#### 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
- 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 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
- Problems Based on Fraction
- Concept of Decimal Numbers
- Comparing Decimal Numbers
- Addition of Decimal Numbers
- Subtraction 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
- Problems Based on Decimal Numbers

##### Data Handling

##### Simple Equations

##### Lines and Angles

- Concept of Points
- Concept of Line
- Concept of Line Segment
- 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

##### 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

##### 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

- 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 Parallelogram
- Area of a Triangle
- Circumference of a Circle
- Area of Circle
- Conversion of Units
- 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

##### 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)

##### 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

**Cost Price(C.P.): **The buying price of any item is known as its cost price.

**Selling Price(S.P):** The price at which you sell any item is known as the selling price.

**Profit: **Profit is the revenue remaining after all costs are paid from selling price.

**Loss: **When the item is sold and the selling price is less than the cost price, then it is said, the seller has incurred a loss.

## Formula

Profit = Selling Price - Cost Price, which means CP < SP.

Loss = Cost Price - Selling Price, which means CP > SP.

Profit Percent = `"Profit"/"Cost Price" xx 100`.

Loss Percent = `"Loss"/"Cost Price" xx 100`.

## Notes

**Profit Or Loss as a Percentage:**

**Cost Price(C.P.): **The buying price of any item is known as its cost price. It is written in short as CP.

**Selling Price(S.P):** The price at which you sell any item is known as the selling price. It is written in short as SP.

**Profit: **Profit is the revenue remaining after all costs are paid from selling price.

**Loss: **When the item is sold and the selling price is less than the cost price, then it is said, the seller has incurred a loss.

You can decide whether the sale was profitable or not depending on the CP and SP.

- If CP < SP then you made a profit = SP – CP.
- If CP = SP then you are in a no-profit no loss situation.
- If CP > SP then you have a loss = CP – SP.

The profit or loss can be converted to a percentage.

It is always calculated on the CP.

- Profit Percent = `"Profit"/"Cost Price" xx 100`.
- Loss Percent = `"Loss"/"Cost Price" xx 100`.

## 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%.