How to Use Histograms

In statistics graphical representation of data by using bar graphs , histograms , frequency curves are important tools.

Introduction to how to use histograms:

Histograms is the method used in the analysis of graphical data. This is most widely applied in summarizing variables distributional information. Intervals or bins are the division of response variables, in equal size. The  occurrence number are calculated for each bin. Histograms is the composition of:  I like to share this Making Histograms with you all through my article.

Vertical axis = frequencies or relative frequencies.

Horizontal axis = response variable – refers to each intervals mid point.

The histogram is a visual representation of data , it shows the location of the measurement of data and how they are spread out.

It shows the highest frequency of the data by having a look at the highest rectangle.

The lengths of bases are same indicates that the class interval is same.

The data available in a table is not easy to interpret while the data on a histogram is easy. Comparing the data from two histograms is easier an less time consuming.

Tables of large sets of data make it complex to use while histograms are graphs , it is easier to organize and summarize using them.


Histograms:


Histogram are used for counting graphs  where number of pixels are calculated for each level between black and white. Black and white are on left and right side correspondingly. Depending on the number of bright pixels the height of graph varies. Lighter and darker images move the graph on right and left accordingly.I have recently faced lot of problem while learning Partial Fraction Example, But thank to online resources of math which helped me to learn myself easily on net.

Types of histograms


Histograms are classified into four types as follows:

1)      Histogram - absolute counts.

2)      Relative histogram - convert counts to proportions.

3)      Cumulative histogram.

4)      Cumulative relative histogram.


Uses of Histograms:


Calculate the range by finding the lowest and highest values from the given data.

Identify the number of bars that are to be used in histogram. The number of bars being used should not be too high or too low in order to provide the pattern in an effective way.

Width of each bar is calculated by dividing the range by the number of bars. Then, start with the low values, to determine the group of values to be contained or represented by each bar.

Compilation table is created, then filling the boundaries for each groups takes place.

Based on the data point counting the compilation table is filled for each bar. Total number of data points in each bar are also calculated.

Draw both the horizontal and vertical axes, and also perform labelling them

Draw in the bars to correspond with the total from the frequency table

The pattern of variation is identified and classified.

Functions Solving Online

Introduction to Functions solving online:

The mathematical idea of a function expresses dependence between two quantities, one of the produced is   independent variable, argument of the function, or it’s "input" and another one of the produced is dependent variable, value of the function, or "output". A function expression is associates a unique output with all input element from a fixed set, such as the real numbers.

The function of expression is f(x) =ax^2 + b x + c


Explain about Functions Solving online using quadratic functions


Quadratic function in online is helpful to know more about the simplest way to solving online functions which under  "ax^2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. While factoring could not always be successful, in online the Quadratic Formula can always find the solution by using the .

The general form is,

ax^2 + bx + c = 0

Where, x represents the variable, and a, b, and c, constants, with a ? 0. (If a = 0, the equation becomes a linear equation.) .

Is this topic Implicit Function hard for you? Watch out for my coming posts.


Examples of Functions solving online using Quadratic functions


Using Quadratic functions solving some example problems are given below,

Ex 1 :   solving qudratic functions f(x) = x^2+6x+9 = 0

Sol :  To factorize the quadratic function, split the middle term (6x) into two terms so that the product of their coefficients is equal to the constant term (9).

Like 6x = (3x) and (3x)

So, 3x + 3x = 6x and

3 * 3 (coefficients of 3x and 3x) = 9 (constant term)

So, now the function becomes

x^2 + 3x + 3x + 6 = 0

Here ‘x’ in first term and 3 in last two terms commonly, by taking both of them commonly out, we get

x(x+3) + 3(x+3)  = 0

Now (x+3) in common

(x+3) (x+3)  = 0

Now x+3 = 0 or x+3 = 0

Ex 2 :  solving qudratic function f(x) =x^2+6x+8 = 0

Sol :    To factorize the quadratic function, we have to split the middle term (6x) into two terms so that the product of their coefficients is equal to the constant term (8).

Like 5x = (2x) and (4x)

So, 2x + 4x = 6x and

2 * 4 (coefficients of 2x and 4x) = 8 (constant term)

So, now the function becomes

x^2 + 2x + 4x + 8 = 0

Here ‘x’ in first term and 4 in last two terms commonly, by taking both of them commonly out, we get

x(x+2) + 4(x+2) = 0

Now we have (x+2) in common,

(x+2) (x+4)  = 0

Now x+2 = 0 or x+4 = 0

Mathematics Program

Introduction to solve algebra homework answers:

Algebra is one of the main branches of mathematics that deals with calculating unknown variables from the help of known values. Homework problem with answers helps us to understand the concept of algebra. It is the study of rules of operation and relations, algebraic expressions, conditions and polynomials. An algebraic expression represents a scale where all the arithmetic operations are carried out on both the sides of the scale. Algebra homework problem contains problems with complex numbers, matrices, vector etc. The solved homework problems with answers are given below.

solve algebra homework answers : Examples


The following are the solved example problems for algebra homework.

Ex 1:

Solve the algebraic expression

6(c -3) + 5d - 2(c -d -2) + 1

Sol:

Given algebraic expression is

6(c -3) + 5d - 2(c -d -2) + 1

Multiplying the integer with above terms

= 6c - 12 + 5d -2c + 2d + 4 + 1

Grouping the above terms

= 6c + 7d – 7

Ex 2:

Solve the algebraic equation.

x 2 - 3x = 0

Sol:

Given equation is
x 2 - 3x = 0

Take X factor as common
x (x - 3) = 0

So the product x (x - 3) to be equal to zero, then we get

x = 0 or x - 3 = 0

Solve the above simple equations to obtain the solutions.
x = 0
or
x = 3

X= 0 or 3 is the solution.

Ex 3:

Solve the algebraic expression

4(a -1) + 2b - 5(a -b -4) + 5

Sol:

Given algebraic expression is

4(a -1) + 2b - 5(a -b -4) + 5

Multiplying the integer terms

= 4a - 4 + 2b -5a + 5b + 20 + 5

Grouping the above terms

= -a + 7b + 21

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Solve algebra homework answers : Practice problems


Find h (4) and g(4) and h(4) / g(4) and the functions g and h is given as

h (x) = 3x - 8 and g (x) = x 2 - 12

Solution:

Calculate h(4)

h(4) = 3(4) - 8 = 4

Calculate g (4)

g (4) = 4 2 - 12
= 16 -12 = 4

h (4) / g (4) =4/4 =1

Solve the algebraic expression

4(c -1) + 2d - 5(c -d -4) + 5

Solution:

Given algebraic expression is

4(c -1) + 2d - 5(c -d -4) + 5

Multiplying the integer terms

= 4c - 4 + 2d -5c + 5d + 20 + 5

Grouping the above terms

= -c + 7d + 21

Solve the equation     5(-2x - 2) - (-2x - 4) = -4(4x + 4) + 15

Sol:

Given the equation

5(-2x - 2) - (-2x - 4) = -4(4x + 4) + 15

Multiplying the integer with above terms.
-10x - 10 + 2x + 4 = -16x - 16 +15

Grouping the above terms.

-8x - 6 = -16x - 1

Add -8x - 6 to both sides, the above equation becomes

-8x = 7

X= -7/8

The practice problems are given below for homework.

1) Solve the algebraic equation     6(-8y - 3) - (-5y - 5) = -8(2y + 4) + 9

Answer: y = 8/27

2) Solve the algebraic equation.   x 2 - 10x = 0

Answer:   x= 0 or 10

Perfect Squares

Introduction:

Square of number is multiply the same number twice.A × A =A2, here square of  A  is written as A2 Here A is called the base and 2 is called the index or the power.Now observe the following examples:

0^2 = 0 ×0 = 0

1^2 = 1 × 1 = 1

2^2 = 2 ×2 = 4

These examples are square the same number.The square of 0,1,2,3,4 are 0,1,4,9,16 respectively. These square numbers are known as perfect squares. I like to share this Transformations Geometry with you all through my article.


Explain perfect square with examples:


Examples:

1). Is 625 a perfect square?

Yes, because 625 can be expressed as the product of two same numbers as 25 × 25.

2). Is 10 a perfect square?

No, 10 is not a perfect square since 10 cannot be written as the product of two same numbers.

3). Is 144 a perfect square?

Yes, because 144 can be expressed as the product of two same numbers as 12 × 12.

4). Is 70 a perfect square?

No, 70 is not a perfect square since 70 cannot be written as the product of two same numbers. Understanding Graphing Calculators is always challenging for me but thanks to all math help websites to help me out.


Perfect square Examples:


(1). Find the Perfect square of 20

Solution:

20 ^2 = 20 * 20

= 400

(2). Find the Perfect square of 111

Solution:

111 2 = 111 * 111

= 12321

(3). Find the Perfect square of 13

Solution:

13^ 2 = 13 * 13

= 169

(4). Find the Perfect square of 81

Solution:

81^ 2 = 81*81

= 6561

(5). Find the Perfect square of 100

Solution:

100 ^2 = 100 * 100

= 10000

Perfect square Exercises:

(1). Find the Perfect square of 32

(2). Find the Perfect square of 15

(3). Find the Perfect square of 09

(4). Find the Perfect square of 723

(5). Find the Perfect square of 40

Answers:

(1). 1024

(2). 225

(3). 81

(4). 522729

(5). 1600

Perfect Number Examples

Definition:

A perfect number is the positive integer in that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently of a perfect number is that a number is half the sum of all of its positive divisors (including itself), or σ(n) = 2n.


Even perfect numbers

In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. However, not all numbers of the form 2p − 1 with p a prime are prime

In the first four perfect numbers are generated by the formula 2p−1(2p − 1), with p a prime number:

for p = 2: 21(22 − 1) = 6

for p = 3: 22(23 − 1) = 28

for p = 5: 24(25 − 1) = 496

for p = 7: 26(27 − 1) = 8128.

Noticing that 2p − 1 is a prime number in each instance, Euclid proved that the formula 2p−1(2p − 1) gives an even perfect number In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes. I have recently faced lot of problem while learning Definition of Rational Numbers, But thank to online resources of math which helped me to learn myself easily on net.


Odd perfect numbers

It is unknown whether there are any odd perfect numbers. Various of results have been obtained, but none of that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has been presented by a heuristic argument in which suggests that no odd perfect numbers exist.[4] Also, it can been conjectured of that there are no odd Ore's harmonic numbers, except for 1. If true in this would to imply that there are no odd perfect numbers.

Mathematics in Daily Life

Introduction to Everyday Mathematics answers:-

Our life is intertwined with mathematics.

We use mathematics in our daily life.

We use mathematics  to measure the distance, the weights, to find volume, to know the time,  to measure temperature, to dilute liquids, to  make a concentrate of a solution etc. Understanding radian and degree measure is always challenging for me but thanks to all math help websites to help me out.


Use of mathematic in every day with answers

As soon as we get up in the morning, we look at the clock to know the time.

We measure the coffee and  count the slices of bread for the family.

When we go to the market to buy vegetables, we calculate the prices of the purchase.

As we walk home after purchase, we make a note of the day's temperature.

Cooking needs time management, weight management and volume management and temperature adjustment.

We need to pay for the ironed dress to wear for the office, pay for petrol for the motorbike  etc.

The office work also involves mathematical calculations, charts and graphs. Having problem with Radian Measure keep reading my upcoming posts, i will try to help you.


Use of mathematics outside home with answers


When we go shopping, we have to pay the bill, which is a  mathematical calculation.

Discount offered at the shops and taxes also involves calculations.

Buying and selling of various commodities involves profit or loss.

When you share your  food with friends, it involves fractions.

We convert money to bigger or smaller units.

We also convert measurements to higher or lower units in our daily life.


Mathematic in managing outside activities with answers


When we  build a house , we use the calculation called "Time and  Work".   This system is also used for many

other activities like mowing a garden, plucking tea leaf, harvesting a crop etc

We use the form "Time and distance" when we travel.

We use  the form 'Time and speed"  when we go up and down the stream and river.

We also use 'Time and speed" in sports activities, especially races.


Perimeter, area and volume in daily life with answers


If we want to fence our garden, we use the perimete formula of various shapes.

If we want to  have a lawn in our garden, we calculate the area of the lawn.

If we want to store our harvested grains, we use the volume formula.

If we want to pack the grains in sacks, we use the break-up of the volume to smaller volumes.

Thus we note that  purchases, sales, profits , losses, tax, discounts, time, weights, volumes, area, perimeter

work, speed etc are  used daily in our life .

Probability with Replacement

Introduction to Probability with replacement :

To calculate the probability of ball which is strained from a population by using explanation of probability and the calculations can be established by using the relative frequency definitions of probability. In a examination can have n equally likely conclusion, and that a 'success' can take place in s ways (from the n).

Then the probability of a 'success' = s / n


Probability with replacement Problem 1


A pitcher contains 7 orange and 4 green balls. Find the matching probabilities if the balls are replaced after every draw.

(a) Mutually orange

(b) a orange and a green

(c) both the identical color.

Solution:-

a.) Since the balls are restored after each draw,

P(s)is 7 in both cases since we replace it with in each case.

This is simply 7/11 * 7/11 = 49/121

b.) This should be 7/11 * 4/11 = 28/121

c.) As for P(c).

P(c) = (P(two orange and two green)) .,

so

P(two orange or two green) = P(two orange) + P(two green) = P(c)

P(two orange) is given by 49 / 121

P(two green) is given by 16 / 121

now by just plugging in it in the given function we get it as follows

= 49/121 + 16/121 = 65/121.

65 / 121 is the finalised answer. Please express your views of this topic Example of Theoretical Probability by commenting on blog.


Probability with replacement Problem 2


A population of 50 red mice, 200 green mice, selections with replacement:

a) Probability of 3 red mice in 3 selections

b) Probability of selecting, in order, red, red and then Green.

c) If, however, we are not interested in the order (i.e. red, red, green) but just the overall outcome (i.e. 2 red, 1 green), the probability is different:0

Solution:-

a) Probability of 3 red mice in 3 selections = (50/250) * (50/250) * (50/250)

= (1/5) * (1/5) * (1/5) = 0.008

b) Probability of selecting, in order, red, red and then green = (50/250) * (50/250) * (200/250)

= (1/5) * (1/5) * (4/5) = 0.032

c) If, however, we are not interested in the order (i.e. red, red, green) but just the overall outcome (i.e. 2 red, 1 green), the probability is different:0

Possible outcome of 3 selections with replacement

Rounding and Addition

Rounding is one of the major concepts in basic mathematics taught in the middle school but used throughout the growing years. Rounding is a process of changing a number by estimating its nearest value. If a number is followed by a number that is greater or equal to 5, the number can be rounded up and if a number is followed by a number that is less than 5, the number can be rounded down.
• Example 1: If he buys a full collection of Yonex badminton racquets, he will pay just Rs.155 per racquet. If he buys a full collection of Yonex badminton racquets , he will pay just Rs.160 per racquet
• Example 2: The mall has given away about 23% of discount on all fashion wear . The mall has given away about 20% of discount on all kid's fashion wear. Addition at times is done after rounding a number. Below are the steps to add a number after rounding.
• Round the number to its nearest value
• Add the numbers

Examples: 1.
She paid Rs.1190 to the trainer for teaching her tricep dips at home . She also paid Rs.2000 for her gym trainer. How much total she paid including trainer for teaching tricep dips at home and gym trainer. Rs.1190 + Rs.2000 Rs.1200 + Rs.2000 (Rs.1190 is rounded to Rs.1200) Rs. 3200 is the total amount she paid including trainers. 2. Mohan is planning to invest 24% of his salary in Gold. Mohan’s wife on the other hand is planning to invest 65% of her salary in gold. What is the total investment the family makes on gold? 24% + 65% 25 + 65% (24% is rounded to 25%) 90% of their total investment is made on gold. These are the basics about rounded and addition.

calculus derivative problems exam

Introduction for calculus derivative problems exam:

Two mathematicians, Namely Gottfried Leibniz and Isaac Newton, developed calculus. Calculus problems can be dividing into two branches: Differential Calculus problems and Integral Calculus problems. Differential calculus is use to measure the rate of change of a given quantity whereas the integral calculus is use to measure the quantity when the rate of change is known. The output of a function will change when we change the input value of a function. The measure of the change in the function is called as Calculus Derivatives.


Calculus derivative example problems:

The following solving problems are based on the derivatives.

Ex 1:

Determine the derivative dy/dx of the inverse of function f defined by

f(x) = (1/8) x - 2

Sol:

The first is used to find the inverse of f and differentiate it. To find the inverse of f we first write it as an equation

y = (1/8) x - 2

Solve for x.

x = 8y + 16.

Change y to x and x to y.

y = 8x + 16.

The above gives the inverse function of f. Let us find the derivative

dy / dx = 8


Ex 2:

Determine the critical number(s) of the polynomial function f given by

f(x) = x 4 - 108x + 100

Sol:

The domain of f is the set of all real numbers. The first derivative f ' is given by

f '(x) = 4 x^ 3 - 108

f '(x) is defined for all real numbers. Let us now solve f '(x) = 0

4 x^ 3 - 108 = 0

Add 108 on both sides,

4x^ 3– 108 108=108

4x^ 3= 108

x^ 3 = 27

x = 3 or x = -3

Since x = 3 and x = -3 are in the domain of f they are both critical numbers. Is this topic Limits of a Function in Calculus hard for you? Watch out for my coming posts.


Calculus derivative Practice Problems exam:


1) Determine the derivative dy/dx of the inverse of function f defined by

f(x) = x/2+ 3x/2 – 2

Answer:  dy / dx = 2


2) Determine the critical number(s) of the polynomial function f given by

f(x) = x^ 3 - 48x + 10

Answer:  X = 4 or X= -4

Five Thirty Eight

Introduction to five thirty eight:

Five thirty eight is represented in number form as 538. Thus five hundred thirty eight has five of hundred and three of ten and eight one. Thus the total sum of five hundred and three ten and eight ones is equal to five thirty eight. In this article we see about five thirty eight with some example problem related to five thirty eight. Is this topic Teaching Distributive Property hard for you? Watch out for my coming posts.

Five Thirty Eight:

Let us see the example problem of five thirty eight.

Example Problem – Five Thirty Eight:

Example 1:

Write the expand form of five thirty eight.

Solution:

We can split the five thirty eight as five hundred and thirty and eight.

Number form of five hundred is 500 and thirty as 30 and eight is 8.

Now we get the expand form of five thirty eight is added the five hundred and thirty and eight.

538 = 500 + 30 + 8

Answer: Expand form of five thirty eight is 500 + 30 + 8

I have recently faced lot of problem while learning Ray Definition, But thank to online resources of math which helped me to learn myself easily on net.

Five Thirty Eight:

Example 2:

Which of the following expression give the result as five thirty eight?

Option:

a)     500 – 30 + 8

b)    30 – 500 + 8

c)     500 + 30 + 8

d)    500 * 30 * 8

Solution:

Check the expressions which give five hundred fifty.

Option a: 500 – 30 + 8, give the result as 478.

Option b: 30 – 500 + 8, give the result as -442

Option c: 500 + 30 + 8, by adding the five hundred and thirty and eight give as result as five thirty eight.

Option d: 500 * 30 * 8, the product of 500 and 30 and 8 give the result as 120000

Hence the expression which has given in option c gives the result as five thirty eight

Answer: Option c - - - > 500 + 30 + 8

Now clear about the five thirty eight.

Multiplication Times Tables

Introduction for multiplication times tables:

Multiplication is the arithmetical operation of calculating one value with another value. This is a one kind of operations in basic arithmetic. Multiplication is nothing but the repeated addition. For example: 3 * 4 = 12 , so it represents 3 + 3 + 3 + 3  = 12, therefore for adding the number 4 simultaneously 3 times we get 12. Multiplication times tables is useful for making the multiplication process more easy. I like to share this Double Digit Multiplication Practice with you all through my article.

Multiplication Times Tables:

x	0	1	2	3	4	5	6	7	8	9	10	11	12
0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6	7	8	9	10	11	12
2	0	2	4	6	8	10	12	14	16	18	20	22	24
3	0	3	6	9	12	15	18	21	24	27	30	33	36
4	0	4	8	12	16	20	24	28	32	36	40	44	48
5	0	5	10	15	20	25	30	35	40	45	50	55	60
6	0	6	12	18	24	30	36	42	48	54	60	66	72
7	0	7	14	21	28	35	42	49	56	63	70	77	84
8	0	8	16	24	32	40	48	56	64	72	80	88	96
9	0	9	18	27	36	45	54	63	72	81	90	99	108
10	0	10	20	30	40	50	60	70	80	90	100	110	120
11	0	11	22	33	44	55	66	77	88	99	110	121	132
12	0	12	24	36	48	60	72	84	96	108	120	132	144

How to Find Answers Using Multiplication Times Tables:

First we have to see the numbers what we have to multiply. Let's take we have to multiply 8 and 8 so the answer can be easily found from the table. First we have to find 8 in the first column, and find 8 in the first row. Then we to see the intersecting cell. In the chart the intersecting cell of 8 and 8 has 64. So 64 is the answer for 8 times of 8. This is how we find the answers form multiplication from the multiplication times tables

Example problems using multiplication times tables

1) 7 x 7 = 49

2) 10 x 11 = 110

3) 8 x 1 = 8

4) 1 x 9 = 9

5) 5 x 6 = 30

6) 4 x 8 = 32

7) 3 x 2 = 6

8) 7 x 3 = 21

9) 8 x 3 = 24

10) 12 x 2 = 24

Distance of Earth from Sun

Introduction to distance of earth from sun:

Earth, as we all know, is the third planet from Sun. Earth revolves around Sun in an elliptical orbit. Due to the orbit of Earth being elliptical and not perfectly circular, the distance between Sun and Earth varies with the position of Earth.

Mean Distance between Earth and Sun
As it is very difficult to be able to continuously measure the distance between Earth and Sun, distances at the farthest and closest points are determined and the mean distance is calculated from them and used as a reference. These two points are:

Aphelion: Earth is farthest from Sun at Aphelion. This occurs during summer in the Northern Hemisphere - around the first week of July. The distance of Earth from Sun at Aphelion is about 152 million km (94.4 million miles).
Perihelion: Earth is closest from Sun at Perihelion. This occurs during winter in the Northern Hemisphere - around the first week of January. The distance of Earth from Sun at Perihelion is about 147 million km (91.3 million miles).
The mean distance between Earth and Sun (calculated from the mean of the distance at Aphelion and distance at Perihelion) is 149.5 million km.

Mathematically it can be expressed as:

Mean distance between Earth and Sun = (Distance at the farthest point or Aphelion + Distance at the closest point or Perihelion) / 2

= (152 + 147)/2 million km

= 149.5 million km

The exact value of the distance from Earth and Sun has been established as 149, 597, 870.7 kilometers (92,955,887.6 miles) and is also called 1 astronomical unit (or AU).

Implications on Distance of Earth from Sun

While we may think that Earth-Sun distance is critical in maintaining average temperatures on Earth and hence using a mean value would be misleading. But this is not true. The variation in distance due to the elliptical orbit is 5 million km. (This can be calculated by subtracting the distance at Perihelion from the distance at Aphelion. i.e. 152 – 147 million km.) This is a very small percentage of the actual distance between Earth and Sun (149.5 million km) and hence does not cause a significant change in the average temperatures on Earth.