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Lesson Posted on 17/03/2021 Learn ICSE Schools

Suvam Banerjee

I am a B.Tech Graduate from the West Bengal University of Technology giving home tuition. I have more...

Expected Utility: TheoremsThe theory of expected utility was developed by the founders of game theory, namely John von Neumann and Oskar Morgenstern, in their 1944 book Theory of Games and Economic Behavior.In a rather unconventional way, we shall first (in this section) state the main result of the... read more

Expected Utility: Theorems

The theory of expected utility was developed by the founders of game theory, namely John von Neumann and Oskar Morgenstern, in their 1944 book Theory of Games and Economic Behavior.

In a rather unconventional way, we shall first (in this section) state the main result of the theory (which we split into two theorems) and then (in the following section) explain the assumptions (or axioms) behind that result. The reader who is not interested in understanding the conceptual foundations of expected utility theory, but wants to understand what the theory says and how it can be used, can study this section and skip the next. Let O be a set of basic outcomes. Note that a basic outcome need not be a sum of money: it could be the state of an individual’s health, or whether the individual under consideration receives an award, or whether it will rain on the day of her planned outdoor party, etc. Let L (O) be the set of simple lotteries (or probability distributions) over O. We will assume throughout that O is a finite set: O = {o1,o2,..., om} (m ≥ 1). Thus, an element of L (O) is of the form o1 o2 ... om p1 p2 ... pm with 0 ≤ pi ≤ 1, for all i = 1,2,...,m, and p1 + p2 +...+ pm = 1. We will use the symbol L (with or without subscript) to denote an element of L (O), that is, a simple lottery. Lotteries are used to represent situations of uncertainty.

For example, if m = 4 and the individual faces the lottery L = o1 o2 o3 o4 2 5 0 1 5 2 5 , then she knows that, eventually, the outcome will be one and only one of o1,o2,o3,o4, but does not know which one; furthermore, she can quantify her uncertainty by assigning probabilities to these outcomes. We interpret these probabilities either as objectively obtained from relevant (past) data or as subjective estimates by the individual.

For example, an individual who is considering whether or not to insure her bicycle against theft for the following 12 months knows that there are two relevant basic outcomes: either the bicycle will be stolen, or it will not be stolen. Furthermore, she can look up data on past bicycle thefts in her area and use the proportion of stolen bicycles as an “objective” estimate of the probability that her bicycle will be stolen.

Alternatively, she can use a more subjective estimate: she might use a lower probability of theft than suggested by the data because she knows herself to be very conscientious and – unlike other people – always to lock her bicycle when left unattended.

The assignment of zero probability to a particular basic outcome is taken to be an expression of belief, not impossibility: the individual is confident that the outcome will not arise, but she cannot rule out that outcome on logical grounds or by appealing to the laws of nature.

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Comments Answered on 20/01/2021 Learn ICSE Schools

Sumit Roy

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Highcourt judges are promoted to that position.

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Answers 2 Comments Lesson Posted on 19/10/2020 Learn ICSE Schools

Bimal Upadhyay

I am a teacher in cbse affiliate school and persuing Msc(Mathematics) from distance .I have been teaching...

Goods and Services Tax (GST)It is an indirect tax levied by the Central Government. In this tax ·there are so many taxesmerged such as Excise Duty, Customs Duty, Service Tax, Value Added Tax,Entertainment Tax and Lottery Tax etc. The purpose of GST is to make one nationand one tax. Important... read more

**Goods and Services Tax (GST)**

It is an indirect tax levied by the Central Government. In this tax ·there are so many taxes

merged such as Excise Duty, Customs Duty, Service Tax, Value Added Tax,

Entertainment Tax and Lottery Tax etc. The purpose of GST is to make one nation

and one tax.

**Important Terms Related to GST**

There are various terms related to GST, which are as follow

(i) Manufacturer The person who produces goods for sale, is called the manufacturer.

(ii) Dealer The person, who purchases goods for resale, is known as a dealer (trader).

(iii) Turnover The total amount received from the sale of goods (excluding tax) by a dealer

during any fixed period, is called turnover.

(iv) Intra State Sales Suppose a person make a business and they sales their goods (items)

and provide their services in the same state (or union territory), then it is said to

be Intrastate sales.

(v) Inter-State Sales Suppose a person make a business and they sales their goods (items)

and provide their services outside the state (or union territory), then it is said to be

Interstate sales.

(vi) Input GST The tax paid by a dealer on his/her purchase of goods and providing

services is called Input GST.

(vii) Output GST The tax charged by a dealer on his/her sales of goods and providing

services is called output GST.

**Different Types of Taxes Used in GST**

In this system, there are three taxes applicable, which are given below

(i) State (or Union Territories) Goods and Service Tax (SGST or UTGST) the State collects this tax (or union territory) Government on an intra-state sale. e.g. Suppose any goods are manufacturing in Uttar Pradesh and sales these goods also in Uttar Pradesh, then tax on this transaction is said to be SGST.

(ii) Central Goods and Services Tax (CGST) This tax is collected by the Central Government on an intrastate sale.

Both SGST (or UTGST) and CGST are levied on intrastate (i.e. same state) sales of goods and services. In intra-state sales, GST is equally divided between Central and State Governments.

e.g. Suppose a dealer of Tamil Nadu sells some goods at the rate of Rs4000 to the consumer in the same state (i.e. Tamil Nadu). Suppose GST is charged at the rate of 18% on that goods, the GST will comprise of CGST at the rate of 9% and SGST at the rate of 9%. Therefore, the seller will collect the CGST amount of Rs 360 (i.e. 9% of4000), and this amount goes to the account of the Central Government and seller will also collect the SGST amount of Rs360 (i.e. 9% of4000) which will go to the account of the State Government of TamilNadu.

Hence, the dealer collects the 18% GST amount on Rs4000 (i.e. Rs720), which will equally (i.e. 360 each)

distribute in State (i.e. Tamil Nadu) and Central

Government

(iii) Integrated Goods and Services Tax (IGST) It is levied on interstate sales of goods and services outside the state. This tax is also levied on the import of goods and services from one country to another country.

This tax is collected only by the Central Government for inter-state sales. e.g. Suppose a dealer from Gujarat sells goods worth of8000 to another dealer in Rajasthan. Suppose the rate of GST is 12% on the goods, then the seller will collect 12% of 8000 (i.e. 960) under as IGST, and the whole amount of IGST will go to the Central Government.

**Objectives and Advantages of GST**

Some of the objectives and advantages are given below

(i) The vital advantage is that to reduce the multiplicity of taxes and make a unified common national market.

(ii) Through GST, the tax system becomes more transparent, regular, and predictable.

(iii) Ease of doing business It means there are some administrative rule set up between the Central

Government and State Government, so that there is no interruption of doing business.

(iv) Reduce Tax Evasion Some of the persons are evading tax. To reduce tax evasion, use GST. In this

procedure, each taxpayer registered under GST to make a GST return file electronically for each transaction (either purchase or sale), and this file should match with the input GST credit against the output GST and lastly paid net GST.

**Basic Terms Required to Solve the Questions**

There are various basic terms, which are as follow

(i) **Cost Price (CP)** The price, at which an article is bought, is called its cost price. All the overhead expenses in the transaction like freight, damage, etc., are added to the cost price.

(ii) **Selling Price (SP)** The price, at which an article is sold, is called its selling price.

(iii) **Profit** When the selling price of a commodity is more than its cost price, then we are in profit, and Profit= SP - CP gives it and Profit%=(profit/cp) x 100

CP

(iv) Loss When the selling price of a commodity is less than its cost price, then we are in loss, and it is given by

Loss = CP- SP and Loss% = (loss/cp)X 100

CP

(v) Marked Price (MP) The price of an article excluding tax, is known as marked price. It is also, known as a list price, quoted price, printed price, catalogue price, an introductory price.

(vi) Discount The reduction in the price of the object given by the shopkeeper to the customer, is known as a discount.

Discount=(rate of discount x market price)/100

or Discount= Marked price - Selling price

**Example 1**. Mr Sharma goes to a shop and buys a Jacket having cost t 1180 (list price). The rate of GST

18%. He tells the shopkeeper to reduce the price to such an extent that he has to pay t 1180 inclusive of GST. Find the reduction needed at the price of the jacket.

** Solution **

Let the reduced price of the jacket bet Rs x.

Then, the amount of GST on Rs x = 18% of x=(18/100)*x

:. Mr Sharma pays the amount for the jacket = x+(18/100)x

=(59/50)X

According to the given condition,

(59/50)X=1180

or, x=1180*50/59

or, x=1000

:. Reduced price of the jacket= Rs 1000

Thus, the reduction needed at the price of the jacket

(1180-1000) = Rs180

**Example 2.** The price of a motorcycle is t 44880 including tax (under GST) at the rate of 18% on its listed price. A buyer asks for a discount on the listed price so that after charging GST, the selling price of the motorcycle becomes equal to the listed price. Find the discount amount in which the seller has to allow for the deal.

**Sol.**

Let the listed price of the motorcycle be Rs x and the discount bet y.

Amount of GST on Rs x = 18% of x=(18/100)x=(9/50)x

The selling price of the motorcycle including tax=x+(9/50)x

=(59/50)x

According to the given condition,

(59/50)x= 44880

or, x=44880*50/59

or, x=38034

:. The list price of the motorcycle = Rs 38034

Now, the reduced price of the motorcycle = (38034-y)

Amount of GST on(38034-y)

=18% of (38034-y)

=(9/50)(38034-y)

The new selling price of the motorcycle including GST

= (38034-y) +(9/50)(38034-y)

=(59/50) (38034-y)

According to the given condition, Selling price of motorcycle including GST = Listed price of the motorcycle

(59/50) (38034-y)=38033

or, 2244006- 59y = 1901650

or, 59y = 342356

or, y = 5803

Hence, the amount of discount is Rs5803.

**EXERCISE **

1. Miss Anjali goes to a mall to purchase a saree whose cost is Rs 885 (list price). She tells the shopkeeper to reduce the price to such an extent that she has to pay Rs 885, inclusive GST which is at the rate of 18%, find the reduction of price needed in the saree.

2. Sandeep purchased a digital camera for Rs 25488, which includes a 10% rebate on the list price and 18% tax (under GST) on the remaining price. Find the marked price of the digital camera.

3. The price of a spider toy is Rs3136 inclusive tax ( under GST) at the rate of 12% on its listed price. A buyer asks for a discount on the listed price so that after charging GST, the selling price becomes equal to the listed price. Find the amount of discount which the seller has to allow for the deal.

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Lesson Posted on 03/09/2020 Learn ICSE Schools

Shruti

Wrong Correct Cousin brother/sister Cousin Good name name Big blunder blunder Bad dream nightmare No place no room read more

Wrong | Correct |

Cousin brother/sister | Cousin |

Good name | name |

Big blunder | blunder |

Bad dream | nightmare |

No place | no room |

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Comments Lesson Posted on 28/06/2020 Learn ICSE Schools

Tiyasha

A Noun is a person, place, thing or idea. A vern is an action bird. A complete sentence must include a subject and a predicate. The only exception to the above rules is the imperative sentences. An Adjective can go directly before the noun they describe, or after it, if separated by a verb. (Eg:... read more

- A Noun is a person, place, thing or idea. A vern is an action bird.
- A complete sentence must include a subject and a predicate.
- The only exception to the above rules is the imperative sentences.
- An Adjective can go directly before the noun they describe, or after it, if separated by a verb. (Eg: The angry bird flew away)
- A compound subject includes two or more simple subjects.
- A compound predicate includes two or more predicate.
- A compound sentence includes more than one subject or predicate.

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Comments Lesson Posted on 23/06/2020 Learn ICSE Schools

All temperatures on the Kelvin scale are in positive figures.

Deleted

0 K or absolute zero is the lowest temperature on the Kelvin temperature scale and considered as the lowest temperature possible. This is the reason that all temperatures on the Kelvin scale are in positive figures. It was finding of J Charles that volume of a gas at -273.15°C will be zero or gas... read more

0 K or absolute zero is the lowest temperature on the Kelvin temperature scale and considered as the lowest temperature possible. This is the reason that all temperatures on the Kelvin scale are in positive figures.

It was finding of J Charles that volume of a gas at -273.15°C will be zero or gas will not exist at this temperature. This temperature is called absolute zero.

This scale is also called a thermodynamic scale of temperature, and degree sign is not used in this scale.

To convert Celsius temperature into Kelvin, we add 273.15 to Celsius temperature.

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Answered on 05/05/2020 Learn ICSE Schools

Mekhala M.

Experienced Preschool Teacher & Home Tutor, with 6+ years of Exp

5 hours per day intensive study

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Answers 11 Comments Answered on 11/09/2019 Learn ICSE Schools

Ravi

Dear Parents Be happy and proud of your little champ's acheivement. Only say him to write and to do practice of Capital and small case letters. Slowly within a month, your champ will be able to copy the letters fast.

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Answers 12 Comments Answered on 28/05/2020 Learn ICSE Schools

Suchan Kamat

Nursery-KG teacher for 5years. MSc with PGdiploma in Pre&primary Teaching

Human being says to the toy we have a heart which helps us stay alive. The toy says to human beings kids love us, which helps us stay alive in their homes.

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Lesson Posted on 06/03/2018 Learn ICSE Schools

How To Solve Compound Interest Sums?

Soumi Roy

I am an experienced, qualified tutor with over 8 years of experience in teaching maths and Science across...

The formula for Compound Amount: P n = P 2n = P 12n Also, A = CI + P Where, P= Principal R= Rate of Interest n=Time (in years) A= Amount CI= Compound Interest Solved Questions Questions 1: Find the amount if Rs 20000 is invested at 10% p.a. for 3 years. Solution: Using the... read more

The formula for Compound Amount:

P [1+ R/100]^{n} [When money is compounded annually]

= P [1+ R/(2*100)]^{2n} [When money is compounded half-yearly]

= P [1+ R/(12*100)]^{12n} [When money is compounded monthly]

Also, A = CI + P

Where,

P= Principal

R= Rate of Interest

n=Time (in years)

A= Amount

CI= Compound Interest

Solved Questions

Questions 1:

Find the amount if Rs 20000 is invested at 10% p.a. for 3 years.

Solution:

Using the formula: A= P [1+ R/100]^{n}

A = 20000 [1 + (10/100)]^{3}

On Solving, we get A = Rs. 26620

Question 2:

Find the CI, if Rs 1000 was invested for 1.5 years at 20% p.a. compounded half yearly.

Solution:

As it is said that the interest is compounded half yearly. So, the rate of interest will be halved and time will be doubled.

CI = P [1+(R/100)]^{n} - P

CI = 1000 [1+(10/100)]^{3}- 1000

On Solving, we get

CI = Rs. 331

Question 3:

The CI on a sum of Rs 625 in 2 years is Rs 51. Find the rate of interest.

Solution:

We know that A = CI + P

A = 625 + 51 = 676

Now going by the formula: A = P [1+(R/100)]^{n}

676 = 625 [1+(R/100)]^{2}

676/625 = [1+(R/100)]^{2}

We can see that 676 is the square of 26 and 625 is the square of 25

Therefore, (26/25)^{2} = [1+(R/100)]^{2}

26/25 = [1+(R/100)]

26/25 - 1 = R/100

On solving, R = 4%

Question 4:

A sum of money is put on CI for 2 years at 20%. It would fetch Rs 482 more if the interest is payable half yearly than if it were payable yearly. Find the sum.

Solution:

Let the Principal = Rs 100

When compounded annually,

A = 100 [1+20/100]^{2}

When compounded half yearly,

A = 100[1+10/100]^{4}

Difference, 146.41 - 144 = 2.41

If difference is 2.41, then Principal = Rs 100

If difference is 482, then Principal = 100/2.41 × 482

P = Rs 20000.

Question 5:

Manish invested a sum of money at CI. It amounted to Rs 2420 in 2 years and Rs 2662 in 3 years. Find the rate percent per annum.

Solution:

Last year interest = 2662 - 2420 = Rs 242

Therefore, Rate% = (242 * 100)/(2420 * 1)

R% = 10%

Important Formula: To find the difference between SI and CI for 2 years, we use the formula Difference = P[R/100]^{2}

Question 6:

The difference between SI and CI for 2 years at 20% per annum is Rs 8. What is the principal?

Solution:

Using the formula: Difference = P (R/100)^{2}

8 = P[20/100]^{2}

On Solving, P = Rs 200

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