How to Solve Anti Derivatives

In mathematics, anti derivative is also called as integration process. Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function f. In this case it is called an indefinite integral. Integral can be classified as definite and indefinite integral. (Source: Wikipedia)

General formula for integration:

∫ `(x^n)` = `(x^(n + 1) / (n + 1))` + c

Example problems for how to solve anti derivatives

Anti derivative example problem 1:

Solve:

Integrate the given function ∫ (14.3x4 + 31.7x3 - 6x) dx.

Solution:

Given ∫ (14.3x4 + 31.7x3 - 6x) dx

Integrate the given function with respect to x, we get

∫ (14.3x4 + 31.7x3 - 6x) dx = ∫ 14.3x4 dx + ∫ 31.7x3 dx - ∫ 6x dx.

= 14.3 (x5 / 5) + 31.7 (x4 / 4) - 6 (x2 / 2) + c.

= `(14.3 / 5)` x5 + `(31.7 / 4)` x4 - 3 x2 + c.

Answer:

The final answer is  `(14.3 / 5)` x5 + `(31.7 / 4)` x4 - 3 x2 + c.

Anti derivative example problem 2`:`

Solve:

Find the value of the integration

`int_0^3(x^10)dx`

Solution:

Integrate the given function with respect to x, we get

`int_0^3(x^10)dx`  = `(x^11 / 11)`30

Substitute the lower and upper limits, we get

= `((3^11 / 11) - (0^11/ 11))`

= `((177147 / 11) - (0 / 11))`

= `(177147 / 11)`

Answer:

The final answer is `(177147 / 11)`

Anti derivative example problem 3:

Solve:

Integration using algebraic rational function ∫ `(15 dx) / (8x + 21)`

Solution:

Using integrable function method,

Given function is ∫ `(15dx) / (8x + 21)`

Formula:

∫ [L / (ax + c)] dx = (L / a) log (ax + c)

From given, L = 15, a = 8, and c = 21

Integrate the given equation with respect to x, we get

= `(15 / 8)` log (8x + 21)

Answer:

The final answer is `(15 / 8)` log (8x + 21).

Practice problems for how to solve anti derivatives

Anti derivative practice problem 1:

Solve:

Integrate the given function using integrable function ∫ `(2 / (12x + 103))` dx

Answer:

The final answer is `(1 / 6)` log (12x +103)

Anti derivative practice problem 2:

Solve:

Integrate the given function using integrable function ∫ `(27 / (21x + 48))` dx

Answer:

The final answer is `(9 / 7)` log (21x + 48)

Algebra is widely used in day to day activities watch out for my forthcoming posts on 3x3 Matrix Multiplication and cbse class 10. I am sure they will be helpful.

Anti derivative practice problem 3:

Solve:

Integrate the given function ∫ (27.39x3 - 17.132x) dx

Answer:

The final answer is `(27.39 / 4)` x4 - `(17.132 / 2)` x2 + c

Reading and Decimal Writing

The numbers left to the decimal point is known as whole number and right to the decimal point is called decimal number. The number in the digit has place value. Depending upon the place value we can read and write the decimals.

For example: 3456.89 Here 3456 is the whole numbers and .89 is the decimal numbers. Let us see about place value of the decimal used for read and write the decimal.

Understanding Decimal Symbol is always challenging for me but thanks to all math help websites to help me out. 

Place Value – Reading and Decimal Writing:

Let take the example decimals 35926.94843 denotes the place value of the following table.

Place value	Decimals Number
Ten thousands	3
Thousands	5
Hundreds	9
Tens	2
ones	6
Decimal point	.
Tenths	9
Hundredths	4
Thousandths	8
Ten thousandths	4
Hundred thousandths	3

Example Problems – Reading and Decimal Writing:

Example 1:
Write the place value of the simple decimals of this 6978.4254
Solution:
6 x 1000 thousands
9 x 100 hundreds
7 x 10 tens
8 x 1 ones
4 x 0.1 tenths
2 x 0.01 hundredths
5 x 0.001 thousandths
4 x 0.0001 ten-thousandths

Example 2:
Convert this mixed fraction of 39 49/100 to decimals and how to read and the decimal form.

Solution:

Mixed Number	Expanded Form		Writing Decimal form	Reading  the decimal (phrase form)
89 `(43)/(100)`	(8 * 10) + 9 * 1	4* (`(1)/(10)` )+  3 * (`(1)/(100)`)	89.43	Thirty nine and four tenths nine hundredths
	Whole number part	Fractional part

Example 3:
How to read and write the following fraction to decimal form?
  `(74)/(100)`, `(87)/(1000)` and `(65)/(10)`

Solution:

Fraction  Number	Reading decimal		Writing Decimal form
`(74)/(100)`	Seven tenths	Four hundredth	0.74
`(87)/(1000)`	Eight hundredth	Seven thousandth	0.087
`(65)/(10)`	Six ones	Five tenth	6.5

Practice Problem – Reading and Decimal Writing:

1.  How to read the decimal form 33. 67?
Answer: Thirty three and sixty seven hundredth
2. Write in decimal form of the phrase seven hundred fifty three and four hundred and forty three thousandths.
Answer: 753.443

Median of Three Numbers

In statistics, the median is the middle value of the set of data after the arrangement of ascending or descending order. If the total of elements in a set is even, median is arithmetic mean of two middle numbers (n and n+1). Divide total number of elements by 2 to get n. In this lesson we will discuss about how to find the median of three numbers.

Median of Three Numbers – Example Problems

Example 1: In a class 3 students’ marks as follows 76, 71, and 65. Find the median of given data.

Solution:

Arrange the numbers in ascending order {65, 71, 76}

Median:

{65, 71, 76}

Total number of elements in a data set is 3 which is an odd number. So find the middle value which is the median.

Here, 71 is the middle value.

Therefore 71 is the median.

Example 2: Find the median of three numbers 69, 72, and 87.

Solution:

Arrange the numbers in ascending order {69, 72, 87}

Median:

{69, 72, 87}

Total number of elements in a data set is 3 which is an odd number. So find the middle value.

Here 72 is the middle value.

Therefore median of given three numbers is 72.

Example 3: Three students’ height in a class as follows 158, 178, and 184. What is the median?

Solution:

Arrange the numbers in ascending order {158, 178, 184}.

Median:

{158, 178, 184}

Total number of elements in a data set is 3 which is an odd number. So find the middle value.

Here, 178 is the middle value.

Therefore 178 is the median.

Example 4: Three students’ weight in a class as follows 88, 75, and 84. What is the median?

Solution:

Arrange the numbers in ascending order {75, 84, 88}.

Median:

{75, 84, 88}


Total number of elements in a data set is 3 which is an odd number. So find the middle value.

Here, 84 is the middle value.

Therefore 84 is the median.

Median of Three Numbers – Practice Problems

Problem 1: Find the median of three numbers 145, 78, and 256.

Problem 2: Three students’ height in a class as follows 192, 178, and 182. Find the median.

Answer: 1) 145 2) 178

Binomial Nomenclature List

Introduction to Binomial nomenclature

The species concept: Species is recognized as the smallest unit of taxonomy or the basic unit of classification. In simpler form every different and recognizable type of organism (animal, plant or microbe) is a different species. In scientific terms, a species could be defined as a group of closely resembling organisms of the same type, in which members can interbreed freely and naturally to produce viable and fertile offspring, whereas they cannot interbreed freely or produce viable and fertile offspring with members outside the group.

Binomial nomenclature

Binomial nomenclature is the scientific method of giving names to every species of living organism. Carolos Linnaeus first introduced it in 1758. In this method every species of organism is given a name consisting of two words. The first word would refer to the name of the genus to which the organism belongs in the existing system of classification. Therefore this is known as the generic name. The second word in the name is called the specific name, since it refers to the exact species to which the organism belongs.

Rules to be followed in giving scientific names to plants and animals on the basis of binomial nomenclature

• The name must consist of two words both in Latin and in other languages in Latinized form.
• The first word should refer to the genus to which the organism belongs and it should start with capital letter.
• The second word should refer to the species to which the organism belongs and it starts with a small letter.
• Names derived by binomial nomenclature should also include reference in abbreviation to the name this is known as author citation.
• The generic and specific names when printed must be in italics and when written otherwise must be separately underlined, indicating their Latin origin. Author citation is not in italics nor is it underlined.
• When the same organism is given more than one scientific name by different authors, the earlier name becomes accepted.
• These rules are framed and presented as International Code for botanical Nomenclature (ICBN) for plants and as International Code for Zoological Nomenclature (ICZN) for animals. They are periodically reviewed and revised by specific committees.
  
  

Advantages of binomial nomenclature

• Names derived from binomial nomenclature gives universal reorganization to the organism.
• Scientific name avoid confusion arising out of the same organism having different common names in different origins of the world.e.g. For brinjal in India (Solanum melongena) is called egg plant in western countries.
• Using binomial nomenclature system automatically establishes taxonomic interrelationships between different taxa.
• Addition of new information, acquiring rare information, exchange of resource materials, and exchange of information etc., become easy when universally accepted scientific names are used.

Geometry Solid Figures

Geometry solid figures are a branch of geometry which mainly focuses on the properties like surface area, and volume of solid figures or three dimensional shapes like Rectangular Prism, cube, cone, cylinder, and sphere. Geometry solid figures generated by the revolution of a two dimensional plane. For example a cylinder is generating by the revolution of a rectangle; the revolution of a circle about its diameter generates a sphere.

Formulas for geometry solid figures:-

Geometry solid figures of Cube:-


Surface area = 6a2 square units, Volume = a3 cubic units.

Geometry solid figures of Cylinder:-


Lateral surface area = 2πrh square units, Total surface area = 2πr (h + r) square units, Volume = πr2h cubic units

Geometry solid figures of Cone:-


Lateral surface area = πrs square units, Total surface area = πrs + πr2 square units, Volume = 1/3 πr2h cubic units

Geometry solid figures of Sphere:-


Surface area of a sphere = 4πr2 square units, Volume = 4/3 πr3 cubic units

Geometry solid figures of Rectangular Prism:-


The equation for find the volume (V) of a rectangular prism is V = wdh.

Example geometry solid figures problems:-

Example problem1:-

Find the volume of cylinder given the radius is 6 cm and 11 cm.

Solution:-

Volume of cylinder = r2 h cubic units.

= (3.14) * 62 * 11

= 3.14 * 36 * 12

= 1 356.48cm3

Example problem2:-

Find the volume of cone given the radius is 4 cm and 8 cm.

Solution:-

Volume of cone = 1/3 r2 h cubic units.

= 1/3 (3.14) * 42 * 8

= 0.33 * 3.14 * 16 * 8

= 132.6336cm3

Example problem3:-

Find the volume of cube with the side length of 8 cm.

Solution:-

Volume of cube = a3

= 83

= 512 cm3

Example problem4:-

A rectangular prism contains at width for 3 inches and depth 6 inches and height 4 inches then finds the volume of rectangular prism?

The faces are known as width (w), depth (d), and height (h).

Solution:-

Volume (V) of a rectangular prism is:

V = wdh.

Now we can calculate,

V= wdh

V= 3 in x 6 in x 4 in

V = 72 in3

Example problem5:-

The sphere has the radius of 4m.find the surface area of the sphere.

Solution:-

Radius (r) = 4 m

Surface area of the sphere = 4 π r2 square unit

= 4 x 3.14 x (4)2

=4 x 3.14 x 16

= 200

Surface area of the sphere =200 m2.

How to Reduce Binomials

Binomial is also an algebraic equation but it has only two terms. (a+b) It’s  a example of binomial. And the binomial theorem commonly used to expand the binomials to any given power without direct multiplication .Here we are going to learn about how to reducing binomials. Let us see about binomial with examples.

More information about binomials

The following binomial shows, how we can write the binomials using powers
(a + b)⁰ is equal to  1
(a + b)¹ can be written as  a + b
(a + b)² can be written as  a² + 2ab + b²
(a + b)³ can be written as a³ + 3a²b + 3ab² + b³
(a + b)⁴ can be written as  a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
(a + b)⁵ can be written as  a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Example problems for how to reduce binomials:

Example 1:
Reduce  `((x+2)^2)/(x+2)`
Solution:

Step 1: We can write it as an .`((x+2)^2)/(x+2)`

Step 2: And the question can be written as  `((x+2)(x+2))/(x+2)`

Step 3: On both the numerator and denominator, the (x+2) will be cancelled.

Step 4: Therefore, the answer is (x+2).

Example 2:
Reduce 2ab+5yz-ab-3yz+9ab.
Solution:

Step 1: First we need to arrange the terms.

Step 2: So, 2ab-ab+9ab+5yz-3yz.

Step 3: Here, we need to add the terms.

Step 4: Therefore the answer is 10ab+2yz.

Example 3:
Reduce  `((x+7) (x+9))/((x+81)^2 (x+49)^2)`
Solution:

Step 1: It can be written as  `((x+7) (x+9))/((x+81)^2 (x+49)^2)`

Step 2: And, we can write the given question is `((x+7)(x+9))/((x+9)(x+9)(x+7)(x+7))`

Step 3: And Here both the sides the (x+7) and (x+9 ) are canceled.

Step 4: Therefore the answer  is`(1)/((x+9)(x+7))`

Example 4:
Reduce 4ts+6gh-ts-2gh+7ts.
Solution:

Step 1: First we need to arrange the terms.

Step 2: So, 4ts+7ts-ts+6gh-2gh.

Step 3: Here, we need to add the terms.

Step 4: Therefore the answer is 10ts+4gh

Example 5:
Reduce:  `((a+b)^3 (5a))/((a+b) (25))`
Solution:

Step 1: The question can be written as `((a+b)(a+b)(a+b)(5a))/((a+b) (25))`

Step 2: On both the numerator and denominator, the (a+b) and 5 will be canceled.

Step 3: Therefore, the answer is `((a+b)^2 (a))/(5)`

These are the examples of how to reducing binomials.

Scale Factor Triangle

A scale area is a digit to scales otherwise multiplies, a few number. In the equation y=Cx, C represents the scale factor x. C is as well as represents the coefficient of x with might be recognized the stable of proportionality of y toward x. For illustration, doubling-up distance communicate to a scale factor of 2 meant for distance, as unkind a block in half consequences in part by a scale factor of ½.


Definition of scale factor triangle:

The ratio of any two corresponding lengths in two similar geometric figures is called as Scale Factor.
The ratio of the length of the scale drawing to the corresponding length of the actual object is

called as Scale Factor.   (source:wikipedia)


More about Scale Factor:

Scale factor triangle is an item that is applies to calculate the size of the entity. The conditions scale factor is utilize to declare the proportions of the geometric form. In geometry, the expression scale factor is frequently used to modify the proportions of the image. Some shapes aspect knows how to be improved otherwise reduce. In online, scholar knows how to study regarding different subject. In online, scholar is able to study on scale factor obviously. Online knowledge is appealing with interactive. Furthermore online knowledge is dissimilar as of group area education in dissimilar method. In online, scholar contain single to ones knowledge.

A scale factor triangle is a form applies as a form through that a further shape is develops in scale.

A scale factor is innovative to scale form in 1 toward 3 sizes.

Scale factor know how to be recognized through the subsequent situation

Examples problem for scale factor triangle:

Example 1:

find the image point p(3,4) order dilation with center(0,0) with scale factor of 3.

Solution:

Step1:   To find the point on the coordinate plane under dilation with center as origin multiply the coordinates with scale     factor.

Step 2:   The points of P(3,4) is P'(3*3, 4*3)

Step 3:   The value is P'(9,12)


Example 2:

find the image point p(2,6) order dilation with center(0,0) with scale factor of 2.

Solution:

Step1:   To find the point on the coordinate plane under dilation with center as origin multiply the coordinates with scale factor.

Step 2:   The points of P(2,6) is P'(2*2, 6*2)

Step 3:   The value is P'(4,12)

Learn Quadrilaterals Pictures

The quadrilaterals are the two dimensional plane shapes which have four surfaces. Therefore it is a polygon with four sides. The sum of interior angle of a quadrilaterals are 360 degrees. There are two diagonals for each quadrilaterals. Since we have two diagonals in a quadrilaterals, both the diagonals will intersect at a common point. The regular quadrilaterals will have four edges and four vertices’. The lines of symmetry of a quadrilaterals are based on what sort of the quadrilaterals are. Let us learn about various quadrilaterals pictures.

Learn Pictures of a quadrilaterals 1:

Learn picture of Square:

• Square has 4 equal sides
• It has 4 equal angles
• Each angle of a square is a right angle
• It has 4 lines of symmetry
• Square is a regular shape

Learn Picture of a Rectangle:

 

• Rectangle has 2 pairs of equal sides
• It has 4 equal angles
• Each angle of a rectangle is a right angle
• It has 2 lines of symmetry
• Rectangle is an irregular shape

Learn Picture of a Parallelogram:

• Parallelogram has 2 pairs of equal sides
• It has 2 pairs of equal angles
• Opposite sides of a parallelogram are parallel
• It has NO lines of symmetry
• Parallelogram is an irregular shape

Learn Pictures of a quadrilaterals 2:

Learn picture of  Rhombus:

• Rhombus has 4 equal sides
• It has 2 pairs of equal angles
• Opposite sides of a rhombus are parallel
• It has 2 lines of symmetry
• Rhombus is an irregular shape

Learn Picture of a Trapezium:

 

• Trapezium has unequal sides
• One pair of opposite sides are parallel for a trapezium
• It is usually has NO lines of symmetry
• Trapezium is an irregular shape

Learn Picture of a Kite:

 

• Kite has 2 pairs of equal sides
• It has 1 pair of equal angles
• Equal sides of a kite are adjacent
• Kite has 1 line of symmetry
• Kite is an irregular shape

Chart Composite Numbers

It is an integer that has more than one prime factors. They can be expressed as unique set of prime numbers. The first composite number is 4. Besides 1 every other number is either prime or composite number. Besides 1 each other natural number is either prime or composite number. In chart composite number all the numbers will the composite numbers. so that we can find the composite numbers easily by seeing the chart.

Chart composite number upto 200:

4	50	91	130	170
6	51	92	132	171
8	52	93	133	172
9	54	94	134	174
10	55	95	135	175
12	56	96	136	179
14	57	98	138	177
15	58	99	140	178
16	60	100	141	180
18	62	102	142	182
20	63	104	143	183
21	64	105	144	184
22	65	106	145	185
24	66	108	146	186
25	68	110	147	187
26	69	111	148	189
27	70	112	150	190
28	72	114	152	192
30	74	115	153	194
32	75	116	154	195
33	76	117	155	196
34	77	118	156	199
35	78	119	158	200
36	80	120	159
38	81	121	160
39	82	122	161
40	84	123	162
42	85	124	164
45	86	125	165
46	87	126	166
48	88	128	168
49	90	129	169

Chart composite number upto 400:

201	243	288	329	370
202	244	289	330	371
203	245	290	332	372
204	246	291	333	374
205	247	292	334	375
206	248	294	335	376
207	250	295	336	377
208	252	296	338	378
209	253	297	339	380
210	254	298	340	381
212	255	299	341	382
213	256	300	342	384
214	258	301	343	385
215	259	302	344	386
216	260	303	345	387
217	261	304	346	388
218	262	305	348	390
219	264	306	350	391
220	265	306	351	392
221	266	309	352	393
222	267	310	354	394
224	268	312	355	395
225	270	314	356	396
226	272	315	357	398
228	273	318	358	399
230	274	319	360	400
231	275	320	361
232	276	321	362
234	277	322	363
235	280	323	364
236	282	324	365
237	284	325	366
238	285	326	367
240	286	327	368
242	287	328	369

90 Degree Right Angle

In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.
A right triangle or right-angled triangle is a triangle in which one angle is a right angle (that is, a 90 degree angle).



The side opposite the right angle is called the hypotenuse (side c in the figure above). The sides adjacent to the right angle are called legs. (Source: from Wikipedia).It is shown in the figure.
Now, we are going to see some of the problems involving 90 degree right angle. From these problems, we can get clear view about 90 degree right angle.

90 degree right angle problems:

Example problem 1:
Using the diagram shown, Find the length of hypotenuse.

Given:

Right angle triangle

 a = 5cm and b = 5cm

Use the Pythagorean Theorem to find c.

c² = a² + b²

Substitute a = 5 and b = 5.

c² = 5² + 5²

c²=25+25

c²=50

Find the square root of each side.

C= √50=5 √2cm

Additional problems in 90 degree right angle:

Example problem 2:
Determine the length of the hypotenuse in the right angle triangle, given that angle θ = 37 degree.

Solution:

Sin θ= opposite side / hypotenuse

Sin 37 = 12 / x

0.6 = 12 / x

x = 12 / 0.6

x = 20

So, the side AC is 20cm.

Example problem 3:
Determine the length of the side x in the diagram, given that angle θ = 60 degree.

Solution:

Here, we use the sin θ. because the sin θ is related to the opposite side and hypotenuse.

Sin θ= opposite side / hypotenuse

Sin 60 = x / 24

0.866 = x / 24

x = 24* 0.866

x = 20.8

So, the side AC is 20.8cm.

Practice problems in right angle:
1) Find the length of hypotenuse when the adjacent side is 5 and the opposite side is 12. (Answer: hypotenuse=13)
2) Determine the length of the opposite side, given that angle θ = 30 degree and hypotenuse is 24. (Answer: opposite side= 12).

Alternate Segment Theorem

Given a circle and two points P and Q on its boundary and a departure to the circle at Q, then angle between the departure and the line PQ is the same as the angle subtended by this chord in segment of the circle on the opposite side of PQ.  Alternate segment theorem states that approach involving a departure and its chord is equal to angle in the alternate segment.Interactive animatronics representatives proof of an alternate segment theorem.

Theorem for alternative segment

         The length of the line segment AB, which joins A (x₁, y₁) and B (x₂, y₂) is given by
d= `|AB|=sqrt((x2-x1)^(2)+(y2-y1)^(2))`  
Proof:
Let A (x₁, y₁) and B (x₂, y₂) be two points in the plane.
Let d = distance between the points A and B.
Draw AL and BM perpendicular to x-axis (parallel to y-axis).
Draw AC perpendicular to BM to cut BM at C.

OL = x₁, OM = x₂ [AC = LM = OM - OL = x₂ - x₁]
MB = y₂, MC = LA = y₁ [CB = MB - MC = y₂ - y₁]
`AB^(2)=AC^(2)+CB^(2)`
`d^(2)=(x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)`
d=`sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1)^(2)))`

Alternate Segment Circle Theorem

           The chord CE divides the circle into 2 segments. Angle CEA and CDE are angles in alternate segments because they are in opposite segments. Circles are easy blocked curves which separate the plane into two regions, an interior and an exterior.The alternating segment theorem states that a position connecting a departure and a chord through the point of contact is equal to the angle in the alternate segment.
         In conditions of the beyond diagram, the alternating segment theorem tells us to facilitate angle CEA and angle CDE are equivalent.



A digression makes an angle of 90 degrees with the radius of a circle,so we know that ∠OAC + x = 90.
The angle in a semi-circle is 90, so ∠BCA = 90.
The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180.
Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90.
But OAC + x = 90, so ∠OAC + x = ∠OAC + y.
Hence x = y.

Fractions Addition

The whole thing is partitioned in two different pieces. Each piece is called fraction of the whole thing.A fraction represents the divided form of two numbers. It takes the general form of `(a)/(b)` . Here a represents the numerator value and b represents the denominator value. `(17)/(5)` is an example for fraction. Fractions are classified according to their values numerator and denominator which plays the important role in fractions addition table

Types of fraction

1.Proper Fraction

2.Improper Fraction

3.Mixed Fraction

4.Equivalent Fraction


Activities involved in fractions addition:

1. When we add two fractions with same denominator, those fractions are added directly.

For example:

`(2/3) + (7/3) = (2+7)/(3) = 9/3`   (here numerator is divided by denominator)

=3

2. When we add two fractions with different denominator, we have to find least common denominator.

`(2/3) + (6/9) = [(2xx3)+6]/(9) = (6+6)/(9) = (12)/(9)` (here numerator is divided by denominator)

=4/3

3. If mixed fraction is involved in fractions addition, we have to convert the mixed fraction to improper fraction. Then we have to check the denominators. If the denominators are different, we have to find least common denominator. Otherwise no problem.

2 ¾ + 5/4

Take the mixed fraction 2 ¾

Convert it to improper fraction as below

` [(2xx4)+3]/(4) = (8+3)/(4) = (11)/(4)`

Now we can do the fractions addition

`(11)/(4) + (5)/(4)= (11+5)/(4) = (16)/(4) = 4`

About Easy Vector Tutorial

Easy vector tutorial is process of tutoring to students in online by the tutors. Easy vector tutorial is the vectors which are specified by magnitude and direction. The examples of vectors are displacement, velocity, acceleration, momentum of force and weight. Easy vector tutorial is method  tutoring on vector problems in the online with basic steps and calculations. Tutor vista is the main website to provide online tutors to give easy vector tutorial.  The following problems are some of the examples of  easy vector tutorial process done by tutor vista.

Easy vector tutorial on example problems

Example problem 1: Add 2`veci` +6`vecj`+4`veck` with 5`veci`+2`vecj`+2`veck`
Solution
Place the vectors according to the magnitudes shown below.

    2`veci` + 6`vecj` +4`veck`

    5`veci` + 2`vecj` +2`veck`

   ---------------------------

    7`veci` + 8`vecj` + 6`veck`

   ----------------------------

Example problem 2: Subtract 5`veci` + 4`vecj`- 8`veck` with 4`veci`- 6`vecj` +3`veck`
Solution
Since we have the magnitudes in opposite direction it will be easy to perform method
    (5`veci` + 4`vecj`- 8`veck`4`veci`- 6`vecj` +3`veck`
    `(13 - 4)veci + (4-(-6))vecj + (-8-3)veck`
    `(13 - 4)veci + (4 + 6))vecj + (-8-3)veck`

    `(9) veci + (10)vecj + (-11) veck`

     `9 veci +10vecj -11 veck`

Example problem 3: Find the magnitude of `2 veci - vecj + 7 veck`
Solution:
Magnitude of `2veci - 3vecj + 7 veck = |2veci - 3vecj + 7 veck|`
                                                   = `sqrt((2)^2 + (-3)^2 + (7)^2)`
                                                   `sqrt(4+9+49)`
                                                  =`sqrt (62)`


Example problem 4: Find the sum of the vectors `veca - vecb + 2 vecc` and `2veca +3vecb - 4vecc` and also find the modulus of the sum.
Solution
Let  `vecx = veca - vecb + 2 vecc` ,      `vecy = 2veca + 3 vecb - 4vecc`
`vecx + vecy = (veca - vecb + 2 vecc) + (2veca + 3 vecb - 4vecc)`
             `= 3 veca +2vecb - 2 vecc`
`|vecx + vecy| = sqrt (3^2 + 2^2 + (-2)^2)`
               ` = sqrt (9+4+4)`
               ` = sqrt 17`

Easy vector tutorial on practce problems

1. Add 7`veca` +2`vecb`+3`vecc` with 2`veca`+3`vecb`+`3vecc`
Answer: 9`veca`+5`vecb`+6`vecc`
2. Subtract 4`veci` + 3`vecj`- 6`veck` with `veci`- 2`vecj` +2`veck`
Answer: `veci`+5`vecj` -8`veck`
3. Find the sum of the vectors `vecp -3 vecq + 2 vecr` and `3vecp +vecq - 2vecr` and also find the modulus of the sum.
Answer: 6

Geometric Mean

     The geometric mean is the mean or average which indicates the central tendency or typical value of a set of numbers. This is similar to arithmetic mean, which is the sum of adding the set of numbers and then dividing the sum of count number in set, n, and the number are multiplied and the nth root of resulting value is taken. The geometric mean only applies to positive numbers.

Formula Of Geometric Mean

Geometric Mean = ((X₁)(X₂)(X₃)........(X_N))^1/N
Where
              X = Individual score
              N = Sample size (Number of scores)
Geometric mean is also used for set of numbers whose values are multiplied together or exponential in nature, such as growth of growth of human population or interest rates of a financial investment. The geometric mean is one of the three classic Pythagorean means, aforementioned arithmetic mean and harmonic mean.

Calculation of Geometric Mean

 The geometric mean of data set [ a₁, a₂, --------, a_n ] is given by
^{$\left(\prod_{i = 1}^{n} a_{i}\right)^{1/n}$} = $ \sqrt[n]{a_{1}a_{2}......a_{n}}$
The geometric mean of a given set is less than or equal to the given set of arithmetic mean (the two mean of equal when all the members of a data set are equal) this allows the definition for arithmetic geometric mean the two values always lie in between the geometric mean is also arithmetic harmonic mean in that if two sequences (a_n) and (h_n) are defined:
a_{n + 1} =  $\frac{an + hn}{2}$ , a_o   = x and
h_{n + 1} = $\frac{2}{1/an + 1/bn}$, h_o = y.
Then a_n is converge to the geometric mean of x. h_n is converge to the geometric mean of y.
The sequence for converge to a common limit ( which can be shown by Bolzano - Weierstrass theorem) the geometric mean is given by
$\sqrt{a_{i}h_{i}}$  = $\sqrt{\frac{a_{i} + h{i}}{1/a_{i} + 1/h_{i}}}$  =  $ \sqrt{a_{i + 1}h_{i+1}}$

Exponential Growth of Geometric Mean

The geometric mean is more approximate than the arithmetic mean for explaining exponential growth. 

How to Calculate Square Feet

How to calculate square feet :Square feet of an area is determined by multiplying length and the width of the given area . The area of regular polygons like square , rectangle , triangle etc... can be calculated by using the formula . If the dimensions of a regular polygon is given in feet then by just substituting the given dimensions in the formula the area in square feet can be calculated , if the measurements are given in any other units such as centimeter or inches or yards then we convert these units to feet and then calculate the area in square feet , similarly for a circle also if the radius is in feet we can just substitute the radius in the area formula and calculate the area in square feet ,if the radius is given in any other unit then convert that to feet and then calculate the area in square feet .

Area of composite shapes , a composite shape is a shape which can be divided into more than one of the basic shapes . An example of a composite shape or figure is shown in the picture .

Here in this picture we can see a square , a rectangle and a triangle . Area of the whole figure can be calculated by calculating the areas of each shape and then adding them . If the measurements are not in feet then we have to convert them to feet and calculate the area in square feet .

How to calculate square feet live situations .

The home we live , the school we study and mostly all other places one or the other time needs the area of floors , walls, ceilings , windows etc. in square feet to get them painted or any other purpose and usually the areas are calculated in square feet as most of the paints or coating materials have the coverage  specification mentioned in square feet like a ' x ' brand paint covers 30 square feet per liter . 

For How to calculate square feet the example here is an interior view of a room which has to be carpeted and walls painted .( all measurements are in feet )

We can see a room with a floor plan of a 

composite figure , that can be split into two rectangles as shown.

                                                          

Area of the floor in square feet to be carpeted is calculated by first learning How to calculate square feet :

Area of the floor  =  Area of rectangle 12 x 8  +  Area of rectangle 9 x 6

Area of the floor  =  96 square feet  +  54 square feet

Area of the floor  =  150 square feet .

There are all together six rectangular shaped walls as shown below .( figures not to scale )

To find the  total wall area , find the area of each wall and add them up .

   

  Area of wall 1. = ( 10 feet x 8 feet ) - ( 7 feet x 3 feet ) [ area of door ]                                                =   80 square feet  -  21 square feet  =  59 square feet

   Area of wall 2. = 10 feet  x  12 feet  =  120 square feet

   Area of wall 3. =  ( 10 feet x 14 feet ) - ( 4 feet x 5 feet ) [ area of window ]                                       =    140 square feet  -  20 square feet  =  120 square feet

   Area of wall 4. =  10 feet  x  9 feet  =  90 square feet

Area of wall 5.  =  10 feet  x  6 feet  =  60 square feet  ,  Area of wall 6.  =  10 feet  x  3 feet  =  30 square feet 

Total area of the wall to be painted  =  59  +  120  +  120  +  90  +  60  +  30  =  479 square feet .

How to calculate square feet unit conversion examples .

How to Calculate square feet  in each diagram .



The dimensions given for this square is in inches , we have to convert the inches to feet so that we can calculate the area in square feet .
12 inches = 1 feet  so   24 inches = 2 feet
Area of square is given by  A  =  s²       where  s  =  2 feet
Area of square  A  =  ( 2 feet )²      =    4 square feet

The dimensions for this triangle are in centimeters , we have to convert centimeters to feet so that we can calculate the area of triangle in square feet .
30 cm = 1 feet    and   15cm = 0.5 feet
Area of triangle is given by  A  =  1/2 b h  ,   where base b = 0.5 feet and height h = 1 feet
Area of triangle  A  =  1/2 x  0.5 feet  x  1 feet  =  0.25 square feet


This a composite shape and we have split that into basic shapes , we have two rectangles , one triangle and a semi circle.
Area of the total shape = 4 x 5 + 4 x 9 + 1/2 x 4 x 4 + 3.14 x 2²                                                                                                                   2
Area = 20 + 36 + 8 + 6.28  =  70.28 square feet


The radius of the circle is given in meters , we have to convert the meters to feet so that we can calculate the area of circle in square feet .
1 meter  =  3 feet   so   3 meters  =  9 feet
Area of circle is given by   A  =  π r²       ,    where     =  3.14 ,  radius r  =  9 feet
Area of circle  A  =  3.14  x  9²    =  3.14   x  81  =  254.34 square feet

Students can learn How to calculate square feet from the above examples and solve problems on similar lines.

Domain all the Real Numbers

In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or "value" for each member of the domain. For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0.(Source: Wikipedia)

Domain all the real values - Example problems:

Example problem: 1
Find the domain function of f that can be defined by f (x) = -1 / ( x + 10)
Solution:
  To find the domain equate denominator to zero
  x + 10 = 0
  x = -10
Therefore the domain function of f is the set of all real values of x in the interval is(-infinity, -10) U (-10, +infinity)
Example problem: 2
 Find the domain function of f that can be defined by f(x) = x² + 6.
Solution:
 The function f(x) = x² + 6 is defined for all real values of x.
 Hence, the domain function of f(x) is the function of  "all real values of x".
 Since x² is never negative, x² + 6 is never less than 2
Example problem: 3
 Find the domain function of f that can be defined by f(t) = (1)/(t + 6)
Solution:
 The function f(t) = (1)/(t + 6) is not defined for t = -6, as this value requires division by zero.
 Hence the domain function of f(t) is the function of all the real numbers except -6
 Also, no matter how large or small t becomes, f(t) will never be equal to zero

Domain all the real values - Practice problems:

Practice problem:- 1
Find the domain of function of f that can be defined by f (x) = sqrt (-x + 16)
[Answer: Domain of function f is the set of all values of x in the interval (-infinity, 16) ]
Practice problem:- 2
Find the domain of function f defined by f (x) = sqrt( -x + 4) / [(x + 2)(x + 8)]
[Answer: domain of function f is the set of all values of x in the interval   (-infinity, -8) U    (-8, -2) U (-2, 4)] 

Properties of the Parallel Lines

Parallel lines
          The distance between the two lines will be equal and they will never intersect, such lines are said to parallel lines.

 Properties of the parallel lines
           The parallel lines will have the same slope (m1=m2)
           Here the m1 and m2 are slopes of the lines.
           When the parallel lines are cut by transversal line, then eight angles are formed

properties :

Here
                     Angle 1 = angle 3 =angle 5=angle 7
                    Angle 2= angle 4 = angle 6= angle 8.

 Adjacent angles properties:
        when  the parallel line is cut by a transversal line, then the adjacent angles sum up to 180 degrees
                   Angle 1 +angle 2= 180 degree
                   Angle 3 +angle 4= 180 degree
                   Angle 5 +angle 6= 180 degree
                   Angle 7 +angle 8= 180 degree
                Here the adjacent angles always sum up to 180 degree.

 Corresponding angles properties
       The angle in the same position around the intersection of two points is said to be corresponding angle.
          when  the parallel line is cut by a transversal line, then corresponding angles are  equal
                    Corresponding angles:
                       Angle 1 = angle 6
                       Angle 2 = angle 5
                       Angle 3 = angle 8
                       Angle 4 =angle 7

Alternate angles properties:
                   when the parallel line is cut by a transversal line then alternate angles are equal
                   So
                    Alternate interior angle
                       Angle 3 = angle 5
                       Angle 4 = angle 6
                    Alternate exterior angle
                      Angle 2 = angle 8
                      Angle 1 = angle 7


Algebra is widely used in day to day activities watch out for my forthcoming posts on hard math problems for 8th graders and cds exam 2013. I am sure they will be helpful.

Model problems:

1.Find the angle 2, when the angle 1 = 120?
Solution:
 Here angle 1 and 2 are adjacent angles
 When parallel lines is cut by a transversal line, then the    adjacent angles sum up to 180 degree
          So angle 1+angle 2 = 180
                    120 +angle 2= 180
                     Angle 2= 180-120
                    Angle 2 =60 degrees
 2.Find the angle 4, when the angle 1 = 120?
Solution:
           Here angle 3 and 4 are adjacent angles
           Angle 1= angle 3 (vertical opposite angles are equal)
           Angle 3 = 120
 When parallel lines is cut by a transversal line, then the adjacent angles sum up to 180 degree
           So angle 3+ angles 4 = 180
                       120 +angle 4= 180
                        Angle 4= 180-120
                       Angle 4 =60 degrees.

Fraction Variable Solver

A fraction (from the Latin fractus, broken) is a number that can represent part of a whole. The earliest fractions were reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions which are still used today (½, ?, ¾, etc.) and which consist of a numerator and a denominator.
Source – Wikipedia.


Fraction variable solver for Addition:

Variable Solver:

A variable solver is the way of thinker that refers to give proper solutions or values for the given fractions problems.

The following are some of the examples for Fraction variable solver in addition.

Simplify the fraction: `(3x)/7+(5x)/6`

Solution:

=` (3x)/7+(5x)/6`

The LCD for the denominators 7 and 6 is 42.

= `(6*3x)/42+(7*5x)/42`

We can express the denominators in the LCD form as

= `(18x)/42+(35x)/42`

We can simplify the numerators.

= `(18x+35x)/42`

Then by adding the numerators.

= `(53x)/42`

Simplify: `(3x)/5+(5x)/3`

Solution:

= `(3x)/5+(5x)/3`

The LCD for the denominators 5 and 3 is 15.

= `(3*3x)/15+(5*5x)/15`

We can express the denominators in the LCD form as

= `(9x)/15+(25x)/15`

We can simplify the numerators.

= `(9x+25x)/15`

Then by adding the numerators.

= `(34x)/15`

Simplify: `(2x)/5+(7x)/6`

Solution:

= `(2x)/5+(7x)/6`

The LCD for the denominators 5 and 6 is 30.

= `(6*2x)/30+(5*7x)/30`

We can express the denominators in the LCD form as

= `(12x)/30+(35x)/30`

We can simplify the numerators.

= `(12x+35x)/30`

Then by adding the numerators.

= `(47x)/30`

Fraction variable solver for Subtraction:

The following are some of the examples for Fraction variable solver in subtraction.

Simplify the fraction: `(6x+3)/12-(4x+5)/8`

Solution:

= `(6x+3)/12-(4x+5)/8`

The LCD for the denominators 12 and 8 is 24.

= `(2(6x+3))/24-(3(4x+5))/24`

We can express the denominators in the LCD form as

= `(12x+6-12x-15)/24`

We can simplify the numerators.

= `(x-9)/24`

Simplify: `(2x+3)/6-(6x+5)/5`

Solution:

= `(2x+3)/6-(6x+5)/5`

The LCD for the denominators 5 and 6 is 30.

= `(5(2x+3))/30-(6(6x+5))/30`

We can express the denominators in the LCD form as

= `(10x+15-36x-30)/30`

We can simplify the numerators.

= `(-26x-15)/30`

Simplify: `(8x+3)/12-(3x+5)/8`

Solution:

= `(8x+3)/12-(3x+5)/8`


The LCD for the denominators 12 and 8 is 24.

= `(2(8x+3)) /24-(3(3x+5)) /24`

We can express the denominators in the LCD form as

= ` (16x+6-9x-15)/24`

We can simplify the numerators.

= `(7x-9)/24`