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
- Profit Percentage
- Loss Percentage
- Formula
- Understanding Profit or Loss Percentage with Example
Formula
Profit = Selling Price - Cost Price, which means CP < SP.
Profit Percent = `"Profit"/"Cost Price" xx 100`.
Loss = Cost Price - Selling Price, which means CP > SP.
Loss Percent = `"Loss"/"Cost Price" xx 100`.
Understanding Profit or Loss Percent with Example
Calculating profit or loss as a percentage, we compare it with the cost price.
- So, if we say there is a 10% profit, it means:
The seller made a ₹10 profit on every ₹100 of the cost price. - If we say there is a 10% loss, it means:
The seller lost ₹10 on every ₹100 of the cost price.
Example:
Abbas bought vegetables worth ₹400 and sold them for ₹650. Balbir bought fruits for ₹300 and sold them for ₹500. Whose transactions were more profitable?
Solution: Abbas made a profit of ₹250, and Balbir’s profit was ₹200.
However, the cost price for each of them was different. To compare, we shall have to find out the percentages of the profits.
Let us suppose Abbas made A% and Balbir made B% profit.
Let us find the ratios of profit to cost price, express them in two forms, obtain equations, and solve them.
`"A"/"100"` = `"250"/"400"`
`"A"/"100"` × 100 = `"250 × 100"/"400"`
A = `"250"/"4"` = `"125"/"2"` = 62`"1"/"2"`
`"B"/"100"` = `"200"/"300"`
`"B"/"100"` × 100 = `"200 × 100"/"300"`
B = `"200"/"3"` = 66`"2"/"3"`
Therefore, Balbir's transactions were more profitable.
Example
The cost of a flower vase is Rs. 120. If the shopkeeper sells it at a loss of 10%, find the price at which it is sold.
Given: CP = Rs. 120 and Loss percent = 10.
Loss of 10% means if CP is Rs. 100, Loss is Rs. 10
Therefore,
SP would be Rs.(100 - 10) = Rs. 90
When CP is Rs. 100, SP is Rs. 90.
Therefore, if CP were Rs. 120 then
SP = `90/100 xx 120` = Rs. 108.
Given: CP = Rs. 120 and Loss percent = 10.
Loss is 10% of the cost price
= 10% of Rs. 120
= `10/100 xx 120`
= Rs. 12
Therefore,
SP = CP - Loss = Rs. 120 - Rs. 12 = Rs. 108
Example
Selling price of a toy car is Rs. 540. If the profit made by the shopkeeper is 20%, what is the cost price of this toy?
Given: SP = Rs. 540 and the Profit = 20%.
20% profit will mean if CP is Rs. 100, profit is Rs. 20.
Therefore, SP = 100 + 20 = 120
Now, when SP is Rs. 120, then CP is Rs. 100.
Therefore, when SP is Rs. 540,
Then CP = `100/120 xx 540` = Rs. 450.
Given: SP = Rs. 540 and the Profit = 20%.
Profit = 20% of CP and SP = CP + Profit
So, 540 = CP + 20% of CP
= CP + `20/100 xx "CP" = [1 + 1/5]"CP"`
= `6/5 "CP"`.
Therefore, `540 xx 5/6` = CP or Rs. 450 = CP
Example
Joseph bought a machine for Rs. 23500. He paid Rs. 1200 for transport and Rs. 300 as tax. If he sold it to a customer for Rs. 24250, what was his percent profit or loss?
Total cost price of machine = 23500 + 1200 + 300 = Rs. 25000
Selling price = Rs. 24250
Cost price greater than selling price. Therefore, loss.
Loss = Cost price - Selling price
= 25000 - 24250
= Rs. 750
Joseph suffered a loss of Rs. 750.
Supposing loss was N%,
`"N"/100 = 750/25000`
∴ `"N"/100 xx 100 = 3/100 xx 100`
∴ N = 3
∴ Loss = 3%
Example
Saritaben bought 18 chairs each at Rs. 700 and sold them all for Rs. 18900. What was the percentage of her profit or loss?
Cost price of one chair Rs. 700.
∴ Cost price of 18 chairs = 700 × 18 = Rs. 12600.
Total selling price of all chairs, Rs. 18900.
Selling price more than cost price.
Therefore,
Profit = Selling price - Cost price
= 18900 - 12600
= 6300
Saritaben made a profit of Rs. 6300.
Supposing profit was N%.
`"N"/100 = 6300/12600`
∴ `"N"/100 xx 100 = 63/126 xx 100`
∴ N = `(63 xx 100)/126`
∴ N = 50
∴ Profit was 50%.
