Category: Maths Notes PPT

Before understanding the determinant for finding the area of a triangle, let us have a quick look over the meaning of determinant first. So, the sum-product which is obtained by the elements of the square matrix is called a determinant. It helps to find the adjoint of the matrix, as well as the inverse of the matrix.

Now, finding the area of a triangle is not that difficult if the given triangle is a right-angle triangle, because the area of such a triangle can easily be found by finding the product, one-half, of the base and the height. But if the triangle is not the right-angle triangle, then finding the area of the triangle is not that easy. 

Hence, there are few other methods of finding the area in such cases and one such method is finding the area of a triangle using the determinants.

 

The determinant is the scalar value which is computed from different elements of a square matrix that has certain properties of a linear transformation. Let us now learn how to use the determinant to find the area of a triangle. Let’s say that (x₁, y₁), (x₂, y₂ ), and ( x₃, y₃ ) are three points of the triangle in the cartesian plane. 

 

Now the area of the triangle of the will be given as: 

 

k = ½ [ x₁ ( y₂ – y₃ ) + x₂ ( y₃ – y₁ ) + x₃ ( y₁ – y₂ ) ]

 

Here, k is the area of the triangle using determinant and the vertices of the triangle are represented by (x₁, y₁), (x₂, y₂ ), and ( x₃, y₃ ).

 

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In order to find the area of a triangle in determinant form, you use the formula given below:  

 

K = ½ [begin{bmatrix} x_{1} & y_{1} & 1\ x_{2} & y_{2} & 1 \ x_{3} & y_{1}  & 1end {bmatrix}]

The value of the determinant is either positive or negative but since here we are talking about the area of the triangle, we cannot have a negative value. Hence, we take the positive and the negative value or the absolute value of the determinant.

 

In case we already know the area of the triangle or the area has been given in the equation, we can use both the positive values of the determinant and the negative value of the determinant. In case three points are colinear, then it forms a line and not a triangle and the area of the triangle that is enclosed in a straight line is equal to 0. Therefore, the value of the determinant to find the area of the triangle would also be equal to zero. Keeping the aforementioned statements in mind, let us use the determinant expansion techniques using minors and cofactors and try to expand the determinant which denotes the area of the triangle. 

 

Hence,

 

k = ½ (x₁ ( y₂ – y₃ ) + x₂ ( y₃ – y₁ ) + x₃ ( y₁ – y₂ ))

 

This is how you apply determinants to make the calculation of the determinant easy. Let us apply this to a matrix and understand the concept much better. 

 

Solved Examples   

1. Find the area of the triangle whose vertices are A ( 1, 1 ), B ( 4, 2 ), and C ( 3, 5)

Solution: Using the formula that we have previously learnt, we can  find out the area of the triangle by joining the point given in the formula

 

K = ½ [begin{bmatrix} x_{1} & y_{1} & 1\ x_{2} & y_{2} & 1 \ x_{3} & y_{1}  & 1end {bmatrix}]

 

When you substitute the given values in the above formula, we get: 

 

K = ½ [begin{bmatrix} 1 & 1 & 1\ 4 & 2 & 1 \ 3 & 5  & 1end {bmatrix}]

k = ½ (1 ( 2 – 5 ) – 4 ( 4 – 3 ) + 3 ( 20 – 3 ))

 

k = ½ (1 ( -3 ) -4 ( 1 ) + 3 ( 17 ))

 

k = ½ (- 3 – 4 + 51)

 

k = ½ (44)

 

k = 22 units.

 

Since the area of the triangle cannot be negative, the value of k = 3 units.

 

2. Find the area of a triangle by determinant method whose vertices are A ( 4, 9 ), B ( – 3, 3 ), and C ( 6, 2 )

Solution: Using the formula that we have previously learnt, we can  find out the area of the triangle by joining the point given in the formula

K = ½ [begin{bmatrix} x_{1} & y_{1} & 1\ x_{2} & y_{2} & 1 \ x_{3} & y_{1}  & 1end {bmatrix}]

When you substitute the given values in the above formula, we get: 

K = ½ [begin{bmatrix} 4 & 9 & 1\ -3 & 3 & 1 \ 6 & 2  & 1end {bmatrix}]

k = ½ (4 ( 3 – 2) – 9 ( -3 – 6 ) + 1 ( – 6 – 18 ))

 

k = ½ (4 ( 1 ) – 9 ( – 9 ) +1 ( – 24 ))

 

k = ½ (4 + 81 – 24)

 

k = ½ (61)

 

k = 61 / 2 units

 

3. Find the area of the triangle whose vertices are A ( 4, 8 ), B ( – 6, 2 ), and C ( 5, 7 )

Solution: Using the formula that we have previously learnt, we can  find out the area of the triangle by joining the point given in the formula

 

K = ½ [begin{bmatrix} x_{1} & y_{1} & 1\ x_{2} & y_{2} & 1 \ x_{3} & y_{1}  & 1end {bmatrix}]

 

When you substitute the given values in the above formula, we get: 

 

K = ½ [begin{bmatrix} 4 & 8 & 1\ -6 &  2  & 1 \ 5 & 7  & 1end {bmatrix}]

 

k = ½ (4 ( 2 – 7 ) – 8 ( – 6 – 5 ) + 1 ( – 42 – 10 ))

 

k = ½ (4 ( – 5 ) – 8 ( – 11 ) +1 ( – 52 ))

 

k = ½ (20 + 88 – 52)

 

k = ½ (56)

 

k = 28 units

We have learned about equations in the earlier classes. An equation is a statement of the equality of two expressions. The two sides of the equality sign are referred to as the left-hand side (LHS) and the right-hand side (RHS) of the equation. 

 

For example, in the equation 3x + 4 = 8, where 3, 4, and 8 are the constants, and x is the variable. The LHS is given by the expression 3x + 4 and the RHS is given by the constant 8. The equation remains unchanged if we carry out the same operation on both sides of the equation.

 

To solve an equation, we carry out a series of identical Mathematical operations on two sides of the equation such that the unknown variable is on one side and its value is obtained on the other side.

 

Equation: An equation is a statement of equality of two algebraic expressions involving constants and variables. 

 

Based on the degree and variable in the equations, they are classified as linear and nonlinear equations.

Linear Equation

An equation in which the maximum degree of a term is one is called a linear equation. Or we can say that a linear equation that has only one variable is called a linear equation in one variable. 

 

A linear equation value when plotted on the graph forms a straight line.

The general form of a linear equation is ax + b = c, where a, b, c are constants and a0 and x and y are variables.

 

For Example: x + 7 = 12, 5/2x – 9 = 1, x2 + 1 = 5 and x/3 + 5 = x/2 – 3 are equations in one variable x. 

 

Here the highest power of each equation is one. 

 

2x + 3y = 15, 7x – y/3 = 3 are equations in two variables x and y.

 

When the linear equation is plotted on the graph we get the below figure.

Nonlinear Equation 

An equation in which the maximum degree of a term is 2 or more than two is called a nonlinear equation.

 

For example [3x^{2}] + 2x + 1 = 0, 3x + 4y = 5, this is the example of nonlinear equations, because equation 1 has the  highest degree of 2 and the second equation has variables x and y.

 

The nonlinear equation values when plotted on the graph forms a curve.

 

The general form of a nonlinear equation is [ax^{2} + by^{2} = c], where a, b, c are constants and a0 and x and y are variables.

 

When plotted on the graph we get the below curve

Difference Between Linear and Nonlinear Equations

Understanding the difference between linear and nonlinear equations is foremost important. Here is the table which will clarify the difference between linear and nonlinear equations. So let us understand exactly what linear and nonlinear equations are.

 

Differentiate Between Linear and Nonlinear Equations

Let us understand what are linear and nonlinear equations with the help of some examples.

Solved Examples

Example1:  Solve the Linear equation 9(x + 1) = 2(3x + 8)

Solution:

9(x + 1) = 2(3x + 8)

Expand each side

9x + 9 = 6x + 16

Subtract 6x from both the sides

9x + 9 – 6x = 6x + 16 – 6x

3x + 9 = 16

Subtract 9 from both the sides

3x + 9 – 9 = 16 – 9

3x = 7

Divide each by 3

3x/3 = 7/3

x= 7/3

 

Example 2 : Solve the nonlinear equation 

[3x^{2}] – 5x + 2 = 0 

Solution: 

[3x^{2}] – 5x + 2 = 0

Factorizing

[3x^{2}] – 3x – 2x + 2 = 0

3x(x – 1) – 2(x – 1) = 0

( 3x – 2)( x – 1) = 0

(3x – 2) = 0 or (x – 1) = 0

x = 2/3 or x = 1

Quiz Time 

Q. Solve the following linear equation and find the value of x

1. 3(5x + 6) = 3x – 2

2. (2x +9)/5 = 5

 

Q. Solve the nonlinear equations

1. [7x^{2} = 8 – 10x]

2. [3x^{2} – 4 = 5x]

 

Knowing basic math is important not only in each of the standards but it is also important that you keep your foundation strong even when you are learning for any higher class Math. Now finding these concepts online is not a problem at all as you can find them all listed on . Here you will get to know what is the Difference Between Linear and Nonlinear Equations and how to distinguish between them! Learning the concepts related to the linear equation and non-linear equation will help you solve a lot of problems in Algebra as well.

What Forms A Linear Equation?

The simplest form of a linear equation can be explained in the form y = a + bx where both a and b represent constants in an equation while there will be two variables that will be present. This forms the backbone of the linear equation. This linear equation when it is plotted on a graph paper will yield you a straight line with the line passing through the origin. It will have a constant slope value throughout the straight line that is passing through the origin. 

Distinguish Between Linear and Non-Linear in A Single Look

When talking about linear and nonlinear equations it is to be understood that the linear equations will have no exponents while the non-linear equations that are present will contain exponents raised to higher powers than 1.

In this article we are going to be about Simple Interest and Compound Interest. It covers the important topics like Simple Interest and Compound Interest and Simple Interest vs Compound Interest.

Simple Interest:

• In Mathematics, Simple Interest is a quick and easy method of calculating the interest charge on a given amount of money or loan. 

• We can determine this by Simple Interest multiplying the daily rate of interest by the principal by the number of days (n) that elapse between payments.

• The formula for Simple Interest is ,

Simple Interest (SI) = [frac{(Ptimes Rtimes T)}{100}]

 

Where, P is equal to the Principal, R is the equal to the Rate of Interest,  T = Time (Period).

 

The time is in years and the rate of interest is in percentage (%). 

 

NOTE : 

• Simple interest is calculated by multiplying the rate of interest by the principal and by the number of days (time period)  that  elapse between  the payments.

• It benefits consumers who pay their loans on time or early every month.

• Auto loans and short-term personal loans are examples of places where Simple Interest is used.

• We can calculate the total amount,  using the following formula:

Amount = Principal  + Interest 

Where, Amount (A) is equal to the total money paid back at the end of the time period (T)    for which the money was borrowed.

 

Compound Interest:

• Compound interest is defined as the interest calculated on the principal and the interest accumulated over the previous period of time.

• Compound interest is different from the Simple Interest. 

• In Simple Interest the interest is not added to the principal while calculating the interest during the next period while in Compound Interest the interest is added to the principal to calculate the interest.

Compound Interest (CI) = [Principal left ( 1+frac{Rate}{100} right )^{n}-Principal]

where, P is equal to principal ,  R is equal to rate of Interest,  T is equal to Time (Period)

 

The Formula to Calculate the Amount is

[Amount=Principal left ( 1+frac{Rate}{100} right )^{n}]

where , P is equal to Principal ,  Rate is equal to Rate of Interest,  n is equal to the time (Period).

 

Applications of Compound Interest:

Some of the applications of Compound Interest are:

1. Increase in population or decrease in population.

2. Growth of bacteria.

3. Rise in the value of an item.

4. Depreciation in the value of an item.

What is the Difference Between Simple Interest and Compound Interest?

• Besides Simple Interest there is another type of interest known as Compound Interest.

• The major difference between Compound and Simple Interest is that Simple Interest is based on the principal of a deposit or a loan whereas Compound Interest is based on the principal and interest that accumulates in every period of time. 

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Here’s the Difference Between Simple Interest and Compound Interest in a Tabular Form(SI vs CI)

 

Relation Between Simple Interest and Compound Interest –

Here’s the relation between Simple Interest and Compound Interest:

 

We already know from the SI vs CI definition that the interest is typically expressed as a percentage, and can be Simple or Compound Interest. Simple interest is generally based on the principal amount of a loan or deposit whereas Compound Interest is based on the principal amount and also on the interest that accumulates on the principal every period. We have already discussed this in the SI vs CI definition.

 

Questions to be solved:

1. Sohan takes a loan of Rs 1000 from the Central bank for a period of one year. The given rate of interest is 10% per annum. Find the interest and the amount Sohan has to pay at the end of one year.

Ans:  Let’s write down the given information,

Here, the loan amount = Principal = Rs 1000

Rate of interest = R = 10%

Time for which it is borrowed = T = 1 year

The formula to calculate the Simple Interest for one year,

Simple Interest (SI) = [frac{(Ptimes Rtimes T)}{100}]

Thus, the Simple Interest for a year, (SI) = [frac{(Ptimes Rtimes T)}{100} = frac{(1000times 10times 1)}{100}]

Now, let’s calculate the amount of money at the end of one year,

Amount = Principal  + Interest

The amount that Sohan has to pay to the bank at the end of one  year = Principal + Interest = 1000 + 100 = Rs 1100.

 

2. Ram borrowed a sum of Rs 5000 for 2 years at the rate of 3% per annum. Find the interest accumulated on the sum of at the end of 2 years and calculate the total amount.

Ans: Let’s write down the given information,

P = Rs 5000

R = 3%

T = 2 years

The formula to calculate the Simple Interest for one year,

Simple Interest (SI) = [frac{(Ptimes Rtimes T)}{100}]

(SI) = [frac{(Ptimes Rtimes T)}{100} = (SI) = frac{(5000times 3times 2)}{100} = Rs. 300]

Now, let’s calculate the amount of money at the end of two years,

Amount = Principal + Interest

The amount that Ram has to pay to the bank at the end of two years = Principal + Interest = 5000 + 300 = Rs 5300.

 

3. Mahi pays Rs 5000 as an amount on the sum of Rs 2000 which he had borrowed for 3 years. What is the rate of interest?

Ans: Let’s write down the given information,

Amount at the end of three years = Rs 5000

Principal= Rs 2000

SI = Amount – Principal = 5000 – 2000 = Rs 3000

Time = 3 years

Rate =?

We know the formula to calculate the Simple Interest,

(SI) = [frac{(Ptimes Rtimes T)}{100}]

R = (Simple Interest × 100) /(Principal× Time)

R = (3000 × 100 /5000 × 3) =0.2% 

Thus, R = 0.2%

 

4. The count of a certain breed of bacteria was found to increase at the rate of 5% per hour. What will be the growth of bacteria at the end of 3 hours if the count was initially 6000.

Ans: 

Since the population of bacteria increases at the rate of 5% per hour,

We know the formula for calculating the amount,

[Amount=Principal left ( 1+frac{Rate}{100} right )^{n}]

Thus, the population at the end of 3 hours = 6000(1 + 3/100)3

= 6000(1 + 0.03)3 = 6000(1.03)3 =  Rs 6556.36.

 

The essential differentiation between Simple Interest and Compound Interest is that Simple Interest is determined on the chief sum alone, while Compound Interest is determined on the chief sum in addition to intrigue accumulated over a period cycle.

 

We as a whole realize that Simple Interest and Compound Interest are two key ideas that are every now and again utilized in different monetary administrations, especially in banking. Straightforward interest is utilized in advances, for example, portion advances, car credits, understudy loans, and home loans. The Compound Interest is utilized by most of the bank accounts to pay the premium. It pays something beyond interest. Allow us to have at the distinction between Simple Interest and Compound Interest inside and out in this post.

 

Straightforward and Compound Interest definitions

Straightforward Interest: Simple premium is characterized as the chief measure of an advance or store made into an individual’s ledger.

 

Build Interest: Simply put, Compound Interest will be interest that amasses and accumulates over the chief sum.

 

What is the Simple Interest equation?

Basic premium is determined by duplicating the period’s financing cost by the chief sum and the residency. The term may be estimated in days, months, or a long time. Accordingly, the loan fee should be deciphered prior to being increased by the chief sum and residency.

 

To process Simple Interest, utilize the accompanying equation:

 

Straightforward Interest = P*I*N

Where

P signifies the chief sum.

I – The time frame’s loan cost

N represents residency.

 

What precisely is Compound Interest?

Build revenue (CI) acquires revenue on the recently procured revenue, rather than Simple Interest, which acquires interest on the primary total. The interest is applied to the head. CI represents Interest on Interest. The entire idea depends on creating critical returns by building interest on the primary sum.

 

As such, CI can possibly yield a better yield than essentially procuring revenue on a venture. Since Compound Interest depends on the essential force of accumulating, ventures rise dramatically.

 

The recurrence of compounding is dictated by the bank, monetary organization, or moneylender. It tends to be done on an every day, month to month, quarterly, half-yearly, or yearly premise. The higher the recurrence of accumulating, the more prominent how much premium gathered. Subsequently, financial backers benefit more from Compound Interest than debt holders.

 

Build revenue is utilized by banks for certain credits. Accumulate revenue, then again, is most regularly used in contributing. Accumulate revenue is likewise utilized by fixed stores, shared assets, and whatever other speculation that considers benefit reinvestment.

What is the Compound Interest Formula?

CI is determined by increasing the chief sum by one or more the interest to the force of the accumulating time frames. At last, the essential sum should be eliminated to compute the CI.

To register Compound Interest, utilize the accompanying equation:

A =[ P left ( 1+frac{r}{n} right )^{nt}-1]

Where

A = Annual Percentage Yield

P signifies the chief sum.

 

What is the Significance of Compounding?

Accumulating is a circumstance wherein premium procures interest. Basically, it demonstrates that when profit are reinvested, both the underlying speculation and the reinvested income ascend at a similar speed. This makes the ventures develop at a higher rate. This is alluded to as the force of compounding. The higher the intensifying recurrence, the better the venture returns. The occasions interest is accumulated in an entire year is alluded to as the building recurrence.

 

Compounding is an interesting thought, and it’s nothing unexpected that Albert Einstein named it the “Eighth Wonder of the World.” Compounding permits you to make your cash turn out more enthusiastically for you. Over the long haul, collecting revenue acquires a higher interest. Furthermore, the more extended measure of time you stay contributed, the greater the profit from the venture. Accordingly, it is ideal to start contributing early on to benefit from the force of compounding.

Studying calculus is an important part of the mathematical skill development of the students. These concepts of differential and integral calculus will be used in various domains of higher studies. Hence, learning the basic and advanced concepts of differentiation rules will create a strong foundation among the students. It will help to grasp the topics of the subjects related to higher mathematics and science applications in the professional courses. Here is the proper elaboration and explanation of the derivative rules you need to understand and learn to apply and solve problems.

What is Differentiation?

In the previous classes, you have studied the different functions that contain two variables. In these functions, a variable depends on the values of the other variable. The relation between these variables is interpreted using a formula/function/mathematical expression. This expression can be algebraic, trigonometric or related to any domain of mathematics.

Differentiation is the mathematical way to find the derivative of a function of two variables. This process is developed to find the instantaneous changes occurring in one of the variables depending on the changes of the other one. For instance, the instantaneous change in the rate of displacement considering time as the prime factor is called velocity.

If we elaborate the process then the changes in variable ‘y’ with respect to another variable ‘x’ is expressed as dy/dx. If y = f(x) then, f’(x) = dy/dx. Here, f’(x) represents the derivative of f(x). There are differentiation rules you will study in the Class 12 Maths syllabus so that you can easily carry out these operations on the functions given.

Differentiation or Derivative Rules

The evaluation of the derivatives should be properly. In fact, the result coming out of differentiating a function will be universal. Hence, there are some differentiation laws or rules that you need to understand and follow. Check the list of such rules mentioned below.

1. Power Rule of Derivatives

This is one of the basic rules of differentiation that you will find easier to understand. Observe the changes in a function when a power rule is applied.

If f(x) = xn,

Then, f’(x) = d/dx (xn) = nxn-1

If we consider an example, you will understand the application properly.

If f(x) = x6

Then, d/dx (x6) = 6x6-1 = 6x5

2. Sum Rule of Derivatives

If a function is represented by the difference or sum of two smaller functions, the sum rule of derivatives suggests the following changes.

If f(x) = m(x) ± n(x)

Then, f’(x) = m’(x) ± n’(x)

This formula shows that the signs of the smaller functions will be retained but these functions will follow the derivative rules. Consider this example.

If f(x) = x2 + x3

Then, f’(x) = 2x + 3x2

This is how the sum rule of derivatives is executed

3. Product Rule of Derivatives

According to this rule, if the function of a variable is the product of two other functions, then the outcome will be as follows.

If f(x) = m(x) × n(x),

Then, f’(x) = m′(x) × n(x) + m(x) × n′(x)

Consider this example to understand this rule better.

If f(x) = x2 × x3

Then, f’(x) = d/dx (x2 × x3)

= x3 × d/dx (x2) + x2 × d/dx (x3)

= x3 × 2x + x2 × 3x2

= 2x4 + 3x4

= 5x4

This will be the outcome of this rule of derivatives.

4. Quotient Rule Derivatives

The quotient rule derivatives suggest how to perform a differentiation of a function where there are two terms in division mode. Here is what the rule suggests.

If f(x) = m(x) / n(x),

Then, f′(x) = [m′(x) × n(x) − m(x) × n′(x)] / (n(x))2

If you follow the rule and put the values of the functions after performing differentiation, you will get the accurate answer.

5. Derivation of Chain Rule

If a function is represented by a function with another variable and this function is represented by the variable of the first function, then the derivation of chain rule suggests the following differential operation.

If f(x) = m(u) and  u = n(x),

Then, f’(x) = d/dx f(x) = d/du m(u) × d/dx n(x)

This formula or rule is quite simple to execute if you observe the terms properly in every step and learn the approaches proving the chain rule.

Learning Differentiation Rules is Fun

Take one step at a time and cover every rule related to the derived fractions. If you look carefully, you will find that the rules are nothing but the representation of the simpler rules such as sum, power, and product of derivative functions. It means that the basic rules are what you need to understand and then proceed to the next ones.

If you follow the rules of differentiation as elaborated in this section, it will become a lot easier to comprehend these concepts. The first segment of calculus will become much easier to understand and study. Your confidence will increase when you understand the basic differentiation rules and use them to execute the operations as required in the exercise sums. Keep practicing after learning these differentiation rules and solve problems.

For the purpose of analysis, data are presented as the facts and figures collected together. It is classified into two broad categories: qualitative data and quantitative data. It is not possible to measure qualitative data in terms of numbers and it is subdivided into nominal and ordinal data. Quantitative data, on the other hand, is one that contains numerical values and uses a scope. It is further classified as discrete data and continuous data.

What is Discrete Data?

Data that can only take on certain values are discrete data. These values do not have to be complete numbers, but they are values that are fixed. It only contains finite values, the subdivision of which is not possible. It includes only those values which are separate and can only be counted in whole numbers or integers, which means that the data can not be split into fractions or decimals.

Discrete Data Examples: The number of students in a class, the number of chocolates in a bag, the number of strings on the guitar, the number of fishes in the aquarium, etc.

What is Continuous Data?

Continuous data is the data that can be of any value. Over time, some continuous data can change. It may take any numeric value, within a potential value range of finite or infinite. The continuous data can be broken down into fractions and decimals, i.e. according to measurement accuracy, it can be significantly subdivided into smaller sections.

Continuous Data Examples: Measurement of height and weight of a student, Daily temperature measurement of a place, Wind speed measured daily, etc.

Discrete vs Continuous Examples

Height is continuous but we sometimes don’t really worry too much about minor variations and club heights into a set of discrete data instead.

On the other hand, if we count large quantities of any discrete entity. We may prefer not to think of 10,00,100 and 10,00,102 as crucially different values, but instead as nearby points on an approximate continuum.

Difference Between Continuous and Discrete Data

Questions on Discrete Data Continuous Data

Q: Classify the Following as Discrete and Continuous Data

1. Ducks in a pond.

2. Height of a student from age 5-15.

3. Number of animals in the Zoo.

4. The result of rolling a dice.

5. The number of books in a rack.

Ans: Ducks in a pond are discrete data because the number of ducks is a finite number.

1. The height of a student from age 5-15 is continuous data because the height varies continuously from age 5-15 which is not a constant for 10 years.

2. The number of animals in a zoo is continuous data because the number of animals varies yearly depending on the reproduction of new animals or the arrival of new animals.

3. The result of the rolling is a dice is 1, 2, 3, 4, 5, 6, 7, and 8. Therefore this is discrete data.

4. The number of books in a rack is a finite countable number. Therefore this is discrete data.

5. The number of books in a rack is a finite countable number. Therefore this is discrete data.

Tips and Tricks to study Discrete and Continuous Data

Statistics is a quite tough subject but for those who know how to use their mind, it’s the best one for them as it involves more practical knowledge and conceptual learning. And this is the reason why most students find it to be an interesting subject. 

In today’s changing times, pursuing a career in this subject is known to provide some of the most sought professional options. However, students shall understand that it is important for them to recognize their interests and based on that, select the best career path possible. Statistics involves organizing, analyzing, interpreting, and representing data. 

For some students, it might feel difficult as it involves logical thinking and mathematics but it’s a great option for those who love solving and putting their minds all into logic. 

Here are a few steps to prepare well for your exam which has statistics as a part of the course. 

Let us understand them. 

It would be great if you already have a bit of knowledge ab
out a course before opting for it. For example, in the case of statistics, algebra is a good option to learn before starting your statistics course. It helps you in saving time later as you already have some knowledge about algebra. Many other courses are available online to master your foundational knowledge which may assist you later. Another example is learning how to use calculators. There are particular calculators for different statistics exams and having good knowledge about their use and implementation would be great. 

Work more on basics as they form the basis for many other concepts. Once you are clear with the basic concept, you shall be able to understand additional concepts easily. Excelling your basic knowledge about the subject promotes conceptual learning and hence, you can easily get familiar with new concepts too.

We all know that time is very important, it doesn’t wait for anyone. So is the case of statistics too. You should know how to manage your time and utilize it in the best way possible. Procrastination isn’t even the last option if you are preparing for a statistics exam. You have to be dedicated. Making a schedule/timetable or studying together with your peers may help you in focusing and keep you active. 

Whenever you face any problem the best solution is consulting the concerned teacher, you should never hesitate to ask for doubts. If you feel like your doubt is stupid and you skip it rather than ask it, it is not the correct approach. You should ask the doubt there and then clear it, analyze it and work on it. 

It’s really normal for you to feel anxious before your exam especially when the subject is statistics. Just know that you have prepared and worked hard for this. So, you will get the result too! Results are the outcome of our efforts and if you believe that you have put your best, then you will surely be able to score good marks. It is understandable that it must be making you feel nervous right now but your emotions are valid! All you can do right now is to just take a deep breath and believe in yourself! If you feel tired at the moment, even that too is fine. Have some sleep, it’s important. You cannot expect your brain to work nicely with hundreds of thoughts, it will only create a mess. Sleep deprivation is of no use, instead, you should take a nap to relax the nerves of your brain in order to learn and study later. It might sound a bit weird but do not burden yourself with new topics before the exam. Instead of that, just revise what you’ve already done.

Did You Know?

Age is discrete data because we could be infinitely precise and use an infinite number of decimal places, rendering age continuous as a result. However, generally, we use age as a discrete variable.

In Mathematics, the four fundamental operations widely used are Addition, Subtraction, Multiplication, and Division. The role of the division is to break the number into equal parts. The four terms used in the division to divide the number into equal parts are dividend, divisor, quotient, and remainder. The number which divides another number is termed as a divisor whereas the number which is being divided is termed as a dividend. A quotient is a result which we get in the process of division whereas the remainder is the portion of the dividend that is left after the division. In this article, we will study division meaning, divisor definition, divisor formula, divisor examples, etc.

Division Meaning

The Division is a basic Arithmetic operation used widely in Mathematics. Division breaks a number into an equal number of parts. It divides a given number of items into different groups.

 

For example, If you take 20 apples and place them into 4 equal-sized groups, there will be 5 apples in each group.

 

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Division Symbol

Although there are numerous division signs that people use, the most common division sign is    ( ÷ ) and in case of fraction backlash (/) is used, where the numerator is written on the top and the denominator is written on the bottom.

 

Division example sign for x divided by z

 

x ÷ z

x/z

Division Formula

Dividend  ÷ Divisor = Quotient

 

Or,

 

Quotient = Dividend/ Divisor

Divisor Meaning

In division, we divide a number by any other number to obtain another number as a result. The number which is getting divided is known as the dividend and the number which divides a given number is termed as the divisor. The number which we receive as a result is known as the quotient. The divisor which does not divide the given number completely is referred to as the remainder.

Divisor Definition

A Divisor is a Number that Divides the Other Number in the Calculation.

 

For example: when you divide 28 by 7, the number 7 will be considered as a divisor, as 7 is dividing the number 28 which is a dividend.

7) 28 (4

–   28

——–

    0

——–

Divisor Formula

Operation division can be written in the following manner:

 

Quotient= Dividend ÷ Divisor

 

The above remainder equation can also be written in the following manner

 

Divisor = Dividend ÷ Quotient

Divisor Example

You can understand the divisor meaning in a better way by looking at divisor examples given below:

 

If 33 ÷ 11= 3, then 33 is the dividend and 11 is the divisor of 33 which divides the number 33 into 3 equal parts

 

If 50 ÷ 5 = 10, then 50 is the dividend and 5 is the divisor of 50 which divides the number  50 into 10 equal parts

 

1÷ 2 = Here divisor 2 is dividing the number 1 into a fraction.

5) 46 (9

–    45

——–

    1

——-

In the above example – 5 is the divisor, 46 is the dividend, 9 is the quotient and 1 is the remainder.

General Form of Division

The division is one of the 4 primary mathematical operations. Division can also be called the process of repetitive subtraction. The division is denoted by a mathematical symbol. The symbol consists of a short horizontal line with a dot, above and below it. The general formula for the division requires the quotient, the dividend, the divisor, and the remainder. Long Division Method is one of the methods of division that is used to divide two given numbers. The general formula of division is: 

Dividend = (Divisor × Quotient) + Remainder

Verification of division can also be done to check if the process done is right or not. Division can also be seen as the reverse of multiplication and the check by substituting the values in the General form of Division.

Special Cases of Division-

• When any number is divided by 1, the quotient is always to equal the dividend and the answer will be the same as the dividend. 

• If the dividend is equal to the divisor, the same numbers (but not 0) then the answer will always be 1.

• No number can ever be divided by 0. That’s why when a number is divided by 0 the answer is always undefined.
  

Divisor in Division

In a division, the divisor is the number that divides another number. It might or might not leave a remainder. In other words, in a division with given numbers, the divisor divides the dividend into equal groups. The number getting divided in a division is called the dividend. Some properties of the divisor of a division are-

• In any division, the remainder is always less than the divisor.

• If the quotient is similar to the dividend, then number 1 will always be the divisor of all the numbers. 

• If the dividend and the divisor are equal in a situation, the quotient will always be 1. 

• If the remainder is 0, that implies that the divisor has divided the dividend, completely.

• If in a division, the divisor is greater than the dividend, then the resultant number will always be a decimal number.

• In a division, a quotient is a number that is obtained upon dividing a dividend by a divisor. When any number is left over after the division. That is called the remainder.
  

Remainder in Division

The remainder is a part of a division. Remainder is the left-over digit that is received while performing division. In case of an incomplete division after certain steps we get a remainder as a result. The remainder is left-over when a few numbers are divided into groups with an equal number of things.

Some properties of the remainder are –

• In any division, the divisor will always be greater than the remainder. If the remainder of a given division is greater than or equal to the divisor, this indicates that the division is performed incorrectly.

Solved Example

1. There are 26 men and Rs. 5876 is distributed equally among all men. How much money will each person receive?

Solution:  Money received by 26 men = Rs. 5876

Money each person will receive = 5876 ÷ 26

Divisor = 26 ,  Dividend = 5876

Quotient = 22, Remainder =0

Therefore, each man will receive = Rs. 22

 

2. If 9975 Kg of rice is packed in 95 jute bags, how much rice will each jute bag contain?

Solution: Quantity of rice packed in 95 jute bags = 9975 kg 

Quantity of rice packed in 1 bag = 9975÷ 95

Divisor = 95 , Dividend = 9975

Quotient =105,  Remainder = 0

Hence, the quantity of rice each bag will contain = 105 kg

 

Interesting Facts

• De Morgan in 1845 introduced the oblique bar used as a sign of that process in the division.

• Division of a number by 0 is not defined

• The term colon is used to represent the division in some non-English speaking countries. The colon was introduced by Gottfried Wilhelm Leibniz in his 1684 Acta eruditorum.

• “÷” also known as obelus was introduced by Swiss Mathematician Johann Rahn in 1659.
  

Quiz Time

1. The given number that we are dividing by is called

Remainder

Divisor

Quotient

Dividend

2. 456 ÷ 7

7

76

70

75

3. In a division sum, the dividend is 336, the quotient is 42 and the remainder is 0. What will be the divisor?                                

7

9

8

2