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
- Steps to Convert a Percentage into a Fraction
- Examples: Converting Percentages to Fractions
- Steps to Convert a Fraction into a Percentage
- Examples: Converting Fractions to Percentages
- Real-Life Applications
Introduction
Have you ever wondered why we can say "half" in three different ways? Look around you – when you eat half a pizza, you can express this as:
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50% (fifty percent)
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`1/2` (one-half fraction)
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0.5 (zero point five decimal)
All three mean exactly the same thing – HALF! It's like speaking the same idea in three different mathematical languages.
Steps to Convert a Percentage into a Fraction
- Step 1: Remove the % symbol
- Step 2: Divide the number by 100 to convert it into a fraction.
- Step 3: Reduce to lowest terms by finding the common factor
Examples: Converting Percentages to Fractions
1.25% = `25/ 100` = `1/4` or, 25% = `25/100` = 0.25
2. 35% = `35/100` = `7/20`
3. 37.5% = `37.5/ 100` = `(375)/(100×10)` = `3/ 8`
or 37.5% = `37.5/ 100` = 0.375
Steps to Convert a Fraction into a Percentage
Method A – Using Division:
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Step 1: Multiply the result by 100
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Step 2: Add the % symbol
Method B – Making Denominator 100:
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Step 1: Find what number multiplied by the denominator gives 100
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Step 2: Multiply both the numerator and the denominator by that number
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Step 3: The numerator becomes your percentage
Examples: Converting Fractions to Percentages
1.`3/4` = `3/4` × 100% = 75%
2. 0.225 = 0.225 × 100% = 22.5%
3. `3/4` = `"3 × 25"/"4 × 25"` = `75/100` = 75%.
4. `2/5` = `"2 × 20"/"5 × 20"` = `40/100` `2/5` = 40%.
Real-Life Applications
Shopping Discount: If a ₹200 shirt has a 25% discount:
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25% = `25/100` = `1/4`
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Discount = `1/4` × ₹200 = ₹50
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Final price = ₹200 - ₹50 = ₹150
Test Scores: If you scored 18 out of 20 in a test:
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Fraction = `18/20` = `9/10`
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Percentage = 18 ÷ 20 × 100 = 90%
Key Points Summary
- "Cent" means 100 – remember "century" = 100 years and "century" = 100 runs
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Quarter relationships: 25% = 1/4 = 0.25
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Half relationships: 50% = 1/2 = 0.50
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For easy fractions: If the denominator divides 100 evenly, multiply both parts.
Common Mistakes to Avoid
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Don't forget to reduce fractions to their simplest form
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Remember the % symbol when writing percentages
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Be careful with decimal placement when converting
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Check your work by converting back to the original form
Example Question 1
A survey of 40 children showed that 25% liked playing football. How many children liked playing football?
Here, The total number of children are 40. Out of these, 25% like playing football.
(i) 25% of 40
Number of children who like playing football = `25/100` × 40 = 10.
(ii) Out of 100, 25 like playing football So out of 40,
Number of children who like playing football = `25/100` × 40 = 10.
Hence, 10 children out of 40 like playing football.
Example Question 2
Convert given percent to decimal: 1%
1% = `1/100` = 0.01.
Example Question 3
Rahul bought a sweater and saved Rs. 200 when a discount of 25% was given. What was the price of the sweater before the discount?
Rahul has saved Rs. 200 when price of sweater is reduced by 25%. This means that 25% reduction in price is the amount saved by Rahul.
(i) 25% of the original price = Rs. 200
Let the price (in Rs.) be x.
So, 25% of x = 200 or `25/100 xx x = 200`
or, `x/4` = 200 or x = 200 × 4
Therefore, x = 800.
(ii) Rs. 25 is saved for every Rs. 100
Amount for which Rs. 200 is saved = `100/25 xx 200` = Rs. 800
Thus both obtained the original price of sweater as Rs. 800.
Given:
Discount = 25%
Amount saved = ₹ 200
If Rahul saved ₹ 200 due to a 25% discount, that means:
₹ 200 is 25% of the original price.
Let the original price be x
25% of x = 200
`25/100 xx x = 200`
`x/4 = 200`
x = 200 × 4
= 800
Example Question 4
For summative evaluation in a certain school, 720 of the 1200 children were awarded A grade in Maths. What is the percentage of students getting A grade?
Suppose the students getting A grade are A%.
`"A"/100 = 720/1200`
∴ `"A"/100 xx 100 = 720/1200 xx 100`
∴ A = 60
∴ 60% students got A grade.
Example Question 5
Write `1/3` as a percent.
`1/3 = 1/3 × 100/100 = 1/3 × 100% = 100/3% = 33 1/3 %`
Example Question 6
Convert the given decimals to percents: 0.2
0.2 = `2/10 xx 100%` = 20%.
Example Question 7
A certain Organization adopted 18% of the 400 schools in a district. How many schools did it adopt?
Here, 18% means 18 schools adopted out of a total of 100.
The total number of schools is 400.
Suppose the number of schools adopted is A.
`"A"/400 = 18/100`
`"A"/400 xx 400 = 18/100 xx 400`
∴ A = 72.
∴ The number of schools adopted is 72.
Example Question 8
Convert `2/40` to percent.
`2/40 = 1/20 = (1 xx 5)/(20 xx 5) = 5/100 = 5%`
Example Question 9
Out of 25 children in a class, 15 are girls. What is the percentage of girls?
Out of 25 children, there are 15 girls.
Therefore, percentage of girls = `15/25 xx 100 = 60`.
There are 60% girls in the class.
Example Question 10
Out of 32 students, 8 are absent. What percent of the students are absent?
`8/32 = 1/4 = (1 xx 25)/(4 xx 25) = 25/100 = 25%`.
