Beginner Multiplication

Multiplication (x) is the arithmetical operation of calculating one value by another value. It is a one kind of essential operations in basic arithmetic (the others operations are addition, subtraction and division). Since the outcome of calculating by whole numbers can be thinking of as including of a few number of copies of the original, whole-number products larger than 1 can be calculated by frequent addition.

Types of multiplication for beginner:

• Multiplication of variable by exponent

• Multiplication of fraction

• Multiplication of unlike signs ( positive(+),negative (-))


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Multiplying Variables with Exponents:

Multiplication of exponent:

1) Exponent of 0:

If the exponent is 0 specifies you are not multiplying by anything and the answer is 1

For example, a0 = 1

x0 = 1

2) Exponent of 1:

If the exponent is 1 specifies you are multiplying the variable with 1. (Example x1 = x)

Rules for multiplication terms on fractions:

• First multiple the value of numerator.

• Then multiple the value of denominator.

• Lastly decrease the fraction (if required).

Example:

`2/5` ×` 3/4`

Step1 Multiply the numerators:

`3 / 4` × `2 / 5` = > 3 × 2 / 4 x 5 = `6/20`

Step2 Multiply the denominators:

` 3/4` ×`2/5` = 3×2 / 4 x 5 = `6/20`

Step3 Lastly decrease the fraction

Therefore solution is `3/10` .

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Multiplication of unlike signs for beginner:

• Positive(+) × Positive(+) = Positive(+)

Ex:  6 × 2   =12
• Positive (+) × Negative (-) = Negative (-).

Ex:   6 × (-2) = -12
• Negative (-) × Positive (+) = Negative (-).

Ex: (-6) × 2 = -12
• Negative(-) × Negative(-) = Positive(+).

Ex: (-6) × (-2)=12

Example:

Multiply by (a + 4) (a -5)

Step 1: multiply by y in the second factor

a (a-5) =  a2- 5a

Step 2: multiply by 5 in the second factor

4(a - 5) = 4a-20

Step 3: add step 1 and step 2

(a + 4) (a - 5) = a2- 5a + 4a - 20

= a2 -1a - 20

Basic multiplication problems for beginner:

1) Multiply the values i) 23 × 2 ii) 4 × (6 × 3) = (4 × 6) ×3

Solution:

i) 23 × 2 = 46

ii)    4 × (6 × 3) = (4 × 6) ×3

4 × 18 = 24 × 3

72 = 72.

2) Multiply the values i) 4 × (2 + 6)   ii) 6 × (5 × 3) = (6 × 5)×3

Solution:

i) 4 × (2 + 6) = 4 × 2 + 4 × 6

= 8 + 24

= 32.

ii)6 × (5 × 3) = (6 × 5) × 3

6 × 15 = 30 × 3

90 = 90.

Isosceles Triangle Hypotenuse

When the two sides of the triangle are said to be equal then the triangle is called as isosceles triangle. When all the sides of the triangle are equal then the triangle is called as the equilateral triangle and when no sides of the triangle are equal then it is said to be scalene triangle. In an isosceles triangle only the sides but also the two angles are said to be equal.
                                                               
Here we will see about the isosceles triangle hypotenuse.

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Isosceles triangle hypotenuse.

The hypotenuse side of the isosceles triangle is the sides that are found opposite to that of the right angle. According to the right angle theorem the height of the isosceles triangle is given by
                         h = √(b² – ¼ a²)
Thus the area of an isosceles triangle is
                        A = ½ ah
                            = ½ a √(b² – ¼ a²)
                            = ½ a² √((b²/a²) – (¼))
An isosceles triangle is also called as the triangle with two congruent sides. The angles which are opposite to these congruent sides are called as the base angles and the angles found between those sides are called as the vertex angle of the isosceles triangle.
Any equilateral triangle can be an isosceles triangle but no isosceles triangle is an equilateral triangle.                           

Properties of the isosceles triangle base:

• According to the right angle theorem the sum of the squares of the hypotenuse side is equal to the sum of the squares of the other two sides. The hypotenuse side is the side which is opposite to the right angle

• The side which is not equal to the other sides of a triangle is called as the base of the isosceles triangle.

• The base angles of the isosceles triangle are found to be equal.

• When the third angle of the isosceles triangle is a right angle then it is called as the right isosceles triangle.

• The perpendicular distance from the base to the vertex of an isosceles triangle is called as the altitude of the isosceles triangle.

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Externally Tangent Circles

Two intersect circles in a single point is known as tangent circles. It can be divided into two types of tangency: internal and external. By using the tangent circles many problems and constructions in geometry are solved. This type of problems have real-life applications. The followings are some real-time applications: trilateration and maximizing the use of materials.

                                             

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Illustration:

A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called Common External Tangent. A common tangent can be any one of the following:External tangent or Internal tangent.

                                                   
          From the above figure, we can see that line PQ is the common external tangent.

                                               
          From the above figure, we can see that line AB is the common external tangent.

Theorem for Externally Tangent Circles:

Theorem:If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
                                                       
Solution:
            If:        Line  AB is a tangent
                       D is point of tangent.
            Then:   OD ┴ AB

Example problems:

Example 1: Find the common external tangent from the below figure.

                                             
Solution:
            A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called common external tangent.
            From the given figure, the line CD touches both the circles on the same side of the line is the external tangent to the circle.
            Therefore, CD is the common external tangent line.

Example 2: Find the common external tangent from the below figure.
                                                                                    
Solution:
            The given figure is concentric circles. Concentric circles having the same center and not tangent.

Example 3: Find the common external tangent from the below figure.
                                                     
Solution:
          From the above figure, we can identify that given is two complete separate circles.
          The given figure have two external tangents.

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Example 4: Find the equation for tangent lines y=3x³ from the point x= 2.
Solution :
Given, y=3x³………….(1)
Step 1: To find the first derivative of y

                y ' = 3x ² ………..(2)
Step  2: To plug x =2 into y ' to find the slope at x.
                y ' = 3(2) ²
                y= 12
               Slope of tangent line =12
Step 3:To plug x=2 into y to find the y value of the tangent point
              y =3(2)³
              y=24 
              Hence tangent point (2,24)
Step 4: To plug the slope=12 and point (2,24) using the point-slope equation to find the equation     for the tangent line.
              y- y₁ =m (x-x₁)
              y- 24 =12 (x-2)
              y =12x- 24 + 24
              y=12x
Example 5: Find the slope of tangent line which is passing through the point (2, 0) and the curve y = 2x – x³
Solution:
            Step 1: Find the derivative y¹
                                y = 2x – x³
                                y¹ = 2 -3x²
            Step 2: To find the slope of tangent line,
                         Given point (2, 0) --- > (x, y)
                         y¹ = 2 -3(2)² ---- > 2 – 3(4) --- > 2 -12 = -10
                         Slope of the tangent line = -10         

Practice problems:

1. Determine the equation of the tangent line passing through the point (− 1, − 2) and having slope4 /7
Answer:   4x − 7y − 10 = 0
2. Determine the equation of the line with the slope 3 and y-intercept 4.
Answer: y = 3x + 4

Power Factor Conversion

To signify a lot of the numbers we have used scientific notation. For example the number 7500 is identified as 7.5E3. Here the letter “E" can be read as "times ten to the power of". Every unit is written out in full before its contraction is used. Some units have superscripts in them. In power factor conversion includes the watt to joule, foot pound to force per second, kilogram to meter per second, horse power to watt.

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Example problems of power factor conversion:

Power factor conversion Problem 1:

Convert watt to joule per second.

25 W

30 W

Solution:

We can convert watt to joules per second by using the following method.

1 watt = 1 joule per second

The initial unit is 25 W and 30 W

Here conversion factor is 1 W/ 1 joule/sec

That is, `25*` `1/1` = 25

`30*` `1/1` =30

Therefore the final unit is 25 and 30 units

Answer: 25 joule/sec and 30 joule/sec

Power factor conversion Problem 2:

Convert watt to joule per minute.

55

Solution:

We can convert watt to joule per minute by using the following method.

1 watt = 60 joule per minute

The initial unit is 55 W

Here conversion factor is 1 watt = 60 joule per minute.

That is, `55*` `60/1` =3300

Therefore the final unit is 3300 units

Answer: 3300 joule/minute

Power factor conversion Problem 3:

Convert watt to horsepower.

25

Solution:

We can convert watt to horsepower by using the following method.

1 watt = 0.001341022 horsepower

The initial unit is 25 watt

Here conversion factor is `(0.001341022 hp)/(1 watt)`

That is, 25*0.001341022= 0.033525552 horsepower

Therefore the final unit is 0.033525552 horsepower

Answer: 0.033525552 horsepower

Practice problems of power factor conversion:

1. Convert watt to joule per second: 75

2. Convert watt to joule per minute: 22

3. Convert watt to horsepower: 45

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Answer:

75 joule per second
1320 joule per minute

3.  0.060345994 horsepower

Prime and Rate

Prime: Definition of prime number is a number which cannot divide by any other numbers. That is ,only the number itself and 1 can divide the prime number.
        Rate: Definition of rate is the ratio between the distance and time. Definition of rate is always in the form of fraction.
Let us see the definition of prime and rate.

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Definition of prime number:

Definition of prime number:

If a number is not allowed to divide by any number then it is called as prime number. Other wise it is called as composite number.

Example: 133

This number can not be divide by any number. Only 1 and 133 can divide it. So, 133 is considered as prime number.

Example:1

Find the prime number among the following numbers.

a. 155

b. 242

c. 141

d. 339

Ans :141

Example: 2

Find the prime number among the following numbers.

 a.122

b.223

c.144

d.166

Ans : 223

Definition of rate:

Definition of rate:
  Formula for finding a rate is,

Rate=Distance/Time

        Rate has a units depends on the units of distance and time. If the distance has a unit in km and the time is in hr then the rate should be written as km/hr.
Example problems for finding a rate:
Example:1

Find the rate of scooter. The distance travelled by a scooter is 89 km and the time taken for that is 3 hours.

Solution:
Given,

Distance = 89km

Time=3 hr

We know the rate formula is Rate=distance/time

Rate=`89/3`km/hr.

Therefore the rate of the given scooter is `89/3` km/hr.

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Example:2

Find the time taken by a cycle. The distance travelled by a cycle is 24 km and the rate is `12/6`km/hr.

Solution:
Given,

Distance = 24km

Rate=`12/6`km/hr

We know the rate formula is Rate=distance/time

Time= Distance/time

         =`24/(12/6)`

        =`144/12`

       =12 hours.

Therefore the time taken by a cycle to travel 24 km is 12 hours.

Fraction Conversion Table

Fraction:
A fraction is a number that can represent part of a whole. The earliest fractions are reciprocals of integers: ancient symbols represent one part of two, one part of three, one part of four, and so on. A much later development was the common fractions which are used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator.

Fraction can be converted into to forms as shown below,

• Fraction conversion into decimal
• Fraction conversion into percentage.

Through this article you can learn about fraction conversion table, how to make conversion of fraction into decimal and how to make conversion of fraction into percentage.

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Example for fraction conversion table:

Fraction conversion table:

         We can express the conversion of fraction in terms of table format.This table is known as fraction conversion table.The following are the two examples for fraction conversion table.

                       

Fig(i) Fraction conversion table 1                                      Fig(ii) Fraction conversion table 2

The fractions that contains the denominator as a  power of  10 called as decimal fractions.
Ex:
0.1 = 1/10
0.01 = 1/100
0.001 = 1/1000

Problems on fraction conversion table:

Pro 1: Convert the fraction 45 / 90
Sol:
Given, 45 / 90 into decimal.
To find the decimal,divide the numerator by the denominator ,
             _________
         90 )  45      (
45 cannot be divided by 90.
So Multiply 45 by 10 and add one decimal place in quotient,
             _________
         90 )  450     ( 0.5
                450     
                 0        
So the decimal of fraction 45/90 is 0.5
Ans: 45/90 = 0.5
Pro 2: Convert the fraction 76 / 43 into decimal.
Sol:
Given, 76 / 43 as decimal
To find the decimal,divide the numerator by the denominator ,
             _________
         43)  76     ( 1.76
               43     
                33      
33 cannot be divide by 43. So multiply 33 by 10 and one decimal place in quotient,
            _________
         43)  76     ( 1.7
               43     
               330    
               301   
                 290
                 258
                   38
The decimal for 76 / 43 = 1.76
Ans:76 / 43 = 1.76

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Pro 3: Convert the fraction 6 / 5 into decimal.
Sol:
Given, 6 / 5 as decimal
To find the decimal,divide the numerator by the denominator ,
           _________
        5 )  6     ( 1
              5     
              1   
1 cannot be divide by 5. So multiply 1 by 10 and one decimal place in quotient,
            _________
           5 )  6     ( 1.2
                 5     
                 10
                 10  
                   0 
The decimal for 6/5 = 1.2
Ans: 6/5 = 1.2

Venn Diagram Example of Disjoint

Let use see about venn diagram example of disjoint.Venn diagrams also called as the set diagram.These are diagrams that show all theoretically possible logical relations between finite groups of set. Venn diagrams were considered approximately 1880 by John Venn. They are used to teach elementary set theory, and show simple set relationships in logic, probability and statistics.


venn diagram example of disjoint - Notations


Curly  braces - {...} - are used to phrase.
These braces can be used in various ways.
For example:

• List the elements of a set: {-3, -2, -1, 0, 1, 2, 3,4}

• Describe the elements of a set: {integers between -3 and 3 inclusive}

• Use an identifier (the letter x for example) to symbolize a typical element, a '|' symbol to stand for the axiom such that', and then the rule or rules that the identifier must follow: {x | x is an integer and |x| < 5}

The Greek letter ∈ is used as follows:

• ∈ means 'is an element of ...'. For example: 3 ∈ {positive integer}

• ∉ means 'is not an element of ...'. For example: Washington DC ∉ {European capital cities}

• The set is a finite: {British citizens}

                      Or

•  infinite: {6, 12, 21, 24, 35, ...}

Sets are usually be represented using upper case letters: A, B,X,Z ...

venn diagram example of disjoint - Example

The following is the diagram representation of disjoint in venn diagram.
Two sets are equally exclusive also called disjoint. If do not have any elements in common and need not together contain the universal set.
The following venn diagram represents the disjoint sets.

Set of all elements of A is also known as difference of set A-B, which do not go to B. In the set planner form, the difference set is :   

 

Example problem for disjoint set.
A={2,3,4,1,8,9}   B={2,3,4,1,10,12} What is the A-B and B-A?
Solution:
A-B=?
Given A={2,3,4,1,8,9}
            B={2,3,4,1,9,10,12}
Here all elements of A  an available in B except 9.
So the A-B is 9.
B-A=?
Here all elements of A  an available in A except 12.
So the B-A is 12.