#### Topics

##### Geometrical Constructions

- Constructing a Bisector of an Angle
- Drawing a Perpendicular to a Line at a Point on the Line
- The Property of the Angle Bisectors of a Triangle
- Perpendicular Bisectors of the Sides of an Acute-angled Triangle
- Perpendicular Bisectors of the Sides of an Obtuse-angled Triangle
- Construction of Triangles
- Construct a Triangle Given Two Sides and the Angle Included by Them
- Construct a Triangle Given Two Angles and the Included Side
- Construct a Right-angled Triangle Given the Hypotenuse and One Side
- Congruence Among Line Segments
- Congruence of Angles
- Congruence of Circles

##### Multiplication and Division of Integers

- Concept for Natural Numbers
- Concept for Whole Numbers
- Negative and Positive Numbers
- Concept of Integers
- Addition of Integers
- Addition of Integers on Number line
- Subtraction of Integers
- Multiplication of a Positive and a Negative Integers
- Multiplication of Two Negative Integers
- Division of Integers
- Multiplication of Two Positive Integers

##### HCF and LCM

##### Angles and Pairs of Angles

##### Operations on Rational Numbers

- Concept of Rational Numbers
- Addition of Rational Number
- Additive Inverse of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers
- Rational Numbers Between Two Rational Numbers
- Decimal Representation of Rational Numbers
- BODMAS - Rules for Simplifying an Expression

##### Indices

- Concept of Exponents
- Concept of Square Number
- Concept of Cube Number
- Laws 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
- Expressing Large Numbers in the Standard Form
- Finding the Square Root of a Perfect Square

##### Joint Bar Graph

##### Algebraic Expressions and Operations on Them

- 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
- Multiplication of Algebraic Expressions
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Binomial
- Multiplying a Binomial by a Binomial
- Equations in One Variable

##### Direct Proportion and Inverse Proportion

##### Banks and Simple Interest

##### Circle

##### Perimeter and Area

##### Pythagorasâ€™ Theorem

##### Algebraic Formulae - Expansion of Squares

##### Statistics

#### definition

**Partnership: **A partnership is an arrangement where parties, known as business, partners, agree to cooperate to advance their mutual interests.

**Partnership deed:** An agreement that contains the details of every aspect and terms that the partners agree upon before starting a partnership firm is termed as partnership deed.

**Capital:** When starting a business enterprise, money is required for an office, raw materials, etc. This amount is called the capital.

#### notes

**Partnership:**

- A partnership is an agreement between two or more persons to share profits and losses of the firm. According to Section 4 of the Indian Partnership Act, 1932, “Partnership is the relation between persons who have agreed to share profits of a business carried on by all or any one of them acting for all.”
- A partnership is an arrangement where parties, known as business, partners, agree to cooperate to advance their mutual interests. The partners in a partnership may be individuals, businesses, interest-based organizations, schools, governments, or combinations.
- An agreement that contains the details of every aspect and terms that the partners agree upon before starting a partnership firm is termed as partnership deed.
- At least two persons are required to form a partnership business. However, according to the Companies Act, 1956, a firm cannot have more than 10 partners in case of banking business and not more than 20 persons in case of any other business.
- Organizations may partner to increase the likelihood of each achieving their mission and to amplify their reach. A partnership may result in issuing and holding equity or maybe only governed by a contract.
- When starting a business enterprise, money is required for an office, raw materials, etc. This amount is called the capital. Often, two or more people put in money for the capital. In other words, these people start a business by investing in the partnership.
- In a business partnership, all partners have a joint account in a bank. The profit made or the loss incurred is shared by the partners in proportion to the money each one has invested.

#### Example

Jhelum and Atharva invested 2100 and 2800 rupees respectively and started a business. They made a profit of 3500 rupees. How should it be shared?

Let us find out the proportion of investments.

2100: 2800 = `2100/2800 = 3/4` = 3: 4.

∴ The proportion of investments is 3: 4.

The profit must also be shared in the same proportion.

Let Jhelum’s profit be 3x and that of Atharva, 4x. Then,

3x + 4x = 3500 as total profit is 3500.

∴ 7x = 3500

∴ x = 500

Jhelum’s share = 3x = 1500 rupees and Atharva’s share = 4x = 2000 rupees.

#### Example

Chinmaya and Sam invested a total of 130000 rupees in business in proportion 3: 2 respectively. What amount did each of them invest? If their total profit was 36000 rupees, what is the share of each?

The proportion of Chinmaya’s and Sam’s investment is 3: 2.

The profit is shared in the same proportion as the investment, hence, the proportion of profit is 3: 2.

Let Chinmaya’s investment be 3y and Sam’s 2y.

3y + 2y = Total investment.

∴ 5y = 130000

`5/(5y) = 130000/5`..... (dividing by 5)

∴ y = 26000

∴ Chinmaya’s investment = 3y = 3 × 26000 = 78,000

Sam’s investment = 2y = 2 × 26000 = 52000 rupees

Let Chinmaya’s profit be 3x and Sam’s 2x.

3x + 2x = Total Profit

5x = 36000

`5/(5x) = 36000/5`

∴ x = 7200

∴ Chinmaya’s profit = 3x = 3 × 7200 = 21600

Sam’s profit = 2x = 2 × 7200 = 14400 rupees

#### Example

Abdul, Sejal, and Soham each gave Sayali 30 rupees, 70 rupees, and 50 rupees respectively. Sayali put in 150 rupees and bought paper, colours, etc. Together they made greeting cards and sold them all. If they made a total profit of 420 rupees, what was each one’s share in the profit?

The capital invested by all four was 300 rupees. Of this Sayali had invested 150 rupees, that is, half of the capital. The total profit was 420 rupees. So, Sayali’s profit was half of that, i.e., 210 rupees. The remaining 210 was shared by Abdul, Sejal, and Soham.

Abdul, Sejal, and Soham’s investment is 30, 70, and 50 rupees. The proportion is 30: 70: 50 i.e. 3: 7: 5. Their share of the profit is altogether 210 rupees.

Let their individual profit be 3k, 7k, 5k.

Then, 3k + 7k + 5k = 210

∴ 15k = 210

∴ k = 14

Abdul’s profit = 3k = 3 × 14 = 42 rupees.

Sejal’s profit = 7k = 7 × 14 = 98 rupees.

Soham’s profit = 5k = 5 × 14 = 70 rupees.