## Types of Fractions

The fractions can be classified in to four types like fractions, unlike fractions, Proper fraction & improper fraction. This differentiation is vital in various mathematical operations.

fractions having the same denominator are called as like fraction. For example, 2/4 and 3/4 are like fractions as the denominator 4 is same for both the fractions.

fractions having different denominators are called unlike fraction. For example, 3/4 and 1/7 are unlike-fractions as the denominator differs for both the fractions.

fractions having denominator greater than the numerator is called proper fraction. For example, 2/5 & 9/13 are proper fractions as the denominator is grater than numerator

fractions having numerator lesser than the denominator is called improper fraction. For example, 5/3 & 9/4 are improper fraction as the numerator is greater than denominator

## Fractions

Fractions are the rational numbers of the form p/q where q is a non zero number. For q = 0 the fraction would be undefined. Numbers for example 1/5, 2/7, 7/9 etc. represents the fraction and called as simple fractions.

Whereas the numbers like 5 3/4 are called as mixed fractions.

## Highest Common Factor (HCF)

The highest common factor is a quantity obtained from the given quantities and which divides each of them without leaving a remainder.

## Least Common Multiple (LCM)

The LCM is defined as that quantity which is divisible by the quantities of which it is the LCM without leaving the remainder.

## Unbounded Intervals

Intervals which expands indefinitely in both the directions are known as unbounded intervals.
1. (a,infinity) is the set of all real numbers x such that a < x.
2. (–infinity, a) is the set of all real numbers x such that x < a.
3. [a, +infinity) is the set of all real numbers x such that a <= x.
4. (–infinity, a] is the set of all real numbers x such that x <= a.

## Bounded Intervals

Let X and Y be fixed real numbers such that X < Y on a number line. Various types of intervals as as fallows
1. Open Interval: open interval (x, y) with end points x and y as a set of all real numbers “n”, such that x < n < y. i.e., the real number n will be taking all the values between a and b. An vital point to consider in this case is the type of brackets used. Generally open intervals are denoted by ordinary brackets ( ).

2. The closed interval [x, y]: closed interval [x, y] with end points x and y as a set of all real numbers “n”, such that x <= n <= y. In this case the real number n will be taking all the values between x and y inclusive of the end points x and y. Generally closed intervals are denoted by [ ] brackets.

3. The half open interval [x, y): a half open interval [x, y) with end points x and y as a set of all real numbers “n”, such that x <= n < y. In this case the real number n will be taking all the values between x and y, inclusive of only x but not y. The half open interval in different (x, y].

## Rules for inequalities

For real numbers X, Y, Z and N

X < Y, if and only if  (Y – X) > 0

If  X < Y and Y < Z , then X < Z

If  X < C then ( X + Z )  <  ( Y + Z )

If  X < Y, then  –  X >  –  Y

If X < Y, then
• N = 0, then X.N = Z.N
• N < 0, then X.N > Y.N
• N > 0, then X.N < Y.N

If  0 < X  < Y, then 0 < 1/X < 1/Y
The basic concept of inequalities is employed in-order to understand the intervals.

## Inequalities

To put it simple, its an alternate way to express statements.

If it is given that a real number ‘p’ is not less than another real number ‘q’ , then either p should be equal q or p should be greater than q. The p and q now can be expressed as p=q or p > q or p >= q, such types of statements are called inequalities.

These are inequalities because p and q may not be equal in every case, if it is so then there would be an equation i.e. p = q.

In the equation the p or q could be algebraic expressions, bearing situation specific values.

## Zero Factor Property of Multiplication

According to this property any real number X if multiplied by zero would yield a zero.

For Example:

A.0 = 0.X = 0
A.B = 0 [then either (A or B = 0)]

## Cancellation property

According to this property a constant quantity when present on both sides of the equation can be cancelled.

If A + P = A + Q then P = Q
If A . P = A. Q then P = Q

provided A is a non zero.

## Inverse property

Addition – According to this property for every element A there exists another element –A such that addition of both returns zero value.

Example: A+0 = A

Multiplication – According to this property for every element B (not being zero) there exists another element 1/B such that multiplication of both results in 1

Example: B.1 = B

## Identity property

For Addition – when 0 (identity element) is added to a real number it returns back the number itself

A + 0 = A

For Multiplication – when 1 (identity element) is multiplied to a real number it returns the same number

## Distributive property

Distributive property, as it name says this property distributes or expands the elements of expression For Example:
X.(Y+Z) = X.Y + X.Z

## Associative property

Associative property says elements can be grouped together in any manner, i.e the result of element will not change, no mater however its group. This property nullifies BODMAS rules.
For Example:

(X + Y) + Z = X + ( Y + Z )

(A .B) . C = A . ( B . C )

So, if an expression contains only the addition or multiplication sign, it can be grouped in any order. Combination of elements is not be applicable under this property

## Commutative Property

According to this property the addition and multiplication can be carried out in any order

A + B + C = B + A + C

A.B.C = C.B.A

So, the order of addition or multiplication will not affect the result in any way.

But for solving the expressions which contain more than one mathematical operator, order of solving becomes vital. Following the order, known as operational hierarchy or BODMAS, should be followed in solving the mathematical expression.

B All the brackets.

M/D Multiplication or Division.

## Properties of Real Numbers

Properties denote the basic characteristics of the real numbers. Without fail that must be followed in any mathematical processing.

It is very vital to know the basic properties which help in solving the expressions as well as adherence to the mathematical conventions.

Following Properties of the real numbers are explained for Addition and Multiplication

* Commutative property
* Associative property
* Distributive property
* Identity property
* Inverse property
* Cancellation property and
* Zero factor property for multiplication.

## Imaginary numbers

However, Real numbers does not include imaginary numbers.

Numbers can be classified in to two parts

1.Real numbers
2.Imaginary numbers

Imaginary numbers - Consider a number (-16)^1/2. The roots of (-16)^1/2 are +4i or -4i where i stands for an imaginary number.

Real numbers can be shown on number line. Number line shows the positive or negative real numbers.

## Real Numbers

All the number sets discussed in previous posts i.e., Natural numbers, Whole numbers, Integers and Rational numbers comprise the set of real numbers. So a set of natural/whole/integers numbers can be termed as a sub set of real numbers.

The numbers used to measure exact quantities such as length, area, volume, temperature; GNP, GDP, growth rate, inflation etc. are called as real numbers.

A Set of real numbers includes set of Natural numbers, Whole numbers, Integers and Rational numbers so it can be represented as "R = { N, W, I, Q }"

## Rational Numbers

Even a set of integer is inadequate, because it does not include rational numbers like, 3/4, -8/9 etc.

Rational numbers are of the form x / y  (p/q) where x (p) and y (q) are integers and second condition is that y (q) must be a non-zero otherwise that number would be undefined. For example, number 0/5 is a rational number but its not the same for the number 5/0 as it is an undefined number.

The set of rational numbers are denoted Q and expressed as

Q = (... -2/5, -1/4, 100/99, 15/7...)

In a set of rational numbers decimal part may be terminating or not terminating or/and repeating. Numbers whose decimals are non terminating and non repeating are included in a set of numbers called Irrational Numbers.

## Integers

Set of whole numbers does not satisfy all requirements, also it does not include negative numbers. To overcome this disadvantage, a set of integers is constituted that also represent negative integers over the set of whole numbers.

This set is definitely superior to natural and whole numbers in the sense that it caters to a larger audience as compared to the other number system dealt so far.  A set of integers is denoted by "I" and represented as

I = { …. -4, -3, -2, -1, 0, 1, 2, 3, 4…. }

## Whole Numbers

One of the type of Number System. natural numbers does not have a zero. This shortcoming is made good when we consider the set of whole numbers. It consists of zero as well over the natural numbers.
The set of a whole number is represented by W and is expressed as

W = {0, 1, 2,……….}

## Natural Numbers

Natural numbers is first type of number system, in practice natural numbers are denoted by "N". The natural numbers are originated by adding 1 to the antecedent number starting from 1. i.e.. N = 1, 2, 3,…. to get an infinite series of the natural numbers.

It is always important to express such numbers as a set. Set is a collection of any well-defined objects. Each well defined individual object is also referred to as an element of that particular set. The concept of a set can be used and popularly used to represent infinite and finite number of elements.

For example: A Set of natural number is represented as

N = {1, 2, 3, 4, 5, 6,……} or {1, 2, 3, 4, 5}

## The Number Systems

The Number System deals with the biography of numbers. It elaborates the various types of numbers and various associated properties embed with it. The fundamental of any mathematical calculations are numbers. It is essential to understand the various types of numbers and their properties. Knowing it helps to strengthen the basics as well as to facilitate the mathematical calculations.

Types of Number system

1. Natural numbers
2. Whole numbers
3. Integers
4. Rational numbers
5. Real numbers.

## Reasons for Depreciation

There are four basic reasons for Depreciation. Those four reasons are
1. Wear & Tear of the Asset
2. Exhaustion
3. To Face Technological Obsolescence and
4. Accident

## Accident

The value of the asset mainly depends upon the efficiency and economy; which gets affected due to the accident.

## Technological Obsolescence

To replace the old machinery with new machinery before the expiry of the economic life period of the asset in order to maintain the efficiency and economy of the asset. The type writer was replaced by
the electronic typewriter during the yester periods of office automation. To replace the old type writer which is not efficient as well as economical, should be replaced by the new electronic typewriter through the depreciation charge on the old one

## Exhaustion

Nothing will be remaining due to the continuous extraction of resources. The resources in the oil wells, mine fields will become nothing due to continuous extraction should be replaced by new exploration. To invest on the new exploration in order to have continuous exploration which requires the depreciation as a charge against the revenues of the fields.

Example, Oil & Natural Gas Corporation Ltd. (ONGC) indulges in the process of new oil exploration projects through research projects. Then the new projects should be identified and invested by huge initial investment outlay through the current revenues out of the existing projects on account of replacement due to depletion of
resources.

## Wear and Tear of the Asset

The long term assets are becoming less efficient and poor quality in operations due to the continuous usage of the asset.

## Depreciation Accounting

The depreciation accounting is mainly based on the concept of income. The concept of income is matching of revenues with expenses. The goods purchased are frequently matched through immediate sale or within a year. The crux of the concept of income is that the expenses are to be matched against the revenues.

The ultimate aim of matching is done in order to determine the volume of profit or loss of the transaction. If the assets are nothing but long term assets procured by the enterprise should be matched against the revenues of them. The matching of expenditure of the assets incurred by the firm at the time of purchase against the revenues is the hard core task of the firm. Why it is being considered as a cumbersome task in matching ? The benefits/revenues of the fixed assets expected to accrue for many number of years but not within a year. The
initial investment on the assets at the time of purchase should be matched against the revenue pattern of the same year after year in order to find out the profitability of the long term investment. To have an effective matching against the revenues on every year, the amount of purchase has to be stretched. The stretching of expenses into many years is known as depreciation.

## Abandoned Option

Where an Option, i.e call option is allowed to run past its expiry date, rather than being sold or exercised. In other words, Where an option is neither sold nor exercised but allowed to lapse at expiry

## Accepting House

An organisation, typically a merchant bank, that accepts or guarantees exchange bills (bills of exchange) and so finances trade deals and goods are being shipped. It accepts the bills by agreeing to pay it at a discounted rate at some point in the future.

## Authorization act

A law or legislation under the jurisdiction of a committee other than the House and Senate Committees on Appropriations that establishes or continues the operation of a federal program or agency, either indefinitely or for a specified period.
An authorization act may suggest a level of budget authority needed to fund the program or agency, which is then provided in a future appropriation act. However, for some programs, the authorization itself may provide the budget authority.

## Appropriation act

A law or legislation under the jurisdiction of the House and Senate Committees on Appropriations that provides authority for federal programs or agencies to incur obligations and make payments from the Treasury.
Each year, the Congress considers regular appropriation acts, which fund the operations of the federal government for the upcoming fiscal year. The Congress may also consider supplemental, deficiency, or continuing appropriation acts (joint resolutions that provide budget authority for a fiscal year until the regular appropriation for that year is enacted).

## Alternative minimum tax

A tax intended to limit the extent to which higher-income people can reduce their tax liability (the amount they owe) through the use of preferences in the tax code.
Taxpayers subject to the Alternative minimum tax(AMT) are required to recalculate their tax liability on the basis of a more limited set of exemptions, deductions, and tax credits than would normally apply. The amount by which a taxpayer’s AMT calculation exceeds his or her regular tax calculation is that person’s AMT liability.

Budget authority provided in an appropriation act that is first available for obligation in a fiscal year after the year for which the appropriation was enacted. The amount of the advance appropriation is included in the budget totals for the year in which it will become available.

All income that is subject to taxation under the individual income tax after "above-the-line" deductions for such things as alimony payments and certain contributions to individual retirement accounts.
Personal exemptions and the standard or itemized deductions are subtracted from adjusted gross income (AGI) to determine taxable income.

## Accrual accounting

A system of accounting in which revenues are recorded when they are earned and outlays are recorded when goods are received or services are performed, even though the actual receipt of revenues and payment for goods or services may occur, in whole or in part, at a different time.

## Gross Domestic Product

In economics, a country's GDP is the total value of goods and services produced within a country in a year, not including its income from investments in other countries. GDP is an abbreviation for `gross domestic product'.

GDP is the standard measure of the size of the economy. It is the total production of goods and services within the country. The total value of a nation's output, income, or expenditure produced within a nation's physical borders. One of the main measures of economic activity.

## Crisis

crisis is a situation in which something or someone is affected by one or more very serious problems. Crisis is an unstable situation of extreme danger or difficulty; "they went bankrupt during the economic crisis".

People use crisis management to refer to a management style that concentrates on solving the immediate problems occurring in a business rather than looking for long-term solutions.

## Budget

Budget is sum of money allocated for a particular purpose. Its a summary of intended expenditures along with proposals for how to meet them for particular span of time.
The budget of an organization or country is its financial situation, considered as the difference between the money it receives and the money it spends. In other words its estimate of the income and expenditures for a future period of time, usually one year

budgeting - The activity of constructing a budget

## Provision for Discount on Creditors

Similar to cash discount allowed to debtors, the firm may have a chance to receive the cash discount from the creditors for prompt payment. Provision for discount on Creditors is calculated at a certain percentage on Sundry Creditors.

Provision for discount on creditors will be shown on the credit side of Profit and Loss account and on the liabilities side of the Balance sheet by way of deduction from Sundry creditors.

## Provision for Discount on Debtors

To motivate the debtors to make prompt payments, cash discount may be allowed to them. After providing provision for bad and doubtful debts, the remaining debtors are called as good debtors. They may pay their dues in time and avail themselves of the cash discount permissible. So a provision for discount on good debtors at a certain percentage may have to be created.

Provision for discount on debtors will be shown on the debit side of Profit and Loss account and on the asset side of the Balance sheet by way of deduction from Sundry debtors (after deducting bad debts written off and provision for bad and doubtful debts).

## Provision for Bad and Doubtful Debts

Every business suffers a percentage of bad debts over and above the debts definitely known as irrecoverable and written off as Bad (Bad debts written off). If Sundry debtors figure is to be shown correctly in the Balance sheet provision for bad and doubtful debts must be adjusted.

This Provision for bad and doubtful debts is generally provided at a certain percentage on Debtors, based on past experience. While preparing final accounts, the bad debts written off given in adjustment is first deducted from the Sundry debtors then on the balance amount (Sundry debtors – Bad debt written off) provision for bad and doubtful debts calculated.

Provision for bad and doubtful debts will be shown on the debit side of Profit and Loss Account and on the assets side of the Balance sheet by way of deduction from Sundry debtors (after Bad debts written off if any).

Debts which cannot be recovered are called bad debts. It is a loss for the business.

Bad debts will be shown on the debit side of Profit and Loss account and on the assets side of the Balance Sheet by way of deduction from sundry debtors.

## Depreciation

Depreciation is the cut in the value of fixed assets due to its use or obsolescence. By and large depreciation is charged at some percentage on the value of fixed asset.

Depreciation will always materialized on the debit side of Profit and Loss(P&L) account and on the assets side of the Balance Sheet by way of deduction from the value of concerned asset.

## Interest on Investment

Interest receivable on investments is an income for the business.

Accrued interest on investments (outstanding interest receivable) will always materialized on the credit side of the Profit and Loss(P&L) account by way of addition to the appropriate interest account and On the assets side of the balance sheet by way of addition to the investments account.

## Interest on Loan - Outstanding

Loans are the accumulated amount which is materialized in liabilities side of Balance sheet, it is derived by summing up all the borrowings from banks, financial institutions and other outsiders.

If loan is taken, it is an liability of the proprietor to pay back it with an Interest, that interest is accounted as expenses for the business.

Interest on loan outstanding will always metalized on the debit side of the Profit and Loss account adding it to the apt interest account and on the liability side of the Balance sheet by adding it to the particular loan account.

## Interest on Drawings

Amount withdrawn from the business by the proprietor for his personal use is booked as drawings. Interest on drawings is treated as an income for the business and by the way it will reduce the capital of the owner. This is accounted like this in order to protect the capital in the business and to discourage drawings.

Interest on drawings will always be materialized on the credit side of Profit and Loss(P&L) account and on the liabilities side of the Balance Sheet by adding it to Drawings a/c which are eventually deducted from the capital.

## Interest on Capital

In progress of assessing the profitability of the business it is required to charge interest on capital at a certain rate. It also on the way helps to benchmark the profit standard at par to prevailing rate of income in the market.

Interest on capital will always be materialize on the debit side of Profit and Loss(P&L) account and on the liabilities side of the Balance Sheet by adding it to the capital.

Income inward during a particular accounting period for the work or service to be done in future period is booked as income received in advance.

Incomes received in advance will always shown on the credit side of the Profit and Loss(P&L) account by subtracting the same from the respective income and on the liabilities side of the Balance sheet.

## Accrued Incomes or Outstanding Incomes

Income which is liable to be received but not received during the accounting period is booked as accrued income.

Accrued income will always be shown on the credit side of Profit and Loss(P&L) account by adding it to respective income and on the assets side of the Balance Sheet.

## Prepaid Expenses

Expenses which are paid in advance are called as Unexpired(prepaid) expenses. That is, the expenses pertaining to upcoming accounting period is booked in present books.

Prepaid expenses will always materialized on the debit side of the Profit and Loss(P&L) account by deducting from the respective expenses and on the assets side of the Balance Sheet.