2 = 10 → 10 is not divisible by 9       ∴ 532 is not divisible by 9.

Divisibility by 10: –

a number is divisible by 10, if its once place is 0.

Ex: – 20 is divisible by 10. 22, 45 are not divisible by 10.

Divisibility by 11: –

A number is divisible by 11, if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or 11.

Ex: – 6545

Sum of the digits at odd places = 5 + 5 = 10

Sum of the digits at even places = 4 + 6 = 10

Now difference is 10 – 10 = 0

∴ 6545 is divisible by 11.