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A₁ = 0.01 m²

A₂ = 1 m²

F₁ = ?

F₂ = 20000 N

\(\frac{F_1}{F_2}=\frac{A_1}{A_2}\)

\(F_1=F_2\times\frac{A_1}{A_2}=20000\times\frac{0.01}{1}=200\,N\)

No. depression occurs in the case of mercury.

Capillarity:

The rise or depression of a liquid in a narrow tube or a minute hole is capillarity. 

Cohesive Force: 

Cohesive force is the force of attraction between molecules of the same type. 

Adhesive Force: 

Adhesive force is the force of attraction between molecules of different types of substance.

Weight of the liquid in the tube is supported by adhesive force. When the diameter of the tube decreases, the weight of the liquid it can contain also decreases. The adhesive force with the tube is greater than the cohesive force of the liquid. So the liquid is able to rise. Capillary rise increases with the decrease in the diameter.

• Hydraulic brake of vehicles 
• Hydraulic jack 
• Hydraulic press 
• Excavator, Hydraulic lift

Viscosity is the characteristic property of liquid to initiate a force that tries to prevent the relative motion between the layers of the liquid.

Density of water is 1000 kg/m³

Digital cash is a form of electronic currency existing in cyber space.-In case of digital cash, on payment of the amount, the bank issues digital cash in favor of the buyer.

It is division of society on the basis of familiarity with digital technology and nonfamiliarity with digital technology. The digital divide causes a significant problem in many struggling parts of the world. “As of 2003, only seven per cent of the world’s 6.4 billion people have had access to the World Wide Web” [Ryder M, 2005]. The parts of the world that have a predominant amount of internet access is the Western World, the United States, Europe and Northern Asia, whereas access is more restricted in the poorer less developed parts of the world such as Africa, India and southern parts of Asia. These poorer nations are unable to afford the initial start up cost to be able to invest into technology to allow their nation to be able to have and maintain internet access. This puts these countries at a competitive and economic disadvantage. This is due to the fact that it impacts on society at many levels. By a country not having internet access, it means that schools are unable to teach IT skills and take advantage of the vast amount of information available on the web. With the lack of IT skills people from these countries are unable to compete at an international level.

(b) the percentage change in quantity demanded is greater than the percentage change in price

(c) An auto producer is interested in determining the response of consumers to the price of cars being lowered by Rs. 100 

E-business and e-commerce are terms that are sometimes used interchangeably and sometimes they’re used to differentiate one vendor’s product from another. But the terms are different, and that difference matters to today’s companies. 

Definition of E-Commerce: E-commerce is “any transaction completed over a computer-mediated network that involves the transfer of ownership or rights to use goods and services,” defines the U.S. Census Bureau. Transactions aren’t required to have a price and include both sales and items like free downloads. E-commerce includes transactions made on the internet, Intranet, Extranet, World Wide Web, by e-mail and even by fax. 

Definition of E-Business: E-business is broader than e-commerce; including the transaction based e-commerce businesses and those who run traditionally but cater to online activities as well. An e-business can run any portion of its internal processes online, including inventory management, risk management, finance, human resources. For a business to be e-commerce and e-business, it must both sell products online and handle other company activities or additional sales offline. 

• E-commerce covers outward-facing processes that touch customers, suppliers and external partners, including sales, marketing, order taking, delivery, customer service, purchasing of raw materials and supplies for production and procurement of indirect operating expense items, such as office supplies. E-business includes e-commerce but also covers internal processes such as production, inventory management, product development, risk management, finance, knowledge management and human resources. 
• E-business strategy is more complex, more focused on internal processes, and aimed at cost savings and improvements in efficiency, productivity and cost savings than ecommerce. 
• Scope of e-business is wider than e-commerce.

(A) Meaning:

Statutory Corporations are autonomous bodies established under special legislative Acts. A statutory corporation is formed under a Special Act of Parliament or State

Legislature. The powers, duties, functions and scope of operations are laid down in the Act. LIC, IFCI, SBI, UTI, Air India are the examples of public corporation.

Statutory Corporation is a body with a separate existence, which can sue and be sued and is responsible for its own finance. It is administered by a board appointed by public authority to which it is answerable.

(B) Features of Statutory Corporation: 

(i) No political Interference : It enjoys freedom from political, parliamentary and government interference in day-to-day management.

(ii) Own Staffing System: They recruit their own employees and they are not government servant. Employees terms and services are not governed by civil services rules.

(iii) No Political Interference : It enjoys freedom from political, parliamentary and government interference in day to day management of its affairs. 

(iv) Financial Autonomy : Statutory Corporations are financially autonomous. After getting the prior permission from the Government, it can even borrow money within and outside the country.

(v) Independent Identity : They have an independent identity different from the government. Though, the overall business policies are formulated by the government, they have administrative autonomy and hence operational flexibility.

(vi) Special Act : They are established under a special Act passed by the Parliament. Its objectives, powers 98and functions are regulated by the Act. 

(vii) Corporate Body : Statutory Corporation is a corporate body. It has a separate legal entity distinct from its members and thereby can enter into contracts and acquire property on its own name. 

(viii) Answerable to the Legislature : A statutory corporation is answerable to Parliament or State Assembly whomsoever creates it. Parliament has no right to interfere. Though the overall business policies are formulated by the government, they have administrative autonomy and hence operational flexibility.

(ix) Legal Status : As a body corporate, it has a separate legal entity, distinct from its members and thereby can enter into contracts and acquire property in its own name. 

(x) Independent Accounting System : They are not subject to budget accounting and audit laws and procedures applicable to government departments. But financial reports are placed in the Parliament for discussion. 

(xi) Public Accountability : It’s accounts are audited by the Comptroller and Auditor General of India. Its annual reports and results are placed in Parliament or Legislative Assembly for discussion and hence answerable for their working and results to the Parliament.

(xii) Objective : It is service oriented and not profit oriented. It works efficiently to earn profit which is used for its day to day functioning.

All those activities that are performed to earn income are collectively called the business. These activities are done with some objective or aim behind them.

 Example: 

A trader sells goods in his shop, or a factory owner produces goods in a factory.

Characteristics of Business: 

(1) It is an Economic Activity – The main aim of the business is to earn a profit. A person does not start a business due to love, affection or care. It is an economic activity. 

(2) Production or procurement of Goods and Services – A businessman undertakes production and distribution of goods and services to earn revenue, and in exchange of this price, he distributes the goods or services to consumers. 

(3) Sale or exchange of Goods/Services – One single transaction of sale or purchase does not constitute a business. Business involves dealing in goods and services on a regular basis. 

(4) Profit Earning – Every businessman, whether small or big, strives to achieve this objective. He invests capital, bears the risks or uncertainties, and endeavours to operate the business successfully. All this is done by him with an expectation of earning a profit. This way, profit is an incentive which motivates him to continue his business. 

(5) Uncertainty of Return – The uncertainty of return here refers to the risk of not getting the return in a proper course of time. Risk is the result of some adverse and unwanted events. Risk is attached to every business. It cannot be separated from the business. Most common reasons that generate risks include a change in fashion, strike, fire, theft, natural disaster, etc. 

(6) An element of Risk – Any business can come into existence when the person who is starting it can bear the risks behind it. Therefore, no business can be started without risk, and thus the businessman is also called risk-taker. 

(7) The satisfaction of Customer – The services of a business depends upon its customers. Therefore, the main aim of every businessman is to fulfil the needs of his customers and satisfy them. 

(8) Service Motto – Other than self profit motto, the aim of every business should also to provide service towards society, therefore, other than marketing, businesses also focus towards earning the goodwill of the society.

In the modern era, business is regarded as the basis for economic progress of any nation. Business is significant from an economic and social perspective.

The importance of business can be studied under the following points:

1. Optimum utilisation of Human Resources – 

Development of business facilitates the establishment of industries. This results in generating and increasing employment opportunities, therefore, making the best use of human resources for productive purposes. 

2. Growth in Employment Sources – 

The pace at which the population is increasing in India, the only unlimited source of employment among people is business. The available human resources can be utilized through business only. 

3. Enhancement of standard of Living – 

Business increases employment and income, which, in turn, increases the purchasing power of the economy. Those goods which are under – produced or not produced in the country, are imported from countries producing them under business. In this way, business increases the standard of living in society. The production of new and innovative goods is possible through the development of business. The consumption of such goods in large quantity helps in improving the standard of living of the members of the society. 

4. Output Specification – 

To survive in this competitive era, a business needs to run accordingly. Good value and qualitative product with low cost are compulsory to make a customer satisfied and because of labour specification, the production of the material specification is been done. 

5. Helpful in Creation of New Objects – 

As the human population is increasing day by day, the needs of people are also increasing at the same speed, hence changing their vision towards goods and services. 

6. Development of Education – 

Complicated business processes paves the way for Vocational Education. Under Vocational Education, in this century, many innovations have taken place, like management, science, accountancy, insurance, law, etc. without which, it is nearly impossible to run any business. 

7. Barometer of Economic Development – 

In true sense, business is the barometer of economic development of any nation. The level of development of business in a nation indicates the level of the economic prosperity of that nation. If we take a glance at the world’s economy, we will find that the prosperous and economically – strong countries like America, Britain, France, etc., have undergone economic development through business only. Keeping their needs in mind, many such institutions are finding out different and new products to keep their customers demand at pace.

Objectives of Business: 

All the activities done by humans are done with some objective or aim behind it. Business is also a human activity, having a predetermined objective. Without an objective, planning and execution of plans become very difficult. Predetermined objectives provide direction to the businessman and the businessman endeavours to achieve them. It will not be rational to say that the sole objective of a business is earning profits, rather it is also social service.

A businessman determines the optimum mix of human objectives along with the economic objective of earning profits, keeping in mind all the situations. The objective of .business is not only to satisfy the economic interest in profit maximization but also to protect the interest of workers, consumers and other sections of society because all these are affected by business and the success of the business depends on the cooperation and support extended by them.

We can classify the objectives of business into the following categories: 

1. Economic objectives 

2. Social or Service objectives 

3. Human objectives

1. Economic Objectives: 

This objective is the primary objective of the business. Economic objectives are intended to maximise profits, accelerate the growth of the business, increase market territories by increasing the number of consumers. 

From an economic point of view, a businessman has the following three objectives:

• Profit Earning 
• Innovation 
• Creation of Customers.

 2. Social or Service Objective: 

Society and business are strongly interrelated. Without society, the business doesn’t find its existence. The capital required by the business is raised from the society, the goods produced are consumed by society and businessman himself lives in society and is an integral part of society. The important objective of a business is to earn profits by considering social aspiration and social beliefs. If any business goes against society’s expectations, then its survival can be threatened. The business should work for social welfare. In today’s consumer market, where the consumer is the king of the market, the social objective holds more importance. 

The important social objectives of the business are explained below:

• Best possible use of scarce resources. 
• Social Welfare and Development work. 
• Availability of desired quality goods. 
• Charging fair and reasonable price. 
• Production and supply of novel and innovative goods. 
• Compliance of government rules and legislations.
• Regular Payment of government taxes.

 3. Human Objectives: 

In the present time, no businessman can look after entire business activities himself alone. With the increasing size of enterprises, people from outside are hired to get the work done. The capital requirements of the business are also not met by its own internal sources, and one has to depend on others for all this.

In this way, at every level of business, a businessman essentially comes across other members of society. It is an important objective of business to treat those persons coming in contact with the business, fairly. Humane behaviour is that which is expected by every person from others. 

The main human objectives of the business are given below:

• Humane behaviour with workers.
• Employee welfare and improvement in working conditions. 
• Fair remuneration to employees. 
• Fair return to investors. 
• To protect the interest of the consumers.

1. (ii) kind and compassionate

2. (iv) without hurt or harm

3. (iii) a strong and young boy

4. (i) asked for blessings for the young boy

Yes, the old lady was grateful to the young boy who had helped her. We know this because that night, in her home, she prayed to God to be kind to him.

It is winter. The time of the day is probably late afternoon or early evening.

The setting is a busy street, covered with a layer of fresh snow. The area is very crowded and there is a school nearby.

1. The woman’s feet were slow because of the snow. False

2. The people around her did not bother about her. True 

3. The schoolboys were happy. True 

4. A boy came immediately to help the old woman. False

Every businessman does his business to receive profit only. If the business is earning a profit, then only he can properly grow and utilize the available resources and protect himself from the upcoming future uncertainties. No business can run on its own, even in the state of momentum, it requires money and the inflow of money depends upon the profit earned by the business. 

From an economic point of view, a businessman has three objectives:

1. Profit Earning 

2. Innovation 

3. Creation of Customers

1. Profit Earning: 

Every businessman, whether small or big, strives to achieve this objective. He invests capital, bears the risks and endeavours to operate the business successfully in order to: 

• fulfil their physical needs. 
• protect themselves against future risks and uncertainties. 
• The growth of a business depends upon the profit it earns. 
• For the growth and expansion of the business.
• For optimum utilization of Resources.

 2. Innovation: 

Innovation refers to doing work with a totally new method or with a totally new idea. Innovation can be done in 2 ways in a business. By innovation in goods and services, or by innovation in supply. In the modern era, no business can be run without innovation. A good businessman should innovate according to the changing trends and fashion, according to the needs of the customers and changing the environment to satisfy their customer needs. 

3. Creation of Customers: 

Production in business is easy, but in the present competitive era, maintaining the utility of products and services is very difficult. If the products and services are good, only then the customer would be attached to them and then only the business would grow. 

According to Peter. F. Druker 

“Business has only one definition, i.e. to satisfy the customers.”

a. The woman was reluctant to cross the street by herself because 

1. the road was slippery because of the snow. 

2. there was heavy traffic of horse carriages on the road.

b. The school boys were in a happy mood because

1. school was over for the day. 

2. the snow was piled high on the road.

c. One of the schoolboys helped the old woman cross the street because

1. she was old and afraid to cross on her own 

2. he hoped someone, sometime, may lend a hand to his own mother when he was not around.

d. We must help those who are in need because

1. we too may need help one day.

2. we must show compassion to those in need.

1. Quick decision making – 

Sole trade refers to that form of business organization which is owned by a single individual, who is the recipient of all profits and bearer of all risks, and whenever he gets any profit, he takes its full benefit. 

2. Secrecy – 

The rights of any decision is kept to himself by any sole trader. Therefore, there is a lot of secrecy maintained in this business. He doesn’t need to reveal his documents. 

3. Direct Incentive – 

Here, only one person is the sole recipient of all the profits. This motivates the proprietor to work harder. 

4. Sense of accomplishment – 

When the sole trader receives profit/gain because of his/her own hard work, he/she feels proud of himself and becomes satisfied. 

5. Full control – 

In sole trade, the proprietor is the all in all of a business which he establishes and manages. As he operates the business himself, thus he has full control over the functioning and working of the business. 

6. Rare information and closure – 

A sole trade business requires hardly any legal formality to set it up. Anybody wishing to set up sole trade can do so without spending much time. Closure of business can also be done easily. 

7. Economic Scenario – 

Sole trader operates his business alone. He fully utilizes the resources available and tries to limit his expenses. 

8. Alertness – 

In case of loss, the sole trader has to bear the entire loss, so he always remains alert and well – versed with the business environment and situations. 

9. Individual Contact – 

To attract the customers, a sole trader behaves very politely and treats them with courtesy. This enables him to maintain personal contact with the customers. 

10 Advantage of ancestral goodwill – 

As a sole trader runs the same business as his forefathers, therefore he enjoys the benefit of their goodwill. 

11. Good Relation with employees – 

A sole trader individually gets in touch with his employees and shares their happy and sad moments. And thus, the employees work with full devotion. 

12. Liberty of business – 

A sole trader is the only responsible person and has full liberty in every decision he takes including when to start and stop the business and make any changes that are needed to be implemented.

Sole trade refers to a form of business organisation which is owned, managed and controlled by an individual who is the recipient of all profits and bearer of all risks.

According to L.H. Haney, 

“The individual entrepreneurs in the form of business organisation on the head of which stands an individual as the one who is responsible, who directs its operations and who alone runs the risks of future.”

Characteristics of Sole Trader: 

1. Formation and closure of business – 

There is no seperate rule for a sole trader. Therefore, a sole trade business requires hardly any legal formality to set it up. Anybody wishing to stop a business can easily do it individually. 

2. Unlimited Liability – 

A sole trader has unlimited liability. And if the business faces any kind of loss and is not able to repay the loans, then he can repay them from his property too by selling his property for the same. 

3. Risk Bearer and Profit Recipient – 

The sole trader is solely responsible for bearing the risk, as well as he is the sole profit recipient. He does not share either of the two. Therefore, he enjoys full gain for his risk-bearing. 

4. Control – 

Insole trade, the proprietor is ‘all in all’ of a business which he establishes and manages. As he operates the business himself, thus he has full control over the functioning and working of the business. 

5. No Separate Entity – 

Both business and businessman are not different from each other. Business is not seperate from its trader. Therefore, the businessman is held responsible for every activity 

6. Lack of Business Continuity – 

Due to the same entity of business and its businessman, the business would discontinue on the death of the businessman, his becoming lunatic, on his becomes insolvent, etc. And the business would suffer from loss if the businessman is put behind the bars, gets sick or because of any other reason.

When two or more persons come together to fulfil a similar aim, it is called partnership. In other words, two or more than two persons mutually agree to invest their capital, skill and labour in some lawful business with an objective to share the profits therefrom, according to business agreement of partnership.

Advantages of Partnership: 

1. Ease in Formation and Closure – 

It is very easy and simple partnership to form due to a virtue of an agreement. The registration of such a firm is also not compulsory. 

2. Balanced Decision Making – 

The businessmen do business in their field of expertise. That helps in balancing their decisions and earn profits in return. 

3. Sharing of Risks – 

The risks are shared equally among all the partners. In this way, no single partner has to bear all the risk. 

4. Secrecy – 

A partnership firm does not need to publish the details of their business. Therefore, information is kept confidential in a partnership firm. 

5. Individual Supervision – 

In partnership, the task of management is performed by partners only and therefore every employee receives individual supervision. 

6. Flexible Nature – 

Partnership is a flexible organisation as its formation is based on agreement, and if needed, partners can jointly come together and invest in the business. Also, the business can be contracted or expanded according to the wish of the partners. 

7. Merits of Specialisation – 

One individual cannot be an expert in everything. Different partners can perform different functions according to their area of specialisation and ability 

8. Increase in working efficiency – 

Every partner works with full efficiency and thinks of every work as his/her own. The workers can also remain relaxed because of this, hence they do their work effectively. 

9. Ease in operation – 

There are better efficiency and continuity in partnership than sole trading. Even if one partner gets sick and is not able to perform regular activities, then other partners can take care of the situation and hence it does not affect the continuity of the business. 

Limitations of Partnership: 

1. Unlimited Liability – 

The partners in this business are liable to repay the debts even from their personal resources in case the business is not efficient enough to meet its debts, which creates a risk to their personal property. 

2. Limited Resources – 

The number of partners is less in partnership. Therefore, their contribution in terms of capital investment is also less. All of this may result in the limitation of business opportunities. 

3. Possibility of conflicts – 

Partnership means a group of people involved in a business who take decisions in their own fields. This might result in personal conflict. 

4. Lack of continuity – 

In case of either death of a partner, or he/she becoming insolvent, becoming lunatic can result in dissolving of the partnership firm. This results in its discontinuation. 

5. Lack of Public confidence – 

A partnership is not legally required to publish its financial reports or make other related information public. So here, the partnership business lacks public faith. 

6. Ban of transfer of shares – 

No partner can transfer his share to a third party without the consent of other partners. Therefore, the partner feels restricted. 

7. Uncertain Existence – 

The existence of a partnership is uncertain; because in case if a partner dies, becomes lunatic or willingly separates himself from the firm, then automatically, the partnership discontinues. 

8. Lack of cooperation – 

Partners start seeing their own interest in a partnership firm. This results at the beginning of the conflict between them. 

9. Loss due to the carelessness of partners – 

It one member shows negligence in his work, it affects every other member including those who do their work effectively. 

10. Lack of Quick Decision – 

All important decisions are taken by consent of partners, so decision – making process becomes time-consuming.

Correct option is (A) 5 : 7

Answer is (B) vector(a.(c x b)) = -3

3x/4 – (x-1)/2 = (x-2)/3

Let us rearrange the equation

3x/4 – (x-1)/2 – (x-2)/3 = 0

By taking LCM for 4, 2 and 3 which is 12

(9x – (x-1)6 – (x-2)4)/12 = 0

(9x – 6x + 6 – 4x + 8)/12 = 0

(-x + 14)/12 = 0

By cross-multiplying

-x + 14 = 0

x = 14

Let us verify the given equation now,

3x/4 – (x-1)/2 = (x-2)/3

By substituting the value of ‘x’ we get,

3(14)/4 – (14-1)/2 = (14-2)/3

42/4 – 13/2 = 12/3

(42 – 26)/4 = 4

16/4 = 4

4 = 4

Hence, the given equation is verified.

x – (x-1)/2 = 1 – (x-2)/3

Let us rearrange the equation

x – (x-1)/2 + (x-2)/3 = 1

By taking LCM for 2 and 3 which is 6

(6x – (x-1)3 + (x-2)2)/6 = 1

(6x – 3x + 3 + 2x – 4)/6 = 1

(5x – 1)/6 = 1

By cross-multiplying

5x – 1 = 6

5x = 6 + 1

x = 7/5

Let us verify the given equation now,

x – (x-1)/2 = 1 – (x-2)/3

By substituting the value of ‘x’ we get,

7/5 – (7/5 – 1)/2 = 1 – (7/5 – 2)/3

7/5 – (2/5)/2 = 1 – (-3/5)/3

7/5 – 2/10 = 1 + 3/15

(14 – 2)/10 = (15+3)/15

12/10 = 18/15

6/5 = 6/5

Hence, the given equation is verified.

x − 2 = 7

Transposing 2 to R.H.S, we obtain

x = 7 + 2 = 9

x – 2x + 2 – 16/3x + 5 = 3 – 7/2x

Let us rearrange the equation

x – 2x – 16x/3 + 7x/2 = 3 – 2 – 5

By taking LCM for 2 and 3 which is 6

(6x – 12x – 32x + 21x)/6 = -4

-17x/6 = -4

By cross-multiplying

-17x = -4×6

-17x = -24

x = -24/-17

x = 24/17

Let us verify the given equation now,

x – 2x + 2 – 16/3x + 5 = 3 – 7/2x

By substituting the value of ‘x’ we get,

24/17 – 2(24/17) + 2 – (16/3)(24/17) + 5 = 3 – (7/2)(24/17)

24/17 – 48/17 + 2 – 384/51 + 5 = 3 – 168/34

By taking 51 and 34 as the LCM we get,

(72 – 144 + 102 – 384 + 255)/51 = (102 – 168)/34

-99/51 = -66/34

-33/17 = -33/17

Hence, the given equation is verified.

(3a-2)/3 + (2a+3)/2 = a + 7/6

Let us rearrange the equation

(3a-2)/3 + (2a+3)/2 – a = 7/6

By taking LCM for 2 and 3 which is 6

((3a-2)2 + (2a+3)3 – 6a)/6 = 7/6

(6a – 4 + 6a + 9 – 6a)/6 = 7/6

(6a + 5)/6 = 7/6

6a + 5 = 7

6a = 7-5

6a = 2

a = 2/6

a = 1/3

Let us verify the given equation now,

(3a-2)/3 + (2a+3)/2 = a + 7/6

By substituting the value of ‘a’ we get,

(3(1/3)-2)/3 + (2(1/3) + 3)/2 = 1/3 + 7/6

(1-2)/3 + (2/3 + 3)/2 = (2+7)/6

-1/3 + (11/3)/2 = 9/6

-1/3 + 11/6 = 3/2

(-2+11)/6 = 3/2

9/6 = 3/2

3/2 = 3/2

Hence, the given equation is verified.

(i) y + 3 = 10

Transposing 3 to R.H.S, we obtain

y = 10 − 3 = 7

(ii) 6 = z + 2

Transposing 2 to L.H.S, we obtain

6 − 2 = z

z = 4

1/2x + 7x – 6 = 7x + 1/4

Let us rearrange the equation

1/2x + 7x – 7x = 1/4 + 6 (by taking LCM)

1/2x = (1+ 24)/4

1/2x = 25/4

By cross-multiplying

4x = 25 × 2

4x = 50

x = 50/4

x = 25/2

Let us verify the given equation now,

1/2x + 7x – 6 = 7x + 1/4

By substituting the value of ‘x’ we get,

(1/2) (25/2) + 7(25/2) – 6 = 7(25/2) + 1/4

25/4 + 175/2 – 6 = 175/2 + 1/4

By taking LCM for 4 and 2 is 4

(25 + 350 – 24)/4 = (350+1)/4

351/4 = 351/4

Hence, the given equation is verified.

(6x+1)/2 + 1 = (7x-3)/3

(6x + 1 + 2)/2 = (7x – 3)/3

By cross-multiplying

(6x + 3)3 = (7x – 3)2

18x + 9 = 14x – 6

18x – 14x = -6 – 9

4x = -15

x = -15/4

Let us verify the given equation now,

(6x+1)/2 + 1 = (7x-3)/3

By substituting the value of ‘x’ we get,

(6(-15/4) + 1)/2 + 1 = (7(-15/4) – 3)/3

(3(-15/2) + 1)/2 + 1 = (-105/4 -3)/3

(-45/2 + 1)/2 + 1 = (-117/4)/3

(-43/4) + 1 = -117/12

(-43+4)/4 = -39/4

-39/4 = -39/4

Hence, the given equation is verified.

5(7x+5)/3 – 23/3 = 13 – (4x-2)/3

By rearranging

(35x + 25)/3 + (4x – 2)/3 = 13 + 23/3

(35x + 25 + 4x – 2)/3 = (39+23)/3

(39x + 23)/3 = 62/3

By cross-multiplying

(39x + 23)3 = 62(3)

39x + 23 = 62

39x = 62 – 23

39x = 39

x = 1

Let us verify the given equation now,

5(7x+5)/3 – 23/3 = 13 – (4x-2)/3

By substituting the value of ‘x’ we get,

(35x + 25)/3 – 23/3 = 13 – (4x-2)/3

(35+25)/3 – 23/3 = 13 – (4-2)/3

60/3 – 23/3 = 13 – 2/3

(60-23)/3 = (39-2)/3

37/3 = 37/3

1 = 1

Hence, the given equation is verified.

3/4x + 4x = 7/8 + 6x – 6

Let us rearrange the equation

3/4x + 4x – 6x = 7/8 – 6

By taking 4 and 8 as LCM

(3x + 16x – 24x)/4 = (7 – 48)/8

-5x/4 = -41/8

By cross-multiplying

-5x(8) = -41(4)

-40x = -164

x = -164/-40

= 82/20

= 41/10

Let us verify the given equation now,

3/4x + 4x = 7/8 + 6x – 6

By substituting the value of ‘x’ we get,

(3/4)(41/10) + 4(41/10) = 7/8 + 6(41/10) – 6

123/40 + 164/10 = 7/8 + 246/10 – 6

(123 + 656)/40 = (70 + 1968 – 480)/80

779/40 = 1558/80

779/40 = 779/40

Hence, the given equation is verified.

(45-2x)/15 – (4x+10)/5 = (15-14x)/9

By rearranging

(45-2x)/15 – (4x+10)/5 – (15-14x)/9 = 0

By taking LCM for 15, 5 and 9 which is 45

((45-2x)3 – (4x+10)9 – (15-14x)5)/45 = 0

(135 – 6x – 36x – 90 – 75 + 70x)/45 = 0

(28x – 30)/45 = 0

By cross-multiplying

28x – 30 = 0

28x = 30

x = 30/28

= 15/14

Let us verify the given equation now,

(45-2x)/15 – (4x+10)/5 = (15-14x)/9

By substituting the value of ‘x’ we get,

(45-2(15/14))/15 – (4(15/14) + 10)/5 = (15 – 14(15/14))/9

(45- 15/7)/15 – (30/7 + 10)/5 = (15-15)/9

300/105 – 100/35 = 0

(300-300)/105 = 0

0 = 0

Hence, the given equation is verified.

3/7 + x = 17/7

Transposing 3/7 to R.H.S, we obtain

x = 17/7 - 3/7 = 14/7 = 2

Now, find the value of x by considering the given equation,=3 + 2x = 1 – x

Transposing 3 to RHS and it becomes -3 and -x to LHS it becomes x

= 2x + x = 1 – 3

= 3x = -2

Multiplying both side by (1/3)

= 3x × (1/3) = -2 × (1/3)

= x = (-2/3)

By substituting -2/3 in the place of x in given equation, we get

LHS,

= 3 + 2 × (-2/3)

= 3 + (-4/3)

= 3 – (4/3)

= (9 – 4) / 3

= 5/3

RHS,

= 1 – (-2/3)

= 1 + (2/3)

= (3 + 2)/ 3

= 5/3

By comparing LHS and RHS

= 5/3 = 5/3

∴ LHS = RHS

Hence, the result is verified.

The above question can be written as,= 2x – 4 + 12x – 3 = 0

Now, find the value of x by considering the above equation,

= 2x + 12x – 4 – 3 = 0

= 14x – 7 = 0

Transposing -7 to RHS and it becomes 7

= 14x = 7

Multiplying both side by (1/14)

= 14x × (1/14) = 7 × (1/14)

= x = (7/14)

= x = 1/2

By substituting 1/2 in the place of x in given equation, we get

LHS,

= (2 × (1/2)) – 4 + (12 × (1/2)) – 3

= 1– 4 + 6 – 3

= 7 – 7

= 0

By comparing LHS and RHS

= 0 = 0

∴ LHS = RHS

Hence, the result is verified.

6x = 12

Dividing both sides by 6, we obtain

6x/6 = 12/6

x = 2 

The above question can be written as,= 10x – 15 – 9x + 21 = 5

Now, find the value of x by considering the given equation,

= 10x – 15 – 9x + 21 = 5

= x + 6 = 5

Transposing 6 to RHS and it becomes -6

= x = 5 – 6

= x = -1

By substituting -1 in the place of x in given equation, we get

LHS,

= 10 (-1) -15 -9(-1) + 21

= -10 – 15 + 9 + 21

= -25 + 30

= 5

By comparing LHS and RHS

= 5 = 5

∴ LHS = RHS

Hence, the result is verified.

t/5 = 10

Multiplying both sides by 5, we obtain

t/8 x 5 = 10x5

t = 50

Now, find the value of x by considering the given equation,= 2x – (1/3) = (1/5) – x

Transposing – (1/3) to RHS and it becomes (1/3) and – x to LHS it becomes x

= 2x + x = (1/5) + (1/3)

= 3x = (3 + 5) / 15

= 3x = 8/15

Multiplying both side by (1/3)

= 3x × (1/3) = (8/15) × (1/3)

= x = (8/45)

By substituting (8/45) in the place of x in given equation, we get

LHS,

= 2 × (8/45) – (1/3)

= (16/45) – (1/3)

= (16 – 15) / 45

= (1/45)

RHS

= (1/5) – (8/45)

= (9-8) / 45

= (1/45)

By comparing LHS and RHS

= (1/45) = (1/45)

∴ LHS = RHS

Hence, the result is verified.

Now, find the value of x by considering the given equation,= (1/2)x- 3 = 5 + (1/3)x

Transposing -3 to RHS and it becomes 3 and (1/3)x to LHS it becomes (-1/3)x

= (1/2)x – (1/3)x = 5 + 3

= ((3-2)/6)x = 8

= (1/6)x = 8

Multiplying both side by 6

= 6 × (1/6)x = 8 × 6

= x = 48

By substituting 48 in the place of x in given equation, we get

LHS,

= (1/2) × (48) – 3

= (48/2) – 3

= 24 – 3

= 21

RHS,

= 5 + (1/3) × (48)

= 5 + (48/3)

= 5 + 16

= 21

By comparing LHS and RHS

= 21 = 21

∴ LHS = RHS

Hence, the result is verified.

Now, find the value of x by considering the given equation,= (x/2) + (x/4) = (1/8)

= ((2 + 1) / 4) = (1/8)

= (3/4)x = (1/8)

Multiplying both side by (4/3)

= (3/4)x × (4/3) = (1/8) × (4/3)

= x = (1/6)

By substituting (1/6) in the place of x in given equation, we get

LHS,

= (3/4)x

= (3/4) × (1/6)

= (1/8)

RHS,

= (1/8)

By comparing LHS and RHS

= (1/8) = (1/8)

∴ LHS = RHS

Hence, the result is verified.

2x/3 = 18

Multiplying both sides 3/2, we obtain

2x/3 x 3/2 = 18 x 3/2

x = 27

The above question can be written as,= 3x + 2x + 4 = 20 – 2x + 5

Now, find the value of x by considering the above equation,

= 5x + 4 = 20 – 2x + 5

Transposing 4 to RHS and it becomes -4 and -2x to LHS it becomes 2x

= 5x + 2x = 20 – 4 + 5

= 7x = 25 – 4

= 7x = 21

Multiplying both side by (1/7)

= 7x × (1/7) = 21 × (1/7)

= x = 21/7

= x = 3

By substituting 3 in the place of x in given equation, we get

LHS,

= 5x + 4

= 5 × (3) + 4

= 15 + 4

= 19

RHS,

= 20 – 2x + 5

= 20 – 2 × (3) + 5

= 20 – 6 + 5

= 25 – 6

= 19

By comparing LHS and RHS

= 19 = 19

∴ LHS = RHS

Hence, the result is verified.

The above question can be written as,= 13y – 52 – 3y + 27 – 5y – 20 = 0

Now, find the value of y by considering the above equation,

= 13y – 3y – 5y – 52 + 27 – 20 = 0

= 13y – 8y – 72 + 27 = 0

= 5y – 45 = 0

Transposing -45 to RHS and it becomes 45

= 5y = 45

Multiplying both side by (1/5)

= 5y × (1/5) = 45 × (1/5)

= y = 9

By substituting 9 in the place of y in given equation, we get

LHS,

= 5y – 45

= 5 × (9) – 45

= 45 – 45

= 0

RHS,

= 0

By comparing LHS and RHS

= 0 = 0

∴ LHS = RHS

Hence, the result is verified.