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