Perfect Squares Chart

Perfect squares chart :

 Perfect squares are defined as expressions or numbers that can be factored into two equal expressions or numbers.

For example,

x² + 2xy + y² = ( x + y) * ( x + y) , so  (x² + 2xy +  y²) is a perfect square of (x + y)

4 = 2*2 , so 4 is a perfect square of 2.

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Properties of Perfect Squares Chart

All perfect squares are non-negative ( at least in the real numbers' set).

• Square of 0 is 0.
• Squares of perfect numbers are the same as the squares of their positive counterparts,  

 For example           -1*-1 = 1*1 = 1,  (-2)²= 2²= 4

Difference between two perfect squares

Difference between a perfect square and the next perfect square is given by :

Square of a number n = n²

The next perfect square is square of the next number, that is, square of n + 1, so ( n + 1) ² = n² + 2n + 1

Difference between the two numbers =  2n + 1

So, if n² is a perfect square, the next perfect square will be n² + ( 2n + 1).

Example : 

100 = 10² is a perfect square.

Here, n = 10

So, the next perfect square will be 100+ 2*10 + 1 = 121 = 11²

Uses of Perfect Squares Chart

Included below is a chart of perfect squares of numbers from 1 to 25

Number n	Number Square n2
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625

Remember: 

• A square number can only end with digits 00,1,4,6,9, or 25
• Squares of even numbers are even, since (2n)² = 4n².
• Squares of odd numbers are odd, since (2n + 1)² = 4(n² + n) + 1.
• It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd.

Uses of Perfect Squares :

Squaring is used in statistics in determining the standard deviation of a set of values. The deviation of each value  from the mean  of the set is defined as the difference . These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the variance and its square root is the standard deviation. In finance, the volatility of a financial instrument is the standard deviation of its values.

Triangle with One Obtuse Angle

Introduction for obtuse angle:

Let us discuss about one obtuse angle triangle.

The measurement of angle between 90° and less 180° is called as an obtuse angle. The angles 95°, 100°, 130°, 162° and 178° are some examples of obtuse angles.

From the law of cosines, it can be given as,

Cos C = a^2+b2-c2/2ab

For an angle to be obtuse, C < 0. Therefore, an obtuse angle which satisfies one of the following a^2+b2
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Surface Area Equation for One Obtuse Angled Triangle:

The triangle with one obtuse angle which can be given as,

A = bh/2 = b/2 √ [a^2-{(c2+a^2+b2)/2b}2

If S = 1/2 (a+b+c),then

A = √ [s(S-a)(S-b)(S-c)].

Problems for Triangle with One Obtuse Angle:
Example1:

In triangle with one obtuse angle, a right angled triangle has one other angle that is 45º. What is the measurement of the third angle?

Solution:

In the triangle with one obtuse angle, A right triangle angle = 90°. Sum of the angles which are known and to be added 90° + 45º = 135°.

The sum of all the angles in a given triangle is 180º. Subtract the sum of known angles from 180°.                    180° – 135° = 45°

The size of the third angle is 45°.

Example2:

Whether it is possible for a triangle in which it consists of an obtuse angle or more?

Solution:

Let the angles of the triangle be f,g and h . Let f be the obtuse angle.

The sum of all the angles in any triangle is 180º. f + g + h = 180º.

If f > 90º then g + h must be less than 90º. Therefore, g and h must be acute angles.

So therefore no triangle with one obtuse angle is present.

Example3:

In triangle with one obtuse angle,if an Obtuse angled triangle has sides 2 units, 4 units and X units. Then how many such triangles exist?

Solution:

For the triangle with one obtuse angle which can be given as follows,

If x^2+2^2=4^2, this implies that its a right angled triangle. Therefore for the triangle to be Obtuse angled triangle, x^2+ 2^2<4 8.="8." a="a" be="be" br="br" can="can" condition="condition" fulfill="fulfill" if="if" less="less" of="of" ow="ow" satisfied="satisfied" sides="sides" sum="sum" than="than" the="the" this="this" to="to" total="total" triangle="triangle" would="would" x="x">
If 2^2 + 4^2 = x^2

x = 4, it is a right angled triangle.

So, x > 4.So, 4

Area of a circle

Introduction to area of a circle:
Area is the measure of surface occupied by an object. The standard unit for measurement of area is metres quare (m2).However the area of smaller dimensions can be expressed in mm2 or cm2. The areas of large dimensions can be expressed in acre or hectare.

We can find the area of a circle by using the formula as, A=πr². Here r is the radius of a circle.

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Formula of Area of a Circle:
Formula:We can find the area of a circle by using the formula as,

        Area A=pr2

Here r is represents the radius of the circle and then the value of p is `22/7`  or 3.14. The radius is squared in an area of circle formula. A distance between the center and any point of a circle is called radius.

A distance surrounding on a circle is called circumference and diameter of a circle is defined as a distance across a circle via the center of a circle.

Steps: How to calculate the Area of circle

 Find out the radius of a circle from a given problem.
 We can apply the formula for Area of circle.
         Area A= pr2

 Multiplying the square of the radius and the value of p.
 Finally we get the solution.

Examples of Area of a Circle:

Ex 1: Find the area of circle with radius equal to 6.5 inch using p = 3.14

Solution:The Area of circle as, A=pr2

Substitute the values of radius and p.

Therefore, A= (3.14 * 6.5*6.5)

Squaring the radius and then multiplying with 3.14 and then we get,

Area=3.14 * 42.25                                                

Area=132.67 square inch.

Ex 2: Find the Area of circle with the radius of 10 feet with taken the value of p = 3.14.

Solution: Formula for Area of circle, A=pr2

Substitute the values of radius and p.

A=3.14*10*10

Squaring the radius and then multiplying with 3.14 and then we get,

Area=3.14*100

Area=314 square feet.

Binomial Random Variable

Introduction to binomial random variable:

Binomial Random variables help us to make a link between probability and numbers that we observe as data.

Binomial Random Variable: A numerical valued function defined on a sample space. A random variable X “maps” an outcome in a sample space to a numerical value. We use capital letters such as X or Y to denote binomial random variables. Let s be an elementary outcome. A value, X(s), of X is denoted x.

A binomial random variable is discrete if it can take on a finite or countable number of values. A continuous random variable takes on an uncountable number of values.

Binomial Random Variable:

General features of a Binomial Random Variable:

The binomial random experiment consists of n identical trials.
Possible outcomes on each trial are of only two. given one outcome by S (for Success) and the other by F (for Failure).

The probability for Success(S) remains the same from trial to trial. Denote it by p=Pr(Success).

The trials of binomial random experiment are independent of each other.

Then we say  Binomial Random Variable X as the number of Successes in n trials.

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Binomial Random Variable:

Ex 1:  What is the probability we roll less than a 5?

Sol :              P(X < 5) = P(X = 2) + P(X = 3) + P(X = 4)

                              = 1/36 + 2/36 + 3/36

                              = 6/36

                  the probability we roll less than a 5  = 1/6

Ex 2 :  What is the probability we roll a number between 7 and 10  (inclusive)?

Sol :                              P(7 `<=` X `<=` 10) = P(X = 7) + P(X = 8) + .......

                                       = 6/36 + 5/36 + 4/36 + 3/36

                                       = 18/36

         the probability we roll a number between 7 and 10 (inclusive) = 1/2

Ex 3 :  What is the probability the sum of two dice will be odd?

Sol :              P(X odd) = P(X = 3) + P(X = 5) + ..........

                     = 2/36 + 4/36 + 6/36 + 4/36 + 2/36

                     = 18/36

              the probability the sum of two dice will be odd = 1/2

Ex 4:  What is the probability we roll a number between 8 and 12 ?

Sol :                            P(8 < X <12 10="10" 11="11" 9="9" br="br" p="p">
                                     = 8/36 + 9/36 + 10/36

                                     = 27/36

         the probability we roll a number between 8 and 12 = 3/4

Introduction to algebra of parallel lines

Introduction to algebra  of parallel lines:

Definition of parallel lines:

The algebra parallel lines do not intersect each other.
Standard form of equation of a line is
                      y = mx + c

           Y = m1 x + c ---- equation of line 1

           Y = m2 x + c ---- equation of line 2

           M1 = slope of line 1

           M2 = slope of line 2

  Slope of lines must be same for the lines to be parallel.
   The condition of a line to be parallel m1 = m2

Concept of Parallel Lines :

 The two distinct lines have a common point, then the lines are called intersecting lines. If the two distinct lines in a plane have no point in common that is intersecting, then the two lines are called non-intersecting lines. Two non-intersecting lines are also called as parallel lines.

Algebra Parallel Lines:examples

1. Y = 4x + 64, 4y = 6x + 1 identify the two lines are parallel or not.

Solution:Condition for the two lines to be parallel the slope of lines must be same

The standard form of the equation of line is   

                              Y = mx + c

By comparing the first equation with the standard form, we get slope of line1 (m1) = 4.

By comparing the equation 2 with the standard form we get the slope of the line2 (m2) = 3/2.

 As per the condition for parallel line m1 must be equal to m2.Here the m1 is not equal to m2.

The two lines are not parallel. The slopes of two lines are not same so the two lines are not parallel lines.

2.Y = 4x + 6, 4y = 16x + 12 identify the two lines are parallel or not.

Solution:Condition if the two lines to be parallel line the slope of lines must be same or equal

The standard form of the equation of line is

                             Y = mx + c

By comparing the first equation with the standard form, we get slope of line1 (m1) = 4.

By comparing the equation 2 with the standard form we get the slope of line1 (m2) = 4

As per the condition for parallel line we get it as m1 = m2.

                                   4 = 4

The slopes of two lines are same so the two lines are parallel lines.

Introduction to prism definition for kids

Introduction to prism definition for kids:-

               The articles generally explain the basics of prism definition based topic and problems also. This article help the kids based diagram and simple kids’ problems shown here.   A prism has two parallel sides with congruent lengths. Those 2 parallel sides form the basis of the prism. Prism is a 3D purpose. Prism is confidential as based on its base shape.

Prism Definition for Kids Types and Formulas:

Types of prism for kids:

Types of prism:

• Rectangular prism
• Triangular prism

• Pentagonal prism

Basic formula on prism definition for kids:- 

Rectangular prism for kids:-

       Rectangular prism = length * width * height

Triangular prism for kids:-

       Triangular prism = (1/ 2) * base * height * length

Pentagonal prism for kids:-

      Pentagonal prism = area of base * height

Example Problems - Prism Definition for Kids:-

Prism definition for kid’s problem 1:

         Get the volumes of the place rectangular prism with the length is 9 cm, width is 30 cm, and height is 10 cm.

Solution:

           Given, length (l) = 9 cm, width (w) = 30 cm, height (h) = 10 cm.

           Formula:

                   Rectangular prism = length * width * height

           Substitute the specified values in the above equation, we get

                                Volume = 9 * 30 * 10 cm³

                                               = 2700 cm³

           The volume of the rectangular prism is 2700 cm³

Prism definition for kid’s problem 2:-

            The base and length of the triangular prism is 12 cm and 26 cm. The height of the triangular prism is 18 cm. get the volume of the triangular prism.

Solution:

            Given, base (b) = 12 cm, length (l) = 26 cm, height (h) = 18 cm

             Formula:

                       Triangular prism = (1 / 2) * base * height * length

            Substitute the given values in the above equation, we get

                                                                   = (1 / 2) * 12 * 26 * 18 cm³

                                                                   = 2808 cm³

            The volume of the triangular prism is 2808 cm³

Prism definition for kid’s problem 3:-

            Area of the base of the pentagonal prism is 20 cm² and height of the pentagonal prism is 12 cm.

Solution:-

            Given, Area of base = 20 cm², height = 12 cm.

            Formula:

               Pentagonal prism = area of base * height

            Substitute the given values in the above formula, we get

                              Volume = 20 * 12 cm³

                                             = 240 cm³

             The volume of the pentagonal prism is 240 cm³

Prism definition for kid’s problem 4:-

            The base and length of the triangular prism is 12 cm and 4 cm. The height of the triangular prism is 7 cm. get the volume of the triangular prism.

Solution:

            Given, base (b) =12 cm, length (l) = 4 cm, height (h) = 7 cm

           Formula:

                    Triangular prism = (1 / 2) * base * height * length

           Substitute the specified values in the above equation, we get

                                             = (1 / 2) * 12 * 4 * 7 cm³

                                             = 168 cm³

           The volume of the triangular prism is 168 cm³

Introduction to equivalent ratios calculator

Introduction to equivalent ratios calculator:

               Let we discuss about calculator of equivalent ratios. Ratio can be used to portraying how two quantities will relate. For example, we will say that apple squash can be mixed with water in ratio 2:5. That means, for each 2 part squash, there will want to add 5 parts of water. If two ratios should have same value when they are simplified in calculation. These ratios are known as Equivalent Ratios Calculator.

More about Equivalent Ratios:

Calculator of equivalent ratios:

• Ratio of apple squash to water in the above example above is 2:5.
• We can write this ratios as but this could be written as 200:500, or 20:50, or 4:10.
• These are the equivalent ratios since the ratios are having the similar meaning.
• Any of the ratio, the meaning is "the amount of water should be equal to the six times amount of squash".

Examples of Equivalent Ratios Calculator:

• 1 / 3 and 3 / 6 both are equivalent ratios. Because they are representing same fraction number.
• Two ratios 6 :42 and 1 : 7 are the equivalent ratios.
• There are 9 dolls for every 54 children in a school. Therefore, ratio of the number of children to the dollars = 54:9 = 6:1.

Example Problems on Equivalent Ratios:

Example 1:

Calculate 120 : 360 in simplest form.

Solution:

Given, 120:360

Dividing of both sides by 4 =>  30:90

Dividing of both sides by 3 => 10:30

Dividing of both sides by 2 => 5:15

Dividing both sides in ration by 5, then we get, 1:3

Therefore, simplest form of 120:360 is 1:3

Example 2:

Cricket game has 2 winners for every 5 losers. Find three equivalent ratio for winners to losers.

Solution:

Ratio of winner to loser = 2 / 5

1. Multiplying 2 in both numerator and also denominator of  ratio 2 / 5

              Ratio will be, 4 / 10

2.Mutiplying 3 in both numerator and denominator of ratio 2 / 5

              Ratio will be, 6 / 15

3.Mutiplying 4 in both numerator and denominator of the ratio 2 / 5

              Then the ratio will be, 8 / 20

Answer:

                Equivalent ratios of 2 / 5 are 4 / 10 , 6 / 15 and 8 / 20