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.