Direct Speech and Indirect Speech

Direct Speech and Indirect Speech
Direct and Indirect Speech is one of the very important concepts in English learning. Speech can be classified into direct speech and indirect speech. Let’s have a look at the same in this post along with some examples for better understanding.

Direct Speech
Direct speech is a type of speech that exactly says what someone has said. In direct speech, the exact saying is repeated or quoted and the words spoken are written within inverted commas (“-----“). Direct speech is therefore also referred as quoted speech.

Examples of Direct Speech
1. Uncle John said, “I have bought an infant play gym for my son. It is not just fun but is also healthy.”
2. Mary asked, “Have you tried online diaper purchase?”

Here, in the first sentence, the speech of Uncle John speaking about his buying an infant play gym for his son is exactly presented. On the other hand, in the second sentence Mary’ question to on trying online diaper purchase is repeated exactly.

Indirect Speech
Indirect speech is a type of speech that encloses what a person has said. It doesn’t use quotations and is not repeated exactly like direct speech. In indirect speech, the tense usually changes as when we use indirect speech, we generally speak about something in the past.

Examples of Indirect Speech
1. Roy said that he enjoys to shop baby online products as it is easy and fast.
2. Mohan asked to Riya whether her kid has started going to school.
Here, in the first sentence, the Roy’s enjoying to shop baby online products is reported indirectly and not repeated. On the other hand, in the second sentence Mohan’s question about Riya’s kid going to school is similarly reported and not repeated.

Change direct speech to indirect speech:
1. Tony said, “I enjoy shopping for kids online.”
Tony said that he enjoys shopping for kids online.
2. Maria asked, “Do you eat non-vegetarian food?”
Maria asked me if I eat non-vegetarian food.
3. Father asked, “How long have you worked here?”
Father asked me how long I was working there.

Active and Passive Voice

Active and Passive Voice are two of the most essential concepts in English grammar. Taught in early middle school, active and passive voice is used extensively while speaking English language. Active voice and passive voice are two special forms of verbs. Let’s have a look at active voice definition and passive voice definition along with examples.

Active Voice
When the subject of a sentence performs the action expressed by the verb, the verb form is in active voice. In other words, the object receives the action of the verb. Active voice is most commonly used while speaking English.

Examples of Active Voice:
1. Mrs. Shah gets two baby dresses from  Little Kangaroo Kids wear collection. (Here, ‘Mrs. Shah’ is the subject, ‘gets’is the verb and ‘two baby dresses from Little Kangaroo Kids’ wear collection’ is the object.)
2. Maya shops baby clothing from online newborn baby clothes India stores. (Here, ‘Maya’ is the subject, ‘shops’ is the verb and ‘baby clothing from online newborn baby clothes India stores’ is the object.)
3. Ram sings a song. (Here, ‘Ram’ is the subject, ‘sings’ is the verb and ‘a song’ is the object.)

Passive Voice
When a subject in a sentence is acted by some other agent or something unnamed, the verb form is in passive voice. Passive voice is generally used when the action is more important than the subject in a conversation.

Examples of Passive Voice:
1. Infant clothes online India collection is explored by Mina. (Here, the subject Infant clothes online India collection’ is acted by the other agent ‘Mina’.)
2. Disney toys are loved by kids. (Here, ‘Disney toys’ is the subject, ‘is loved’ is the verb and ‘by kids’ is the object.)
3. Fish are eaten by cats. (Here, ‘Fish’ is the subject, ‘are eaten’ is the verb and ‘by cats’ is the object.

Active to Passive Voice:
1. Rama loves Sita
Answer: Sita is loved by Rama
2. Everybody believes in God
Answer: God is believed by everybody
3. Mother shops baby essentials from online stores
Answer: Baby essentials from online stores are shopped by mother
These are the basics about active and passive voice.

Scientific Notation Definition

Scientific Notations are standard way of writing number which are too small or too big.
In this standard notation all numbers are written in the form of:
m * 10n
here exponent n is an integer, m is mantissa which is a real number.
Scientific Notation Rules are as follows:
The value of mantissa holds a real number from 1 to 9 and exponent n contains the place value to get the original number.
If the decimal shifts to left by ‘n’, then the value of exponent increases by ‘n’. if decimal shifts to right the exponent decreases by 1 for each shift.
For addition or subtraction of numbers in scientific notations form their exponents should be same.
For multiplication of two such numbers, mantissa is multiplied and exponents are added.  
For division exponents are subtracted.

Scientific Notation Problems are as follows:
Q.1) Write the following numbers in scientific notations form:
1) 56788
2) 67.345
3) -6890
4) 0.000009888
5) 978.0 * 10-5
Sol.1)
1) 56788 = 5.6788 * 104. As we know that the mantissa can be real number from one to 9 so we shifted decimal to left four times and hence the exponent of 10 increases by 4.
2) 67.345 = (67.345)*100, in this number, value of mantissa is greater than 1 hence we shift the decimal to left and increase exponent by 1. So its scientific notations form is: 6.7345 * 101
3) -6890 = (-6.890)*103 by using rule used in above solution.
4) 0.000009888 = (0.000009888)*100. This number is less than one hence we will shift the decimal to the write and decrease the decimal by 1 for each right shift.
(9.888)* 10(0-6) = 9.888*10-6.
5) 978.0 * 10-5 is in exponential form but not in standard scientific notations as mantissa is greater than 1. So, shift it to let two time and increase exponent by 2 to get 9.78 * 10-5+2 = 9.78*10-3

Q.2) Solve the following:
1) (7.0  x  102)+(9.4 x 106)
2) (9.4 x 104)(3.5 x 10 –5)
3) (3.5  x  10 – 2)/(9.6 x 10 – 4)
Sol.2)
1) As exponents are not same. 9.4x106 = 94000 x 102
    (3.0  x  102)
  + (94000.0  x  102)
      94003.0 x 102 = 9.4003 x 106
2) (9.4 x 104) (3.5 x 10 –5)=(9.4 x 3.5) x 10(4+(-5))  (exponents are added as multiplication operation)
= 32.9 x 10-1
= 3.29 x 100
3) (3.5  x  10 – 2) / (1.5 x 10 – 4)
= (3.5/1.5) x 10 (-2-(-4)) (exponents are subtracted as division operation)
= 2.33 x 102

Present Tense and its types

Tense in English refers to the time of the verb’s action or state of being. There are three types of tenses – Present tense, Past tense and Future tense. Present tense refers to present action or state of being; past tense refers to past action or state of being while future tense refers to a future state of action or being. These three major types of tenses can be further classified into sub-categories. Let’s have a look at present tense and its classifications in this post.
Present Tense: Present tense is a type of tense that refers to some action or expression in the present time. For example: I am exploring online baby stores for new born shopping for my niece. Here, the sentence is referring to an action of exploring online baby stores for new born shopping in the present time and therefore, it is in present tense.

Types of Present Tense:
•         Present Continuous
•         Present Perfect
•         Present Perfect Continuous

Present Continuous Tense
Present continuous tense talks about an action that is still going on in the present time i.e.  continuing. For example: My cousin is using mustard seeds pillow for her baby. Here, mustard seeds pillow is still on use and therefore, the sentence is in present continuous tense.
Present Perfect Tense
Present perfect tense is that which speaks about something that began in the past and completes in the present. For example: John has solved the jigsaw puzzle. Here, John started solving the jigsaw puzzle in the past and finished it in the present and therefore, the sentence is in present perfect tense.
Present Perfect Continuous Tense
Present perfect tense is that which speaks about something that began in the past and is continuing in the present. For example: Jane has been buying baby essentials from online shopping baby stores from quite some time. Here, Jane started buying baby essentials from online shopping baby stores in the past and continuing in the present and therefore, the sentence is in present perfect continuous tense.
These are the basics about present tense and its types.

Subject and Predicate

Subject and Predicate are the most important concepts of English grammar. Introduced in the middle-school studies, subject and predicate plays an important role throughout the usage of English language. Every complete sentence conveying a meaning has two parts – the subject and the predicate. Let’s have a look at these two parts, its definitions and examples for better understanding.
Subject:
The subject is what or whom, the sentence is about. In simple terms, a person or thing that is discussed, described or dealt with in a sentence is called a subject of a sentence. For example:
• Mina has bought Pigeon products for babies. (Here, ‘Mina’ is the subject because the sentence talks about her buying Pigeon products for babies.
• Pigeon has wide range of products like baby diapers, shampoo, soap etc. (Here, ‘Pigeon’ is the subject as the sentence is talking about the brand.)
Predicate:
Predicate is a part of sentence that speaks something about the subject. It is a verb that states something about the subject. For example:
• John’s parents usually buy baby products India collection from online stores. (Here, ‘usually buy baby products India collection from online stores’ is the predicate of the sentence as it talks about the subject.)
• My uncle has recently started buying accessories from baby product online India stores. (Here, ‘has recently started buying accessories from baby product online India stores’ is the predicate of the sentence.)
More examples:
• I love my pet a lot. (‘I’ is the subject while ‘love my pet’ is the predicate)
• Hari is a busy person. (‘Hari’ is the subject while ‘is a busy person’ is the predicate)
• Online baby stores have made shopping easy for parents. (‘Online baby stores’ is the subject and ‘have made shopping easy for parents’ is the predicate.)
• Jack sings well. (‘Jack’ is the subject and ‘sings well’ is the predicate.)
• India is our motherland. (‘India’ is the subject and ‘is our motherland’ is the predicate.)

Introduction to parallel line extender:

Properties of parallel lines are the under the base of Euclid's parallel property. Two lines in a single plane that not even intersect or meet at any point then it is called as parallel lines. In other words Parallel lines are nothing but “A pair of lines in a plane which do not intersect or meet each other” then they are called parallel lines.

i.e. m1 = m2,

Where, m1 = slope of the first line.

m2 = slope of the second line.

Parallel Line Extender Problem:

Example for parallel line extender: Define the equation parallel to 4y + 4x = 8 and the line extender, with point (6.5,4).

Solution:

Given 4y - 4x = 8 and the point (6.5, 4)

To detect the perpendicular line, we have to find slope first.

For finding the slope, we need to change the given equation into slope intercept form.

4y + 4x = 8

Add 4x on both side,

4y + 4x = 8

- 4x = -4x

4y = -4x + 8

Divide by 4 on both sides,

y = (-x + 2)

The obtained equation is in the form, y = mx + b

So, the slope from the obtained equation m = -1

From generally we know that the slope of parallel lines are equal i.e. m1 = m2

Here, m1 = -1

So, m2 = -1

The line equation is,

(y - y1) = m(x - x1)

(y - (4)) = -1 (x - ( 6.5))

(y - (4)) = (-x + 6.5)

Y - 4 = - x + 6.5

Subtract 4 on both sides,

y = -x + 2.5

Answer: Thus, the parallel line extender is given through the line y = -x + 2.5

Example for Parallel Line Extender:

Example for parallel line extender: Define the equation parallel to 2y - 2x = 4 and the line extender is gets through the point (-2, 0.5).

Solution:

Given 2y - 2x = 4 and the point (4, 3)

To detect the perpendicular line, we have to find slope first.

For finding the slope, we need to change the given equation into slope intercept form.

2y - 2x = 4

Add 2x on both side,

2y - 2x = 4

+ 2x = +2x

2y = 2x + 4

Divide by 2 on both sides,

y = (x + 2)

The obtained equation is in the form, y = mx + b

So, the slope from the obtained equation m = 1

From generally we know that the slope of parallel lines are equal i.e. m1 = m2

Here, m1 = 1

So, m2 = 1

The line equation is,

(y - y1) = m(x - x1)

(y - (3)) = 1 (x - (4))

(y - (3)) = (x - 4)

Y - 3 = x - 4

Subtract 3 on both sides,

y = x - 1

Answer: Thus, the parallel line extender is used to find the line y = x - 1

Introduction to positive divisors

In math divisor is defined as “the number that you are going to divide by”. Divisor of a given integer can also be called as the divisor of a given integer.

Dividend /Divisor = Quotient.

Positive number/ Negative number = negative number

Negative number/Positive number = negative number

Positive/Positive  = Positive

Let us have an example 16/2 = 8 here 2 is a divisor.

16/4= 4 here 4 is a divisor.

16/8= 2 here 4 is a divisor.

In the above shown example the divisors are positive and also known as positive divisors.

The positive divisors when the dividend is 16 are 8, 4, and 2.

Example Problems for Positive Divisors:

Example problem 1:

Find the positive divisors of 28

Solution:

Positive Divisors of 28 are 1, 2, 4, 7, 14, and 28

Where 28 /1 = 28.

28 /2 = 14.

28 /4 = 7.

28 /7 = 4.

28 /14 = 2.

28 / 28 = 1

Hence for 28 we have 6  Positive divisors

Moreover 28 is a composite number

Example problem 2:

Find the positive divisors of 52

Solution

Positive   Divisors of 52 are 1, 2, 4, 13, 26, and 52.

Where 52/1=52.

52/2=26.

52/4=13.

52/13=4.

52/26=2.

52/52=1.

Hence for 52 we have 6   Positive divisors

Example problem 3:

Find the positive divisors of 30

Solution

Positive Divisors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.

Where 30/1=30

30/2= 15.

30/3=10.

30/5=4.

30/6=5.

30/10=3.

30/15=2.

30/30=1

Hence for 30 we have 8 Positive divisors

Example problem 4:

Find the positive divisors of 100

Solution

The positive divisors of  100 are  1,2,4,5,10,20,25,50,100

Where 100/1=100

100/2= 50.

100/4=25.

100/5=20.

100/10=10.

100/20=5.

100/25=4.

100/50=2.

100/100=1.

Hence for 100 we have 9 Positive divisors

Example problem 5:

Find the positive divisor of 31

Solution

Given that 31 is a prime number

All the prime numbers have only two divisors

The two divisors are 1 and the numbers itself

So the positive divisor of the number 31 is 1, 31 alone.

Finally the divisors of the prime numbers are 1 and the number itself.

Practice Problems for Positive Divisors:

Problem 1:

Find the positive divisor of 63

Solution:

The divisors of 63 is 1, 3,7,9,21,63

So the number 63 has 6 divisors.

Problem 2:

Find the positive divisor of 37

Solution:

The divisors of 37 are 1, 37

Problem 3:

Find the positive divisor of 15

Solution:

The divisors of 15 is 1, 3, 5, 15

Problem 4:

Find the positive divisor of 35

Solution:

The divisors of 35 is 1, 5, 7, 35

Understanding how to graph parabolas

A parabola (math) in co-ordinate geometry is defined as the locus of a point whose distance from a fixed point in the plane and perpendicular distance from a fixed line in the plane (not passing through the fixed point) are equal. The fixed point is called the focus of the parabolic and the fixed line is called the directrix of the parabolic.

These have many applications. Parabola help us to understand the trajectory of a projectile. The reflectors in car head lights or speakers in a sound system or mirrors in a telescope are all parabolic in shape. Parabolic mirrors are also used to harness solar energy. These are only a few examples where parabolas are useful. Parabolic forms find application in various other sciences field as well. Therefore it is of prime importance that we know how to graph a parabola.

Graph the parabola:

Parabola graphs are of two types: (1) horizontal and vertical. The horizontal form is again sub categorized into two types: (a) the ones that open left and (b) the ones that open right. Similarly the vertical forms are also of two types: (a) ones that open up and (b) the ones that open down.

The general equation of a horizontal form is: x = a(y-k)^2 + h, where (h,k) is the vertex of the parabolic. Whether the parabolic form opens left or right would depend on the sign of the term ‘a’. If a is positive, then that opens right and if a is negative the same opens left.

The general equation of a vertical parabolic form is y = a(x-h)^2 + k, where again (h,k) is the vertex of the same. If the parabolic form opens up then the ‘a’ in the above equation would be positive. If ‘a’ in the above equation is negative then the parabolic form would open down.

As we can see from the equation of the para-bola stated above, these equations are quadratic. The standard form of a quadratic equation in x is f(x) = ax^2 + bx + c. This would be a vertical parabolic form. If we were to find the vertex from this equation it would be: h = -b/2a and k = f(h). There to if ‘a’ is negative, these open down and if ‘a’ is positive, the same opens up.

Adding and Subtracting Matrices

A matrix is a rectangular array of symbols, numbers or expressions which were arranged in a sequence of rows and columns. These having only one row (1 x n) is known as a row vector and if it is having only one column (m x 1) is called as a column vector. The forms of these which are of the same size can be subjected to both addition and subtraction by element to element.

Matrix Addition and Subtraction
If in case when two matrices are needed to be either added or subtracted, then there lays a condition. It is the condition that both of these should be of the same order. This means that both of the forms involved in the operation should have the same number columns and rows. The expression generally used to represent the addition and subtraction is as follows.

Zij = Aij  +  Bij . This means that the elements of the ‘i’ row and column ‘j’ of the matrice A and the elements of the ‘i’ row and ‘j’ column of the B are added or subtracted. When we do this, then it will result in a new matrix of the same order in which addition or subtraction was done. This is nothing but Z form which has ‘i’ rows and ‘j’ columns.

Matrix Addition Rules
As we saw earlier about the order of matrices that is to be added or subtracted should have the same order there lays one more rule or necessity for doing addition. When ‘m’ is the total number of rows and ‘n is the total number of columns; then ‘i’ will be greater than or equal to 1 and less than or equal to ‘m’. Also ‘j’ will be greater than or equal to 1 and less than or equal to ‘n’.

Let us see an example for both Matrix addition and Matrix Subtraction.

• Let us consider of A and B matrices having two rows and three columns. The matrice A has elements 1, 2 and 3 in the first row and 4, 5 and 6 in the second row. While the B has elements 3, 4 and 5 in the first row and 7, 8 and 9 in the second row. Now for Addition, we do first row as 1+3, 2+4 and 3+5. Also for second row, we do as 4+7, 5+8 and 6+9. Hence we will get a final part of first row as 4, 6 and 8 and second row with elements as 11, 13 and 15.
• Similarly subtracting instead of adding from the above step, we will get a matrice with first row elements as -2, -2, -2 and second row as -3,-3,-3. This is better known as its subtraction form.

Real numbers

Real Number Definition– As the name says “Real”, real numbers are actually numbers that really exists. Any number that we think of is considered as a real number, be it positive or negative, fraction or decimal. Real numbers are numbers those includes both rational and irrational number. A real number has to have a value. If there is no value to any number then we can call that number as an imaginary number. All integers like -75, 89, and 84 etc. are considered as real numbers. All fractions like 3/5, 7/2, -9/7 are considered as real numbers too.

Decimals along with repeating decimals are also considered as real numbers. A real number can be any positive or negative number. We can plot All Real Numbers on the number line too.  Therefore we can order the real numbers and that we cannot do in case of imaginary numbers. The name of imaginary numbers itself says that they are imaginary so we can just imagine them; they do not have a specified value.
The real numbers can be plotted the same way we plot the integers that is smaller numbers on left and larger numbers on right. So greater the number, more it will be towards right side of number line.So we can call real numbers as all those numbers which are present on number line are termed as real numbers.
Some Real Numbers Examples are pi, 34/7, 5.676767, -1034, 45.87 etc. Some examples of imaginary numbers are square root of -34 or square root of -2, as there is a negative under the root, so the value of this number cannot be found.

Similarly value of infinity cannot be determined too .Hence these numbers are not real numbers and are considered as an imaginary numbers. So real numbers are all integers, fractions decimals and repeating decimals numbers.

We can add, subtract, multiply and divide real numbers just like another numbers. We can perform the operations on real numbers same way as we do them on other numbers.

Now the question arises that Is 0 a Real Number– Zero is considered as an integer and all integers are real numbers. Therefore zero is considered to be as a real number. All Real Numbers Symbol is R which is used by many mathematicians. The symbol R is used to represent the set of real numbers. All real numbers can be seen on the number line but we cannot find imaginary numbers on that.

Learning to divide Mixed Numerals

We know that a number in the form of P/Q is called a fraction. Both P and Q are integers. The fractions can be proper fractions, improper fractions and sometimes even mixed fractions. Mixed fractions are also known as mixed numerals or mixed numbers. The operations of addition, subtraction, multiplication and division can be performed on these fractions. Now we will learn the process of dividing mixed fractions to get the final answer. This can be done in a few simple steps. The first step would be to convert the mixed fraction into an improper fraction. Improper fraction is nothing but a fraction in which the denominator must be less than numerator. After this step again the question arises how to divide mixed fractions to arrive at the final answer. The second step is to get the divisor. Divisor is the number or the fraction by which the mixed fraction has to be divided. After getting the divisor, the numerator of divisor is made its denominator and vice versa and multiplied with the improper fraction we got from the mixed fraction given.

The product got after the multiplication is the answer. Now the question how do I divide fractions would have been cleared in the minds of many. This is a very simple process as we understand it. We get division fractions in the process. After finding the reciprocal of the divisor and multiplying with the improper fraction we get a product. This has to be simplified to the lowest terms possible. Only then we get the right answer. Simplifying to lowest terms can be done by dividing the numerator and the denominator by a common number so that both the numerator and denominator are fully divisible by that number. This process has to be continued till the numerator and denominator in the lowest forms and they are not further divisible. This process is done to simplify the fraction and not keep it as large numbers. This will help in the calculations as we need deal with large numbers. Also the time consumed would be less. Lesser the time consumed more will be efficiency. The process of division takes bit more time than other operations of addition, subtraction and multiplication. This is because division involves multiplication as well. This can be overcome by doing more and more practice. Once we have enough practice we can bring down the time required to solve a problem.

Introduction to Boolean logic calculator

Boolean logic algebra is the algebra of two values. These values are usually taken to be 0 and 1, as we shall do here, although F and T, false and true, etc. Boolean logic algebra are also a common uses of Boolean logic calculator. More generally Boolean algebra is algebra of values from any Boolean algebra as a model of the laws of Boolean algebra. In Boolean logic calculator first enter the expression then then enter the values of x and y. now press the calculate button then the answer will be produced.

Basic Operations Involved in Boolean Logic Calculator:

in Boolean logic calculator the operations mentioned below are involved automatically ,    

After values, the next ingredient of any algebraic system has its operations. Whereas elementary algebra was based on numeric operations multiplication x • y, addition x + y, and negation −x, Boolean logic algebra calculator is customarily based on logical counterparts to those operations, namely conjunction x • y (AND), disjunction x + y (OR), and complement or negation ¬x (NOT). In electronics, the multiplication is represented by a AND, an addition is represented by OR, and the NOT is represented with an over bar.

Conjunction was the closest of these three to its numerical counterpart, in fact on 0 and 1 it is multiplication. As the logical operation conjunction of two propositions is true when both propositions are true, and otherwise is false.

Disjunction works almost similar to addition in the boolean logic calculator, with one exception: the disjunction of 1 and 1 is neither 2 nor 0 but 1. Thus the disjunctions of two propositions are false when both propositions are false, and otherwise are true.

Logical negation does not work like numerical negation in the calculator. Instead it corresponds to incrementation: ¬x = x+1 mod 2. It shares the common numerical negation the property that applying it twice returns the original value: ¬¬x = x, just as −(−x) = x.

Some of the identities and operations involved in boolean logic is also in boolean logic calculator as,

(1a) x • y = y • x                                     (1b) x + y = y + x                   (1c) 1 + x = 1

(2a) x • (y • z) = (x • y) • z                      (2b) x + (y + z) = (x + y) + z

(3a) x • (y + z) = (x • y) + (x • z)             (3b) x + (y • z) = (x + y) • (x + z)

(4a) x • x = x                                          (4b) x + x = x

(5a) x • (x + y) = x                                  (5b) x + (x • y) = x

(6a) x • x1 = 0                                        (6b) x + x1 = 1

(7) (x1)1 = x

(8a) (x • y)1 = x1 + y1                            (8b) (x + y)1 = x1 • y1

Problem Based on Boolean Logic Calculator:

Problem 1: Find the value for x.y = y.x

x y x.y y.x
0 0 0 0
0 1 0 0
1 0 0 0
1 1 1 1

Addition and Its Properties

Addition is one of the four basic mathematical operations that refer to combination of collections of objects towards forming a larger number. For example: Mother bought two Pigeon baby products for Rs. 220 and three baby toys from Fisher Price India brand for Rs. 350. How much she paid in total for Max online shopping. Adding, Rs. 220 + Rs. 350 = Rs.  570. Thus, mother paid Rs. 570 in total for Max online shopping for the baby. This is called addition operation in mathematics. There are five properties of addition, namely

• Identity Property
• Commutative Property
• Associative Property
• Property of Opposites
• Property of Opposite of a Sum

Identity Property: According to the identity property of addition, there is a unique real number zero for every real number a, b or c etc. Zero is referred as the identity elements in addition. For example: Mother bought one powder (suppose a) from pigeon baby products. Mathematically implementing identity property of addition, a + 0 = a and 0 + a = a.

Commutative Property: According to the commutative property of addition, for all real numbers, the order of addition does not change the result. For example: Mother bought 3 baby toys (suppose a) from Fisher Price India brand and 7 baby products (suppose b) from Pigeon baby brand. Mathematically, 3 + 7 = 10 and 7+3 = 10. Therefore, a + b = b + a

Associative Property: This property of addition states that when three real numbers are added, the grouping or association doesn’t change the result. Example is similar to the previous one. Therefore, (a + b) + c = a + (b + c).

Property of Opposites: This property of addition states that against every real number, suppose a, there is a unique real number, i.e, -a. A number and its opposite are called additive inverses and the addition of additive inverses is equal to zero. Therefore, a + (-a) = 0.

Property of Opposite of a Sum: This property of addition states that the opposite of a sum of real numbers is equal to the sum of the opposites. For example: - (2+3) = (-2) + (-3) = 5. That is, - (a + b) = (-a) + (-b).
These are the properties of addition.

Double Bar Graphs

Double Bar Graphs, also called as Double Bar Chart, is the type of bar graph that helps in finding the relationship between two set of data. These are used to compare the data between same value and within the group of value from time to time. Double Bar Graphs also has all the advantage and attributes as it like the regular bar graph. Double Bar Graphs may be created in horizontal or vertical position as per the given data.

Double Bar Graphs Properties

• We have to follow the instruction to construct the double bar graphs. The instructions are given below.
• The frequency scale must start from zero and go up in equal steps in x-axis and y-axis.
• Bars should have equal width.
• Two types of data should have different colors and the abbreviation for the color was given separately in the graph sheet.
• Both of the axes were labeled and they should name as per the given data.

Advantage and Difference between Single Bar Graph and Double Bar Graphs

• Double Bar Graphs are easy to construct with the help of given set of data.
• It is easy to understand and easy to find the difference between two set of data.
• Data group in the double bar graph is represented by two bars as comparing with single bar graph

Illustration for Double Bar Graphs

Let us see in detail about the double bar graph with an example. Construct a double bar graph for the given set of data.


               Mathematics   Science
Student 01      85          70  
Student 02      95          75  
Student 03      65          80  
Student 04      82          70  
Student 05      88          85  
Student 06      90          80  

• Double bar graph was drawn for six students to compare their marks in mathematics and science.
• To construct the graph, first we have to decide about the title of the graph.
• Then choose the vertical or horizontal bars.
• We name the x-axis as students and y-axis as marks.
• Choose the scale in the y-axis as per the given marks.
• Put label on the graph and finally draw the bars.

Sine graph generator

Sine graph is the periodic with period 2`pi` .They wriggle back and forth between -1 and 1 in a smooth way. Sine starts at 0 and goes up to 1 .Sine graph generator produce the sinusoidal wave. The sine function graph usually how to find phase sift, period, amplitude for the equation

Sine graph generator following conditions produced y = sinx ,  Domain (− ∞, ∞) , Range [− 1, 1] , Principal domain (−π / 2,π / 2)

Sample Problem for Sine Graph Generator:

Example 1:

 Draw the amplitude for given question y=a sinx for various values of  a.

Solution:

For sine is how far the graph gets away from the x-axis, so it is called the amplitude.

Y=a sinx ,  where a=1,2,3 0.5,0.2

See the given  Sine graph  generator

For sine is how far the graph gets away from the x-axis, so it is called the amplitude.

Y=a sinx    a=2 , -2

See the given  Sine graph  generator

y= 2 sinx, -2sinx

Example 2:

 Draw the amplitude for given question y= sinx for various values of  a.

Solution

y=sinx where x values follows
 x =1, -1
see the sine graph generator

Example 3:

Show that Periods other than 2pi is a little harder; because there is a lot that the graph does in the course of its period. P=2`pi` /b

Y=sinbx where b=1,2,3,0.5

solution:

See the given  Sine graph  generator

Practice Problem for Graphs of Sine Generator:

1. Graphing the amplitude for given question y=d sin2x for various values of d

Answer: d values 1,2,3,1.5,4

2. Graphing the period of given question y= sinbx for various values of b

Answer: b values 1,2,3,1.5

3. Graphing sine periods for y= sinbx values of b and d are;

Answer: b=1,2,3,0.5

4. Graphing the amplitude for given question y=t sin3x for various values of t

Answer: t values 1,2,1.5 0.5,0.3.

Binary System and Introduction

What are Binary Numbers
In the word Binary, ‘Bi’ means two, so a system of numbers which involves only two digits is called binary system. The two binary numbers used are ‘1’ and ‘0’. The binary system works in the same way as the decimal system which consists of ten digits which are 0 to 9, in case of binary system there are only two possible digits 1 and 0. Any number which is expressed in the binary system would be a combination of the two digits, 1 and 0. It is mainly useful in the field of technology.  In an electrical circuit we have only two options which are ‘on’ and ‘off’ and hence based on this the binary system plays an important role in all the communications which take place inside a computer. In a decimal system the place values are ones, tens, hundreds, thousands, ten thousands and so on. These values can be written as power of 10 which would be 10^0, 10^1, 10^2, 10^3, 10^4 and so on. In the same way the binary system starts from the right and proceeds towards left the difference being the base is 2.

Binary Numbers Chart can be given as below
2 raised to power zero = 2^0=1
2 raised to power one = 2^1=2
2 raised to power two= 2^2=4
2 raised to power three=2^3=8
2 raised to power four = 2^4=16
2 raised to power five= 2^5=32
2 raised to power six = 2^6=64 and so on, in general it can be considered as 2 raised to power x = 2^x
For example, let us consider a binary number 1011 and convert it into decimal form
Binary conversion: 8   4   2    1
      Decimal form:  1   0   1    1
1011(base 2) = 1x(8) + 0x(4) + 1x(2) + 1x(1) = 8+0+2+1=11(base 10)

List of Binary Numbers are as follows
0000=0; 0001=1; 0010=2; 0011=3; 0100=4; 0101=5; 0110=6; 0111=7; 1000=8; 1001=9; 1010=10 and so on.
Subtracting Binary Numbers
While subtracting binary numbers the following are the facts to be remembered:
0 – 0 =0 ;  1 – 0 = 1; 1 – 1= 0 and 10 – 1 is equivalent to 2 – 1 = 1 in binary system
Subtract 1011 from 10101.

0110 0110 1
-  1   0 1 1
__________
          1  0 1 0        
(1 -1 =0; 10 – 1 =1;0 – 0 = 0; 10 – 1 = 1 and 0 = 0 which gives 1 0 1 0)
Binary number      10101   -    1011 = 1010
Decimal equivalent   21       -       11   =10

Absolute Values

In algebra and in vectors we come across numbers in negative directions although the magnitudes of such quantities have no signs or supposed to be positive.  Let us take a closer study.



What is Absolute Value in Math
Let us consider the numbers 5 and -5. Algebraically, they are not same. But when you plot these numbers on a number line, they are no doubt on the opposite sides of the origin 0. But they are at the same distance from the origin. That is -5 is also 5 units away from the origin on a number line. Hence considering only the fact of ‘how far’, we can say that the absolute value 0f -5 is 5.
Thus we can define that an absolute value of a number is only its physical value and hence it is always referred as positive. Symbolically the absolute value of a variable ‘x’ is denoted as lxl and it is always equal to + x.

Finding Absolute Value
As explained earlier an absolute value of a number cannot be negative. Thus, the easiest method of finding absolute value of a number is just consider the number only, ignoring the sign before it. That is for finding absolute value of -10, just consider only the number 10. That is l-10l = 10.

Limit of Absolute Value
Since the absolute value of a variable is always positive, the upper limit of the absolute value is infinity. However, because an absolute value can never be negative, it cannot cross below 0. Therefore, the lower limit of absolute value is 0. Thus the limit of absolute value of a variable can be expressed as [0, ∞).

Absolute Value Practice
The concept of absolute value is important. For example, A is situated 10 miles from his office and B is situated 5 miles from the same office but exactly in the opposite direction. Now to calculate the distance between A and B, we cannot algebraically say 10 + (-5) = 5 miles or 5 + (-10) = -5 miles. Here you need to apply the absolute value practice and say the distance is l10l + l5l = 10 + 5 = 15 miles.

Properties of Absolute Values
The important properties of absolute values are,
lxl = x for x ≥ 0, but = -x for x < 0.
If lf(x)l = a then there can be two cases. That is f(x) = a and f(x) = -a.
Suppose lf(x)l + a = b and if a > b, then there is no real solution to the equation.

How to Draw Line Plot?

What is a Line Plot in Math?
The definition of a line plot states that it is a graphical representation of frequency of occurrence of each and every data on a data line. Line plot is graphically represented using some marks such as x or dot.

How to do a Line Plot?
A line is drawn with the data values marked in it as we mark values in the x axis of the graph. The number of occurrences of the data values is marked above the number line individually as the variable x to make a Line Plot.  So the number of “x” marked vertically above each data value shows the number of occurrences of the data value in the given data set. Thus this gives the details of how frequently a data value occurs in the data set.

Example of a Line Plot
Let us consider the marks scored by Students of grade VII in their exam.  Let us assume that there are 10 students in the class and the maximum mark they can score is 100. The marks scored by the ten students are 90, 99, 76, 89, 55, 86, 98, 68, 40, and 35 respectively.

Note that the school follows the grade system as shown below:
• The marks 90-100 will lie in grade A
• The marks 80-89 will lie in grade B
• The marks 70-79 will lie in grade C
• The marks 60-69 will lie in grade D
• The marks 50-59 will lie in grade E
• The marks below 50 are considered as ‘fail’ and they will lie in grade F.

Now let us map the students’ marks in the above grade range:
• The marks lying in grade A are 90, 99, and 98
• The marks in grade B are 89, and 86
• The mark in grade C is 76
• The mark in grade D is 68
• The mark in grade E is 55
• The marks in grade F are 40, 35

From this analysis, we can see that three students have scored grade A, two of them have scored grade B, one student each have scored the grade C,D and E and two students have failed and obtained grade F.  Now, let us mark the line plot with the values A, B, C, D, E, and F:

x
x x x
x x x x x x

<--------------------------------------------------------->
| | | | | |
A B C D E F
Grade
The line plot drawn above gives the graphical representation of the frequency of the occurrence of data in every grade.

Examples of Complementary Angles

What are Complementary Angles?
Two angles are said to be complementary angles, if the sum of the two angles measure 90 degrees.  Thus we can say that two complementary angles together from a right angle measuring 90 degrees. But these two angles need not necessarily be adjacent angles i.e. next to each other.

Example of Complementary Angles
One of the best examples of complementary angles can be seen in right angled triangle.  In a right angled triangle, one angle is the right angle. It is a fact that the sum of all the three angles in a right angle is equal to 180 degrees. Thus, it is clear that the sum of the other two non-right angles will be equal to 90 degrees.  This means that the two non-right angles in a right angled triangle are complementary angles. In other words, we can say that these two angles complement each other.

Some of the other examples of complementary angles are:
55 degrees, 35 degrees
40 degrees, 50 degrees
67 degrees, 23 degrees

How to Solve Complementary Angles?
Now let us see how to find complementary angles.  As we know the definition, finding complementary angles is very easy by applying the definition. If the value of one angle is given, then another angle complementary to that angle is found out by subtracting the given value from 90 degrees.

Example 1:
Consider two angles which are complementary to each other.  If one of the angles is 48 degrees, find the other angle.

Solution: If x is the unknown angle, then 48 added to x will give 90 degrees.  Therefore, x is given by subtracting 48 from 90. i.e., X = 90-48 = 42 degrees.
Therefore, the other angle is 42 degrees.

Some Complementary Angles Problems will be in the form of slightly confusing word problems.

Example 2:
If one of the complementary angles is six more than twice the other angle, find the angles.

Solution:  If the variable Y is considered as one angle measure, then, as per the given statement, the other angle is given by 2Y+6. We know that the sum of the two complementary angles is 90 degrees.  So,
Y+2y+6 = 90
3y = 90-6
Therefore the value of Y will be obtained if 84 are divided by 3. Thus, the value of Y will be 28. If y = 28, then the second angle will be (2*28) + 6 which results in the value 62. Thus, the measures of the angle are 28 and 62.

Free Math Help

Plenty of students find it necessary to find good math help in order to understand the subject and get good grades. Math does not come easily to all and while the subject itself is fairly simple, different students understand it in different ways. Basic math skills are necessary since people use them all the time. Many occupations require an understanding of math or at least specific areas of math.

To learn math effectively, study well right from the beginning. Understanding concepts as and when they are covered, helps students retain them longer. Since each lesson in math builds on what was taught earlier, this ensures that students understand what they are taught as they move further into each topic or chapter.

Going through the lesson before class enables students to get the most out of each class. Even if you don't understand what you are reading, hearing the same thing again will make it much clearer in your mind. It will also enable you to ask better questions and jot down good points. Make it  a habit to go back and study what you have covered in class the very same evening, when it's still swimming around in your memory.

There are a lot of math resources at students' disposal, especially on math help websites. Make use of free math help to learn concepts and get ahead in the subject. Math resources online consist of free worksheets, games and quizzes. There are any number of tutorials covering all the topics in school and college level math, written in simple terms and illustrating each concept with plenty of examples.

Online calculators are very useful tools which can give you the answer for just about any type of question. If you're stuck with a problem, try using any of the free online calculators to get the answer immediately. It is also a great way to cross check your homework and make sure you have everything right. Students can also enlist the help of live tutors who will solve the problem and send you the steps to the solution as well. Practice is key to getting good grades in math and online worksheets provide students with plenty of options to choose from. All the sites provide the answers to their worksheets, with some providing the complete solution too.

Practice of statistics

Statistics in the plural are statistical facts systematically together with some definite object in view, in any field of enquiry, whatsoever of observation, measurement or experiment; for example, statistics of the population of a country, males and females, refugees, births and deaths, heights and weights, income and expenditure, food production, etc. The statistics deals with every aspects of this, consists not only the set, analysis and interpretation of data. The statistics is a technique used to obtain, analyze, summaries, compare and present the numerical data. In this article, example problems and practice problems for learning practice statistics exam is given.

Example Problems for Practice of Statistics:-

Example Problems for learning practice Statistics exam are given below:

1) In statistics the daily maximum temperature recorded in degree C. At New york during the first week of July, 2005 was as under, 39, 37, 38, 28, 30, 35, and 36. Find the mean temperature recorded.

Sol:-

Mean temperature = sum of observations / no of observations

= `(39 + 37 + 38 + 28 + 30 + 35 + 36) / 7`

= `243 / 7`

= 34.7 degree C.

2) find the arithmetic mean of the numbers 3, 0, -1, 7, 11 in statistics.

Sol:-

Mean = sum of observations / no of observations

= `(3 + 0 + (-1) + 7 + 11) / 5`

= `20 / 5`

= 4.

3) In statistics what is the median weekly salary of worker in a firm whose salaries are Rs. 84, Rs. 60, Rs. 50, Rs. 40, Rs. 45, Rs. 42, Rs. 38, Rs. 65, Rs. 71?

Sol:-

1) first arrange the salaries in order: Rs. 84, Rs. 71, Rs. 65, Rs. 60, 50, 45, 42, 40, 38.

2) Next, count the number of salaries. It is 9.

The fifth salary (Rs. 50) has the four salaried which are less than it and four salaries above it. Therefore, Rs. 50 is the middle or median salary.

4) Find the mode for the following data 52, 58, 58, 58, 65, 73, 73, 73?

Sol:-

Here 58 and 73 repeated three times.

So 58 and 73 are two modes.

Practice Problems of Statistics:-

Practice problems for learning practice statistics exam are as follows:

1) Find the arithmetic mean of the numbers 4, 2, 1, 0, 7, 10.

Answer: 4

2) Find the median salary of the following salaries of worker: Rs. 56, Rs. 89, Rs. 121, Rs. 38, Rs. 98, Rs. 70, Rs. 70, Rs. 72.

Answer: Rs. 71.

3) Find the arithmetic mean of the numbers -2, -1, 0 , 1, 4 , 10.

Answer: 2

4) Find the mode of the numbers 10, 11, 12, 12, 14, 15, 15.

Answer: 12 and 15

Exterior angles of polygons

Definition:
For any polygon, the angle formed between any one side of the polygon extended and the next consecutive side is called an exterior angle of that polygon.  See the figure below:

The sum of an internal angle and its corresponding exterior angle is always 180 degrees. In other words the interior angle and the exterior angle of any polygon are supplementary to each other.

A polygon has as many numbers of exterior angles as interior angles. The following figure shows the exterior angles of a pentagon.

Exterior angles theorem:

The exterior angles theorem can be stated as follows:
“The sum of exterior angles of a polygon is always 360 degrees. “
The sum of exterior angles does not depend on the number of sides of the polygon. The sum of interior angles would be different for different polygons. But the sum of exterior angles of a polygon is always 360 degrees.

Proof of the above theorem:

We know that the sum of the interior angles of a triangle = 180 degrees
The sum of interior angles of a quadrilateral = 360 degrees
The sum of the interior angles of a pentagon = 540 degrees
Therefore the sum of interior angles of a polygon of n sides = (n-2)*180 degrees
Therefore the measure of each interior angle = (n-2)*180/n degrees
Since each interior and its corresponding exterior angles are supplements of each other,
Therefore the measure of corresponding exterior angle = 180 - (n-2)*180/n
Therefore the sum of n such exterior angles
= n*[180 - (n-2)*180/n]
= n*[180n – 180n + 360]/n
= 180n – 180n + 360
= 360 degrees
Hence it is proved that irrespective of the number of sides of the polygon, the sum of exterior angles is always 360 degrees.

Exterior angles of a polygon formula:
The formula for the sum of interior angles of a polygon is
= (n-2)*180 degrees

Therefore each of the interior angle would be
= (n-2)*180/n degrees

Therefore the corresponding exterior angle for that polygon would be:
= 180 - (n-2)*180/n degrees

Thus the formula for finding the measure of each exterior angle of an n sided regular polygon is
180 - (n-2)*180/n degrees

Frequency table math

In statistics frequency table refers to the tabular representation of frequencies of a sample. The frequency distribution of a sample can be represented in three ways: (a) Textual representation, (b) tabular representation and (c) diagrammatic representation.

(a) Textual representation: This method comprises presenting data with the help of a paragraph or a number of paragraphs. The official report of an enquiry commission is usually made by textual presentation. The merit of this mode lies in its simplicity and even a layman can present data by this method. This method is however not preferred by statisticians simply because it is dull monotonous and comparison between different observations is not possible in this method.

(b) Tabular representation: We can define frequency table as a systematic presentation of data with the help of a statistical table having a number of rows and columns and complete with reference number, title, description of rows as well as columns and foot notes, if any. This method is any day better than the textual representation because it facilitates comparison between rows and columns. Complicated data can also be represented using a frequency table. To be able to make a diagram for the distribution we first need the data in tabular form. Only then can we convert it to a bar chart or a pie chart. All other parameters such as mean, median, mode, standard deviation, regression analysis etc are not possible unless we have the frequency distribution table.

Example of a frequency table:
The production of rice in a particular region for each year starting from the year 2001 to the year 2007 is given in the frequency table below:

Year	Quantity in MT (Metric tons)(frequency)
2001	25
2002	30
2003	32
2004	36
2005	35
2006	35
2007	37

In the above frequency table example we see that it is relative easy to understand the data. By merely looking at the table we can say that the production in the year 2004 was 36 MT. Also it is fairly simple to calculate the mean, median mode etc as well. Thus we see that representing frequency distribution in tabular has many merits.

(c) Diagrammatic representation of data: Another alternative and attractive representation of statistical data is provided by charts, diagrams and pictures. The various forms of diagrammatic representations are bar chart, histogram, pie chart, etc.

Concepts of Trigonometric Equations

In this article we have to discuss about various trigonometric equations. Before this we have to know about trigonometric identities. In mathematics trigonometric identities means which involves trigonometric functions and they show right value for every single variable. In other way we can also say that, geometrically these are the identities which involve some functions with one or more angles. They are also triangle identities which involves angle as well as side length of triangles. These identities are very useful when we solve the trigonometric-equations. One example of these identities is integration by substitution method.

Now we come to trigonometric-equations. An equation means algebraic functions written by using sum and difference formula. Similarly trigonometric-equations also contains algebraic equations but with trigonometric functions of the variables. If they contain only such functions and variables, then in solution we have to find an unknown constant which is an argument to a trigonometric function. Basically there are fundamental forms of trigonometric-equations, they are

1. Sin X=n
2. Cos X=n
3. Tan X=n
4. Cot X=n
5. Cosec X=n
6. Sec X=n

Where (n) is a constant. Now the equation sin X=n, solved only when (n) lies between the interval [-1, 1]. Suppose n comes under the interval then we need angle. After using some identities we can solve the equation. Similarly for cos X=n solved only when (n) lies between [-1, 1]. Next procedure is same. For equation tan X=n and cot X=n, (n) should be any real number. For equations cosec X=n and sec X=n, these equations can be solved by converting in other basic equations such as (cosec x=1/sin x) and (sec x=1/cos x).

Some other basic form of trigonometric-equations is (Asin x+ bcos x=0). To solve trigonometric equations using various relations between the trigonometric functions, we have to convert trigonometric-equations in a form such that value of one of the trigonometric functions of the desired variable can be determined. The roots of these trigonometric functions can be found by using inverse trigonometric functions. For solving trigonometric equations like (Asin x+bcos x=0), we have to convert this equation into basic form of equation and then using various trigonometric identities we can solve the equation. We separate this equation into real part and angle part.

Trigonometric equations solver means a process by which result can be displayed.  It can find all the solutions. The result shown first in the form of radians and then in decimal form. They show the results also in form of degrees. By using trigonometric equations solver we can solve equations easily.

Mean Median and Mode

What is the definition of mean median and mode?
Mean is a parameter of central tendency measure. The central tendency measure indicates the average value of data, where the term average is a generic term used to indicate a representative value that describes the general centre of data. Mean for raw data and for grouped data. For grouped data it is known as frequency distribution. Mean is the central value of the distribution in the sense that positive and negative deviations from the mean balance each other. It is a quantitative average.

Median is the central value of the distribution in the sense that the number of values less than the median is equal to the number of values greater than the median. So, median is a positional average. Median is the central value in the sense different from the arithmetic mean. In case of the arithmetic mean it is the numerical magnitude of the deviations that balances but, for the median it is the number of values greater than the median which balances against the number of values of less than the median.

Mode is defined as the value of the variable which occurs most frequently. Mode is for both raw data and grouped data. In raw data the most frequently occurring observation is the mode that is data with highest frequency is mode. If there is more than one data with highest frequency then each of them is a mode. Thus we have unimodal means single mode, bimodal means two mode and trimodal means three modes data sets. In grouped data mode is that value of (x) for which the frequency is maximum. If the values of (x) are grouped into the classes such that they are uniformly distributed within any class and we have a frequency distribution than we calculate maximum frequency by using formula.

What is the mean median mode in math?
Now we discuss about what does mean median mode mean in math. There are some formulas in math for mean median and mode.
For raw data mean= [∑x/n], x is refers to value of observation, n is the number of observations.
For grouped data mean= [∑fx/∑f]
Median= [(n+1)/2]
Mode=L+ [f0-f1/2f0-f1-f2]*h, L is lower limit, f0 is largest frequency, f1 is preceding frequency, f2 is next frequency and h is width.
Now, How to find the mean median and mode. Suppose we have given some problem then to find mean median and mode we can use above formula, by using formulas we can get the results.

Understanding the box and whisker plot

Box and whisker plot definition:
The basic assumption in statistics is that a set of data has a central tendency. That means the number of the data are distributed around some central value. The box of the box-whisker plot takes care of the middle half observations of the data. So we can define the box-whisker plot as a diagram that represents the tendency of data to centre on the median.

Box and whisker plot examples:
The following pictures show box and whisker plot of various data sets.

Shown above is a box-whisker plot of weight in pounds of players.

Shown below is a box- whisker plot of the length of fish in a particular lake:

From the plot we can see that the median length is 12 cm. The smallest fish has a length of 5 cm, the longest fish is 20 cm long. Most of the fishes lie between 8.5 to 14 cm in length, where 8.5 is the lower quartile and 14 cm is the upper quartile.

How to make a box and whisker plot:
Step 1: Arrange the data in ascending order from the lowest to highest value.

Step 2: Find the median of the data. For a data of n observations, median = ((n+1)/2)th observation if n is even and if n is odd then median = average of (n/2)th and ((n+2)/2)th observation. So if number of observations = 11 then (11+1)/2 = 12/2 = 6th observation is the median. But if number of observations is 14, then average of 14/2 = 7th and (14+2)/2 = 16/2 = 8th observation would be the median.

Step 3: The median that we found above has now divided the data into two halves. Now to further divide each of the half into two quarters, we need the 1st and the 3rd quartile values. Suppose the median is the mth observation. Then the middle value between the 1st and the mth observation would be the 1st quartile and the middle value between the mth and the last observation would be the 3rd quartile. They are denoted by Q1 and Q3.

Step 4: Now  you have these three values, Q1, M (= median) and Q3. We also have the lowest and the highest observation, X1 and Xn. So we now have 5 points. We make a number line using suitable scale and mark these 5 points and draw a box around the Q1 to Q3. That is our box and whisker plot.

Step by step math solution

Mathematics is used throughout the world and it is an essential tool in many fields like natural science, engineering, medicine, and the social sciences. "Mathematics" is the Greek word its gives the meaning of learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.

Step by Step Math Solutions - Method to Solve:

There are many formulae abounded for solving math problem step by step One of the most important formula is P.E.M.D.A.S. It is nothing but the operations done in the problems solving in math. They are

• Parenthesis
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

Step 1: First perform the operations inside a parenthesis.
Step 2: Now solve the exponents
Step 3: Then multiplication and division is done from left to right
Step 4: Then addition and subtraction is done from left to right

Step by Step Math Solutions - Example Problems:

Step by step math - Problem 1:

Solve the equation step by step

X + 6 = 8

Solution:

It is one step equation

Subtract 6 on both sides

X + ( 6 - 6 ) = (8 - 6)

X = 2

Step by step math - Problem 2:

Solve the equation step by step

4x + 2=18

Solution:

Subtract 2 on both sides

4x + (2 - 2) = 18 - 2

4x = 16

Divide by 4 on both sides

`(4x)/4` = `16/4`

X = 4

Step by step math - Problem 3:

Find the value of x step by step

6x - 8 = 4 x -10

Solution:

Subtract 4x  from both sides of the equation

2x - 8 = - 6

Add 6 to both sides of the equation

2x = 2

Divide both sides by 2:

x = 1

The answer is x =  1

Step by Step Math Solutions - Probability Problems:

Step by step math - Problem 1:

A fair coin is tossed two times. What is the probability of getting at least one heads.

Solution:

Let A = be the event of getting at least two heads

Let S = Sample Space which refers to the total number of probable outcomes.

S = (HH, HT, TH, TT) =4

A = (HT, TH) =2

P (A) = `2/4` = `1/2`

The probability is `1/2` .

Step by step math - Problem 2:

Consider a die is rolled; calculate the probability of in receipt of odd numbers?

Solution:

There are six different outcomes 1,3,5,7,9,10,12,14,16,17,18

n (A) = 6, total number of odd numbers occur 6.

n (S) = 11, total number of outcomes is 11.

Probability of the event A happen = P (A) = n (A) / n(S) = `5 / 11` = 0.45.

Probability of the event A does not happen = P (A') = 1 – P (A) = 1 - 0.45 = 0.55.

Circle Division

A circle is a closed curve. The points on the circle are equidistant from the center of the circle. A circle creates 2 regions, the interior of the circle is one region and the exterior is another region.

A circle can be divided into equal or unequal parts. The division can be accomplished by straight lines or curved lines. With every nth line the circle gets divided into n+1 parts, meaning with one line circle can be divided into 2 parts, with nth line circle can be divided into n+1 parts assuming that the lines do not intersect.

Circle Division Term in Maths:

A circle division can be accomplished by using a straight line or a curved line. As we are not considering the area calculation for this discussion, let us restrict ourselves to straight lines for the purposes of this study.

The straight line that joins a point on the circumference of the circle to another point on the circumference of the same circle is called a chord. A chord divides the circle into 2 parts.

Consider several chords on the circle and suppose some or all of the chords intersect. Then it is found that the number of area regions that the chords cut the circle into is always more than the number of chords passing through the circle.

Here is an example

The above is popularly known as the cake cutting example.

In the first picture, one chord divides the circle into 2 parts.in the second picture, two chords divide the circle into 4 parts, in the third three chords divide the circle into 3 parts and the forth diagram 4 chords divide the circle into 11 parts.

Here you see that the first cut creates 1 new region (1+1)

The second cut creates 2 new regions (2+2)

The third cut creates 3 new regions (4+3) and the 4th cut creates 4 new regions (7+4)

This is mathematically represented as

F (n) = n + f (n-1)

Conclusion for Circle Division:

Circle division is an important study in geometry that gives several clues for practical applications. There are several theorems and postulates that delve deeper into circle division and provide new insights into this area of study.

Frequency and Frequency Table

Frequency table, it comes under the category of statistics. First we see about frequency. The frequency of a given table or data is that how many times the value occurs in table or data. For example, suppose in a class five students are score sixty marks in mathematics then the score of sixty is said to have frequency of five. Frequency means range of values also and the frequency of any data is denoted as (ƒ).

What is a frequency table?
A frequency table contains sets of collected data values. The arrangement is such that the magnitude of collected data values is in ascending order along with the corresponding frequencies. Now frequency table definition in other way is that it is a list of quantity in ascending order and list shows the numbers how many times each value occurs.

How to make a frequency table?
Here we understand the procedure of preparing A frequency table. For this we follow some steps. In step one; we make a table with three columns. In these three columns, first columns shows the marks and marks are arranged in ascending order means start from the lowest value. The second column shows tally marks. It means put corresponding tally in front of marks. When all values listed then make horizontal lines for all the values. Third column shows frequency, count the value of frequency for each mark and write in third column. Finally the frequency table is constructed.

The frequency table is different for different types of problems. Sometimes in problems range values is given with no of students and cumulative numbers. When we make frequency table for this type of problem then in first column we write range value in ascending order. In second column we write number of student which comes under the corresponding range. In third column we write cumulative numbers then we solve the problem.
Always i find probability distributions is very hard for me. If you do feel the same watch out for my coming posts.

How to do a frequency table?
We start with simple example. Suppose we have a frequency table with two columns. In first column marks(20-30, 30-40,40-50,50-60, 60-70) is given and in second column frequency(2,4,3,5,7) is given. Frequency column represent the number of students who scored marks in particular range of frequency. We have to calculate the number of student who gets fifty plus marks. Fifty plus marks comes in 50-60 and 60-70 range and their corresponding number of student is 5 and 7. So the answer is 12 students.