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
Example
Solve: `(x + 1)/(2x + 3) = 3/8`.
`(x + 1)/(2x + 3) = 3/8`.
We multiply both sides of the equation by (2x + 3),
`((x + 1)/(2x + 3)) xx (2x + 3) = 3/8 xx (2x + 3)`
(2x + 3) gets cancelled on the LHS we have then,
`x + 1 = (3(2x + 3))/8`
Multiplying both sides by 8,
8(x + 1) = 3(2x + 3)
or 8x + 8 = 6x + 9
or 8x = 6x + 9 - 8
or 8x = 6x + 1
or 8x - 6x = 1
or 2x = 1
or x = `1/2`
The solution is x = `1/2`.
Check:
Number of LHS = `1/2 + 1 = (1 + 2)/2 = 3/2`
Denominator of LHS = 2x + 3 = 2 × `1/2` + 3 = 1 + 3 = 4.
LHS = numerator ÷ denominator = `3/2 ÷ 4 = 3/2 xx 1/4 = 3/8.`
LHS = RHS.
Example
Present ages of Anu and Raj are in the ratio 4: 5. Eight years from now the ratio of their ages will be 5: 6. Find their present ages.
Let the present ages of Anu and Raj be 4x years and 5x years respectively.
After eight years, Anu's age = (4x + 8) years;
After eight years, Raj's age = (5x + 8) years.
Therefore, the ratio of their ages after eight years = `(4x + 8)/(5x + 8)`
This is given to be 5: 6
Therefore,
`(4x + 8)/(5x + 8) = 5/6`
Cross-multiplication gives
6(4x + 8) = 5(5x + 8)
or 24x + 48 = 25x + 40
or 24x + 48 - 40 = 25x
or 24x + 8 = 25x
or 8 = 25x - 24x
or 8 = x
Therefore,
Anu's present age = 4x = 4 × 8 = 32 years.
Raj's present age = 5x = 5 × 8 = 40 years.
If you would like to contribute notes or other learning material, please submit them using the button below.
Shaalaa.com | Problems on Linear Equations in One Variable
to track your progress
Series: Equations Reducible to the Linear Form
0%
