Topics
Integers
- Natural Numbers
- Whole Numbers
- Negative and Positive Numbers
- Integers
- Representation of Integers on the Number Line
- Ordering of Integers
- Addition of Integers
- 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
- Multiplication of Integers with Zero
- Multiplicative Identity of Integers
- Associative Property of Multiplication of Integers
- Distributive Property of Multiplication of Integers
- Making Multiplication Easier of Integers
- Division of Integers
- Properties of Division of Integers
Fractions and Decimals
- Concept of Fraction
- Types of Fractions
- Concept of Proper and Improper Fractions
- Concept of Mixed Fractions
- Concept of Equivalent Fractions
- Like and Unlike Fraction
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of a Fraction by a Whole Number
- Using Operator 'Of' with Multiplication and Division
- Multiplication of Fraction
- Division of Fractions
- Concept of Reciprocals or Multiplicative Inverses
- Problems Based on Fraction
- The Decimal Number System
- Comparing Decimal Numbers
- Addition of Decimal Fraction
- Subtraction of Decimal Numbers
- Multiplication of Decimal Numbers
- Division of Decimal Numbers
- Problems Based on Decimal Numbers
Data Handling
Simple Equations
Lines and Angles
The Triangle and Its Properties
- Basic Concepts of Triangles
- Classification of Triangles based on Sides
- Classification of Triangles based on Angles
- Median of a Triangle
- Altitudes of a Triangle
- Exterior Angle of a Triangle and Its Property
- Some Special Types of Triangles - Equilateral and Isosceles Triangles
- Basic Properties of a Triangle
- Right-angled Triangles and Pythagoras Property
Comparing Quantities
- Ratio
- Concept of Equivalent Ratios
- Proportion
- Unitary Method
- Basic Concept of Percentage
- Estimation in Percentages
- Interpreting Percentages
- Conversion between Percentage and Fraction or Decimal
- Ratios to Percents
- Increase Or Decrease as Percent
- Basic Concepts of Profit and Loss
- Profit or Loss as a Percentage
- Calculation of Interest
Congruence of Triangles
- Similarity and Congruency of Figures
- Congruence Among Line Segments
- Congruence of Angles
- Congruence of Triangles
- Criteria for Congruence of Triangles
- Criteria for Similarity of Triangles
- SAS Congruence Criterion
- ASA Congruence Criterion
- RHS Congruence Criterion
- Exceptional Criteria for Congruence of Triangles
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
Perimeter and Area
- Basic Concepts in Mensuration
- Concept of Perimeter
- Perimeter of a Rectangle
- Perimeter of Squares
- Perimeter of Triangle
- 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 Parallelogram
- Area of a Triangle
- Circumference of a Circle
- Area of Circle
- Conversion of Units
- Problems based on Perimeter
- Problems based on Area
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)
Algebraic Expressions
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
- Crores
Symmetry
Visualizing Solid Shapes
- Introduction
- Case 1: Division of Decimal by a Natural (Whole) Number
- Case 2: Division of a Decimal Number by a Decimal Number
- Case 3: Division of Decimal by 10, 100, 1000, etc.
- Key Points Summary
Introduction
Dividing decimal numbers is an essential skill for many everyday tasks, such as budgeting, cooking, and measuring. It may seem tricky at first, but once you understand the basic rules, it's just like dividing whole numbers—with a little extra attention to where the decimal point goes! Let’s break it down step by step.
Case 1: Division of Decimal by a Natural (Whole) Number
Step 1: Choose your method
-
Method A (Standard Division): Use long division like you would with whole numbers.
-
Method B (Fraction Method): Convert the decimal to a fraction and divide.
Step 2: Simplify the Result (if needed)
Example 1:
`83.6/2` = 41.8
41.8
`2")"overline(83.6)"("`
8
3
2
16
16
×
Example 2:
6.2 ÷ 2 = `62/10` ÷ `2/1`
= `62/10` × `1/2`
= `31/10`
= 3.1
Case 2: Division of a Decimal Number by a Decimal Number
Step 1: Convert the decimals to whole numbers
Step 2: Simplify the fraction
Example:
`5.065 / 0.05` = `"5.065 × 100"/ "0.05 × 100"`
= `5.065 / 5`
= 101.3
Case 3: Division of Decimal by 10, 100, 1000, etc.
Step 1: Move the decimal point to the left
Step 2: Simplify the fraction
Example:
- `48.7 / 10` = 4.87
- `937.3 / 100` = 9.373
- `520.81 / 1000` = 0.52081
Key Points Summary
-
Decimal as Fraction: 6.2 = `62/10`, 0.68 = `68/100`
-
Point Slide: Division by 10 → slide one place left; by 100 → two places left, etc.
Example Question 1
Find: 6.6 ÷ 2.2
6.6 ÷ 2.2
`= 66/10 ÷ 22/10`
`= 66/10 xx 10/22`
`= (66 xx 10)/(10 xx 22)`
`= (10 xx 66)/(10 xx 22)`
`= 10/10 xx 66/22`
`= 1/1 xx 3`
= 3
Example Question 2
Find: 33.73 ÷ 2
33.73 ÷ 2
`= (33.73)/2 xx 1/1`
`= (33.73)/2 xx (1 xx 100)/(1 xx 100)`
`= (33.73)/2 xx 100/100`
`= (33.73 xx 100)/(2 xx 100)`
`= (3373)/(2 xx 100)`
`= (3373 xx 1)/(2 xx 100)`
`= (3373)/2 xx 1/100`
`= 1686.5 xx 1/100`
`= (1686.5)/100`
= 16.865
Example Question 3
Find: 0.0125 ÷ 0.05
0.0125 ÷ 0.05
`= 125/10000 ÷ 5/100`
`= 125/10000 xx 100/5`
`= (125 xx 100)/(10000 xx 5)`
`= (100 xx 125)/(10000 xx 5)`
`= 100/10000 xx 125/5`
`= 1/100 xx 25`
`= (1 xx 25)/100`
`= 25/100`
= 0.25
Example Question 4
Find: 4.449 ÷ 4
4.449 ÷ 4
`= (4.449)/4 xx 1/1`
`= (4.499)/4 xx (1 xx 1000)/(1 xx 1000)`
`= (4.449)/4 xx (1000)/(1000)`
`= (4.449 xx 1000)/(4 xx 1000)`
`= (4449 xx 1)/(4 xx 1000)`
`= (4449)/4 xx 1/1000`
`= 1112.25 xx 1/1000`
`= (1112.25)/(1000)`
= 1.11225
Example Question 5
Find: 233 ÷ 0.5
`= 233/0.5 xx 1/1`
`= 233/0.5 xx 10/10`
`= (233 xx 10)/(0.5 xx 10)`
`= (233 xx 10)/5`
`= (233)/5 xx 10`
= 46.6 x 10
= 466
Example Question 6
Find: 0.375 ÷ 1.5
0.375 ÷ 1.5
`= 375/1000 ÷ 15/10`
`= 375/1000 xx 10/15`
`= (375 xx 10)/(1000 xx 15)`
`= (10 xx 375)/(1000 xx 15)`
`= 10/1000 xx 375/15`
`= 1/100 xx 25`
`= (1 xx 25)/100`
`= 25/100`
= 0.25
Example Question 7
Each side of a regular polygon is 2.5 cm in length. The perimeter of the polygon is 12.5 cm. How many sides does the polygon have?
The perimeter of a regular polygon is the sum of the lengths of all its equal sides = 12.5 cm.
Length of each side = 2.5 cm.
Thus, the number of sides = `12.5/2.5 = 125/25 = 5`.
The polygon has 5 sides.
Example Question 8
Find: 525 ÷ 0.015
525 ÷ 0.015
`= 525/0.015 xx 1/1`
`= 525/0.015 xx (1 xx 1000)/(1 xx 1000)`
`= 525/0.015 xx (1000)/(1000)`
`= (525 xx 1000)/(0.015 xx 1000)`
`= (525 xx 1000)/15`
`= 525/15 xx 1000`
`= 525/15 xx 1000`
= 35 × 1000
= 35000
Example Question 9
A car covers a distance of 89.1 km in 2.2 hours. What is the average distance covered by it in 1 hour?
Distance covered by the car = 89.1 km.
Time required to cover this distance = 2.2 hours.
So distance covered by it in 1 hour = `(89.1)/2.2 = 891/22` = 40.5 km.
