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Understanding random variables

If S is an example space with a probability calculate and X is actual valued meaning defined above the basics of S, next X is known as random variable. A random changeable is also called a chance variable or a stochastic variable. Random variables are used in the learn of probability. They were developed to help the study of games of option, stochastic actions, and outcome of technical experiments. By capture only the arithmetical property required to answer probabilistic questions.

Definition Discrete Random Variable
If a random variable takes single a finite or a countable integer of values, it is called a discrete random variable, to use variable calculator. You can also get help with formula for area of a trapezoid
Types of Random variables

Discrete Random variable
Continuous Random variable

limit calculator

Calculus is based, in general, on the idea of a limit. By using online limit calculator we can solve any type of problem.So,first we will discuss about basic concept of limit.In calculus we will use a symbol ε , to represent a small positive quantity,however small it may be.

Let A be a point on the number axis and A₁ and A₁^' are taken as close to A as possible.If we now represent A by a, then A₁ and A₁^' will be a + ε , a - ε respectively. So, we can say that x lies in (a - ε , a + ε ) and x approaches to a, (x ≠ a). Now the condition that x approaches to a (denoted by x→0) is that for any positive number ε ,however small it may be, the absolute value of ( x - a) should be less than ε. i.e 0<| x - a | < ε the value of x = a is excluded or, a - ε <>
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Understanding Mathematics Form

Introduction to mathematics form 1
The numbers are the basic source of math. The numbers can be written in many several forms such as word form, expanded form, and standard form and also in place value form. The word form can be written for the numerals or numbers in English words. For example: 1 can be written as one, etc in math. The numbers are the symbolic representation or abstract object of math. Let us see about what is word form for math in this article.

Representation of Word Form from 1 to 10 for Math

1 – The number 1 can be written as one.
2 – The number 2 can be written as two.
3 – The number 3 can be written as three.
4 – The number 4 can be written as four.
5 – The number 5 can be written as five.
6 – The number 6 can be written as six.
7 – The number 7 can be written as seven.
8 – The number 8 can be written as eight.
9 – The number 9 can be written as nine.
10 – The number 10 can be written as ten.

Conic Sections

Introduction:

Algebra is the one of the most important chapter in mathematics subject. Here the Conic section is one of the topics in algebra chapter. In mathematics a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. The conic sections were named and studied as long ago as 200 BC.

Conic Sections:

Conic section is generally defined as intersection of a plane and a cone, it will  depend on How the plane is oriented, the curve will be one conic sections. Or a conic section is a curve formed by the intersection of a cone with a plane.
 Four types of conic sections are there,

• Circle,
• Parabola,
• Ellipse, and
• Hyperbola.

Conics: Ellipses: Introduction

Conics: Ellipses: Introduction
An ellipse, informally, is an oval or a "squished" circle. In "primitive" geometrical terms, an ellipse is the figure you can draw in the sand by the following process: Push two sticks into the sand. Take a piece of string and form a loop that is big enough to go around the two sticks and still have some slack. Take a third stick, hook it inside the string loop, pull the loop taut by pulling the stick away from the first two sticks, and drag that third stick through the sand at the furthest distance the loop will allow. The resulting shape drawn in the sand is an ellipse.
Each of the two sticks you first pushed into the sand is a "focus" of the ellipse; the two together are called "foci" (FOH-siy). If you draw a line in the sand "through" these two sticks, from one end of the ellipse to the other, this will mark the "major" axis of the ellipse. The points where the major axis touches the ellipse are the "vertices" of the ellipse. The point midway between the two sticks is the "center" of the ellipse.

Equation of Ellipse with Standard Positions

Equation	X2/a2+y2/b2=1, a > b	X2/a2+y2/b2=1, a <>
Centre	(0,0)	(0,0)
Coordinate of centre.	(a,0),(-a,0)	(0,b),(0,-b)
Length major axis	2a	2b
Length of minor axis	2b	2a
Equation of major axis	Y = 0	X = 0
Equation of minor axis	X = 0	Y = 0
Equation of directrix	X = ± a/e	Y = ± b/e
Eccentricity Latus rectum	‘e’ = √(1-b2/a2) 2b2/a	‘e’ = √(1-a2/b2) 2a2/b
Focal distance	a ± ex	b ± ey
Coordinates of foci.	(ae , 0), (- ae , 0)	(0 , be). (0 , - be)

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Translating Word Problems

Translating Word Problems: The first step to effectively translating and solving word problems is to read the problem entirely. Don't start trying to solve anything when you've only read half a sentence. Try first to get a feel for the whole problem; try first to see what information you have, and what you still need.

The second step is to work in an organized manner. Figure out what you need but don't have, and name things. Pick variables to stand for the unknows, clearly labelling these variables with what they stand for. Draw and label pictures neatly. Explain your reasoning as you go along. And make sure you know just exactly what the problem is actually asking for. You need to do this for two reasons:
The third step is to look for "key" words. Certain words indicate certain mathematical operations. Below is a partial list.
Addition- increased by more than combined, together total of sum added to
Subtraction- decreased by minus, less difference between/of less than, fewer than
Multiplication- of times, multiplied by product of increased/decreased by a factor of (this type can involve both addition or subtraction and multiplication!)
Division- per, a out of ratio of, quotient of percent (divide by 100)
Equals- is, are, was, were, will be gives, yields sold for

Translating Word Problems: Examples

• Translate "the sum of 8 and y" into an algebraic expression.
  This translates to "8 + y"

• Translate "4 less than x" into an algebraic expression.
  This translates to "x – 4"
  
  Remember? "Less than" is backwards in the math from how you say it in words!

•  Translate "x multiplied by 13" into an algebraic expression.
  This translates to "13x"

• Translate "the quotient of x and 3" into an algebraic expression.
  This translates to " x/3"

• Translate "the difference of 5 and y" into an algebraic expression.
  This translates to "5 – y"

• Translate "the ratio of 9 more than x to x" into an algebraic expression.
  This translates to "(x + 9) / x"

• Translate "nine less than the total of a number and two" into an algebraic expression, and simplify.
  This translates to "(n + 2) – 9", which then simplifies to "n – 7"
  
  Here are some more wordy examples:

• The length of a football field is 30 yards more than its width. Express the length of the field in terms of its width w.
  
  Whatever the width w is, the length is 30 more than this. Recall that "more than" means "plus that much", so you'll be adding 30 to w.

         The expression they're looking for is "w + 30".

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Translating words into Algebraic Expression

Definition: Translating words into algebraic expression is the process of translating the word problems into an algebraic expression which can be used to solve the word problem and produce the solution for the given words problem.

Problems of translating words into algebraic expression:

Problem 1:

Flowers shop has thirty Roses and forty Lilly. How many pieces of flowers does flowers shop have?

Solution:

Let a = Total number of Flowers in the flowers shop.
The sum of thirty Roses and forty Lilly is equal to the total number of flowers in the flowers shop. It translates the words problem into an algebraic expression.
a = 30 + 40

Solve this expression.

Let a = Total number Pieces of Flowers in the flowers shop
a = 70.

There are 70 Pieces of Flowers in the flowers shop.

Problem 2:

The sum of twice a number plus 20 is 86.Find the number.

Solution:

In words problem, the word is means equals and the word and means plus.
Translating words problem into an algebraic expression
the sum of twice a number and 20 equals 86.
Write an expression

2X + 20 = 86

Solve this expression using the variable.

2X + 20 = 86 (Expression)

2X + 20 – 20 = 86 - 20 (subtract by -20 on both sides)

2X= 66

X = 33 (Divided by 2 on both sides we get the result)

Solution to the problem is 23.

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Ratio and Proportion

Ratio: Ratio is the numerical relationship between two quantities of the same kind. The first quantity is called the antecedent and the second quantity is called the consequent.

Proportion: a, b, c and d are said to be proportion if a : b = c : d.
a and d are called the extremes, b and c are called the means. a, b, c and d are called first proportion, second proportion, third proportion and fourth proportion respectively.

Types Of Ratios and Proportion:

Ratios are classified in to six types:

1) Duplicate ratio:The ratio of the squares of the two numbers.Ex: 9:16 is the duplicate ratio of 3:4.

2) Triplicate ratio:The ratio of the cubes of the two numbers.Ex: 27:64 is the triplicate ratio of 3:4.

3) Sub-duplicate ratio:The ratio between square root of the two numbers.Ex: 4:5 is the sub-duplicate ratio of 16:25.

4) Sub-triplicate ratio:The ratio between the cube roots of the two numbers.Ex: 4:5 is the sub-triplicate of 64:125.

5) Inverse ratio:If the two terms in the ratio interchange their places,then the new ratio is inverse ratio of the first.Ex: 9:5 is the inverse ratio of 5:9.

6) Compound ratio:The ratio of the product of the first term to that of the second term of two or more ratios.Ex: 3/4,5/7,4/5 and 3/5 is 3/4 x 5/7 x 4/5 x 3/5 = 9/35.

Proportions are classified in to four types:

1) Continued proportion:In the proportion 8/12=12/18, 8,12,18 are in the continued proportion.

2) Fourth proportion:If a:b=c:x,then x is called forth proportion of a,b and c.The fourth proportion of a,b.c =bc/a

3) Third proportion:If a:b=b:x,then x is called third proportion of a and b.Third proportion of a,b =b^2/a

4) Second or mean proportion: If a:x=x:b,then x is called second or mean proportion of a and b.Therefore mean proportion of a and b =Root of(ab)

Simple Interest and Compound Interest

Interest: Interest is the amount of money we pay for the use of some amount of money

There are two types of interests,

a) simple interest: Simple interest is the Interest paid / compensated only on the original principal, not on the interest accrued

b) compound interest: interest means that the interest Which includes the interest calculated on principal amount

Algebra Formula to find simple interest :

Simple interest I = PRT

Where, P is the Principal amount, R is the Rate of interest, N is Time duration.

When we knows interest I we can find p, n or r using the same formula ,

Different forms of algebra simple interest formula

Algebra formula to find Compound Interest:

FV = PV (1+r)ⁿ

PV is the present value

r is the annual rate of interest (percentage)

n is the number of years the amount is deposit or borrowed for.

FV = Future Value is the amount of money accumulate after n years, including interest.

Elementary Algebra, ratio word problems

In Elementary Algebra practice , a ratio expresses the magnitude of quantities relative to each other rather. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient. Mathematically, a proportion is defined as the equality of two ratios. However in common usage the word proportion is used to indicate a ratio, especially the ratio of a part to a whole.Let's see an example from 4th grade math algebra word problems with solution

Question:-

Find two numbers such that their difference ,sum and product are in the ratio 1:4:15 respectively.

Answer:-

This solution will explain how to understand ratio word problems

Let the numbers be x and y

By the problem

(x-y) : (x+y) : (xy) = 1 : 4 : 15

x-y = k ----- 1

x+y = 4k ----- 2

xy = 15k -------- 3

Where k is the constant of proportionality

From 1+2 , we get x = 5k/2

Substitute this in eq 2 , we get y = 3k/2

By substituting this x and y values in eq 3

We get k = 4

If we put this k value in x = 5k/2 , we get x = 10

If we put k value in y = 3k/2 , we get y = 6

So the numbers are 10,6

How to find Slope of parallel lines

Slope is used to describe the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line or pair of straight lines. It is also always the same thing as how many rises in one run.The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.The larger the absolute value of a slope, the steeper the line. A horizontal line has slope 0, a 45° rising line has a slope of +1, and a 45° falling line has a slope of -1. A vertical line's point slope form
is undefined meaning it has "no slope."

Question:-

How to find the slope of the line parallel to x+2y=10

Answer:-

Point to remember:- Parallel lines will have same slope
So we just have to find the slope of the given line.

x+2y = 10

subtract 'x' on both sides

2y = -x+10

divide by 2 on both sides

y = -x/2 + 5

So the slope is -1/2 for both the lines.

same way,we can also find all points having an x-coordinate of 2 whose distance from the point 2 1 is 5

What is a Natural number

In mathematics, from number theory tutorial there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century.

Natural numbers have two main purposes: counting ("there are 3 apples on the table") and ordering ("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively. (See English numerals.) A more recent notion is that of a nominal number, which is used only for naming.

Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics.

When we add two natural numbers

we also get a natural number

Let's have a look at a simple problem from numeric and algebraic operations



2 is a natural number

5 is also natural number

2+5=7

7 is also a natural number.

We represent natural numbers with W

w={ 0,1,2,3.....}

If you know the basics of this you can get any math answers

How to convert Decimal into Fraction

Topic: Fraction

A fraction is a mathematical expression relating two quantities or numbers, one divided by the other. The numbers may be whole numbers integers- this is a rational number. For example, 1/2 is a fraction. They can also be polynomials - this is a rational function.

Let's have a look at the answer which have done by online algebra tutor
The process of decimal to fraction conversion involves the use of the fundamental rule of fractions; the fraction should be written in its lowest terms. The following examples demonstrate how to convert decimals to fractions.

Here is example of a 7th grade math equations




For similar problems of this type purple math will guide you.

Rules of Exponents

Topic: Exponents
Exponentiation is a mathematical operation, written aⁿ, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication:

Here are some examples:

1.)Addition:An exponent can be added only when they have the same number.

For example: 2x1 + 2x1 = 4x1

   3x2 - 2x2 = 5x2


2.)Subtraction:The rule is same for subtraction.


3.)Multiplication:While multiplying, multiply the numbers and add the exponents.

For example: 3x1 * 2x1 = 6x2 (Add)

    4x2 * 3x3 = 12x5 (Add)


4.)Division:While dividing the exponents, the denominator takes the opposite sign and m

and mover to the numerator. The numbers should be divided as usual.

For example:

    36x8-4
 ___________ = 6x4
      64


     48x2+4
 ___________ = 8x6
     6x-4           

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problem on simple interest

Topic:- simple interest

Interest is a fee paid on borrowed assets. It is the price paid for the use of borrowed money or, money earned by deposited funds. Assets that are sometimes lent with interest include money, shares, consumer goods through hire purchase, major assets such as aircraft, and even entire factories in finance lease arrangements.

Let's see a problem on this.

Question:

Greg invests $1200 at an annual rate of 6.5%. How long will it take until Greg earns $195 in interest?

Solution:

Formula to find the interest is

I = PTR/100

here p = profit=1200
   t = time
   r = rate=6.5

Substitute the values in formula,so that

        (1200 x T x 6.5)
195 = -----------------
             100

         T = 2.4

Greg will earn $195 in interest in 2.5 years.

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additoin of exponentials

Topic:- exponentials

Let's see what is exponent first, generally

BaseExponent

The exponent tells us how many times the base is used as a factor.

For example, to write 2 as a factor one million times, the base is 2, and the exponent is 1,000,000. We write this number in exponential form as follows:

2

1,000,000

read as two raised to the millionth power

This math help gives a example problem on addition of exponents.

Question:-

-3 z6(b3y2z2)

Answer:-

-3 z6(b3y2z2)
                   
Here we have 'z' for twice ,
So we can put them as 1 term by
additing the exponentials

we can add them by using exponent rule

formula of exponent

    am * a n = a m+n
So,
    -3 (z6+2*b3*y2)

    -3 (z8*b3*y2)

      -3z6+2*b3*y2 is the Answer

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Topic:Probability

Probability, is a way of expressing knowledge or belief that an event will occur or has occurred.

Here is some probability problems ,which help you to understand the concept much better.

Questions & Answers

1- Mr. Johnson taught a music class for 25 students under the age of ten. He randomlychose one of them. What was the probability that the student was under twelve?

Ans:-

The probability that the student was under twelve was 25/25, so P = 1.

2- The compact disk Jane bought had 12 songs. The first four were rock music. Tracks
number 5 through 12 were ballads. She selected the random function in her Compact
Disk Player. What is the probability of first listening to a balla?

Ans:-

The probability of listening to a ballad is 8/12, so P = 0.67.

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Problem on Solving a Function and Show Graphical Representation for the Function

Relations and functions are the dependent concepts of math where the value of a variable is dependent on the other variable in the given function. It is expressed in terms of graphical representation as shown in the below example

Topic : Solving a Function and expressing graphically

In the given function variable 'x' is the independent value and y is the dependent one.

Problem : a) If y = √x −1, Find the values for x = 0, 1, 2, 3, 4.Round to three decimal places where necessary.
b) Explain why no negative values are chosen as values to substitute
in for x.
c) Draw a graphical representation for the values.

(a)Solution :

When x = 0
y = √x – 1
= √0 – 1
= 0 – 1
= -1

When x = 1
y = √x – 1
= √1 – 1
= 1 – 1 and y = -1 - 1
= 0 and y = -2

When x = 2
y = √x – 1
= √2 – 1
= 1.414 – 1 and y = -1.414 - 1
= 0.414 and y = -2.414

When x = 3
y = √x – 1
= √3 – 1
= 1.732 – 1 and y = -1.732 - 1
= 0.732 and y = -2.732

When x = 4
y = √x – 1
= √4 – 1
= 2 – 1 and y = -2 - 1
= 1 and y = -3

(b) Solution :
No negative values are chosen for x because square root of a negative
number is imaginary and is not defined in real space.

(c) Solution :

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