Topics
Rational Numbers
- Rational Numbers
- Closure Property of Rational Numbers
- Commutative Property of Rational Numbers
- Associative Property of Rational Numbers
- Distributive Property of Multiplication Over Addition for Rational Numbers
- Identity of Addition and Multiplication of Rational Numbers
- Negative Or Additive Inverse of Rational Numbers
- Concept of Reciprocal or Multiplicative Inverse
- Rational Numbers on a Number Line
- Rational Numbers Between Two Rational Numbers
Linear Equations in One Variable
- Variable of Equation
- Concept of Equation
- Expressions with Variables
- Balancing an Equation
- The Solution of an Equation
- Linear Equation in One Variable
- Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Some Applications Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Solving Equations Having the Variable on Both Sides
- Some More Applications on the Basis of Solving Equations Having the Variable on Both Sides
- Reducing Equations to Simpler Form
- Equations Reducible to the Linear Form
Understanding Quadrilaterals
- Concept of Curves
- Different Types of Curves - Closed Curve, Open Curve, Simple Curve.
- Concept of Polygons
- Classification of Polygons
- Properties of a Quadrilateral
- Interior Angles of a Polygon
- Exterior Angles of a Polygon and Its Property
- Concept of Quadrilaterals
- Properties of Trapezium
- Properties of Kite
- Properties of a Parallelogram
- Properties of Rhombus
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Property: The adjacent angles in a parallelogram are supplementary.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Property: The diagonals of a rhombus are perpendicular bisectors of one another.
- Property: The Diagonals of a Rectangle Are of Equal Length.
- Properties of Rectangle
- Properties of a Square
- Property: The diagonals of a square are perpendicular bisectors of each other.
Practical Geometry
- Introduction to Geometric Tool
- Constructing a Quadrilateral When the Lengths of Four Sides and a Diagonal Are Given
- Constructing a Quadrilateral When Two Diagonals and Three Sides Are Given
- Constructing a Quadrilateral When Two Adjacent Sides and Three Angles Are Known
- Constructing a Quadrilateral When Three Sides and Two Included Angles Are Given
- Some Special Cases
Data Handling
- Concept of Data Handling
- Interpretation of a Pictograph
- Interpretation of Bar Graphs
- Drawing a Bar Graph
- Interpretation of a Double Bar Graph
- Drawing a Double Bar Graph
- Organisation of Data
- Frequency Distribution Table
- Graphical Representation of Data as Histograms
- Concept of Pie Graph (Or a Circle-graph)
- Interpretation of Pie Diagram
- Chance and Probability - Chance
- Basic Ideas of Probability
Squares and Square Roots
- Concept of Square Number
- Properties of Square Numbers
- Some More Interesting Patterns of Square Number
- Finding the Square of a Number
- Concept of Square Roots
- Finding Square Root Through Repeated Subtraction
- Finding Square Root Through Prime Factorisation
- Finding Square Root by Division Method
- Square Root of Decimal Numbers
- Estimating Square Root
Cubes and Cube Roots
Comparing Quantities
- Concept of Ratio
- Basic Concept of Percentage
- Increase Or Decrease as Percent
- Concept of Discount
- Estimation in Percentages
- Basic Concepts of Profit and Loss
- Sales Tax, Value Added Tax, and Good and Services Tax
- Calculation of Interest
- Concept of Compound Interest
- Deducing a Formula for Compound Interest
- Rate Compounded Annually Or Half Yearly (Semi Annually)
- Applications of Compound Interest Formula
Algebraic Expressions and Identities
- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
- Like and Unlike Terms
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Multiplication of Algebraic Expressions
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Binomial
- Multiplying a Monomial by a Trinomial
- Multiplying a Binomial by a Binomial
- Multiplying a Binomial by a Trinomial
- Concept of Identity
- Expansion of (a + b)2 = a2 + 2ab + b2
- Expansion of (a - b)2 = a2 - 2ab + b2
- Expansion of (a + b)(a - b) = a2-b2
- Expansion of (x + a)(x + b)
Mensuration
Visualizing Solid Shapes
Exponents and Powers
Direct and Inverse Proportions
Factorization
- Factors and Multiples
- Factorising Algebraic Expressions
- Factorisation by Taking Out Common Factors
- Factorisation by Regrouping Terms
- Factorisation Using Identities
- Factors of the Form (x + a)(x + b)
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Dividing a Polynomial by a Polynomial
- Concept of Find the Error
Introduction to Graphs
- Concept of Bar Graph
- Interpretation of Bar Graphs
- Drawing a Bar Graph
- Concept of Double Bar Graph
- Interpretation of a Double Bar Graph
- Drawing a Double Bar Graph
- Concept of Pie Graph (Or a Circle-graph)
- Graphical Representation of Data as Histograms
- Concept of a Line Graph
- Linear Graphs
- Some Application of Linear Graphs
Playing with Numbers
Notes
Associative Property of Rational Numbers:
1. Associativity of Addition of Rational Numbers:
We have,
`(- 2)/3 + [3/5 + ((-5)/6)] = (-2)/3 + ((-7)/30) = (-27)/30 = (-9)/10`.
`[(-2)/3 + 3/5] + ((-5)/6) = (-1)/15 + ((-5)/6) = (-27)/30 = (-9)/10`.
`(-2)/3 + [3/5 + ((-5)/6)] = [(-2)/3 + 3/5] + ((-5)/6)`
`(-1)/2 + [3/7 + ((-4)/3)] and [(-1)/2 + 3/7] + ((-4)/3)`
If the two sums are equal. We find that addition is associative for rational numbers. That is, for any three rational numbers a, b and c, a + (b + c) = (a + b) + c.
2. Associativity of Subtraction of Rational Numbers:
You already know that subtraction is not associative for integers, then what about rational numbers.
`(- 2)/3 - [(- 4)/5 - 1/2] = (- 2)/3 - [(- 8 – 5)/10] = (-2)/3 + 13/10 = 19/30`
`[2/3 - ((-4)/5)] – 1/2 = [(10 + 12)/15] – 1/2 = 22/15 – 1/2 = 29/30`
Subtraction is not associative for rational numbers i.e., for any three rational numbers a, b and c, a - (b - c) ≠ (a - b) - c.
3. Associativity of Multiplication of Rational Numbers:
`(-7)/5 xx (5/4 xx 2/9) = (- 7)/3 xx 10/36 = (- 70)/108 = (- 35)/54`.
`((-7)/3 xx 5/4) xx 2/9 = (- 35)/12 xx 2/9 = (- 70)/108`
`(-7)/5 xx (5/4 xx 2/9) = ((-7)/3 xx 5/4) xx 2/9`
Multiplication is associative for rational numbers. That is for any three rational numbers a, b and c, a × (b × c) = (a × b) × c.
4. Associativity of Division of Rational Numbers:
Let us see if `1/2 ÷ [(-1)/3 ÷ 2/5] = [1/2 ÷ ((-1)/3)] ÷ 2/5 `
We have, LHS = `1/2 ÷ [(-1)/3 ÷ 2/5] = (- 30)/10`
RHS =` [1/2 ÷ ((-1)/3)] ÷ 2/5 = (-15)/4`
We say that division is not associative for rational numbers. That is, for any three rational numbers a, b and c, a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c.
Example
Find: `3/7 + ((-6)/11) + ((-8)/21) + (5/22)`.
`3/7 + ((-6)/11) + ((-8)/21) + (5/22)`.
`= 198/462 + ((-252)/462) + ((-176)/462) + (105/462)` .....(462 is the LCM 7, 11, 21, and 22)
`= (198 - 252 - 176 + 105)/(462)`
`= (-125)/462`.
`3/7 + ((-6)/11) + ((-8)/21) + (5/22)`.
`= [3/7 + ((-8)/21)] + [(-6)/11 + 5/22]` .....(by using commutativity and associativity)
`= [(9 + (-8))/21] + [(-12 + 5)/22]` .....(LCM of 7 and 21 is 21; LCM of 11 and 22 is 22)
`= 1/21 + ((-7)/22)`
`= (22 - 147)/462`
`= (-125)/462`.
Example
Find: `(-4)/5 xx 3/7 xx 15/16 xx ((-14)/9)`.
`(-4)/5 xx 3/7 xx 15/16 xx ((-14)/9)`
`= (- (4 xx 3)/(5 xx 7)) xx ((15 xx (-14))/(16 xx 9))`
`= (-12)/35 xx ((-35)/24)`
`= (-12 xx (-35))/(35 xx 24)`
`= 1/2`.
`(-4)/5 xx 3/7 xx 15/16 xx ((-14)/9)`
`= ((-4)/5 xx 15/16) xx [3/7 xx ((-14)/9)]` .....(Using commutativity and associativity)
`= (-3)/4 xx ((-2)/3)`
`= 1/2`.
