Read BBA, BMS, B.Com Syllabus wise Notes
Let A₁, A₂, …, An, n arithmetic means are inserted between two numbers ‘a’ and ‘b’ such that a, A₁, A₂, …, An, b from an AP.
Here, total number of terms are (n + 2) and common difference be d
b = (n + 2)th term = a + (n + 2 – 1) d
d = (b – a)/ (n + 1)
Let A1, G2, G3, G4……Gn be N geometric Means between two given numbers A and B . Then A, G1, G2 ….. Gn, B will be in Geometric Progression .
So B = (N+2)th term of the Geometric progression.
Then Here R is the common ratio
B = A*RN+1
RN+1 = B/A
R = (B/A)1/(N+1)
Now we have the value of R
And also we have the value of the first term A
G1 = AR1 = A * (B/A)1/(N+1)
G2 = AR2 = A * (B/A)2/(N+1)
G3 = AR3 = A * (B/A)3/(N+1)
The equality of any two ratios is called a proportion. For example, if we have any four numbers or quantities that we represent as ‘a’, ‘b’, ‘c’, and ‘d’ respectively, then we may write the proportion of these four quantities as:
16 : 9
a:b = c:d or a:b :: c:d. From this, we will now define the proportionals. Let us begin by defining the fourth proportional.
Similar to the f=definition of the fourth proportional, we define the term known as the third proportional. The third proportional of a proportion is the second term of the mean terms. For example, if we have a:b = c:d, then the term ‘c’ is the third proportional to ‘a’ and ‘b’.
If a : b :: c:d or in other words a:b = c: d, then the quantity ‘d’ is what we call the fourth proportional to a, b and c.
For example, if we have 2, 3 and 4, 5 are in the proportion such that 2 and 5 are the extremes, then 5 is the fourth proportional to 2, 3, and 4.
Inversely Proportional: when one value decreases at the same rate that the other increases.
Speed and travel time are Inversely Proportional because the faster we go the shorter the time.
This: y is inversely proportional to x
Is the same thing as: y is directly proportional to 1/x
Which can be written:
y = k / x
When we talk about the speed of a car or an airplane we measure it in miles per hour. This is called a rate and is a type of ratio. A ratio is a way to compare two quantities by using division as in miles per hour where we compare miles and hours.
A ratio can be written in three different ways and all are read as “the ratio of x to y”
X to Y
X : Y
X / Y
A proportion on the other hand is an equation that says that two ratios are equivalent. For instance, if one package of cookie mix results in 20 cookies than that would be the same as to say that two packages will result in 40 cookies.
20/1 = 40 2
A proportion is read as “x is to y as z is to w”
X / y= z / w
Where y, w≠0
If one number in a proportion is unknown you can find that number by solving the proportion.
Find a percentage or work out the percentage given numbers and percent values. Use percent formulas to figure out percentages and unknowns in equations. Add or subtract a percentage from a number or solve the equations.
How to Calculate Percentages
There are many formulas for percentage problems. You can think of the most basic as X/Y = P x 100. The formulas below are all mathematical variations of this formula.
Let’s explore the three basic percentage problems. X and Y are numbers and P is the percentage:
Read on to learn more about how to figure percentages.
How to calculate percentage of a number.
Use the percentage formula: P% * X = Y
Example: What is 10% of 150?
There are concepts you need to understand in duplicate ratios. One is duplicate ratios itself and the other is a sub-duplicate ratio. In duplicate ratios, when the ratio p/q is compounded with itself, the resulting ratio which is p²/q² is called as the duplicate ratio. For example, 16/9 is the duplicate ratio of 4/3.
The duplicate ratio of the ratio of a:b is also defined as the compound ratio of a:b and a:b
=> (a × a) : (b × b) => a² : b²
So, the duplicate ratio of 6:7 = 6²:7² = 36:49
Similarly, for the sub-duplicate ratio, √a/√b is the sub-duplicate ratio of a/b or a:b.
For example 3:4 is the sub-duplicate ratio of 9:16.
Triplicate ratio: The triplicate ratio is the compound ratio of three equal ratios.
The triplicate ratio of the ratio a : b is the ratio a^3: b^3
In other words,
The triplicate ratio of the ratio m : n = Compound ratio of m : n, m : n and m : n
= (m × m × m) : (n × n × n)
= m^3 : n^3
Therefore, the triplicate ratio of 4 : 7 = 4^3: 7^3 = 64 : 343.
Bill Discounting is an important short-term financing service provided by banks and financial institutions. It enables businesses to obtain immediate funds against bills of exchange, trade bills, or promissory notes before their maturity date. In commercial transactions, sellers often allow credit to buyers and receive bills as evidence of debt. Instead of waiting until the due date for payment, the seller can approach a bank and get the bill discounted. This facility improves liquidity, supports working capital requirements, and ensures the smooth functioning of business operations.
Meaning of Bill Discounting:
Bill Discounting is a financial arrangement in which a bank or financial institution purchases a bill of exchange before its maturity date and pays the holder the bill amount after deducting a discount or service charge.
The bank recovers the full amount from the acceptor of the bill on the maturity date.
Bill discounting is the fee or the ‘discount’ that a bank charges a seller of the bill in exchange of releasing the funds to him before the due date of the bill. Essentially, bill discounting is the exchange of the bill for money, either from a bank or any third party.
Present Value:
To fully understand the concept of bill discounting, we need to learn about a few more important terms. One of these terms is present value (PV). Present Value is the current value of a sum of money in the future. So by discounting this future sum of money by a fixed discount rate, we arrive at its present value.
Hence, the higher the discount rate, lower the present value of the sum of money. It is an inverse proportion. Present Value indicates that an ‘x’ amount of money is worth more in the present than the same amount is in the future.
True Discount
This is also an important concept to learn in the discounting of bills. Now the total sum of money due at the end is known as the “Amount (A)”. The present worth or value of this sum is the PV.
The difference between the two is what we call the “True Discount (TD)”. Basically, the interest accrued on the Present Value of the sum is the True Discount. Let us learn its formula.
TD = Amount/Future Value – Present Value
Now while True Discount is the interest amount on the Present Value, there is another term known as the Bankers Discount. This is actually the Simple Interest on the face value of the sum from the date of the discounting to the due date of the bill.
Hence, the difference between the true discount and the bankers discount (fee for discounting the bill early) is known as the Bankers Gain.
Objectives of Bill Discounting:
One of the primary objectives of bill discounting is to improve the cash flow position of businesses. When goods are sold on credit, payment is received only after the credit period expires. This can create liquidity problems and affect daily operations. Bill discounting enables businesses to convert trade bills into immediate cash by obtaining funds from banks before the maturity date. The availability of cash helps businesses meet operational expenses and financial commitments without delay. Thus, bill discounting ensures a steady flow of funds and strengthens the overall financial health of an organization.
Bill discounting aims to provide adequate working capital for business operations. Every business requires funds to purchase raw materials, pay wages, settle utility bills, and manage routine expenses. Waiting for customers to make payments can create shortages of working capital. Through bill discounting, businesses receive immediate funds against accepted bills of exchange. This financing facility helps maintain uninterrupted production and trading activities. As a result, organizations can operate efficiently and fulfill their short-term financial obligations without depending heavily on long-term borrowings or expensive sources of finance.
Another important objective of bill discounting is to encourage and facilitate credit sales. In competitive markets, businesses often need to offer credit facilities to attract customers and increase sales. However, extending credit can delay cash inflows. Bill discounting solves this problem by allowing sellers to obtain immediate funds against bills arising from credit sales. This enables businesses to offer attractive credit terms while maintaining liquidity. Consequently, bill discounting promotes trade, improves customer relationships, and supports increased sales volume without creating financial strain on the seller.
Bill discounting is designed to eliminate the need for businesses to wait until the maturity date of a bill for receiving payment. Normally, sellers must wait for the entire credit period before obtaining cash from buyers. This delay can affect business operations and growth plans. Through bill discounting, banks provide immediate payment after deducting a discount charge. This objective helps businesses access funds quickly and efficiently. By reducing the waiting period, bill discounting improves financial flexibility and enables firms to utilize funds productively without unnecessary delays.
Enhancing liquidity is a major objective of bill discounting. Liquidity refers to the ability of a business to meet its short-term financial obligations. Insufficient liquidity can result in delayed payments, operational disruptions, and loss of business opportunities. Bill discounting converts receivables into cash and improves the availability of liquid funds. This allows businesses to maintain adequate cash reserves and manage unforeseen expenses effectively. Improved liquidity also strengthens financial stability and enhances the confidence of suppliers, creditors, and investors in the organization.
Bill discounting aims to support business growth and expansion by providing timely financial assistance. Growing businesses often require additional funds to increase production, enter new markets, purchase inventory, or invest in business development activities. Delayed customer payments can restrict growth opportunities. By converting bills into immediate cash, bill discounting provides the financial resources needed for expansion. This objective enables businesses to seize market opportunities, improve competitiveness, and achieve sustainable growth without facing liquidity constraints caused by outstanding receivables.
An important objective of bill discounting is to promote smooth trade and commercial activities. Credit transactions play a vital role in business and industrial operations. Bill discounting supports these transactions by ensuring that sellers receive funds promptly while buyers continue to enjoy credit facilities. This arrangement benefits both parties and contributes to the efficient functioning of markets. By facilitating the movement of goods and services through easy financing, bill discounting encourages commercial development and strengthens the overall business environment.
Bill discounting aims to provide businesses with a secure and reliable source of short-term finance. Since financing is backed by legally accepted trade bills arising from genuine transactions, the risk for financial institutions is relatively controlled. Businesses can obtain funds quickly without complex procedures associated with long-term loans. The systematic nature of bill discounting makes it a dependable financing option for managing temporary cash shortages. Therefore, it serves as a practical and trusted method of financing working capital requirements and maintaining financial stability.
Advantages of Bill Discounting:
No entrepreneur can avail conventional working capital loans without meeting the eligibility criteria set by the lending institutions. Many lending institutions even require additional time to process and disburse small business loans. Hence, many business owners opt for bill discounting to avail funds without lengthy approval process. A number of NBFCs even enables borrowers to avail cash in 72 hours by discounting their unpaid invoices.
Often small businesses have to sell goods in credit to expand customer base. When they sell goods on credit, it becomes difficult for entrepreneurs to maintain positive cash flow. The invoice discounting services provided by lending institutions help entrepreneurs to improve cash flow quickly. They can even shorten the working capital cycles by converting unpaid invoices into cash.
As noted earlier, bill or invoice discounting enables business owners to fund working capital needs without increasing liabilities. The business owner can opt for this option to avail cash quickly by releasing the funds locked in unpaid invoices or bills. He can even meet working capital needs simply by converting current assets into liquid assets.
Many enterprises explore ways to credit sales to maintain a positive cash flow position. But small businesses cannot acquire new customers and retain existing customers in the long run without combining cash and credit sales. The bill discounting services make it easier for enterprises to sell goods in credit by liquidating current assets and boosting cash flow.
Disadvantages of Bill Discounting
The lending institutions discount bills or invoices by charging a fee. The fee normally includes interest charges, administrative expenses and maintenance expenses. The percentage of fee or discount also differs from one lender to another. Hence, the business owners have to sacrifice a percentage of the bill value. The fees charges by the lender will even impact the business’s profitability.
An entrepreneur cannot avail funds by discounting all his unpaid bills or invoices. Many lending institutions discount only commercial bill. Also, they evaluate the bills or invoices based on a number of parameters before providing funds. Hence, entrepreneur cannot rely on bill discounting as a consistent or long-term working capital funding solution.
Both banks and NBFCs provide bill discounting services only to existing customers or established enterprises. Some lending institutions even provide discount bills only if the business is generating profit. Hence, new business owners may not fund working capital needs through bill discounting service. Also, the fees charged by the lending institutions will impact their profitability in the short run.
Most banks do not provide collateral free business loans to small business owners. They require the borrowers to use their personal and business assets as collateral to avail credit. Each time a business owner discounts an invoice or bill, his working capital declines accordingly. Hence, the business owner may find it challenging to avail other working capital loans.
The type of number we normally use, such as 1, 15.82, −0.1, 3/4, etc.
Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers.
They are called “Real Numbers” because they are not Imaginary Numbers.
The LCM of a set of two or more numbers is the smallest of their common multiples. Multiples mean the numbers which follow as the result of multiplying the number with numbers like 1,2,3 etc. To find the common multiples all we need to do is see what numbers end up to be the common multiple for all the given numbers.
For example, when we find the LCM of 9 and 12 we need to find the common multiples. The common multiples of 9 are 36,72,108 etc… The smallest of these is 36, hence 36 shall be the LCM of 9 and 12.
To find the LCM using prime factorization method we need to follow the below-mentioned steps:
Let us use the steps in the following example: find the LCM of 8 and 24.
Step 1: First find the prime factors of the numbers 8 and 24
Step 2: Choose out the number occurring a maximum number of times. The number 2 occurs 3 times and 3 occurs 1 time. number occurring the maximum number of times is 2×2×2×3.
Step 3: The product of these numbers is 24. So the LCM of 8 and 24 is 24.
For calculating the LCM by division method we need to follow the below mentioned steps:
The HCF or Highest Common Factor of two or more numbers is the greatest common factor of the given set of numbers. In other words, HCF is the greatest number which exactly divides two or more given numbers.
The listing method involves the process of listing the factors of the given numbers. For example, find the HCF of 20 and 35.
The common factors of the given numbers are : 1,2,4,5,10,20. The greatest among all other numbers is 20, so it shall be the HCF of both the numbers.
Before finding HCF by prime factorization we need to know the concept of the same. Let’s take a number say, 45. Now the factors of 45 are 1,3,5,9,15 and 45 itself. Now, apart from 3 and 5 the other numbers 9 and 15 are composite numbers. We hence further factorize them with 9= 3×3 and 15=3×5.
So the factors of 45 shall be only 1,3,3, and 5. This is prime factorization. We now define prime factorization as the process of expressing the number as the product of its prime factors. The prime factors include only prime numbers and not composite numbers.
When we find HCF by prime factorization method, we are finding the greatest common factor among the prime factors or numbers. Steps to be followed for the method are:
Let us use these steps in the example below: find the HCF of 36 and 48.
Step 1: Finding prime factors individually:
Step 2: Choose out the common factors: 2×2×3
Step 3: Multiply all the common factors to get the HCF of the given numbers:
Here the given numbers are 36 and 48. The product of the common factors: 2×2×3 = 12. So the HCF for the numbers 36 and 48 is 12.
A Rational Number can be written as a Ratio of two integers (ie a simple fraction).
Example: 1.5 is rational, because it can be written as the ratio 3/2
Example: 7 is rational, because it can be written as the ratio 7/1
Example 0.333… (3 repeating) is also rational, because it can be written as the ratio 1/3
But some numbers cannot be written as a ratio of two integers they are called Irrational Numbers.
π (Pi) is a famous irrational number
π = 3.1415926535897932384626433832795… (and more)
We cannot write down a simple fraction that equals Pi.
The popular approximation of 22/7 = 3.1428571428571… is close but not accurate.
The natural numbers are the counting numbers. It goes like 1, 2, 3, 4, …, and so on. It is interesting to know that if we subtract 1 from any natural number, we get its predecessor (previous number). If we add 1 to any natural numbers, it gives its successor (next number).
The predecessor of 5 is 5 − 1 = 4. The successor of 5 is 5 + 1 = 6. Is there any natural number that has no predecessor? The predecessor of 2 is 1. What is the predecessor of 1? Does that predecessor is also a natural number? No, no natural number is the predecessor of 1.
Suppose you have 5 chocolates and you distribute them among your friends. How many chocolates do you have? Zero. Zero is denoted by the symbol 0. When we add 0 to the group of natural numbers, we get whole numbers. The predecessor of 1 is 1 − 1 = 0. 1 has the predecessor which is a whole number and not a natural number.
0 + Natural Numbers = Whole Numbers
The whole numbers are said to consist of two types of numbers – even numbers and odd numbers.
A whole number exactly divisible by 2 is called even numbers.
For example:
2, 4, 6, 8, 10, 12, 14, 16……………………..are even number. Or a number having 0, 2, 4, 6, 8 at its units place is called an even number.
246, 1894, 5468, 100 are even number.
Any two even numbers which differ from one another by 2 are called consecutive even number.
Odd numbers are the numbers which are not completely divisible by 2. The odd numbers leave 1 as a remainder when divided by 2. They have 1, 3, 5, 7, and 9 as their unit digit. 1, 3, 5, 7, 9, 11, 13, 15, etc. are odd numbers. The sets of odd number are expressed as Odd = {2n + 1: n ∈ integer (number)}.
Check Odd & Even
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