Smoothing Function in Math

Introduction for smoothing function in math:

In smoothing function in math, let f(x) be a real-valued function of a real variable x and let a be some fixed point in the domain of f. The number L is the limit of f (x) as x approaches a if, given any positive number `epsi`, we can find a positive number `delta` such that f (x) is in the range

|f(x) – L| `<` `epsi`

Whenever x is in the domain

0 `<` |x – a| `<` `delta`

We write

`^lim_(x -> a)` f(x) `=` L

More about for Smoothing Function in Math:

In smoothing function in math, the notation

`^lim_(x -> a)` f(x) `=` L

is read “the limit of F(x) as x approaches a is L.

f(x) `->` L  as x `->`a;

This is read as “f(x) approaches L as x approaches a” or “f(x) goes to L as x goes to a”. I like to explain limits by writing

f(x) `~~` L when x `~~`a (but x `!=` a).

The notation A `~~` B means A is approximately equal to B. In ? We will study a limit

f’(a) `=` `^lim_(x -> a)` `(f(x)-f(a))/(x-a)`

called the derivative. The derivative is a limit, but the concept of limit is more general. Note that in the definition of limit we consider values of x close to a but never plug in x = a. In the interesting examples (like the derivative) setting x = a in f(x) leads to something undefined like `0/0` for smoothing function in math. Is this topic 8th grade math practice problems hard for you? Watch out for my coming posts.

Example for Smoothing Function in Math:

In smoothing function in math,

`^lim_(x -> 1)` 4x `-` 2 `=` 2

`^lim_(x -> 1)` `x^2` `+` 3 `=` 12

Some limits are more difficult to find if they contain a denominator that goes to zero as x `->` a. These limits must be treated carefully using, for example, L Hopitals rule.

Proof that a limit exists requires manipulation of inequalities to determine the relationship between `epsi` and `delte`. It is also necessary to verify that the inequalities |f(x) `-` L| `<` `epsi` and 0 `<` |x `-` a| `<` `delta` are satisfied as required by the definition of a limit.

Suppose we have the limits

`^lim_(x -> a)` f(x) `=` L

And

`^lim_(x -> a)` g(x) `=` M

Then

`^lim_(x -> a)` [ f(x) `+` g(x)] `=` L + M

That is, the limit of the sum of two functions is the sum of the limits.

`^lim_(x -> 1)` [`x^2` `+` 3 `+` 4x `-` 2 ] `=` 2 `+` 12

Answer is     `^lim_(x -> 1)` [`x^2` `+` 4x `+` 1 ] `=` 14

Statistics of Small Numbers

Introduction to Statistics of Small Numbers:
Statistics means that the formal science that are generating learn with the effective use of arithmetical terms. We are also help with the concepts like mode, median, mean and range. They are help with interpretation and also analysis of the terms. In this article, we are going to see examples with using of small numbers in statistics. Having problem with Mode of Numbers keep reading my upcoming posts, i will try to help you.

Mean and Median in Statistics of Small Numbers

Let see the examples of mean and median that are having the small numbers.

Mean:

Mean of the given numbers could be equivalent to the adding up of all the terms and total numbers of terms in the given list. The formula is,

Mean = Average value of the given numbers

Example problem:

Find the mean of the numbers, 3,7,1,4,9,12,6.

Solution:

Given numbers are, 3,7,1,4,9,12,6.

The formula for mean is,

Mean = Average value of the given numbers

Here, the adding up of all the numbers = 3+7+1+4+9+12+6

= 42

Total number of given numbers = 7

Therefore, Mean = `(42)/(7)`

= 6

Answer: Mean = 6

Median:

For measuring the median value for collection of numbers, we must arrange those numbers in an order.

i) When the total number of terms = Even number

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ii) When the total number of terms = Odd number

Median = Middle value of the ordered list.

Example problem:

Find the median of the numbers, 5,2,9,13,4,7.

Solution:

Given numbers are, 5,2,9,13,4,7.

The ascending order of the given group of numbers is,

2,4,5,7,9,13

Here, total numbers = 6

This 6 should be an even number.

Therefore, Median = `(5+7)/(2)`

= `(12 )/(2)`

= 6

Answer: Median = 6

These are the examples of mean and median that is having the small numbers.

Mode and Range in Statistics of Small Numbers

Let see the examples of mode and range that are having the small numbers.

Mode:

The mode value of the collection of numbers could be a number that should occur most frequently in the given list.

Example problem:

Find the mode of the numbers, 4,1,3,9,5,1,2,1.

Solution:

Given numbers are,

4,1,3,9,5,1,2,1

The ascending order of these numbers = 1,1,1,2,3,4,5,9.

Here, the number ‘1’ is coming three times in the group of given numbers.

Therefore, the mode of those collection of numbers = 1

Answer: Mode = 1

Range:

The differentiation among the values like largest value and smallest value is said to be range.

Example problem:

Find the range of the numbers, 4,8,6,2,1,9,5.

Solution:

Given group of numbers are,

4,8,6,2,1,9,5

The formula for range is,

Range = Biggest value – Smallest value

Here, Biggest value = 9

Smallest value = 1

Therefore, Range = 9 – 1

= 8

Answer: Range = 8

These are the examples of mode and range that is having the small numbers.

That’s all about the statistics of small numbers.

Deference in Math

Introduction to difference in math:

In mathematics, difference or subtraction is one of the four basic arithmetic operations; it is the inverse of addition operation, meaning that if we start with any number and add any number and then subtract the equal number we added, we return to the number and we started with. Subtraction or difference is denoted by a minus sign in infix notation.

The traditional names for the parts of the formula,

a − b = c,

are minuend (a) − subtrahend (b) = difference (c). Instead we say that a and −b are terms, and treat subtraction as addition of the opposite. The answer is called difference.

(Source from Wikipedia)

Example Problems for Difference in Math:

Example 1:

Find the difference value of the given numbers, using subtraction operation, 100 and 80.

Solution:

Given numbers using subtraction operation for, 100 - 80

We are going to subtract the two numbers,

100 - 80 = 20

The difference of given number is 20

Example 2:

Find the difference value of the given numbers, using subtraction operation, 550 and 200.

Solution:

Given numbers using subtraction operation for, 550 - 200

We are going to subtract the two numbers,

550 - 200 = 350

Finally we get the difference answer for given numbers are 350.

Example 3:

Find the difference value of the given numbers, using subtraction operation, 235 and 647.

Solution:

Given numbers using subtraction operation for, 235 - 647

We are going to subtract the two numbers,

235 - 647 = -412

The difference of given number is  -412.

Example Decimal Problems for Difference in Math:

Example 4:

Find the difference value of the given decimal numbers, using subtraction operation, 6.5 and 2.6.

Solution:

Given decimal numbers using subtraction operation for, 6.5 – 2.6

We are going to subtract the two numbers,

6.5 – 2.6 = 3.9

The difference of given decimal number is 3.9

Example 5:

Find the difference value of the given decimal numbers, using subtraction operation, 20.865 and 15.358.

Solution:

Given decimal numbers using subtraction operation for, 20.865 – 15.358

We are going to subtract the two numbers,

20.865 – 15.358 = 5.507

The difference of given decimal number is 5.507.

Example 6:

Find the difference value of the given decimal numbers, using subtraction operation, 3.4 and 9.7.

Solution:

Given decimal numbers using subtraction operation for, 3.4 – 9.7

We are going to subtract the two numbers,

3.4 – 9.7 = -6.3

The difference of given decimal number is -6.3.

Regression Slope Coefficient

Introduction to regression slope coefficient:
In mathematics, regression is one of the interesting topics in statistics. Linear regression is defined as the process of determining the relationship between two variables. It is a statistical analysis method which can be used to assessing the association between the two different variables. Let us see some example problems using regression slope coefficient.

Regression Slope Coefficient - Formula for Regression:

Formula for Regression:

Regression Equation (y) = a + bx
Regression Slope coefficient  (b) = `(NsumXY - (sumX) (sumY)) / (NsumX^2 - (sumX)^2)`
Intercept(a) =`(sumY - b(sumX)) / N `

Where
x and y are the variables.
b = the slope of the regression line (also called as regression slope coefficient)
a = the intercept point of the regression line and in the y-axis.
N = Number of values or elements
X = First Score
Y = Second Score
`sumXY` = Sum for the product of the first and Second Scores
`sumX` = Sum of First Scores
`sumY` = Sum of Second Scores
`sumX^2` = Sum of square First Scores. I like to share this Adding and Subtracting Significant Figures with you all through my article.

Regression Slope Coefficient - Example Problem:

Example 1:

Find the regression equation by using the regression slope coefficient value.

For the given data set of data, solve the regression slope coefficient.
Solution:

Let us count the number of values.
N = 6

Determine the values for XY, X2

Determine the following values `sumX` , `sumY` , `sumXY` , `sumX^2` .
`sumX` = 444
`sumY` = 21.4
`sumXY` = 1589.1
`sumX^2` = 32890

Substitute values in the slope formula
Regression Slope coefficient (b) = `(NsumXY - (sumX) (sumY)) / (NsumX^2 - (sumX)^2)`
= `((6)*(1589.1)-(444)*(21.4))/((6)*(32890)-(444)^2)`
= `(9534.6 - 9506.1)/ (197340 - 197136)`
=`28.5/204`
= 0.14
Substitute the values in the intercept formula given.
Intercept (a) = `(sumY - b (sumX)) / N `
=` (21.4 - 0.14(444))/6`
= `(21.4 - 62.16)/6`
= -`40.76/6`

= -6.79

Substitute the Regression coefficient and intercept values in the regression equation
Regression Equation(y) = a + bx
= -6.79 + 0.14x.

Solution:

Regression Slope coefficient (b) = 0.14

Intercept (y) = - 6.79

Regression equation y = - 6.79 + 0.14x.

Determine the approximate value for y:

When x = 75

Substitute the x value into the regression equation

Regression Equation(y) = a + bx
= - 6.79 + 0.14x.

= - 6.79 + 0.14(75)

= - 6.79 + 10.5
y = 3.71

Solution:

y = 3.71

Solve Math Equation Solutions

Introduction to solve math equation solutions:
An equation is a mathematical statement that asserts the equality of two expressions. Equations consist of the expressions that to be equal on opposite sides of an equal sign. (Source: From Wikipedia).

The following rules are used to solve the equation and the equation does not change: Add or subtract any variable or number to the both sides of the equation. Multiply or divide any variable or number to the both sides of the equation. I like to share this Solving Equations Using Addition and Subtraction with you all through my article.

Now, we are going to see some of the problems on how to solve math equations solutions.

Problems to Solve Math Equations Solutions:
Example problem 1:

How to solve an equation for the variable a: a + 23 = 26

Solution:

a + 23 = 26

Subtract 23 on both sides of the equation

a + 23 – 23 = 26 – 23

a = 3

So, the answer is a = 3.

Example problem 2:

How to solve an equation for the variable t:` t / 8 ` = 10

Solution:

`t / 8` = 10

Multiply 8 on both sides of the equation

`(t / 8)` * 8 = 10 * 8

t = 80

So, the answer is t = 80. Is this topic Mathematical Induction hard for you? Watch out for my coming posts.

Few more Problems to Solve Math Equations Solutions:

Example problem 3:

How to solve an equation for the variable m:  5m - 7 = -8

Solution:

5m – 7 = -8

Add 7 on both sides of the equation

5m – 7 + 7 = -8 + 7

5m = -1

Divide by 5 on both sides of the equation

`(5m) / 5 = -1 / 5`

m =` -1 / 5`

So, the answer is m = `-1 / 5` .

Example problem 4:

How to do solve an equation for the variable c:  -2c - 86 = -136

Solution:

-2c - 86 = -136

Add 86 on both sides of the equation

-2c - 86 + 86 = -136 + 86

-2c = -50

Divide by -2 on both sides of the equation

`(-2c) / -2 = (-50) / -2`

c = 25

So, the answer is c = 25.

Practice Problems to Solve Math Equations Solutions:

1)      Solve the equation for the variable x: 3x + 41 = 59 (Answer: x=6).

2)      Solve the equation for the variable b: b + 35 = 26 (Answer: b=-9)

3)      Solve the equation for the variable n: n - 13 = 3 (Answer: n=16)

Rounding Numbers Practice

Introduction to rounding numbers practice:

In this article we discuss to find the rounding numbers practice solving problems. The rounding numbers or digits are only approximate. An exact solutions are generally cannot be obtain using rounding numbers or digits. If the number 0, 1, 2, 3, 4, do not change the rounding numbers. All the numbers are right side of the requested rounding number will become 0.The number 5, 6, 7, 8, 9 rounding number round up by one number. Let us see to rounding numbers practice solving problems are given below.I like to share this Rounding Decimals with you all through my article.

Rounding Numbers Practice-example Problems:

Example 1:

Round off 3247 to the nearest ten.

Solution:

We note that 3247 lies between 3240 and 3250

3247 is closer to 3250 than to 3240

(Since 3250 – 3247 = 3, while 3247 – 3240 = 7)

Hence, 3247 when rounded off to the nearest ten becomes 3250.

Example 2:

Round off 3247 to the nearest hundred

Solution:

We further note that 3247 lies between 3200 and 3300

3247 is closer to 3200 than to 3300

(Since 3300 – 3247 = 53, while 3247 – 3200 = 47)

Hence, 3247 when rounded off to the nearest hundred becomes 3200.

Example 3:

Round off 3247 to the nearest thousand

Solution:

Finally, we note that 3247 lies between 3000 and 4000

3247 is closer to 3000 than to 4000

(Since 3247 – 3000 = 245, while 4000 – 3247 = 755)

Hence, 3247 when rounded off to the nearest thousand becomes 3000.

Example 4:

Round off 5257 to the nearest thousand

Solution:

Finally, we note that 5257 lies between 5000 and 6000

5257 is closer to 5000 than to 6000

(Since 5257 – 5000 = 257, while 6000 – 5257 = 743)

Hence, 5257 when rounded off to the nearest thousand becomes 5000.I have recently faced lot of problem while learning translation in math, But thank to online resources of math which helped me to learn myself easily on net.

Rounding Numbers Practice Problems:

Problem 1: Round off 4247 to the nearest ten

Problem 2: Round off 3247 to the nearest hundred

Problem 3: Round off 3247 to the nearest thousand

Rounding numbers practice problems-answer key:

Problem 1: 4250

Problem 2: 4200

Problem 3: 400.

Solve Math Median

Introduction to median in math:-

Median is another measure of central tendency. It is the middle value of a series of numbers.  It divides the series into two equal parts. One half contains values less than the median and the other half contains values more than the median.  So we arrange the series in ascending or descending order to find the median.  If the series contains 'n' numbers and if 'n' is odd, then the median is found by the formula `(n+1)/2` . If n is even then we must find the mean value of the `(n/2)` th and `(n + 1)/2` th  numbers to find the median.

If we have many frequencies  of the given numbers in a grouped data, then we use the formula
Median =` l +((N/2-m)/f)xxc`                                                                                                                                 f
where l is the lower limit of the median class,
m is cumulative frequency up to l,
f is the frequency of the median class,
N is total frequency and
c is the length of the class interval.

To Find Median when Frequency is not Given.
Let us find the median of the following numbers  25,29, 32, 46, 73, 25, 28
Solution:

Step 1: We must arrange the numbers in order.  So let us arrange them in ascending order 25,25,28,29,32,46,73

Step 2: Here n=7 so it is odd

Step 3: So median is `(7+1)/2` th number = 4th number = 29

Step 4: Let us count the numbers before 29.  There are 3 numbers.  25, 25, 28

Step 5: Let us count the numbers after the median.  There are 3 numbers 32,46,73

Step 6: Hence we find that 29 is the middle number.

Answer: Median is 29

Find the median of the numbers  12,29, 24, 36, 32,48,59,73
Solution:

Step 1: Let us arrange them in ascending order .12, 24, 29, 32,36,48,59, 73

Step 2 : Let us count the numbers. They are 8.  n = 8 is even

Step 3 : The formula is to find the median of `n/2` th number and `(n +1)/2` th number . That is `8/2` th number and `(8+1)/2` th number

That means we must find the median of 4th number and 5th number.

4th number is 32 and 5th number is 36

Hence median is` (32+36)/2` = `68/2` = 34

Answer: Median of the 8 given numbers is 34.Between, if you have problem on these topics adding exponents, please browse expert math related websites for more help on cbse model question papers for class 9.

To Find Median when Grouped Data is Given:

A grouped data is given.  Let us find the median using the formula for the grouped data.
The income of  the people in a factory is as follows:-
__________________________________________________________________
Income $                 80 - 100     100- 150     150- 180      180-200      200-250   250-300
__________________________________________________________________
Number of persons       16              24                 26                30                 20              7
__________________________________________________________________
Find the median.
Solution:-
Step 1 : We must find the cumulative frequency first. CI is class Interval
_____________________________________________
Income $        Number of persons Cumulative frequency
CI                            f                                  cf
_____________________________________________
80 -100                     16                            16
100 - 150                   24                            40  m
(l)       150 - 180         26 (f)                        66
180 - 200                   30                            96
200 - 250                   20                           116
250 - 300                    7                           123 -> N
Step 3 : N=123 is odd
Hence we must find `(N+1)/2` th item  = `(123 + 1)/2` th number = `124/2` th number  =  62

Step 4 : The 62nd number lies in the Class Interval 150-180 . Hence l is the lower limit 150,so  l = 150
Step 5 : The frequency of f is 150-180 is 26  f = 26
Step 6 : C is the length of the class interval = 150-180 = 30
so C = 30
Step 7 : m is the cumulative frequency up to l that is the previous Class Interval cf  m  = 40
Step 8 : Median =` l + ((N/2 - m)/f )C `
=`150 + (62-40)/26 xx 30 `
= `150 + 22 xx 30 `
= `150 + 25.38 `
= `175.38`

Answer : Median of the grouped data is 175.38

Hence we can find the median by finding the middle value of a grouped data.

Solving Geometry Triangle Problems

Introduction of solving geometry triangle problems:

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ∆ABC.

In Euclidean geometry any three non-collinear points determine an unique triangle and an unique plane (i.e. a two-dimensional Euclidean space).

Learning Angles for Solving Geometry Triangle Problems:

A geometry triangle is a form of polygon having three sides and, therefore, three angles. The geometry triangle is a blocked figure formed from three straight line segments joined at their ends. In geometry triangle the point at the tops can be called the corners, angles, or vertices of the triangle. Since any given triangle lies totally within a plane, triangles are often treated as two-dimensional geometric figures.

Problems for Solving Geometry Triangle:

solving geometry triangle problems is an essential one. problems for solving geometry triangle are given below:

1) An isosceles triangle has angle A 60 degrees greater than angle B. Find all angles of the geometry triangle.

Sol:

An isosceles triangle has the two angles equal in size. Angle A is 60 greater than angle B then A = B + 60 o. The sum of all angles in a triangle is equal to 180o.

(B+60) + B + B = 180o

•    Solve the above equation for B.

B = 40o

•    The sizes of the three angles are

A = B + 60 = 100o

C = B = 40o

2. The apex angle B of isosceles triangle ABC is 120 degrees. Find the quantity determines of each base angle on geometry triangle.Please express your views of this topic help online with math by commenting on blog.

Sol:

•    Let x = the compute of each base angle.

•    Arrangement and equation and solve for x.

•    base angle + base angle + 120 degrees = 180 degrees

x + x + 120 degrees = 180 degrees

2x + 120 = 180

2x = 180 - 120

2x = 60

x = 30 degrees

3. A geometry triangle has a perimeter of 56. If 2 of its sides are equal and the third side is 5 more than the equal sides, what is the length of the third side?

Sol:

•    Let y = length of the equal side

•    Perimeter = sum of three sides.

•    Plug in the values from the question.

56 = y + y + y + 5

•    Combine like terms

56 = 3y + 5

3y = 56 – 5

3y = 51

y =17

So, the length of third side = 17 + 5 =22

Ellipse Parabola Hyperbola

Introduction to ellipse parabola hyperbola
The equation like Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 represents either a (non–degenerate) conic or a degenerate conic.

If it is a conic, then it is

a parabola if B2 - 4AC = 0.
an ellipse if B2 - 4AC < 0
a hyperbola if B2 - 4AC > 0.

Introduction to Ellipse Hyperbola Parabola

Ellipse:

An ellipse is a conic obtained on slicing across obliquely one nappe of a cone.If P is any point on the ellipse and
F1and F2 its foci, the angle subtended by F1P and F2P with the tangent at P are equal and if a source of light or sound is
placed at one focus of an ellipsoidal reflector (surface generated by revolving an ellipse about its major axis) all the
waves will be reflected so as to pass through the other focus



Hyperbola:

A hyperbola is a conic obtained on slicing a double napped cone by a plane parallel to the axis of the cone. The lines from the foci to any point of a hyperbola make equal angles with the tangent at that point. Hence if the surface of a reflector is generated by revolving a hyperbola about its transverse axis, all rays of light converging on one focus are reflected to the other.



Parabola:

A parabola is generally defined as the conic section obtained by the operation on slicing a right circular cone by a plane parallel to the line joining vertex and any other point of the cone.Between, if you have problem on these topics Formula for Volume of a Trapezoidal Prism, please browse expert math related websites for more help on 12th cbse sample papers.

If P is any point on the parabola with focus F and vertex V, the angles subtended by FP and PX with the tangent at P are equal where PX is parallel to the axis VFA of the parabola.




Problems Based on Ellipse Hyperbola Parabola



If e < 1, then the conic is an ellipse.  From the figure 1we observe that F2Pi is always less than PiMi.
i.e., F2Pi / PiMi =  e < 1, (i = 1, 2, 3…)

If e = 1, then the conic is a parabola. From the figure 2  we observe that FPi is always equal to PiMi.
i.e., FPi / PiMi = e = 1, (i = 1, 2, 3…)

If e > 1, then the conic is a hyperbola. From the figure 3 we observe that F1Pi is always greater than PiMi.
i.e., F1Pi / PiMi = e > 1, (i = 1, 2, 3…)

Arc Length and Sector Area

Introduction to arc length and sector area:

Consider a circle whose center is O and radius r. Let A and B be two points on the circle in such a way that arc AB = r.

The length of the arc AB of the circle is equal to the product of the radius of the circle and angle in radians subtended at the center of the circle.

That is, arc length AB = r × ?, here ? is in radians.

The area of the sector AOB is `(1/2) r^2theta` .

We can also write the above two formula as follows:

(i) arc length AB = 2?r × ` theta/360`

(ii) The area of the sector AOB is `(theta/360)` `Pir^2` .



Now let us see few problems on this topic “arc length and area sector”.

Example1 on Arc Length and Area Sector

An arc of a circle subtends 150 at the center. If the radius of the circle is 4cms, find the length of the arc and area of the sector formed.

Sol: Here ? = 15^0, r = 4cms.

Let us convert the degree value of ? to radians.

? = 15^0 = `(( Pi xx 15)/180)`

= `Pi/12` .

The length of the arc is given by r × ? = 4 × `pi/12`

= `pi/3` cm.

The area of the sector formed = `(1/2)`` r^2` ?

= `(1/2)` ` 4^2` `(pi/12)`

= `(2 pi )/3 ` sq cm.I like to share this solve linear system of equations with you all through my article.

Example2 on Arc Length and Area Sector

A wire of length 10cm is bent so as to form an arc of a circle of radius 4cm. What is the angle at the center in radians?

Sol: Given: The length of the arc is given by r × ? = 10

? = `10/r`

? = `10/4` = 2.5 radians.

Example3 on Arc Length and Area Sector

Ex 3: If the distance between the sun and the earth is 1.495 × 1068 km and the angle subtended by the sun at a point on the earth is half a degree, find approximately the diameter of the sun.

Sol: Given: ? = (1/2)^0 = (`Pi/180` ) × `(1/2)`

= `Pi/360` rad.

If d km be the diameter of the sun, then we have: `(s/r)` = ?

` (d/ (1.495 xx 1068))` = `Pi/360`

Therefore, d = (1.495 × 1068) × `Pi /360`

= 1.304637 × 10^6 km.

Fraction Math Worksheets

Introduction on fraction math worksheet:

In Math, Fraction is a part of an object.

Fraction Math Worksheets:

Worksheet contains questions. So the fraction worksheet contains only questions related to fractions.

Worksheet 1:

Let us see equivalence fraction and the simplifying fractions in this worksheet no 1.

1) `(1)/(4)` = `(?)/(12)`

2)`(3)/(18)` = `(1)/(?)`

3)`(2)/(3)` = `(?)/(6)`

4) `(6)/(12)` = `(1)/(?)`

5) `(12)/(15)` = `(?)/(5)`

Answer:

1) 3

2) 6

3) 4

4) 2

5) 4

Fraction Math Worksheet 2:

Worksheet 2 contains the questions on comparing fractions.

1) Which is greater 5/2 or 4/2?

2) Which is smaller 12/2 or 14/7?

3) Which is greater 32/43 or 13/12?

4) Which is smaller 9/2 or 5/3?

5) Which is greater 4/1 or 3/4?

6)Which is smaller 23/12 or 14/15?

7) Which is greater 1/2 or 1/3?

8) Which is smaller 4/2 or 4/3?

9) Which is greater 4/5 or 4/6?

10) Which is smaller 3/2 or 3/4?

Answer:

1) 5/2

2) 14/7

3) 13/12

4) 5/3

5) 4/1

6) 14/15

7) 1/2

8) 4/3

9) 4/5

10) 3/4

Please express your views of this topic free math solver by commenting on blog.

Fraction Math Worksheet 3:

Worksheet 3 contains the questions on addition,subtraction,multiplication and division of fractions:

1) Add 1/2 and 1/2.

2)Add 3/2 and 3/4.

3) Add 4/5 and 1/5.

4)subtract 1/2 and 1/2.

5)subtract 3/2 and 3/4.

6)subtract 4/5 and 1/5.

7)multiply 1/2 and 1/2.

8)multiply 3/2 and 3/4.

9)multiply 4/5 and 1/5.

10)divide 4/5 by 1/5.

Answers:

1) 1

2) 9/4

3)1

4) 0

5) 3/4

6)3/5

7)1/4

8)9/8

9)4/25

10) 4

These are the fraction math worksheets.

Modified Box Plots

Introduction to box plots:

In statistics a box plot can be defined as a diagram explains the summary of data sets. For easy understanding the plots are modified in such a manner that a common man can examine the box plots. Generally creation of box plots involves in finding out the parameters such as median, quartiles, greatest and least observations.

Modified box plots explains the step by step procedure for creating box plots and with the help of it we can evaluate where the observations are concentrated and where they are spread out. As modified box plots are simple to understand they are widely used by many persons in different profession.

Examining Box Plots

Examining box plots is nothing but a process of reading the elements of the plot.

Actually, there are several parameters are available. Namely,

Median
Quartiles
Minimum and maximum values


The following figure shows a typical modified box plots.

Modified box plots



Median:

The median is the value that divides a data set into two parts and it is denoted as Q2.

Quartiles:

As said earlier the median divides the data set into two parts, the first part is known as Lower quartile and later is called as upper quartile.

These quartiles are considered as separate data sets and the median is evaluated.

Lower median is denoted as Q1 whereas upper median is denoted as Q3.

Maximum & Minimum values:

The greatest and the least values of the data set are called as maximum and minimum values respectively.

Outliers:

Upper outlier is the value which is more than 3/2 times of upper quartile and Lower quartile is the value which is less than 3/2 times of lower quartile.Is this topic Greatest Common Factor Word Problems hard for you? Watch out for my coming posts.

Example

Consider the following data set.

Data set => (12, 4, 9, 66, 8, 53, 37)

Solution:

Step 1:

Rearrange the data set in order.

(4, 8, 9, 12, 37, 53, 66)

As there are 7 elements in the data set. The fourth element lies exactly in the center.

So, median Q2 = 12

Step 2:

Quartiles:

Lower quartile = (4, 8, 9)

Median of lower quartile Q1= 8

Upper quartile = (37, 53, 66)

Median of the upper quartile Q3= 53

Step 3:

Minimum value= 4

Maximum value= 66

So, min=4, Q1=8, Q2=12, Q3=53, max=66

Step 4:

Draw a straight line which extends in both directions.

Step 5:

Draw the modified box plots using the parameters.

Modified box plot

Solving Basic Statistics Problems

Introduction to solving basic statistics problems:   

Everyday we come across a wide variety of information’s in the form of facts, numerical figures, tables, graphs, etc. These figures are the numerical data collected with a definite purpose, are called data. Data is a word in a plural form of the Latin word datum. Every part of our lives utilizes data in one form or the other. This extraction of significant information is studied in a branch of mathematics called Statistics.

Solving Basic Statistics Problems-statistics Definition

Basically the word ‘statistics’ appears to have been taken from the Latin word ‘status’ meaning ‘a state (related to a political)’. In its origin, statistics is simply the collection of data on different aspects of the life of people, useful to the State. Basic Statistics deals with collection, organization, analysis and Interpretation of data. The word ‘statistics’ has held different meanings in different contexts, so that the calculation process is based on the meaning and data’s.
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Solving Basic Statistics Problems

Example for basic stastistics problems:

Consider the marks obtained by 10 pupils in a mathematics test as given below:

55 36 95 73 60 42 25 78 75 62

The data in this form is called raw data.

By looking in this way, can you find the highest and the lowest marks?

Did it take you some time to search for the maximum and minimum marks? Wouldn’t it be a lesser amount of time consuming if these scores were arranged in ascending or descending order? So let us arrange the marks in ascending order as

25 36 42 55 60 62 73 75 78 95

Now, we can clearly watch that the lowest marks are 25 and the highest marks are 95.

The difference of the highest and the lowest marks in the basic stastistics data is called the range of the data. So, the range in this case is 95 – 25 = 70.

Arrangement of data in ascending or descending order can be quite time consuming, particularly when the number of observations in an experiment is large.

Variance of a Data Set

Introduction to variance of a data set:

Data is nothing but a set of information or collection of facts. We can analyze the data’s with the help of charts and graphs. The method of collecting or preparing the data’s is the data collection.
Variance is the term that  explains how average value of the data set vary from the measured data.

Variance of Data Set:

Following are the steps involved in finding the variance of data set:    

Step 1: First step is to find the Average, that is the arithmetic mean for the given set of numbers.

Mean = Sum of all values / Total number of elements

Step 2: To find the sum of squares for all the given values in the given data set.

Step 3: Divide the result we obtained in Step 2 over the total number of elements in the given data set.

Step 4: Find the difference between the square of mean and the result we obtained from step 3

Step 5: For finding the variance, take the square root of the number that  we obtained as a result from step 4.

Thus, using the above steps, we can easily calculate the variance of the given data set.

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Solved Examples for Finding Variance of a Data Set:

Ex1: Find the variance  of the given data set: { 5, 6, 9, 13, 19 }

Sol:

Step 1: Arithmetic mean             =  (5 +6 + 9+ 13 + 22) / 5

= 55 / 5

=  11

Step 2: Sum of squares               =  52 + 62 + 92 + 132 + 222

=  25 + 36 + 81 + 169 + 484

= 795

Step 3: Sum of squares / Total number of elements

= 795 / 5

= 159

Step 4: Result from Step 3 – square of mean value

= 159 – 112       

= 159 – 121

= 38

Step 5: Variance                        = square root ( 38 )

= 6.16

The variance of the given data set is 6.16

Ex 2: Find the variance  of the given data set: { 2, 3, 4, 5,16 }

Sol:

Step 1: Arithmetic mean             =  (2 +3 + 4+ 5 + 16) / 5

= 30 / 5

=  6

Step 2: Sum of squares               =  22 + 32 + 42 + 52 + 162

=  4 + 9 + 16 + 25 + 256

= 310

Step 3: Sum of squares / Total number of elements

= 310 / 5

= 62

Step 4: Result from Step 3 – square of mean value

= 62 – 62       

= 62 – 36

= 26

Step 5: Variance                        = square root ( 26 )

= 5.1

The variance of the given data set is 5.1

Percentile Definition Statistics

Introduction to percentile definition statistics:

The definition of percentile is dividing the data into 100 equal parts in statistics. We can also refer the percentile as quartiles. The percentile formula of statistics is using the samples. In statistics, the percentile is determined from ordered data. Now we are going to see about percentile definition statistics.

Explanation for Percentile Definition Statistics

Definition of percentile in statistics:

The percentile definition is the position of score that is part of scored values. In other words, it indicates the particular frequency percentage that is percent of scored values. It defines the score position in statistics.

Percentile formula in statistics:

The individual population is calculated by percentile and the statistics use the formula in different forms. They are,

If the ‘x’ values are used in statistics means we can use the percentile formula as `(B + 0.5E)/n` x 100.
If we does not use the ‘x’ values means we can use the percentile formula as` (Number of below x values)/n` x 100.
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More about Percentile Definition Statistics

Example problems for percentile definition statistics:

Problem 1: The scores of student are 22, 24, 25, 27, 30, 31, 32, 33, 33, 34, 36, 37, 39, 40, 42, 44, 45, 47, 50, 52.

Solution:

Total number of scores is 20 and number of below 33 is 7.

Percentile is `(B + 0.5E)/n` x 100 =` (7 + 0.5(2))/20` x 100 = `8/20` x 100 = 40.

Percentile of score 33 is 40.

Problem 2: Student scores are 12, 14, 15, 16, 17, 19, 21, 21, 22, 24, 26, 27, 29, 30, 31, 34, 35, 37, 38, 40. What is the percentile value of 21?

Solution:

Total number of score is 20 and the below 21 is 6.

The calculation for percentile is` (B + 0.5E)/n` x 100.

P = `(6 + 0.5(2))/20` x 100 = `7/20 ` x 100 = 35.

Percentile of score 21 is 35.

Exercise problems for percentile definition statistics:

1. The score of Alex is below 77 is 5 out of 20. What is the percentile value?

Solution: Percentile of Alex is 25.

2. Student’s scores are 2, 3, 4, 4, 6, 7, 9, 10, 12, 13. Determine the percentile of score 4.

Solution: The score 4 is at 30th percentile.

Decimal Place Value System

Introduction:
Decimal place value system is slightly differing from normal place value for numbers. This decimal place value system has a decimal point and numbers on both sides of decimal point. There are different names for the numbers before and after the decimal system. We are going to see how the place value system for decimals.

Explanation to Decimal Place Value System:

General representation of decimal place value system:

Let us see the place value system for the decimal 123456 . 789

Before the decimal point:

123456

6 - ones

5 - tens

4 - hundreds

3 - thousands

2 - ten thousands

1 - hundred thousands

After the decimal point:

789

7 -tenths `1/10`

8 -hundredths `1/100`

9 - thousandths `1/1000`

The above number 123456.789 can be obtained as,

(1 x 100000) + (2 x 10000) + (3 x 1000) +(4 x 100) + (5 x 10) x (6 x 1) + ( 7 x `1/10`) + (8 x `1/100` ) + (9 x `1/1000`)

Example Problems to Decimal Place Value System:
Example: 1

Which of the following is the place value of 5 in decimal 9.5?

a) Ones

b) Tenths

c) Hundreds

d) Decimal point

Solution:

Given, 9.5

5 lie after the decimal point. So, the place value of 5 is tenths.

Answer: b

Example: 2

Which of the following is the place value of 9 for decimal 98.42?

a) Tenths

b) Ones

c) Hundreds

d) Hundredths

Solution:

9 lie before the decimal point.

98

8 – ones

9 – hundreds

Answer: c

Example: 3


How will you write the expanded form the decimal 3.36?

Solution:

Given 3.36

Before the decimal point: 3

3 - Ones

After the decimal point: 36

3 - tenths

6 - hundredths

3.36 = (3 x 1) + (3 x `1/10`) + ( 6 x `1/100`)

Practice Problems to Decimal Place Value System:
Problem: 1

Which of the following is the place value of 8 in decimal 2.08?

a) Ones

b) Tenths

c) Hundredths

d) Hundreds

Answer: c

Problem: 2

Which of the following is the place value of 1 in decimal 0.001?

a) Ones

b) Thousandths

c) Hundredths

d) Hundreds

Answer: b

Algebra Ratio and Proportion

Introduction of algebra ratio and proportion:

Define of algebra ratio:

In mathematics, a ratio expresses the magnitude of quantities relative to each other. 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

Define of algebra proportion:

If the two or more Ratio Proportion encompass all of the quantities in a particular situation.

Source: Wikipedia


Definition of Algebra Proportion:

The ratio `25/5` may be simplified and written as `5/1`.

`25/5`=`5/1`

An equation that states that two ratios are equal is called an algebra proportion. The preceding algebra proportion may also be written in the form 25:5=5:1.

Concept of algebra proportion:

A proportion is an equation that states that two ratios are equal: `m/n`=`p/q`  or  m:n=p:q (provided n?0 and q?0)

Each term of proportion is given a special name according to its position in the proportion.

`m/n`=`p/q`

Where

m is first proportional

n is second proportional

p is third proportional

q is fourth proportional

the pair of terms that form the 1st and 4th  proportional’s are called as the extremes of a proportion; the 2rd  and 3rd proportional of a proportion are called the means of a proportion.

The pair of terms m and p are called as the extremes of the proportion; the pair of terms n and q are referred to as the means of the proportion.

Example for algebra proportional:

Find the first proportional if the remaining terms of a proportion are 1,3 and 9.

Solution:

Let x=first proportional. Then

x/1=3/9

x=`1/3`

x`xx`3=`1/3``xx`3

3x=1

x=`1/3`

The first proportional is `1/3`.

Sometimes we may be able to work with lesser numbers by simplify an arithmetic ratio in the innovative proportion before cross-multiplying.
Algebra is widely used in day to day activities watch out for my forthcoming posts on Degree of Polynomial and What is the Dependent Variable in an Experiment. I am sure they will be helpful.
Definition of Algebra Ratios:

The ration of two numbers a and b(b?0) is the quotient of the numbers. The numbers a and b referred to as the terms of the ratio.

Example problem for algebra ratios:

Example 1:

Solve `25/125`

Solution:

=`1/5`    (numerator and denominator are divided by 25)

=`1/5` 

Answer is `1/5`or 1:5

Example 2:

Solve`(30/15)``xx``(12/2)`

Solution:

=`(30/15)``xx``(12/2)`

= `360/30`(numerator and denominator are divided by 10)

=`36/3`  (numerator and denominator are divided by 3)

=12 or `12/1`

Answer is `12/1`  or 12:1

Absolute Values Algebra Help

Introduction to absolute values algebra help:

Absolute values in algebra help are nothing but we are going to get the help on the absolute values in algebra. In this we are going to solve the problems based on absolute value rules. For example the absolute values of the function or numbers are always positive. So if we solve the absolute value functions we will get the two values for the absolute value functions. We will see some example for absolute values in algebra.

Example Problems for Absolute Values Algebra Help:

Absolute values algebra help example 1:

Solve the variable x where |4x + 2| = |2x + 3|

Solution:

The given absolute value function is |4x + 2| = |2x + 3|

From this we can divide this into (4x + 2) = 2x + 3 ………. (1) – (4x + 2) = 2x + 3 ………… (2)

Equation 1:

(4x + 2) = 2x + 3

Add - 2 on both sides

4x + 2 – 2 = 2x + 3 - 2

4x = 2x + 1

Add – 2x on both sides

4x – 2x = 2x + 1 – 2x

2x = 1

So x = 0.5

Equation 2:

-(4x + 2) = 2x + 3

-4x -2 = 2x + 3

Add + 2 on both sides

-4x - 2 + 2 = 2x + 3 + 2

-4x = 2x + 5

Add – 2x on both sides

-4x – 2x = 2x + 5 – 2x

-6x = 5

So x = - `(5 / 6)`

We will see some more examples for absolute values in algebra. It is better to getting help on absolute values in algebra.

Algebra is widely used in day to day activities watch out for my forthcoming posts on How to Simplify Polynomials and The Binomial Theorem. I am sure they will be helpful.

Absolute Values Algebra Help Example 2:

Solve the variable x where |2x + 1| = |x - 1|

Solution:

The given absolute value function is |2x + 1| = |x - 1|

From this we can divide this into (2x + 1) = x - 1 ………. (1) – (2x + 1) = x – 1 ………… (2)

Equation 1:

(2x + 1) = x – 1

Add - 1 on both sides

2x + 1 – 1 = x – 1 - 1

2x = x - 2

Add – x on both sides

2x – x = x – 2 - x

So x = -2

Equation 2:

-(2x + 1) = x - 1

-2x -1 = x - 1

Add + 1 on both sides

-2x -1 + 1 = x -1 + 1

-2x = x + 0

Add – x on both sides

-2x –x = x - x

-3x = 0

So x = 0

These are some of the examples for absolute value problems in algebra. This is better to getting help on absolute values.

Permutation and Combination Online Study

Introduction for permutation and combination online study:
Permutation:

It is the rescheduling of symbols into clear series. When we set things in order, we say we have made an array. When we alter the order, we declare we have changed the arrangement.

Combination:

It is a non sequence collection of unique sizes. In a permutation the order of occurrence of the objects or the arrangement is essential but in combination the order of occurrence of the objects is not important.

Online:

In general, "online" indicates a state of connectivity, In common usage, "online" often refers to the Internet or the World Wide.For the study of these topic , the following example will be more useful

Web.(Source: wiki)

Formulas for study of  permutation and combinations:

Permutation = nPr =` (n!) / ((n - r)!)`

Combination = nCr =` (_nP_r) /(r!) (or) (n!)/(r!(n-r)!)`

Permutation and Combination Online Study - Examples:

Solve online permutation and combination – Example 1:

What is the number of permutations and combinations: n=5; r=3.

Solution:

Formula for permutation:

Permutation = nPr = `(n!) /((n - r)!)`

Step 1: Find the factorial of 5.

5! = 5 × 4 × 3 × 2 × 1 = 120

Step 2: Find the factorial of 5 - 3.

(n - r)! = (5-3)! = 2! =2

Step 3: Divide 120  by 2

Permutation = `120/2` = 60

Combination = nCr = `(_nP_r) /(r!)`

Step 4: Find the factorial of 3.

3! = 3×2×1 = 6

Step 5: Divide 60  by 6.

Combination =` 60/6 ` = 10

The before example will assist you to get the Permutation and Combination manually.

Solve online permutation and combination – Example 2:

What is the number of permutations and combinations: n=8; r =3.

Solution:

Formula for permutation:

Permutation = nPr = `(n!) /((n - r)!)`

Step 1: Find the factorial of 8.

8! = 8 × 7 × 6 × 5×4×3×2×1 = 40 320

Step 2: Find the factorial of 8-3.

(n - r)! = (8-3)! = 5! = 120

Step 3: Divide 40 320 by 120.

Permutation = `40320/120` = 336

Combination = nCr = `(_nP_r) /(r!)`

Step 4: Find the factorial of 3.

3! = 3×2×1 = 6

Step 5: Divide 336 by 6.

Combination = `336/6` = 56

The before example will assist you to get the Permutation and Combination manually.

Solve online permutation and combination – Example 3:

What is the number of permutations and combinations: n= 15; r =2.

Solution:

Formula for permutation:

Permutation = nPr = `(n!) /((n - r)!)`

Step 1: Find the factorial of 15.

15! = 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5×4×3×2×1 = 1 307 674 368 000

Step 2: Find the factorial of 15-2.

(n - r)! = (15-2)! = 13! = 6 227 020 800

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Step 3: Divide 1 307 674 368 000 by 6 227 020 800.

Permutation = `(1 307 674 368 000)/( 6 227 020 800)`

=  210

Combination = nCr = `(_nP_r) /(r!)`

Step 4: Find the factorial of 2.

2! = 2×1 = 2

Step 5: Divide 210 by 2.

Combination = `210/2 ` = 105

The before example will assist you to get the Permutation and Combination manually.

Permutation and Combination Online Study - Practice Problems:

Practice Problem 1:

What is the number of permutations and combinations: n= 10; r =5.

Answer:

Permutation =30240

Combination = 252

Practice Problem 2:

What is the number of permutations and combinations: n= 50; r =3.

Answer:

Permutation = 117600

Combination = 19600

The Sum of Two Odd Numbers Is

Introduction to the Sum of Two Odd Numbers:

Odd number is the number which cannot divisible by as perfect (without getting remainder). But when we add or sum the two odd numbers, then we get result of the sum of the two odd numbers is equal to even number.

For Example: We know that the number 3 and 5 is the odd number. Now add the odd number 3 and 5, 3 + 5 = 8 where 8 is the even number.

Example Problem – Sum of Two Odd Numbers:

Example 1:

Which of the following form of number is the result when adding the two odd numbers?

Option:

a)     Odd number

b)    Even Number

c)     Zero

d)    Negative Number

Answer: Option b

Explanation:

When we add or sum the two odd numbers, then we get result of the sum of the two odd numbers is equal to even number.

For Example: We know that the number 5 and 7 is the odd number. Now add the odd number 5 and 7, 5 + 7 = 12 where 12 is the even number (12 can divisible 2 as perfect).

Example 2:

Which of the following is the result of the adding the two odd numbers 9 and 11?

Option:

a)     11

b)    9

c)     20

d)    21

Answer: Option c

Explanation:

We know that, when we add or sum the two odd numbers, then we get result of the sum of the two odd numbers is equal to even number.

Here the given option a, b, d are the odd numbers. Option c is the even number, therefore the option c is the correct answer.

Check: 11 +9 = 20

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Practice Problem – Sum of Two Odd Numbers:

Problem 1:

What is the sum of the two odd numbers 45 and 53?

Answer: 98

Problem 2:

What is the sum of the two odd numbers 75 and 51?

Answer: 126

Elementary Math Work

Introduction to elementary math work:
The elementary math work consists of mathematics topics taught in the primary and secondary schools. The most important in elementary mathematics are arithmetic, geometry, algebra, number work and special functions. In secondary school the topics are trigonometry, Calculus etc. Now we are going to see about the elementary math work.

Elementary Math Work:

1. Number work:  The number work consists of the natural, decimal, fractional, rational numbers etc. Let us see one example for the rational number in number work.

Ex  :  Determine 0.55 in the form of a rational number.

Sol :  The expression can be simplify as,

0.55 = 5 tenths + 5 hundredths

= 5 × (1/10) + 5 × (1/100)

= (5/10) + (5/100)

= 55/100

= 11/20.

2. Arithmetic:   In arithmetic, we can see about the housing finance, speed, time, distance, ratio etc. Let us see an example for the arithmetic.

Ex :  Determine the ratio of 3 Kg to 750 g

Sol : The ratio can be calculated as follows,

3 Kg = 3 × 1000 = 3000 g

Thus the required ratio given as

∴Required ratio = 3000 : 750

= 300 : 75

= 60 : 15

= 12 : 3

= 4 : 1

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Other Elementary Math Works:

1. Algebra:  Algebra is the important branch of mathematics consists of many expressions such as variables, constants and coefficients. It also contains some operations as addition, subtraction, multiplication and division.

Ex :   Expand the expression: (2x + 2y + 2z)^2

Sol :  Let us take the equation with (a + b + c)^2

a = 2x, b =2y, c = 2z

(2x + 2y + 2z)^2 = (2x)^2 + (2y)^2 + (2z)^2 + 2(4xy + 4xz + 4yz)

= 4x^2 + 4y^2 + 4z^2 + 8xy + 8xz + 8yz

2. Geometry:  Geometry is about the learning of various size, shape and properties of the geometrical solid figures. The geometry also consists of some constructional properties for all the solid figures.

Ex : The circumference of a circle is given as 10 centimeters. Determine the diameter of the circle?

Sol : The circumference of the circle can be given as,

C = π × d

10 = 3.14 × d

d = 10/3.14

d = 3.18cm.

The diameter of the circle is 3.18 centimeters.

Pyramid Definition Math

Introduction to pyramid definition math:

In this section we will discuss about pyramid definition math. In math the pyramid is comes under the concept of Euclidean geometry. The faces of pyramid are mostly isosceles triangle. The properties of the pyramid and their images are given below. The pyramid has been named as the basis of the base of the pyramid. Let we see about pyramid definition math.

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Definition – Pyramid Definition Math:

Formulas:

Formula for lateral surface area of the pyramid = `(tP)/(2)`

Formula for total surface area of the pyramid = `(tP)/(2) + B`

Formula for volume of the pyramid = (Bh)/(3)

Where, t = lateral height, B = area of base, P = perimeter of base.

Triangular Pyramid

A triangular based pyramid should be a shape that having the base shape as triangular. Simply a triangular based pyramid is a shape of tetrahedron. A tetrahedron is a polyhedron that self-possessed of four numbers of triangular sides. Three of them should meet at every vertex. It also a kind of pyramid, that having flat polygon shaped base as well as triangular faces that linking base at a common point.

Properties of Triangular based pyramid:

This shape having many properties analogous to those of triangles, include in sphere, circumspect, medial tetrahedron as well as exospheres.
The triangular based pyramid should have respective centers namely, in center, circumference, shrieker center, ex centers as well as points like centroid.