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
Distributive Property of Multiplication Over Addition for Rational Numbers:
1. Distributivity for multiplication over Addition:
To understand this, consider the rational numbers `(-3)/4, 2/3, and (-5)/6`.
`(-3)/4 xx {2/3 + ((-5)/6)} = (-3)/4 xx {((4) + (-5))/6}`
`= (-3)/4 xx ((-1)/6) = 3/24 = 1/8`
Also `(- 3)/4 xx 2/3 = (- 3 xx 2 )/(4 xx 3) = (- 6)/12 = (- 1)/2`
And `(- 3)/4 xx (-5)/6 = 5/8`
Therefore `((-3)/4 xx 2/3) + ((-3)/4 xx (-5)/6) = (-1)/2 + 5/8 = 1/8`
Thus, `-3/4 xx {2/3 + (-5)/6} = ((-3)/4 xx 2/3) + ((-3)/4 xx (-5)/6)`
For all rational numbers a, b, and c, a(b + c) = ab + ac.
2. Distributivity for multiplication over subtraction:
Consider the rational numbers `3/8, (- 2)/5, 6/7`.
`3/8 xx ((-2)/5 - 6/7 ) = 3/8 xx (-14 - 30)/35 = 3/8 xx (- 44)/35 = (- 33)/70`
Also , `3/8 xx (- 2)/5 - 3/8 xx 6/7 = (- 6)/40 - 18/56 = (- 3)/20 - 9/28 = (- 33)/70`
For all rational numbers a, b, and c, a(b – c) = ab – ac.
To understand this, consider the rational numbers `(-3)/4, 2/3, and (-5)/6`.
`(-3)/4 xx {2/3 + ((-5)/6)} = (-3)/4 xx {((4) + (-5))/6}`
`= (-3)/4 xx ((-1)/6) = 3/24 = 1/8`
Also `(- 3)/4 xx 2/3 = (- 3 xx 2 )/(4 xx 3) = (- 6)/12 = (- 1)/2`
And `(- 3)/4 xx (-5)/6 = 5/8`
Therefore `((-3)/4 xx 2/3) + ((-3)/4 xx (-5)/6) = (-1)/2 + 5/8 = 1/8`
Thus, `-3/4 xx {2/3 + (-5)/6} = ((-3)/4 xx 2/3) + ((-3)/4 xx (-5)/6)`
For all rational numbers a, b, and c, a(b + c) = ab + ac.
2. Distributivity for multiplication over subtraction:
Consider the rational numbers `3/8, (- 2)/5, 6/7`.
`3/8 xx ((-2)/5 - 6/7 ) = 3/8 xx (-14 - 30)/35 = 3/8 xx (- 44)/35 = (- 33)/70`
Also , `3/8 xx (- 2)/5 - 3/8 xx 6/7 = (- 6)/40 - 18/56 = (- 3)/20 - 9/28 = (- 33)/70`
For all rational numbers a, b, and c, a(b – c) = ab – ac.
Example
Find: `2/5 xx (-3)/7 - 1/14 - 3/7 xx 3/5`.
`2/5 xx (-3)/7 - 1/14 - 3/7 xx 3/5`
`= 2/5 xx (-3)/7 - 3/7 xx 3/5 - 1/14` .......(by commutativity)
`= 2/5 xx (-3)/7 + ((-3)/7) xx 3/5 - 1/14`.
`= (-3)/7(2/5 + 3/5) - 1/14` .......(by distributivity)
`= (- 3)/7 xx 1 - 1/14`
`= (- 6 - 1)/14`
`= -1/2`.
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Related QuestionsVIEW ALL [50]
Four friends had a competition to see how far could they hop on one foot. The table given shows the distance covered by each.
| Name | Distance covered (km) |
| Seema | `1/25` |
| Nancy | `1/32` |
| Megha | `1/40` |
| Soni | `1/20` |
Who walked farther, Nancy or Megha?
