Math Division Chart

Definition of Dividend, Divisor ,Quotient ,Remainder

Dividend →
 
                   A quantity which is to be divided by another quantity is known as dividend.
e.g. 6 divided by 3, 6 is the dividend.

Divisor →
                  The quantity by which the dividend is to be divided is known as divisor.

e.g. 6 divided by 3, 3 is the divisor.

Quotient  →

                    The number resulting from the division of another number.
              Dividend ÷ Divisor = Quotient

e.g.  2 is the quotient of 6 divided by 3.
Remainder →
                           When an integer m is divided by a positive integer n, and a quotient q is obtained for which  m=nq+r

 Then r is the remainder.

    Math Division chart

Greek Alphabet In Math

Introduction :

The Greek alphabet is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early 8th century BCE. It is the first and oldest alphabet. The Greek alphabet is descended from the Phoenician alphabet, and is not related to Linear B or the Cypriot syllabary, earlier writing systems for Greek.


Now we are going to see about the Greek alphabet in math.

About Greek alphabet in math:
                    Now we will see about the Greek alphabets and its symbols which are used in math.
In math, various symbols are required to represent the various functions such as in the set theory, numbers, functions and spaces.

However at the time of introduction, many of them seem to use some of the notations and the letters in the math.

Now it is familiar in all countries and many formulas in math and in other subjects are using the Greek alphabets only.

The lower case numbers are used often for variables, complex numbers etc.

Greek alphabets and its symbols in math:
                      Now we see Greek alphabets and symbols in math.

1.    α is the symbol used to represent the number and the name is alpha(Α)

2.    β is the symbol used to represent the number and the name is beta(Β).

3.    γ is the symbol used to represent the number and the name is gamma(Γ).

4.    δ is the symbol used to represent the positive number and the name is delta(Δ).

5.    ε is the symbol used to represent a positive number and the name is epsilon(Ε)

6.    ζ is the symbol used as seldom and the name is zeta(Ζ).

7.    η is the symbol used as seldom and the name is eta(Η)

8.    θ is the symbol used for an angle and the name is theta(Θ)

9.    ι is the symbol used hardly and the name is diota(Ι)

10.     κ is the symbol used to represent the kappa(Κ).

11.     λ is used to represent a constant multiplier and the name is lambda(Λ).

12.     μ is used to represent a constant multiplier and the name is mu(Μ).

13.     ν is used to represent number and called as nu(Ν)

14.     ξ is used and the called as xi(Ξ)

15.    ο is used hardly and called as do micron(Ο)

16.    π is used to denote invariably and called as pi(Π)

17.    ρ is used for radius rho(Ρ)

18.    σ is used for and called as sigma(Σ)

19.    τ is used and called as tau(Τ)

20.    υ is used and called as dupsilon(Υ)

21.    φ is used for an angle, called as phi(Φ)

22.    χ seldom used as chi(Χ)

23.    ψ is used for an angle psi(Ψ)

24.    ω is used rare and called as omega(Ω)

Coordinate Plane Slope Math

 Slope in a coordinate plane:

Slope:

In mathematics the slope of the line shows the steepness of the line. Slope is also defined as rise by run. We can find the slope from the equation of the line or form the coordinates. The positive slope has + sign and negative slope has – ve sign. Here we are going to see about slope in coordinate plane.

The formula to find slope is

Slope m = `(y2 - y1) / (x2 - x1)`

Slope in the coordinate plane is determined by rise in y axis by run in x axis.

consider the following figure.



Here the rise in y axis is -15 units

run in y axis is 30 units

Slope = `-15/30`

slope = `-1/2`

`and it can be also calculated using the coordinates from the above formula.`


Slope in a coordinate plane - Examples:


1). Find the slope of the line containing the coordinates (5, 12) and (7, 10).

Solution:

Given:

x1 = 5     x2 = 7

y1= 10    y2 = 12

Slope m = `(y2 - y1) / (x2 - x1)`

= `(10-12) / (7 -5)`

=` -2 / 2`

m = 1

Hence the slope of the given points are m = 1

2). Find the slope of the line containing the coordinates (14, 8) and (18, 9)

Solution:

Given: x1 = 14    x2  = 18

y1 = 8      y2 = 9

Slope m = `"(y2 - y1) / (x2`

=`" ( 9 - 8 ) / ( 18 - 14 `

= `1/4`

m = `1/4`

Hence the slope of the given points m =` 1/4 `

1)Find the slope of the line having the coordinates (6, 10) and (9, 11)
Solution:

Given:            x1 = 6     x2 = 9

y1 = 10    y2 = 11

Slope   m =` (y2-y1) / (x2-x1)`

= `(11 -10) / (9 - 6)`

= `1/3`

2) Find the slope of the line having the coordinates (7, 11) and (12, 14)

Solution:

Given:            x1 = 7     x2 = 12

y1 = 12    y2 = 14

Slope   m = `(y2-y1) / (x2-x1)`

= `(14 -11) / (12 - 7)`

= `3/5`

3) Find the slope of the line having the coordinates (11, 14) and (16, 23)

Solution:

Given:            x1 = 11     x2 = 16

y1 = 14    y2 = 23

Slope   m =` (y2-y1) / (x2-x1)`

= `(23 -14) / (16 - 11)`

= `9/5`.

What Does Area Mean in math

Introduction to calculate area of building:

Area is the measure of surface occupied by an object. The standard unit for measurement of area is meter square (m2)however the areas of smaller dimensions can be expressed in mm2or cm2.The areas of large amount of dimensions can be expressed in acre or hectare. Here we have study about how to calculate the area of the building shape image and its example problems.

Calculate area of the building example problems:

Example: 1

Calculate the area of the building which is in the form of square with side length is 6.

Solution:

First we have to draw the image and mark the side length.



Here the building shape is square

We know that area of the square = a 2

a = 6

Substitute a value in the formula we get

Area = (6)2

=36 meter square

Area of the given building is 36 meter square.

Example: 2

Calculate the area of the building which is in the form of rectangle with base is 8 meter and the height of the rectangle is 7 meter.

Solution:

Figure:



Mark the all measurements.

Here the shape is rectangle so we have to find the area of the rectangle

Formula = base *height

Here the given b = 8 meter and h= 7 meter

Substitute the b and h value in the above formula we get

Area = 8*7

= 56 meter square

Example: 3

Calculate the area of the building which is in the form of triangle with base is 12 feet and the height of the rectangle is 9 feet.

Solution:

Draw the diagram for the given measurement and mark the all data.


Here the building shape is a triangle so we have to find the area of the triangle

Formula:

Area = 1/2 (b*h)

Here base is 12 feet and height is 9 feet substitute the above formula we get

Area = 1/2 (12*9)

= 108 / 2

The simplest form is 54

The area of the given triangle shaped building is 54 feet square

Bar Diagrams for Math

Introduction :

In math a bar diagram is a pictorial representation of numerical data in the form of rectangles or Bars  of equal widths and various heights. These rectangles are drawn either horizontally or vertically.It should be remembered that bar diagram is of one Dimension.The height of bar represents the frequency of the corresponding observation.The gap between two bars is kept the same.


Steps for drawing bar diagram of math:


STEP 1 :- If the given frequency distribution is in inclusive form, convert it into an exclusive form.

STEP 2 :- Taking suitable scales, mark the class-intervals along x-axis and frequencies on y axis.

Note that the scales chosen for both the axes need not be the same.

STEP 3 :- Construct rectangles with class intervals as bases and the corresponding frequencies as heights.


Example for bar Diagram of math:


Draw the bar diagram following data that is the monthly expenditures of jack’s family on various items

Items                     Expenditure (In Rs.)

House Rent                  3000

Food                               4500

Education                       600

Electricity                       350

Transport                       800

Miscellaneous              500

Procedure to Draw a Bar diagram




Step: 1

First we have to take a paper and draw the horizontal line at the bottom of the paper the horizontal line is called x axis. And draw another line perpendicular to the horizontal line which is called y axis.

Step: 2

With  the horizontal axis, we take the information ‘item’ and in the vertical axis, we take the corresponding other information namely ‘the Expenditure

Bar diagram:

The following diagram is the bar diagram for the given data.

Bar diagram is showing the monthly expenditures of jack’s family on various items.

Questions for the bar diagram of math:

1) What is the information given by the bar diagram?

2) What is the number of items shown in the bar diagram and list them.

3) In which Item the maximum amount was spent.

4) In which Item the minimum amount was spent

Solution:

1) The information given by the bar diagram is the monthly expenditure of jack’s family on various items.

2) There are 6 items listed in the bar diagram. The six items are house rent, food, education, Electricity, transport and miscellaneous.

3) Food is the item where the maximum amount is spent.

4) Electricity is the item where the minimum amount is spent.

Ancient Symbols In Math

The two cultures that laid the groundwork for modern western civilization left us a legacy of symbols that is woven through every aspect of modern life. Many of these symbols were even used to build our modern world. 

Where would math, science and engineering be today without the Greek alphabet? There are 25 Greek letters that are used today to represent individual concepts. Here are just a few.
Alpha is used to represent the angle opposite of side A in a triangle, one root of a quadratic equation, an alpha particle, and angular acceleration.

Delta is used to represent a finite difference. Theta is used to represent a plane angle and potential temperature. Kappa is used to represent thermal conductivity. Mu is used to represent dynamic viscosity. Pi is used to represent Archimedes' constant, the ratio of a circle's circumference to its diameter.

From just these few samples, it's clear that modern mathematical and scientific expression is highly dependent on Greek symbols.

Greek mythological symbols have also come down through the ages.

Owl: Athena was the Goddess of wisdom. One of her symbols was an owl. Today, the owl is probably the most widely recognized symbol for wisdom.

Grape vine: Dionysus was the god of wine. One of his symbols was the grape vine. We still use the grape vine today as a symbol of wine.

Winged boots: Hermes was the messenger of Zeus. One of his symbols, the winged boot has made its way through the ages as a symbol for messengers.

Thunderbolt, eagle and oak: These are all symbols that have survived the ages to represent power and strength, attributes of Zeus.

But what about our Roman legacy? Science as we know it would not have developed without Roman nomenclature. It is the single most organizing factor in modern science.

Armor: Mars (god of war) and Minerva (goddess of war) were both represented by armor, a symbol still used today to represent war.

Scales: Them is was the goddess of justice. She actually originated as Ma' at from Egyptian mythology. Her scales symbolize the weighing of the hearts of man. She now stands as a universal symbol of justice blindfolded and holding a sword in one hand and scales in the other.

Ancient Greek and Roman symbols surround us in the modern world. They are not likely to go away any time soon. Our written language is deeply rooted in ancient symbols that represent sounds. Our ideas are nearly parallel to the ideas that inspired the symbols used by the ancients. Why wouldn't they be? Their cultures laid the groundwork for modern western civilization.

Velocity Problems And Solutions

Introduction:

Normally rate of change of position in math is known as velocity.  It is one of the vector quantities because when we mentioned the velocity it will indicate the direction also. Velocity is normally defined by the ratio between the distance and time. In velocity problems we can measure the time and distance. Velocity problems are used to find the solutions for the distance problems. We can find the solutions for maximum height of the object traveled. velocity problems and solutions are one of the main application in physics.

Velocity problems and solutions:


Problem 1:

    A submarine descends 90 miles in 10 seconds. Find the velocity of the submarine.

Solution:

Step1:

    We know the velocity formula V = Distance (d) / Time (t)

Step2:

   Here distance d = 90 miles

  And the time t = 10 seconds.
Here the submarine is descending so the velocity will be negative velocity.

Step 3:

 So velocity = - 90 / 10 miles / sec

 Velocity V = - 9 miles / sec.

We can find the distance travelled by the object using the velocity problems.


Problem 2:


   A ball is thrown upwards in the air from the ground. The distance travelled by the ball is

 h = 16t2 - 64t + 6. Find the maximum height of the ball.

Solution:

 The distance traveled by the ball is h = 16t2 - 64t + 6

 We want to find the maximum height. When the ball is maximum height the velocity will be zero. To find the velocity function we have to differentiate the distance function with respect to t.
 V = d h / d t = d / d t (16t2 - 64t + 6)

   V = d h / d t =32t - 64

   If the ball is at maximum height velocity v = 0         

   32t - 64 = 0

    32t = 64

   t = 64 / 32

   t = 2

   So when time t = 2 the ball will beat maximum height.

   h = 32 (2)2 -64(2) + 6

  h = 128 – 128 +6

 h = 6           

From the above problem we used the velocity to find the solutions for the maximum height.

Middle Grade Math Pizzazz Book e Answer Key

Solving Step of Math Pizza
  Middle grade math pizzazz book e answer key is a solver for math problems. It used to solve the problems in online. The pizzazz worksheet gives the answer and explanation‘s to the work. It used to give exact answers in online without the help of humans. Here we want to enter the questions in the pizzazz worksheet and clicking the concerned buttons to solve the answer and it gives the answer. It consists of all symbols and functions of math.

This answer key provide solution to problem related to determine equivalent relationships between fractions, decimals, and percents, including use of scientific notation and  problems related to simplify expressions that contain rational numbers and the positive exponents, using  order of the operations and mental mathematics and provides the investigate and solve problems involving the volume and surface area of the rectangular prism and cylinders, using practical situations to develop formulas.

This answer key investigate and describe the difference between the probability of an event found through simulation versus the theoretical probability of that same event and gives  solutions to problems  using graphical methods like frequency distributions, line plots , histograms , stem and leaf plots, box and whisker plots and scatter grams


Solving Step of Math Pizza 1

We can directly solve the problem and check the solution by clicking the solution button in Pizzazz’s answer key worksheet or we can click the button “solve step” by clicking it will solve a step or we can solve the problem directly by clicking it “solve all” by clicking the whole problem is solved. In this we can plot graphs in the Pizzazz’s answer key worksheet. Then we can solve the worksheet and save the problem. Then we can do the next problem. It is useful for solving all math problems and even we can plot graphs.

How to Multiply Percentage

Introduction :

Multiply percentages are apply to communicate how large/small individual number is, comparative to a further number. The initial number typically corresponds to a division of, otherwise vary in, the following number, that must be greater than zero. For exemplar, enlarges of dollar 0.05 and dollar 2.5 is an increase in a fraction of 0.05 / 2.5 = 0.02. Expressed while a percentage, this is consequently a 2% increase.

How to Multiply percentage:

Even though percentages are frequently used to know how to communicate numbers among zero also one, some dimensionless proportionality know how to be expressed while a percentage.

Uses of Percentage:

Commission
Discount
Markup
Sales tax
Price with sales tax
Shipping and handling
Simple interest
Simple interest and principal

Consequences of one numeral are separated in a further multiply percentage. Ratios are the simplest mathematical tools to expose significant relationships indefinite in group of data by permit important comparisons. An only some expressed like fractions with some like percentage.


Examples for multiply percentage:

Example 1:

In an election between two canditates 80% of the voters cast their votes, out of which 2% of the votes were declared invalid. A canditate get 972 votes which weer 60% of the total valid votes. how to solve the total number of votes enrolled in the election

Solution:

Step 1: let the total numebr of votes enrolled be x then number of votes cast = 80% of x

Step 2: valid votes = 98% of (60% of x)

Step 3: 80% of [98% of (60% of x)] = 972

Step 4: `(80)/(100) * (98)/(100) * (60)/(100) * x = 972`

Step 5: `x = (972 * 100 * 100 * 100)/(80 * 98 * 60)`

Step 6: the answer of the percentage is x = 12397.

Example 2:

20% of the inhabitants of a village having died of cholera a panic get in during which 50% of the remaining inhabitants left the village. the population is then reduced to 5021. how to find the number of original inhabitants.

Solution:

Step 1: let the total number of original inhabitants be x, then (100 - 50)% of (100-20)% of x = 5021.

Step 2:  50% of 80% of x = 5021

Step 3: `(50)/(100) * (80)/(100) * x = 5021 `

Step 4: `x = (5021 * 100 * 100)/(50 * 80)`

Step 5: x = 12553.

Example 3:

how to Find the 5% of 10

Solution:

Step 1: given that 5% from 10.

Step 2: change the 5 % into percentage form

Step 3: 5% out of 100 so, 5/100

Step 4: `(5)/(100) * 10`

Step 5: So the solution 0.5

Example 4:

how to Find the 15% of 50

Solution:

Step 1: given that 15% from 50.

Step 2: change the 15 % into percentage form

Step 3: 15% out of 100 so, 15/100

Step 4: `(15)/(100) * 50`

Step 5: So the solution 7.5

Middle School Math Proportions : 2

Introduction :

            Proportions are nothing more than two ratios or fractions set equal to one another. Proportions have several very handy properties that make working with them much easier to manage. In proportions correctly and how to solve the problems created with the proportions.

          
 A proportion is an equation involving two fractions. The ratio of the numerators and denominators of the fractions must be equal. A proportion is a statement saying that two fractions are equivalent or equal in value. The fractions can be reduced in the normal way – and they can be reduced in some rather unique ways, too. This property and several others to solve for unknown parts of a proportion.

                        Given the proportion a / b = c / d, the following also

                                                             a x d = b x c

Definition and examples problems on middle school math proportions:

Definition proportion(Middle School Level):

                    If two ratios are not like, then we declare that they are not in proportion. In a declaration of proportion, the four quantities concerned when taken in order are known as respective terms. First and fourth terms are known as extreme terms. Second and third terms are known as middle expressions.If two ratios are equal, we say that they are in proportion and exercise the symbol ‘::’ or ‘=’ to equate the two ratios.

Example problems for middle school math proportion:

Problem 1:

              Are the ratios 20g: 24g and 40 kg: 48 kg in proportion?

Solution:

                  20 g: 24 g = 20 / 24

                                    = 5: 6

                 40 kg: 48 kg = 40 / 48

                                   = 5: 6

                  So, 20: 24 = 40: 48.

Therefore, the ratios 20 g: 24 g and 40 kg: 48 kg are in proportion,

                i.e. 20 : 24 :: 40 : 48

The middle terms in this are 24, 40 and the extreme terms are 20, 48.

Problem 2:

               Are 15, 20, 45 and 60 in proportion?

Solution:

              Ratio of 15 to 20 = 15 /20

                                            = 3: 4.

             Ratio of 45 to 60 = 45 / 60

                                           = 3: 4.

             Since, 15: 20 = 45: 60.

Therefore, 15, 20, 45, 60 are in proportion.

Applications on middle school math proportions:

      If we want to draw a plan of a house, it is not possible to draw the plan in actual dimensions (or size).

      Let the ratio between drawn length and actual length be a: b

                      1) If a is less than b, we get a reduced figure.

                     2) If a = b, we get equal scale figure (same size figure)

                     3) If a is greater than b, we get an enlarged figure.

      
For example,

                  1) A house plan map – a reduced figure

                  2) Geometrical figure drawn in note book – equal scale figure

                  3) Bacteria seen through microscope – enlarged figure

Note: The ratio a / b is called the representative fraction (a and b must be in the same unit).

De Morgan's Law venn Diagram

Venn Diagram is given by an English Logician Mr.Venn. He used diagrams to show the relations between sets. De Morgan's Law Venn diagram is used in Set Language.

A set is a collection of well defined distinct objects.

De Morgan's Law

(i)      (A`uu`B)' = A'`nn` B'

(ii)     (A`nn`B)' = A'`uu`B'

(iii)   A`-` (B`uu` C) = (A`-` B)`nn`(A`-` C)

(iv)   A`-` (B`nn` C) = (A`-` B)`uu` (A`-` C)

(i) and (ii) are w.r.t.complementation

(iii) and (iv) are w.r.t. to set difference.

Venn Diagrams

Let A and B are two given sets. Then the set of all elements common for both A and B is called intersection of A and B and is denoted by A.

Math Knowledge

Introduction of math division sign:

In mathematics, especially in elementary arithmetic, division (÷) is the arithmetic operation that is the inverse of multiplication.

Specifically, if c times b equals a, written:
       c x b = a
where b is not zero, then a divided by b equals c, written:
       `a/b= c`

For instance,
      `6/3 =2`
since
      2 x 3=6
In the above expression, a is called the dividend, b the divisor and c the quotient. (Source: Wikipedia)

Concept of math division sign:

The size of each group formed, quotient of a,b and c. Quantitative division contain a set of size a and forming groups of size b. The number of groups size that can be formed, in c, is the quotient of a and b.

Math Division sign algorithm

The division sign algorithm is the theorem that accurately expresses the output of the division process of integers. The theorem has integers and the quotient q,remainder r that are exist and it has the unique a and divisor d, with d ≠ 0.

The theorem as follows: There exist exclusive integer’s q and r such that a = q d + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d.

Examples problems for math division sign:

Problem 1:
Solve `30/2`
Solution:
         `30/2 ` = 15

Problem 2
Solve `( x/y) /(p/ r)`
Solution:
         `( x/y) /(p/ r) =(x/y) x (r/p) = (x r) / (y p)`

Problem 3:
Find the value for `(x + i y) / (r + i s)`
Solution:
      `(x + i y) / (r + i s) = (x r + y s) / ( r ^2 + s^2 ) + i( y r - x s) / ( r^2 + s^2 )`
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Practice problem for math division sign:

Problem 1:-

Find the value for 45/5.

Answer: 9

Problem 2:-

Solve `( pqr )/ (pq)`

Answer: r

Quadratic Inequalities

In this page we are going to discuss about  quadratic inequalities concept.A quadratic inequality in one variable is in the form of an expression:

ax2 + bx + c ≤ 0  or   ax2 + bx + c < 0
ax2 + bx + c ≥ 0  or   ax2 + bx + c < 0

where a, b and c are real numbers, a ≠ 0. The values of x which satisfy the given inequality are called the solutions of the inequality.

A quadratic equation has only two roots. But a quadratic inequality has many roots.


Methods to solve quadratic Inequalities


There are two methods to solve quadratic inequalities -

Method 1: Finding the solution by dividing the given polynomial into factors. This method is called the 'Algebraic method'.

Method 2: Finding  the solutions by drawing the graph of the inequality. This method is called the 'Graphical method'.

Note:

• The trick in solving a quadratic inequality is to replace the inequality symbol with an equal sign and solve the resulting equation.  The solutions to the equation will allow us to establish intervals that will let you solve the inequality.
• Plot the solutions on number line creating the intervals for investigation. Pick any number from each interval and test it in original inequality. If the result is true, that interval is the solution to the inequality.

Solving quadratic inequalities


Below you can see the example on solving quadratic inequalities-

Example:1 Find the solution set of x2 – 4x – 21 ≥ 0 in

1. Algebraic Method
2. Graphical Method

Solution:

1.  Algebraic Method:

X2 - 4x - 21 ≥ 0
=> ( x + 3) ( x - 7 ) ≥ 0
Here the coefficient of x2 is > 0 and the sign of the quadratic expression is non-negative.
So, x ≥ -3 and x ≤ 7 or there are two cases for the product ( x + 3 ) ( x - 7 ) to be non-negative.

Case (1):
x + 3 ≥ 0 and x - 7 ≥ 0
=> x ≥ -3 and x ≥ 7
So it is necessary that  x≥ 7. Notice the line marked below with x ≥ -3 and x ≥ 7 and their intersection.



From the figure above, it is clear that the intersection is x ≥ 7.

Case (2):
x + 3 ≤ 0 and x - 7 ≤ 0
=>   x ≤ -3 and x ≤ 7
So it is clear x ≤ -3.
i.e., the values of x for which x2 - 4x - 21 ≥ 0 are given by these two cases.

All values of x satisfying the in equations x ≤ -3 or x ≥ 7 become the solution set for the given in equations. This solution set can  be shown in the graph given below.



From the figure above, it is clear that the intersection is x ≥ 7.


2. Graphical Method:

The given polynomial function is f(x) = x2 - 4x - 21

x	-3	-2	-1	0	1	2	3	4	…
f(x)	0	-9	-16	-21	-24	-25	-24	-21	…

Using the above chart, we can draw a graph given below





Now, we require the values of x satisfying x2 - 4x - 21 ≥ 0 , i.e., f(x) ≥ 0 ( y ≥ 0 ). That is we require the values of x for which f(x) lies on x-axis and above. But it is clearly understood from the figure that the values of  x are -3, left side to -3, +7, and right side to +7. Thus all values of x such that x ≤ 3 and x ≥ 7 become the solution set.

Review Geometry Learning

Geometry is learning a part of mathematics which deals with the dimensions of solids and surfaces, shapes, angles and lines etc., It reviews the deduction concepts and consequence logic's, which can be applied through out your life time . In this article, you can review the basic terms and shapes in geometry.

Learning Basics Terms in Geometry:

Learning the following definitions helps to review concepts in geometry.

Point:
A point have no dimensions, it just denotes a position.

Line:
A line can be defined as a straight curve which connects many points. A line has only one dimension, i.e., length.

Collinear points:
If two or more points lie on same line, then they are said to be as colinear points.

Plane:
A Plane can be defined as a set of points and line segments combined infinitely to form a flat surface, it can be extended infinitely in any directions. A plane has infinite length and width, but it doesn't have any height.

Midpoint:
A midpoint can be defined as a point, which divides a line segment equally from both the end points.

Line segment:
A line segment can be defined as a part of a line. It consists of two end points and it was represented by its end points.

Ray:
A ray can be defined as a line having one fixed end point and an infinite extension on the other end.

Angles:

An Angle can be formed by two rays having a common end point.These are the reviews of the basic terms in geometry learning.

Learning Basic Shapes in Geometry:


Square:
A Square can be defined as a regular quadrilateral with equal sides and 4 right angles.

Rectangle:
A Rectangle is a four sided geometrical figure with two pairs of same side lengths.

Triangle:
In geometry, only shape which has three sides and angles is a triangle.

Circle:
A circle is a geometric figure, formed by a locus of points which are all equidistant from a common point called as its center. The distance between the locus of points to the common point is known as radius of the circle.

These are the reviews of the basic shapes in geometry learning.

Long Integer Definition

Integer is a whole number which is not a fraction, that is it can be either positive, negative or zero. Thus the numbers -20, -10, 0, 10, 50, 98, 540 are integers. It can not have decimals. It is commonly used in computer programming as a data type. It can also be used to find out item's location in an array. If two integers are added, subtracted or multiplied, then the output is also an integer. Where as if one integer is divided with other integer, then the result may be an integer or a fraction.

When we think of the number, we first of all think of natural numbers: 1, 2, 3, 4, 5, ...... which are also called positive integers. By applying the operation of subtraction, we get the number zero and negative integers. The set {.....-4, -3, -2, -1, 0, 1, 2, 3, 4,........} is called the set of integers. By division operation, we get the positive and negative fractions. The integers and fractions together constitute the class of rational number p/q, where p and q are integers but q is not equal to 0.

Integers and long integers:


Numbers which includes positive integers{ 1, 2, 3, 4, .....} and negative integers{-1, -2, -3, -4,......} with zero{0} are called integers. There is no fractions or decimal points are included in the integers. They are represented by Z.

Therefore,

Z={ … -4, – 3, – 2, – 1, 0, + 1, + 2, + 3, +4 ….}

+ 1, + 2, + 3, … are positive integers.

– 1, – 2, – 3, … are negative integers.

Note:

Positive numbers can be written even without the ‘+’ sign.

For example, +5, +4, +3, +2, +1 are  written as 5, 4, 3, 2, 1.

Thus, Z = {…… – 3, – 2, – 1, 0, 1, 2, 3, ……}

The integers are represented on the number line as follows:



On the number line, the numbers which are to the right of zero are called positive integers. The numbers which are to the left of zero are called negative integers.

(1)  Every natural number or a whole number.

(2)  Every whole number is an integer.

Order in long integer:

• On the number line, we find that the number value increases as we move to the right and decreases as we move to the left.
• If we represent two integers on the number line, the integer on the right is greater than the integer on the left.
• In other words, the integer on the left is lesser than the integer on the right.

For example, consider the following points marked on the number line in the figure given below:


In the above figure,

4 is to the right of 2                `:.` 4 > 2

2 is to the right of -1               `:.` 2 > -1

-5 is to the left of  -1                 `:.` -5 < -1


Example problems:


Example 1: Which is smaller? -2 and -5

Solution:

First, mark the integers -2 and -5 on the number line.



On the number line, -5 is on the left side of -2.

Therefore, -5 is smaller than -2

That is -5 < -2.

Example 2: Write the following integers in ascending order.

3,-2,0,-4,-1,5

Solution:

First mark these integers on the number line.



Now arrange the integers from left to right to get them in ascending order

-4 < -2 < -1 < 0 < 3 < 5

Therefore, the ascending order is -4, -2, -1, 0, 3, 5.

Which Fraction is Larger

 Yes, which fraction is larger or smaller? The question has great importance. As in many of the competitive exams, we are asked to arrange a set of given fractions. Before learning, how to compare the fractions, we'll learn the basics of fractions. Then we'll learn two methods, that will help us to tell "which fraction is larger".

Fraction: A number of the form `a/b` , where a and b are integers and b`!=` 0,  is known as fraction. Here, a is called numerator and b is called denominator.

Example: `3/5` is a fraction, where Numerator = 3, and Denominator = 5.

7 is also a fraction, because it can be written as `7/1`

1st Method to find which fraction is larger


Method to compare two fractions: This method is known as Cross multiplication method. Steps are as follows:

Let `a/b` and  `c/d` be the two given fractions, that we have to compare and decide which fraction is larger?

Find the product of numerator of the first fraction and the denominator of the 2nd fraction. So, we get ad and bc.

i) If ad > bc, then `a/b`  >  `c/d` .

ii) If ad = bc, then   `a/b` = `c/d` .

iii) If ad < bc, then `a/b`  <  `c/d`

Ex: Compare, which fraction is larger `3/5` and `5/8`?

Products are 3X8= 24 and 5X5=25.

Here, 24 < 25,  So `3/5`` < ``5/8`

Now, we can say very easily, which fraction is larger.

Hence, `5/8` is larger fraction among the two.


2nd Method to know which fraction is larger:


This method is used to compare more than two fractions:

Step I: Find the LCM or LCD of the given fractions. Let it be m.

Step II: Convert all the given fractions into like fractions, each having m as denominator.

Step III: Now we compare the numerators. The fraction having greater numerator is greater and vice- versa.

Ex: Arrange the fractions  `2/5` , `3/10` and `9/14` in ascending order.

LCM of 5, 10 and 14 is 70.

Now, let us change each of the given fractions into an equivalent fraction having 70 as denominator.




Practice problems of which fraction is larger


1. Which fraction is larger: `5/6` or `6/7` ?

2. Which fraction is larger: `21/25` or `3/5` ?

Study graph functions

Graphs of functions f is the collection of the all ordered pairs (x, f(x)). In particular, if x is a real number, a graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on the Cartesian plane is sometimes are called curve sketching. If the function input x is an ordered pair (x₁, x₂) of real numbers, the graph is the collection of all ordered triples (x₁, x₂, f(x₁, x₂)), and its graphical representation is a surface .

Example problems for study graph functions :

Study linear functions graph:
If a function f   : R → R is defined in the form f(x) = dx + e then the function is called a linear function. Here d and e are constants.

Problem 1: 

Draw the graph of the linear function f : R → R defined by f(x) = 7x + 1.
Solution:
Draw the table of some pairs (x, f(x)) which satisfy f(x) = 7x + 1.

x	-2	-1	0	1	2
f(x)	15	8	1	-6	-13

Plot the points and draw the curve passing through these points. Note that, the curve is a straight line.



Study functions of graph:

Graph of a functions: The graph of a function f is a graph of the equation y = f(x)

Problem 2:
Draw the graph of the function f(x) =2x²

Solution:
Draw a table of some pairs (x, y) which satisfy y = 2x²

x	-3	-2	-1	0	-1	-2	-3
f(x)	18	8	2	0	2	8	18

Plot the points and draw the smooth curve passing through the plotted points.



Note:

Note that if we draw a vertical line to the above graph, it meets the curve at only one point
i.e. for every x there is a unique y

Example problem for study logarithmic graph functions :

Draw the graphs of the logarithmic functions (1) f(x) = log ₅2x (2) f(x) = log ₁₀ex (3) f(x) = log ₅3x
Sol:

The logarithmic function is the defined only for the positive real numbers. i.e. (0, ∞)
Domain: (0, ∞) Range: (− ∞, ∞)

Study Online Exponentiation

Exponentiation should be the operation, which is written as the form of an. Where a and n is said to be base and exponent as well as n is any positive integer. In general, exponentiation means that repetitive multiplication. Otherwise, exponentiation an is the product of n factors of a. Online study should be the topic-oriented or else technical help for clarifying doubts that are convey through the computer software. Let us study properties and example problems for exponentiation.


Properties - Study Online Exponentiation

We are having seven number of exponentiation properties that used for solving problems. In this properties, a, m and n are any integer values.

Product of like bases:

The product of powers with the same base means we can add the powers and keep the common base.

am an = am+n

Quotient of like bases:

To divide the powers with similar base, we can subtract the exponents and remain the common base.

`a^m/a^n` = am-n

Power to a power:

In case of raising the power to power, we need to keep same base and multiply the exponent values.

`(a^m)^(n)` = amn

Product to a power:

In case of raising the product to power, we need to raise each factor to the power value.

(ab)m = am bm

Quotient to a power:

In case of raising the quotient to power, we need to raise the numerator and denominator to the power value.

`(a/b)^n` = `a^n/b^n`

Zero exponent:

Any number that is raised with zero power should be equivalent to ‘1’.

a0 = 1

Negative exponent:

a-n = `1/a^n` or `1/a^(-n)` = an

These are the properties that are used for exponentiation problems in study math online.


Example Problems - Study Online Exponentiation


Example 1:

Solve 32 34.

Solution:

Given, 32 34.

This is in the structure of am an, so we need to use am an = am+n property.

Here, m = 2 and n = 4 and a = 3.

Thus, 32 34 = 32+4

= 36

= 3 × 3 × 3 × 3 × 3 × 3

= 9 × 9 × 9

= 729

Hence, the answer is 32 34 = 729.

Example 2:

Shorten the following `6^7/6^4` .

Solution:

Given, `6^7/6^4` .

This is in the structure of `a^m/a^n` , so we need to use `a^m/a^n` = am-n property.

Here, m = 7 and n = 4 and a = 6.

Thus, `6^7/6^4` = 67-3

= 64

= 6 × 6 × 6 × 6

= 36 × 36

= 1296

Hence, the answer is `6^7/6^4` = 1296.

That’s all about the study online exponentiation.

Simple Algebra Problems

Algebra is a branch of mathematics dealing with variables, equations , expressions.Here we are going to solve some of the simple algebra problems including some equations, expressions and algebraic identities.

Solve Simple algebra problems on numbers:

Ex 1: Subtract  – 4 from – 10.

Sol :
=– 10 – (– 4)

= – 10 + (additive inverse of – 4)

= – 10 + 4

= – 6 (see addition of two integers)

Ex 2: Find the product of  (a) (+ 6) × (– 5) (b) (– 12) × (+ 12) (c) (– 15) × (– 4)

Sol : 

(+ 6) × (– 5) = – 30 (positive × negative= negative)

(– 12) × (+ 12) = – 144 (negative × positive = negative) 

(– 15) × (– 4)= + 60 (negative × negative = positive) 

Ex 3: Find the sum of given algebraic expression 2x4 – 3x2 + 5x + 3 and 4x + 6x3 – 6x2 – 1.

Sol:

Using the associative and distributive property of the real numbers, we obtain (2x4 – 3x2 + 5x + 3) + (6x3 – 6x2 + 4x – 1)

= 2x4 + 6x3 – 3x2 – 6x2 + 5x + 4x + 3 – 1

= 2x4 + 6x3 – (3+6)x2 + (5+4)x + 2

= 2x4 + 6x3 – 9x2 + 9x + 2.

  The following scheme is helpful in adding two polynomials 2x4 + 0x3 – 3x2 + 5x + 3 0x4 + 6x3 – 6x2 + 4x – 1 2x4 + 6x3– 9x2 + 9x + 2 


Ex 4: Find the subtraction of given simple algebraic expression 2x3 – 3x2 – 1 from x3 + 5x2 – 4x – 6.

Sol: Using associative and distributive properties, we have (x3 + 5x2 – 4x – 6) – (2x3 – 3x2 – 1)

= x3 + 5x2 – 4x – 6 – 2x3 + 3x2 + 1

= x3 – 2x3 + 5x2 + 3x2 – 4x – 6 + 1

= (x3 – 2x3) + (5x2 + 3x2) + (–4x) + (–6+1)

= –x3 + 8x2 – 4x – 5.

The subtraction can also be performed in the following way:

Line (1): x3 + 5x2 – 4x – 6.

Line (2): 2x3 – 3x2 – 1.

Changing the sign of the polynomial in Line (2), we get Line (3): –2x3 + 3x2 + 1.

Adding the polynomial in Line (1) and Line (3), we get –x3 + 8x2 – 4x – 5 

Ex 5: Find the product of   given simple algebraic expression x3 – 2x2 – 4 and 2x2 + 3x – 1.

Sol:

(x3 – 2x2 – 4) (2x2 + 3x – 1)

= x3 (2x2 + 3x – 1) + (–2x2) (2x2 + 3x – 1) + (–4) (2x2 + 3x – 1)

= (2x5 + 3x4 – x3) + (–4x4 – 6x3 + 2x2) + (–8x2 – 12x + 4)

= 2x5 + 3x4 – x3 – 4x4 – 6x3 + 2x2 – 8x2 – 12x + 4

= 2x5 + (3x4 – 4x4) + (–x3 – 6x3) + (2x2 – 8x2) + (–12x) +4

= 2x5 – x4 – 7x3 – 6x2 – 12x + 4.


Algebra Identities to solve some simple problems :


 Algebraic identity for (x + a)(x + b)

 By using the distributive properties of numbers,

(x + a )(x + b ) = x(x + b) + a(x + b) = x2 + xb + ax + ab

= x2 + ax + bx + ab= x2 + (a + b)x + ab.

 (x + a)(x + b) ≡ x2 + (a + b)x + ab

(x – a)(x + b) ≡ x2 + (b – a)x – ab

(x + a)(x – b) ≡ x2 + (a – b)x – ab

(x – a)(x – b) ≡ x2 – (a + b)x + ab

(a + b)2 ≡ a2 + 2ab + b2

(a – b)2 ≡ a2 – 2ab + b2

(a + b)(a – b) ≡ a2 – b2

(a + b)3 ≡ a3 + 3a2b + 3ab2 + b3

(a − b)3 ≡a3−3a2b+ 3ab2−b3

a3 + b3 ≡ (a + b)3−3ab(a + b)

a3 − b3 ≡ (a −b)3+ 3ab(a−b)

Solve the following simple algebra problems using above identities:


Pro 1: Find the product:  (x + 3) (x + 5)

Sol: (x + 3) (x + 5) = x2 + (3 + 5) x + 3 × 5 = x2 + 8x + 15.


Pro 2:  Find the product:  (p + 9) (p – 2)

Sol: (p + 9) (p – 2) = p2 + (9 – 2) p – 9 × 2

= p2 + 7p – 18.


Solve simple algebra problems on the cubic identity


Algebra problems on cubic identity:

(i) (3x+2y)3 = (3x)3 + 3(3x)2(2y) + 3(3x)(2y)2 + (2y)3

= 27x3 + 3(9x2)(2y) + 3(3x)(4y2) + 8y3

= 27x3 + 54x2y + 36xy2 + 8y3.

 (ii) (2x2– 3y)3 = (2x2)3 – 3(2x2)2 (3y) + 3(2x2) (3y)2 – (3y)3

= 8x6 – 3(4x4)(3y) + 3(2x2)(9y2) – 27y3

= 8x6– 36x4y + 54x2y2 – 27y3


Pro 1: If the values of a+b and ab are 4 and 1 respectively, find the value of a3 + b3.

Solution:a3+ b3 = (a+b)3 – 3ab(a+b) = (4)3 – 3(1)(4)

= 64 – 12

= 52.


Pro 2: If a – b = 4 and ab = 2, find a3 – b3.

Solution:a3 – b3 = (a–b)3+3ab(a–b)

= (4)3+3(2)(4) =64+24

=88.

Median Test Learning

Median Definition :If the generally number of values within the case is even, next the median is the mean of the two center numbers. The median is a supportive number within cases wherever the distribution contain extremely large huge values which would or else twist the data. Represent the center value of a given set of values is called median.

To locate of values set within lower value toward higher value (rising order). If the collections of data contain an odd number of ways; the median is the middle value of the place behind arrangement the listing into rising order.


Median test learning


Method for median computation

`M=(n+1)/2`

Wherever,

M- Median,

n – Entirety number of set

But here is an odd number of values, then the median is the center value of the set.

If an even number to find median first

For example the given set{2,6,8,10}

Total number of value (n) =4

Median`M=(n+1)/2`

`M=(4+1)/2`

`M=(5)/2`

M=2.5

Consequently median must appear among second and third value.

Second value=6

Third value =8

Mean of second and third value is the Median of this set

Median`M=(14)/2`

`M=(7)/2`

M=3.5


Examples for median test learning


Example 1 for median test learning

Compute the median of this set {20, 10, 12, 40, 60,50,30}.

Solution:

First values can be arranged in order

Therefore the given set={10, 12, 20, 30, 40, 50, 60}

Here, total number of value (n) =7.

Median`M=(n+1)/2`

`M=(7+1)/2`

`M=(8)/2`

M=4

Therefore the fourth value is median that is 30.

Example 2 for median test learning

Compute the median of this set {10, 12, 13, 17, 18, and 20}

Solution:

Total number of value (n) =6

Median`M=(n+1)/2`

`M=(6+1)/2`

`M=(7)/2`

M=3.5

Consequently median must appear among Third and Fourth value.

Third value=13

Fourth value =17

Mean of third and fourth value is the Median of this set

Median`M=(13+10)/2`

`M=(30)/2`

M=15

Example 3 for median test learning
Compute the median of this set {26, 32, 36, and 55}

Solution:

Total number of value (n) =4

Median`M=(n+1)/2`

`M=(4+1)/2`

`M=(5)/2`

M=2.5

Consequently median must appear among second and third value.

Second value=32

Third value =36

Mean of second and third value is the Median of this set

Median`M=(32+36)/2`

`M=(68)/2`

M=34

Learning Weighted Median

In calculation of weighted median, the importance of all the items was considered to be equal. However, there may be situations in which all the items under considerations are not equal importance. For example, we want to find average number of marks per subject who appeared in different subjects like Mathematics, Statistics, Physics and Biology. These subjects do not have equal importance. Learning the formula to find weighted median by giving Median.


Learning Median Definition:


The arithmetic median computed by specific importance of every object is known weighted arithmetic median. To consider the every importance object, we can assume number known as weight to every object is directly proportional to its specific importance.

Weighted Arithmetic Median is computed by using following formula:

`sum` Yw = `(sum ty) / (sum t)`

Where:
Yw Stands for weighted arithmetic median.
y   Stands for values of the items and
t   Stands for weight of the item

Learning the important three methods of weighted median:

1.Arithmetic Median

2. Weighted Average

3. Average speed.

The formula for weighted average is:

Weighted Average = Sum of weighted objects / total number of objects


Learning to solve examples of Weighted median:


Ex 1:A student obtained 30, 40, 50, 60, and 35 marks in the subjects of Math, Statistics, Physics, Chemistry and Biology respectively. Assuming weights 1, 3, 5, 4, and 2 respectively for the above mentioned subjects. Find Weighted Arithmetic Mean per subject.

Solution:

Subjects	Marks Obtained       y	Weight    t	`sum` ty
Math	30	1	30
Statistics	40	3	120
Physics	50	5	250
Chemistry	60	4	240
Biology	35	2	70
Total		`sum` t = 15	`sum` y= 710

Now we will find weighted arithmetic mean as:
`sum` Yw = `( sum ty)/(sum t) = 710/ 15`   =  47.33marks/subject

Ex 2: A class of 30 students took a math test. 15 students had an average (arithmetic median) score of 90. The other students had an average score of 70. What is the average score of the whole class?

Solution:Step 1: To get the sum of weighted terms, multiply each average by the number of students that had that average and then sum them up.

90 × 15 + 70 × 15 = 1350 + 1050 = 2400

Step 2: Total number of objects = Total number of students = 30

Step 3: Using the formula

Weighted Average = Sum of Weighted objects / Total Numbers of objects.

=  `2400 / 30`

= 80.

Answer: The average score of the whole class is 80.

Learn Linear Functions in Real Life

In real life learning, linear function has single degree polynomial in its equation. The degree of the linear function always one. It cannot be increased. Linear function is  very useful in real life. In learning, linear function does not dependent any variable degrees. The linear function with the degree of one, always gives the straight line. In real life, linear function has the different variable equations. 'In learning of linear function in real life', we learn about linear equations and its degrees.

Forms of Linear function in real life


The function, which has straight line graph and that line function is known as linear function. In learning, linear functions are in three main forms.

1. Slope-Intercept Form is given by y = mx +b.

Example: y = 3x + 2

Here, slope m = 3 and y intercept b = 2

2. Point Slope Form is given by m = (y - y1) / ( x – x1)

Example: (y – 3) = 2(x – 1)

3. General Form is given by ax + by + c = 0

Example: 2x + 3y -1 = 0

Examples of set of real life linear function:

Example 1:

Solve for x and y. where, y = 3x + 4 and y = 7x.

Solution:

Plug, y = 3x + 4 in y = 7x

3x + 4 = 7x

x = 1

Now plug, x = 1 in y = 3x + 4

y = 3 + 4

y = 2

Answer:

The solutions are x = 1 and y = 2.

Example 2:

5x - y + 11 = 0 and x - y - 5 = 0, solve for x and y.

Solution:

5x - y + 11 = 0     (1)

x - y - 5 = 0         (2)

Subtract Equation (2) form Equation (1), we get,

4x + 16 = 0

4x = - 16

x = - 4

plug x = - 4 in equation (1)

- 4 – y + 13 = 0

- y + 9 = 0

y = - 9

The solutions are x = - 4 and y = - 9.


Learn linear function in real life - Practice problems:


Problem 1:

3x - y = 9 and 4x + y = 5. Solve for x, y.

A) (2, 3) B) (- 2, 3) C) (2, -3) D) (3, -2)

Answer: C

Problem 2:

2x + y = 12 and y = 3x - 3. Solve for x, y.

A) (3, 6) B) (3, - 6) C) (4, 5) D) (6, - 3)

Answer: A

Problem 3:

Solve the linear function 2x - y = 12 and y = 4x - 2

A) (- 5, 22) B) (5, - 22) C) (- 5, - 22) D) (5, 22)

Answer: C