Category: Maths Notes PPT

Mark up is the total profit or gross profit earned on a specific commodity or service. It is denoted as a percentage over a cost price. For example, the cost of a good is Rs. 100 and the good sold is of Rs. 150, so the markup will be 50%. 

The cost of a good or the cost price of the commodity is the price at which the buyer purchases the goods from the shopkeeper. The cost price is often represented as (CP). On the other hand, the selling price of a good or SP is the price at which the shopkeeper sells the goods to the buyer. The term markup is widely used in business studies. Markup is defined as the difference between the selling price and the cost price of a good. The profit and loss of a business are easily determined through markup.

 

Markup Formula

As we know, markup is the difference between the selling price and the cost price of the product. Hence, the markup formula is represented as :

 

What is the Markup Price?

Markup pricing is the method of adding a certain percentage of markup to the cost price of the product to estimate the selling price of a product. 

To make use of mark-up, the companies initially determine the cost price of the product and further decide the amount of profit to be earned over the cost of the goods sold and then include or add that markup in the cost.

Let us understand the concept of markup pricing through the markup pricing example.

 

Markup Pricing Example

Suppose, there is a mobile manufacturing company that has the following cost and sales expectations.

Variable cost per unit – Rs. 30

Fixed cost – 5,00,000

Expected Unit Sales – Rs. 50,000

The unit cost is Variable cost + Fixed cost / Unit sales

Hence, the unit cost = 30 + 500000/ 50000 = RS. 40

Once the cost is estimated, the manufacturer decides to add a 20% markup on sales. The markup price formula for the above markup pricing example is given as 

Markup price – Unit cost / 1- desired return on a product = 40/ 1-0.2 =50

Hence, the manufacturer should ask Rs. 50 from a buyer to earn a  desired profit of Rs. 10

 

Markup Price Formula

As we know, the markup price is the additional price or profit earned by the seller over and above the total cost of the product or service. Mark up price is also defined as the difference between the average selling price per unit and the average cost price per product.

Hence, the markup price formula = Sales Revenue- Cost of goods sold/ Number of units sold.

Markup price formula is also derived as the average selling price per unit – Average cost price per unit.

 

Markup Percentage

Markup percentage is a percentage markup over the cost price of a product to determine the selling price of a product. It is calculated as a ratio of gross profit to the cost price of the unit. Most of the time, the company sells their product during the process of making decisions for the selling price, they take the cost price and use markup which is generally a small factor or a percentage of the cost price, and make use of that as a profit margin and decide the selling price.

 

Markup Percentage Formula

To calculate the markup percentage, we use the following markup percentage formula

Selling  Price = Cost Price x (1 + Markup)

or

Markup = (selling price/cost price) – 1

Markup = (Sale Price-Cost)/Cost

 

Difference Between Margin and the Markup

The difference between margin and markup is such that margin is the difference between sales and cost of goods sold while markup is the price by which the cost of a good is increased to determine the selling price. The margin is also known as gross margin. A mistake in markup and magin can lead to the price determination being substantially too low or too high resulting in fewer sales or less profit. It can also have adverse effects on market shares as an excessively high price or low price may be beyond the price imposed by the competitors. 

We can easily calculate the profit margin of a product in the following way if we know the markup.

The selling price of a product – Cost price of a product  = Selling Price of a product × Profit Margin

Hence, 

Profit margin = (Selling Price – Cost Price)/Selling Price

Margin = 1 – (1 /(markup +1))

Or

Margin = markup/1+markup

For example,  if the markup is 50%, then profit margin;

Margin = 50/(1+0.5) = 50/1.5 = 33.33%

 

The difference Between Markup and Margin can also be Determined from the Following Point.

• To achieve a gross margin of 10%, the company mark up price percentage should be 11.1%

• To achieve a gross margin of 40%, the company mark up price percentage should be 80%

• To achieve a  gross margin of 50%, the company mark up price percentage should be 100%

 

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Compound Interest

Compound interest is the type of interest method where the interest is paid on both the principal and interest together which compounds at regular intervals. The interest collected on a principal over a while is also accounted for under the principal. The interest calculation for the next period is on the collected principal value. Compound interest is a relatively new technique of calculation of interest used for almost all financial and business dealings across the world. Compound interest is popular for its power of compounding. This can be understood, when we observe the compound interest values accumulated across consecutive periods. The compound interest is calculated at regular intervals like annually, quarterly, etc.

To calculate the new principal, the sum of the initial principal and the interest accumulated by that time is calculated. 

Compound Interest can be calculated with the formula

Compound Interest = (Interest on the Principal) + (Compounded Interest in Regular Intervals)

 

Compound Interest Formula

The compound interest is calculated, after getting the total amount over some time( depending on the rate of interest and the initial principal). To calculate the Compound Interest on a given amount, the following formula is used-

A=P (1 + r/n)^(nt)

(where ‘P’ is the principal amount, ‘r’ is the rate of interest, ‘n’ is frequency or no. of times the interest is compounded yearly and t is the overall term)

The above formula denotes the entire amount at the end of the period and contains the compounded interest and the principal. From this formula, the formula of compound interest can be extracted by subtracting the principal from this amount. The formula for calculating the compound interest is-

CI=P(1 + r/100)^t – P

The above formula is to find Compound Interest when the given principal is comp
ounded yearly and the amount after the period at percent rate of interest ‘r’.

Solved Example

1. If the Selling Price of the Chocolate Box is Rs. 500 and the Cost Price of the Chocolate Box is Rs. 150. Find the Markup Percentage.

Solution: Given, Selling price of the chocolate box = Rs. 500

The cost price of the chocolate box  = Rs. 150

Markup percentage formula = 100 × (Selling price – Cost Price)/Cost price

Markup percentage = 100 × ( 500 – 150)/ 150

= 100 × 350/ 150

= 233.33%

2.  If the Markup Rate Used by a Shopkeeper on a Toy Car is 50%, if the Cost Price of a Toy Car is Rs.1000, Find the Selling Price of a Toy Car?

Solution: Markup = 50% of cost price 

Markup = 50% of 1000

= 50/100 × 1000

= 500

Selling price = cost price + markup

= 500 + 1000

= 1500

Selling Price = Rs.1500

Hence, the selling price of a toy car -= Rs. 1500

3. The Overall Sales Revenue of a Company X is ＄20000. The Cost of the Goods Sold by the Company is ＄10000. The Number of Units Sold by the Company is 1000. Find the Markup Price for Company X. 

Solution: Let us use the markup price formula to calculate the markup price for company X.

Markup price- (Sales Revenue – cost price of the unit sold) / Number of units sold.

Markup Price = (＄20000 – ＄10000)/1000

Markup Price = ＄10000/ 1000

Markup price = ＄10 for each unit.

 

Quiz Time

1. Which of the Following is the Type of Term Most Probably Answer to the Question? What is the Markup on this Item?

1. 3 bits

2. ＄1000

3. It depends

4. 50%

2.  A Shopkeeper Pays its Wholesaler $40 for a Certain Item, and Sells the Item for ＄75. What is the Markup Rate?

1. 81%

2. 55%

3. 60%

4. 87.5%

3.  An Item Originally Priced at Rs. 55 is Marked 25% Off. Find the Selling Price.

1. Rs. 42

2. Rs.60

3. Rs.76

4. Rs. 41.25

Before discussing the operations of the matrix, let’s discuss what a matrix is.

• A matrix is a rectangular (2D) array of numbers or symbols which are generally arranged in rows and columns. One can think of it as a table of numbers. 

• The order of the matrix is defined as the number of rows and columns.

• The entries are the numbers in the matrix and each number is known as an element. 

• The plural of matrix is matrices.

• The size of a matrix is referred to as ‘n by m’ matrix and is written as m×n, where n is the number of rows and m is the number of columns.

• For example, we have a 3×2 matrix, that’s because the number of rows here is equal to 3 and the number of columns is equal to 2.

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The dimensions of a matrix can be defined as the number of rows and columns of the matrix in that order. Since matrix A given above has 2 rows and 3 columns, it is known as a 2×3 matrix.

What are the Different Types of Matrix?

There are different types of matrices. Here they are –

1) Row matrix

2) Column matrix

3) Null matrix

4) Square matrix

5) Diagonal matrix

6) Upper triangular matrix

7) Lower triangular matrix

8) Symmetric matrix

9) Anti-symmetric matrix

Adding Matrices

Two matrices must have an equal number of columns and rows in order to be added. The sum of any two matrices suppose A and B will be a matrix which has the same number of rows and columns as do the matrices A and B. The sum of A and B, can be denoted as A + B, is computed by adding corresponding elements of A and B.

A + B = [begin{bmatrix} a_{11} & a_{12} & cdots  & a_{1n}\ a_{21} & a_{22} & cdots & a_{2n}\ vdots & vdots   & ddots  & vdots\ a_{m1} & a_{m2} & cdots & a_{mn}  end {bmatrix}] + [begin{bmatrix} b_{11} & b_{12} & cdots  & b_{1n}\ b_{21} & b_{22} & cdots & b_{2n}\ vdots & vdots   & ddots  & vdots\ b_{m1} & b_{m2} & cdots & b_{mn}  end {bmatrix}]

 

= [begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & cdots  & a_{1n} + b_{1n}\ a_{21} +b_{21} & a_{22} + b_{21}& cdots & a_{2n} + b_{2n}\ vdots & vdots   & ddots  & vdots\ a_{m1} + b_{m1} & a_{m2} + b_{m2} & cdots & a_{mn} + b_{mn} end {bmatrix}]

Matrix Sums and Answers

Let us suppose that we have two matrices A and B.

Both the matrices A and B have the same number of rows and columns (that is the number of rows is 2 and the number of columns is 3), so they can be added. In order words, you can add a 2 x 3 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 2 x 2 matrix. However, you cannot add a 3 x 2 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 3 x 3 matrix. 

A = [begin{bmatrix} 1  & 2 & 3\ 7 & 8 & 9end {bmatrix}]  B = [begin{bmatrix} 5  & 6 & 7\ 3 & 4 & 5end {bmatrix}] 

 

A + B = [begin{bmatrix} 1 + 5  & 2 + 6 & 3+ 7\ 7 + 3& 8 + 4 & 9 + 5end {bmatrix}]

 

A + B = [begin{bmatrix} 6  & 8 & 10\ 10 & 12 & 14end {bmatrix}] 

Note:  Keep in mind that the order in which matrices are added is not important; thus, we can say that  A + B = B + A.

Properties of Matrix Addition

Applications of Matrices

• Matrix is used in many branches of mathematics, for example, for calculations related to vectors like finding the derivative, integration, the integral of a matrix, etc. 

• Matrices are widely used in matrix and linear algebra, in particular, to represent and solve linear systems of equations.

• A matrix is also used in solving eigenvalue problems, symmetric and eigenvectors, linear regression, optimization problems, etc.

Solved Example Problems

Question 1. Add the following matrices:

[A = begin{bmatrix}1 & 2&3 \7 & 8&9 end{bmatrix}] [B =begin{bmatrix}5 & 6&7 \3& 4&5 end{bmatrix} ]

Solution: We have two matrices A and B.

Both the matrices A and B have the same number of rows and columns (that is the number of rows is 2 and the number of columns is 3), so they can be added. In other words, you can add a 2 x 3 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 2 x 2 matrix. However, you cannot add a 3 x 2 matrix with a 2 x 3 matrix or a 2 x 2 matrix with a 3 x 3 matrix. 

[A = begin{bmatrix}1 & 2&3 \7 & 8&9 end{bmatrix}] [B =begin{bmatrix}5 & 6&7 \3& 4&5 end{bmatrix} ]

[ A + B  =begin{bmatrix}1+5 & 2+6&3+7 \7+3& 8+4&9+5 end{bmatrix} ]

[ A+B = begin{bmatrix}6 & 8&10 \10 & 12&14 end{bmatrix} ]

Question 2. Add the following matrices.

[ A = begin{bmatrix}3 & 4&9 \12& 11&35 end{bmatrix}] [ B = begin{bmatrix}6 & 2 \5 & 8 end{bmatrix} ]

Solution: Let’s add the following two matrices A and B. As we know that matrices are added entry-wise, we have to a
dd the 3 and the 6, the 12 and 5, the 4 and the 6, and the 11 and the 8. But what do I add to the entries 9 and 35? There are no corresponding entries in the second matrix that can be added to these entries in the first matrix. So here’s the answer:

We can’t add these matrices A and B, because these matrices are not of the same size.

Question 3. Suppose X, Y, Z, W, and P are matrices of the given order 2 × n, 3 × k, 2 × p, n × 3, and p × k, respectively. The restriction on n, k, and p so that PY + WY can be defined as-

1. k is arbitrary, p = 2

2. p is arbitrary, k = 3

3. k = 2, p = 3

4. k = 3, p = n

Solution: In this, the order of matrix P = p × k, order of W = n × 3, order of matrix Y = 3 × k. Thus, the order of PY = p×k, when k  is equal to 3. And the order of WY = p × k, where p = n. Thus, option (D).

Statistics is a living subject full of challenging problems and exciting developments. Through statistics, you would have the thrill of discovering, learning, and challenging your own assumptions. With statistics, new knowledge is created by pushing the frontier of what is known.

The Power of Data

Data is all around us. The number of people in a country, sales figures of an organization, and the number of hits on a website are data that lets a business or a nation make informed decisions. 

The field of statistics is all about learning from data. It is the process of converting raw data into a meaningful, organized, and informative form. Statistical knowledge is the basis on which proper methods of collecting data, employing the right analysis, and effectively presenting the results are built. The discoveries in science and many predictions are all based on statistical methods. If you want to understand a subject deeply, you need to get into the statistics of it all.

If you are looking to learn and gather statistics notes, you must go through this article where you will learn the meaning and definition of statistics and the classification of statistics.

Meaning and Definition of Statistics

Statistics means studying, collecting, analyzing, interpreting, and organizing data. Statistics is a science that helps to gather and analyze numerical data in huge quantities. With the help of statistics, you can measure, control, and communicate uncertainty. It allows you to infer proportions in a whole, derived from a representative sample. In other words, statistics could be described as the feature or characteristic of a sample and is generally used to estimate a population parameter’s value.

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The Need for Statistics

The early impetus for statistical data arose from the government’s needs for data like census and the need for information about various economic activities. In modern times, the necessity to turn large volumes of data available in many applied fields into meaningful and useful information has resulted in the evolution of both practical and theoretical statistics.

Terms Used in Statistics

Statistics is applied in many real-life situations to make it easy to understand data when data is represented in a particular number (that represents all numbers). This number is termed as a measure of central tendency, and a few common central tendencies are:

This is the average of given numbers and is measured by adding all the numbers and then dividing by the total count of numbers. So if a1, a2, a3, a4,…., an are n numbers then their mean [bar{a}] = (a1 + a2 + a3 + …+ an)/n = [sum_{i=1}^{n}] ai/n

If all the n numbers are arranged either in ascending or descending manner, then the middle number of that series denotes the median of the group. In the case of n being an odd number, the median is the observation at the ((n + 1)/2)th position. In the case of n being an even number, we get the median by taking the average of both the middle numbers i.e. the average of observations at (n/2)th and ((n + 1)/2)th observations.

The mode of n numbers is the number that has the highest frequency in the given sample. In case there are no numbers that are repeated in the list, then that sample has no mode.

The range of observations is the difference between the highest and the lowest number in the list.

Example Problem on Finding Mean, Median, Mode, and Range

From the list of values given below, let us find their mean, median, mode, and range.

12, 16, 13, 14, 13, 18, 14, 21, 13

Mean – 12 + 16 + 13 + 14 + 13 + 18 + 14 + 21 + 13/9 (since there are 9 numbers in the list) = 14.88

Median – To find median we will first write down the list in ascending order: 12, 13, 13, 13, 14, 14, 16, 18, 21

Here n = 9 so it is an odd number. So the median is the number at the position (9+1)/2 = 10/2 = 5. So, the median is 14.

Mode – Since the number 13 is repeated the maximum number of times (3 times) in the list, 13 is the mode of this sample.

Range – In the list the highest value is 21 and smallest value is 12 hence range = 21 – 12 = 9

Classification of Statistics

Statistics has three broad categories as outlined below:

This is the methodology where data is effectively collected, organized, and described.

In this process, conclusions are drawn about unknown sample features taken from a population. This involves an interpretation of the descriptive stats.

This is the process where future values are predicted based on historical data.

Application of Statistics

Statistics provides a clear picture of the work we do on a day-to-day basis, and it has wide applications in the following areas:

We use statistical methods like probability and dispersion, to get more accurate information.

Many economic parameters like the inflation of a country, employment status, exports, and imports, etc. are all heavily dependent on statistical methods.

Any drug is prescribed after it has been analyzed through statistics. Statistics measure the effectiveness of a drug. 

Psychologists use statistics for figuring out things like peer pressure amongst youngsters.

In schools and colleges, lecturers use statistics to interpret which course their students are more interested in.

Business nowadays uses statistics to gauge customer preferences, product quality, target market, etc. Statistics in business is used to analyze past performances of business firms and markets, and hence, in turn, predict the future business strategies and practices. This helps businessmen to lead organizations and business firms very effectively. Description of markets, information about advertising, and price information about various things, customer demands, and responses are all recorded in statistical form. These are the statistical data used in business.   

Let us come to descriptive analysis for example. This takes a look at everything that has happened and gives an explanation for everything. Managers of various business sectors gather all the historical data to analyze failures and successes of the past. The proper term for this is “cause and effect analysis”. Sales, finance, marketing, operations are a few of the many areas where descriptive analytics is put to use. 

Let us now come to predictive analytics. Modeling and data mining are two of the statistical procedures that are a part of predictive analytics.  These statistical procedures are used for the prediction of future probabilities and trends. Besides, reporting the historical data also helps in the creation of the best estimates for future happenings. Detection of fraud, security, marketing, operations, and risk assessment are a few of the many areas where predictive analytics is
put to use. 

Last but definitely not least; let us talk about prescriptive analytics. Prescriptive analytics determine the best course of action in a certain situation. For example, a certain business situation is described; prescriptive analytics will be used here to determine the best course of action in that given situation. The data of prescriptive analysis include future situations, the cause of those situations, and a way to navigate those situations. In prescriptive analytics, information is constantly updated. The constant update may change the course of action and the way to navigate it. The constant update helps managers to maintain and update their action plans according to the situation. 

All the products a company produces go through a quality check by employing statistical tools.

The size, distance, and other parameters of objects in the universe are measured using statistical methods. The statistical methodology has deep roots in probability theory. This helps with a lot of analysis and quantitative procedures. Statistics is the best way to test astrophysical theories. This is because statistics and probabilities are the best ways to analyze theories. The quantitative procedures provided by probability theory are very useful in extracting scientific knowledge from astrophysics data. This knowledge and data are further used to analyze theories. 

There are several statistical methods that are used in astronomy for several purposes. Astrophysics is an area where statistical methods come in handy the most. 

To predict the upcoming weather, a statistical tool is used to compare previous and current weather. Statistics play a huge role in Weather forecasting. Easy interpretability is one of the most important benefits of statistics usage in Weather forecasting. The visualization of results is easily interpreted in weather forecasting with the help of statistics. Statistics in weather forecasting also help laymen understand weather forecasts in a simpler and better way.  In fact, the very word forecasting is related to statistics. It refers to the prediction of future data with the help of data from the present and the past. The data collected is then analyzed to predict the weather. 

Let us take an example to understand this better. Let us consider the estimation of temperature in this case. For a specified date in the future, past temperature data and present temperature data are collected and then analyzed to predict future temperature. Forecasting and prediction are basically the same, but the term prediction is used here to generalize it in a better way. 

Climatology, finance, foreign exchange, are a few of the many areas where statistical forecasting methods have been used. The statistical prediction methods have been used and applied in several areas of the world for a lot of purposes. This leads to better prediction and simulation. The main thing is that, with the help of collected data, that is, statistics, we can make better predictions in every area and weather forecasting is no different. If proper rules are followed and proper methods are used, then these predictions can become pretty easy. 

The Seven Millennium Prize Problems are the most well-known and important unsolved problems in mathematics. A private nonprofit foundation Clay Mathematics Institute that is devoted to mathematical research, famously challenged the mathematical community in the year 2000 to solve these unique seven problems, and a sum of US $1,000,000 reward was established for the solvers of each of the seven problems. Out of the seven Millennium prize problems, one of the problems has been solved, and the other six are a great deal of current research.

With the spin of the century, the timing of the announcement of the Millennium Prize Problems was a homage to a famous speech of the famous David Hilbert to the International Congress of Mathematicians in the year 1900 in the city of Paris. The 23 unsolved problems that were posed by Hilbert were studied by countless 20th century mathematicians, which led not only to solutions to some of these difficult problems but it also led to the development of new ideas as well as new research topics. There are some of Hilbert’s problems that still remain open– namely the famous Riemann hypothesis. 

These seven problems encompass a diverse group of topics, which include theoretical computer science as well as physics, as well as topics of pure mathematical areas such as number theory, algebraic geometry, as well as topics of topology.

7 Millennium Prize Problems

1. Yang-Mills and Mass Gap

Computer simulations as well as various experiments suggest the existence of a “mass gap” in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known. A Yang-Mills theory is known to be a theory in quantum physics that is a generalization of Maxwell’s work on electromagnetic forces to the strong as well as weak nuclear forces. It is a key ingredient in the Standard Model of particle physics. This Standard Model is said to provide a framework for explaining electromagnetic as well as providing nuclear forces and also classifying subatomic particles. 

In particular, successful applications of the theory to experiments as well as simplified models have involved a “mass gap,” which can be formally defined as the difference between the default energy in a vaccum as well as also the energy in the next lowest energy state. So this quantity is also known as the mass of the lightest particle in the theory. A solution to the Millennium Problem will include both a set of formal axioms that characterize the theory as well as will show that it is internally logically consistent.

2. Riemann Hypothesis

The prime number theorem determines the average distribution of the prime numbers. Whereas the Riemann hypothesis basically describes the deviation from the average. It was formulated in Riemann’s 1859 paper, which asserts that all the ‘non-obvious’ zeros of the zeta function are complex numbers with real part 1/2.

3. P vs NP Problem

If it is easy to check that a solution to a problem is right, can you say that it is also easy to solve the problem? This is said to be the exact essence of the NP question vs P question. Typical of the NP problems is that of the Hamiltonian Path Problem: let’s suppose given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that the problem is correct, but it is difficult to find a solution.

4. Navier–Stokes Equation

The Navier-Stokes equation is said to be the equation that governs the flow of fluids such as water as well as air. However, there is no proof for the most basic questions one can ask: do solutions exist as well as are they unique? Why ask for proof? Because proof gives not only certitude but proof also gives understanding.

5. Hodge Conjecture

Hodge conjecture, the answer to this determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture comes in picture in certain special cases, for example, when the solution set has a dimension less than four. But in dimension four it is unknown.

This conjecture is also known to be a statement about geometric shapes cut out by polynomial equations over complex numbers. These are also known as complex algebraic varieties. An extremely useful tool in the study of these varieties was the construction of groups which is also known as cohomology groups, which contained information about the structure of the varieties. 

6. Poincaré Conjecture

The French mathematician Henri Poincaré in the year 1904. He was the one who asked if the three-dimensional sphere is characterized as the unique simply connected three-manifold. The Poincaré conjecture is known as a special case of Thurston’s geometrization conjecture. This Poincaré conjecture proof tells us that every three-manifold is built from a set of standard pieces, each with one of eight well-understood geometries.

7. Birch and Swinnerton-Dyer Conjecture

This z conjecture is basically supported by much experimental evidence that relates the number of points on an elliptic curve mod p to the rank of the group of rational points.

Being one of the most exciting chapters of mathematics, Multiples and Factors enables students to get into the fundamentals of calculating and simplifying equations. The chapter establishes a base for students which they can make use of while understanding more difficult areas of this subject. The chapter mainly contains the definition of multiples, how these are important in Mathematics, activity based on common factors and common multiples, etc., along with some examples.

The chapter also contains a detailed description and usage of factors and the differences between common factors and common multiples. These two are vital concepts related to each other; therefore, students should go through both these topics simultaneously to better understand them.

What are Multiples?

A multiple is a number that you get by multiplying other numbers together. And, multiples are a set of numbers where you have a base number that has been multiplied by other natural numbers.

In simpler terms, the multiples are obtained by multiplying one whole number with other numbers. Class 6 maths common factors and common multiples worksheets are the best resource students can go through to get hold of both Multiples and Factors. However, another better way to understand these series is by looking at the multiplication table, and the following are some multiples of natural numbers you can refer to.

You can go through the multiples of 3 for instance, 

3 x 0 = 0

3 x 1 = 3

3 x 3 = 9

3 x 4 = 12

Here, you can see that the multiples of 3 are 0, 3, 6, 9, 12, and the list is never-ending. Since any number can be multiplied with any other number, it inevitably has an infinite number of multiples. To delve deeper regarding the basics of Multiples, students should consider going through the worksheet of common multiples and factors for class 5th.

What is a Factor?

Factors can be specified as whole numbers that can evenly divide another number. Precisely, when a number is considered as a factor of another second number, the first one has to divide the second number completely without keeping any remainder behind.

In other words, if the dividend is evenly divisible by a number or also called as a divisor, that particular divisor will be known as a factor of that dividend. Every number is supposed to have a common factor, such as one (1) and also the number itself. The highest common factor and lowest common multiple TNPSC questions generally include questions related to these; therefore, students need to go through a few examples to comprehend them.

For instance,

4 is considered as a factor of 24, as it divides 24 evenly and leaves 6 as quotient and zero (0) as remainder. On the other hand, 6 is also considered as the factor of 24, which leaves 4 as quotient and zero (0) as remainder.

Consequently, it is apparent that 24 has multiple factors, such as 1, 24, 4, 6, and 2, 3, 8, including 12, since all these numbers can divide 24 evenly without leaving any remainder.

As discussed earlier, the factor is a number that generally keeps no remainder behind after dividing a particular number. On the other hand, multiple is a number obtained by multiplying a whole number by some other numbers. Following is a list of key differences between common factors and multiples class 4:

Difference Between Common Factors and Common Multiples

Along with these differences, students should also be aware of the process of common factors and common multiples to compare them. Mathematics mainly deals with numbers and different operations, and factors and multiples are one of those kinds. The concepts are pretty basic and should be grasped properly to understand higher-level mathematics in a better way.

Furthermore, with assistance from e-learning platforms like , one can elevate their knowledge of multiples. The model exam question papers, expert notes, along with online classes, and doubt clearing sessions, students can better their final exam preparations.

Multiplication and division are important concepts of algebra in mathematics. Multiplication and division operations are quite similar to each other. The division is the inverse operation of multiplication. In division, we divide the numbers into equal parts. In multiplication, we group up numbers together. In multiplication, the answers we get are known as the product and the numbers which we multiply are called factors. The multiplication is denoted by ‘×’. In division, the answer we get after dividing is called the quotient, the number which is being divided is called the dividend, and the number which divides it is called the divisor. The division is denoted by ‘÷’.

Example 1: 4×5= 20

Here 20 is the product, 4, and 5 are the factors.

Example 2: 10÷2= 5

Here 5 is the quotient, 10 is the dividend, and 2 is the divisor.

Decimals

The numbers with a decimal point are called decimals. They can also be represented in the form of a fraction.

For example, the number 3.556 is the decimal number because it has a decimal point in it.

As we add, subtract, multiply, divide the whole numbers similarly we can add, subtract, multiply, and divide the decimals also. Compare to addition and subtraction, multiplication and division are quite difficult.

Multiplication Of Decimals:

In mathematics when we multiply 5 by 2, we get 10 similarly when we add 5 two times that is 5 + 5 we get 10.

In the case of decimals numbers, we get the same value if we multiply 1.2 by 2, we get 2.4 similarly when we add 1.2 two times that is 1.2 + 1.2 we will get 2.4.

So, 1.2 × 2 = 1.2 + 1.2

Example- Find the multiplication of 2.56 and 3.5.

1. So, in 2.56, the decimal is before two digits and in 3.5 it is before one digit.

2. Now we will multiply these two numbers without decimals.

3. 256 × 35 = 8960

4. Now we will place the decimal here the decimal will be placed before three digits because 2+1=3.

5. The final answer is 8.960.

Division Of Decimals

Dividing two numbers with decimals is quite difficult and confusing. You can follow these steps in the multiplication of decimal numbers.

There are two methods to do the division of decimals. We will discuss how to do decimal division.

Method 1:

In this method, if we have two decimal numbers, we multiply the numerator and denominator by such a number which on multiplication gives us the whole number in the denominator. So that the calculation becomes easier.

Suppose we have two numbers 30.5 and 0.5 we have to divide it.

So, we will multiply the numerator and denominator by 2 so that we get the whole number in the denominator.

(30.5×2)/ (0.5×2)

= 61/1

= 61

Method 2:

As in method 1, we converted the decimal number into a whole number similarly by using this method also, we can convert a decimal number into a whole number. 

In this method, we will multiply the numerator and denominator by 10, 100, 1000, etc. That is the powers of 10 

We will consider the above example that is 30.5 and 0.5 

Now in 30.5, we have one digit after the decimal point so we will multiply it by 10 and similarly the denominator also.

(30.5×10)/ (0.5×10)

= 305/5

= 61

So, by both the methods, we got the same answer after division.

If there are two numbers after the decimal, we multiply by 100. If three numbers then multiply by 1000 and so on.