Topics
Integers
- Concept for Natural Numbers
- Concept for Whole Numbers
- Negative and Positive Numbers
- Concept of Integers
- Representation of Integers on the Number Line
- Concept for Ordering of Integers
- Addition of Integers
- Addition of Integers on Number line
- Subtraction of Integers
- Properties of Addition and Subtraction of Integers
- Multiplication of a Positive and a Negative Integers
- Multiplication of Two Negative Integers
- Product of Three Or More Negative Integers
- Closure Property of Multiplication of Integers
- Commutative Property of Multiplication of Integers
- Associative Property of Multiplication of Integers
- Distributive Property of Multiplication of Integers
- Multiplication of Integers with Zero
- Multiplicative Identity of Integers
- Making Multiplication Easier of Integers
- Division of Integers
- Properties of Division of Integers
Fractions and Decimals
- Concept of Fractions
- Types of Fraction
- Concept of Proper Fractions
- Improper Fraction and Mixed Fraction
- Concept for Equivalent Fractions
- Like and Unlike Fraction
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of a Fraction by a Whole Number
- Fraction as an Operator 'Of'
- Multiplication of a Fraction by a Fraction
- Division of Fractions
- Concept for Reciprocal of a Fraction
- Concept of Decimal Numbers
- Multiplication of Decimal Numbers
- Multiplication of Decimal Numbers by 10, 100 and 1000
- Division of Decimal Numbers by 10, 100 and 1000
- Division of a Decimal Number by a Whole Number
- Division of a Decimal Number by Another Decimal Number
Data Handling
Simple Equations
Lines and Angles
- Concept of Points
- Concept of Line
- Concept of Line Segment
- Concept of Intersecting Lines
- Concept of Angle - Arms, Vertex, Interior and Exterior Region
- Complementary Angles
- Supplementary Angles
- Adjacent Angles
- Concept of Linear Pair
- Concept of Vertically Opposite Angles
- Concept of Intersecting Lines
- Parallel Lines
- Pairs of Lines - Transversal
- Pairs of Lines - Angles Made by a Transversal
- Pairs of Lines - Transversal of Parallel Lines
- Checking Parallel Lines
The Triangle and Its Properties
- Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Classification of Triangles (On the Basis of Sides, and of Angles)
- Equilateral Triangle
- Isosceles Triangles
- Scalene Triangle
- Acute Angled Triangle
- Obtuse Angled Triangle
- Right Angled Triangle
- Median of a Triangle
- Altitudes of a Triangle
- Exterior Angle of a Triangle and Its Property
- Angle Sum Property of a Triangle
- Some Special Types of Triangles - Equilateral and Isosceles Triangles
- Sum of the Lengths of Two Sides of a Triangle
- Right-angled Triangles and Pythagoras Property
Congruence of Triangles
Comparing Quantities
- Concept of Ratio
- Concept of Equivalent Ratios
- Concept of Proportion
- Concept of Unitary Method
- Concept of Percent and Percentage
- Converting Fractional Numbers to Percentage
- Converting Decimals to Percentage
- Converting Percentages to Fractions
- Converting Percentages to Decimals
- Estimation in Percentages
- Interpreting Percentages
- Converting Percentages to “How Many”
- Ratios to Percents
- Increase Or Decrease as Percent
- Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
- Profit or Loss as a Percentage
- Concept of Principal, Interest, Amount, and Simple Interest
Rational Numbers
- Rational Numbers
- Equivalent Rational Number
- Positive and Negative Rational Numbers
- Rational Numbers on a Number Line
- Rational Numbers in Standard Form
- Comparison of Rational Numbers
- Rational Numbers Between Two Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers
Practical Geometry
- Construction of a Line Parallel to a Given Line, Through a Point Not on the Line
- Construction of Triangles
- Constructing a Triangle When the Length of Its Three Sides Are Known (SSS Criterion)
- Constructing a Triangle When the Lengths of Two Sides and the Measure of the Angle Between Them Are Known. (SAS Criterion)
- Constructing a Triangle When the Measures of Two of Its Angles and the Length of the Side Included Between Them is Given. (ASA Criterion)
- Constructing a Right-angled Triangle When the Length of One Leg and Its Hypotenuse Are Given (RHS Criterion)
Perimeter and Area
- Mensuration
- Concept of Perimeter
- Perimeter of a Rectangle
- Perimeter of Squares
- Perimeter of Triangles
- Perimeter of Polygon
- Concept of Area
- Area of Square
- Area of Rectangle
- Triangles as Parts of Rectangles and Square
- Generalising for Other Congruent Parts of Rectangles
- Area of a Triangle
- Area of a Parallelogram
- Circumference of a Circle
- Area of Circle
- Conversion of Units
- Problems based on Perimeter and Area
- Problems based on Perimeter and Area
Algebraic Expressions
- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Like and Unlike Terms
- Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Evaluation of Algebraic Expressions by Substituting a Value for the Variable.
- Use of Variables in Common Rules
Exponents and Powers
- Concept of Exponents
- Multiplying Powers with the Same Base
- Dividing Powers with the Same Base
- Taking Power of a Power
- Multiplying Powers with Different Base and Same Exponents
- Dividing Powers with Different Base and Same Exponents
- Numbers with Exponent Zero, One, Negative Exponents
- Miscellaneous Examples Using the Laws of Exponents
- Decimal Number System Using Exponents and Powers
- Expressing Large Numbers in the Standard Form
Symmetry
Visualizing Solid Shapes
- Plane Figures and Solid Shapes
- Faces, Edges and Vertices
- Nets for Building 3-d Shapes - Cube, Cuboids, Cylinders, Cones, Pyramid, and Prism
- Drawing Solids on a Flat Surface - Oblique Sketches
- Drawing Solids on a Flat Surface - Isometric Sketches
- Visualising Solid Objects
- Viewing Different Sections of a Solid
definition
Unitary Method: The method in which first we find the value of one unit and then the value of the required number of units is known as Unitary Method.
notes
Unitary Method:
- Two friends Reshma and Seema went to the market to purchase notebooks. Reshma purchased 2 notebooks for Rs. 24. What is the price of one notebook?
Cost of 2 notebooks is Rs. 24.Therefore, cost of 1 notebook = Rs. 24 ÷ 2 = Rs. 12.Now, if you were asked to find the cost of 5 such notebooks. It would be = Rs. 12 × 5 = Rs. 60
- A scooter requires 2 litres of petrol to cover 80 km. How many litres of petrol is required to cover 1 km?
We want to know how many litres are needed to travel 1 km.For 80 km, petrol needed = 2 litres.Therefore, to travel 1 km, petrol needed = `2/80 = 1/40` litres.Now, if you are asked to find how many litres of petrol are required to cover 120 km?Then petrol needed = `1/40 xx 120` litres = 3 litres.
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The method in which first we find the value of one unit and then the value of the required number of units is known as the Unitary Method.
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Find the cost of one article from that of many, by division. Then find the cost of many articles from that of one, by multiplication. This method of solving a problem is called the unitary method.
Example
A motorbike travels 220 km in 5 litres of petrol. How much distance will it cover in 1.5 litres of petrol?
In 5 litres of petrol, motorbike can travel 220 km.
Therefore, in 1 litre of petrol, motorbike travels = `220/5` km.
Therefore, in 1.5 litres, motorbike travels
= `220/5 xx 1.5 "km" = 220/5 xx 15/10` km = 66 km.
Thus, the motorbike can travel 66 km in 1.5 litres of petrol.
Example
If the cost of a dozen soaps is Rs. 153.60, what will be the cost of 15 such soaps?
We know that 1 dozen = 12
Since, cost of 12 soaps = Rs. 153.60
Therefore, cost of 1 soap = `153.60/12` = Rs. 12.80
Therefore, cost of 15 soaps = Rs. 12.80 × 15 = Rs. 192.
Thus, cost of 15 soaps is Rs. 192.
Example
Cost of 105 envelopes is Rs. 350. How many envelopes can be purchased for Rs. 100?
In Rs. 350, the number of envelopes that can be purchased = 105.
Therefore, in Rs. 1, number of envelopes that can be purchased = `105/350`
Therefore, in Rs. 100, the number of envelopes that can be purchased
= `105/350 × 100 = 30`.
Thus, 30 envelopes can be purchased for Rs. 100.
Example
A car travels 90 km in 2 1/2hours.
(a) How much time is required to cover 30 km with the same speed?
(b) Find the distance covered in 2 hours with the same speed.
(a) In this case, time is unknown and distance is known. Therefore, we proceed as follows:
`2 1/2 "hours" = 5/2 "hours" = 5/2 × 60 "minutes" = 150 "minutes"`.
90 km is covered in 150 minutes.
Therefore, 1 km can be covered in `(150)/(90)` minutes.
Therefore, 30 km can be covered in `(150)/(90) × 30` minutes i.e., 50 minutes.
Thus, 30 km can be covered in 50 minutes.
(b) In this case, distance is unknown and time is known. Therefore, we proceed as follows:
Distance covered in `2 1/2 "hours (i.e.," 5/2` hours ) = 90 km.
Therefore, distance covered in 1 hour = 90 ÷ `5/2 "km" = 90 × 2/5` = 36 km.
Therefore, distance covered in 2 hours = 36 × 2 = 72 km.
Thus, in 2 hours, the distance covered is 72 km.
