Pictures of Fractions

Introduction to pictures of fractions:

Whole and part of a whole - A whole is one which represent a complete unit, and when this unit is divided into two or more equal sub-units then it becomes parts of a whole.

Picturing what is a Fraction -  If a unit is divided into any number of equal parts, then  one or more parts of that unit is called  a fraction of unit. A  fraction represents a ratio of the number of parts taken to form a fraction with the total number of equal parts into which a unit is divided into.  It has a numerator and denominator  which are the Whole Numbers, its denomonator  can never be Zero.

Types of Fractions: -

There are two types of fractions

Proper Fraction - A fraction whose numerator is less than denominator . Ex: 3 / 7  , 5 / 23 etc.
Improper Fraction - A fraction whose numerator is more than its denominator .  An improper fraction is a combination of a whole number  with  a proper fraction. Ex: 7 / 3   , 23 / 5   etc. Here 7/3 is a combination of  2 + 1/3   and 23/5 is a combination of  4 + 3/5.
Like Fractions - When two or more fractions are equal then they are called like fractions. For instance 1/2,2/4,3/6 are like fractions as all of them are equal .
Unlike fractions -  when two or more fractions are not equal then they are called unlike fraction. For instance 1/2 and 3/7 are unlike fractions.

Between, if you have problem on these topics all the prime numbers to 1000, please browse expert math related websites for more help on proof of prime number theorem.

Pictures of Fractions-examples:

Ex 1: A box contains 4 similar toys and we have taken 3 toys from it. What fraction is taken out from the box.  

Sol: The fraction is 3/4

Ex 2: A circle is divided into 8 equal parts and out of it 5 parts are shaded in red. Which fraction represents the shaded region.  

Sol: Picturing the above problem, the fraction is 5/8

Ex 3:. I want to share a cake among 20 friends of mine and I have divided the cake into 25 pieces . What fraction is divided amongst us.

Sol: 20 / 25  , which is also equal to 4 / 5 so 20/25 and 4/5 are said to be like fractions as  20/25 = 4/5.

Prime Number Greater Than

Introduction to Prime Numbers

Prime numbers are those numbers that cannot be divided by any other number other than 1 and the number itself without leaving a remainder. This means prime numbers have exactly two divisors the number itself and one.

Examples: 2,3,5,7,11,13,17,19 etc .

To explain this further let us take 17. Number 17 cannot be divided by any other number other than 17 or 1. If we divide 17 by any other number, there will be a reminder or 17 can be expressed only in one form

17 = 17 x 1

Consider number 18. This number 18 can be divided by 1, 2, 3,6 and 9 without leaving a reminder. Hence, 18 is not a prime number. Numbers that are not prime numbers are called composite numbers. 18 is a composite number and can be expressed in many forms like

18 = 1 x 18

18=  2 x 9

18= 3 x 6 etc

Note: Number 1 is considered as neither prime nor composite

Method to Find Prime Numbers

Now we will see how to find prime numbers greater than a particular number. Here, we must note that there is no fixed method of solving this. We need to find out whether the number can be divided by another number other than the number and one. However, a few thumb rules can of help to identify the numbers fast:

Rule 1: All even numbers are composite numbers except 2. If we find a number to be even other than 2 then it cannot be a prime number as it is divided by 2
Rule 2: If the sum of the digits of the number is divisible by 3 then the number itself is divisible by 3 and it is not a prime number. Exception is number 3 itself. Number 3 is a prime number. This rule checks for the divisibility of three.
Rule 3: If the number ends with 5, it is not a prime number. Exception is number 5. Number 5 is a prime number. This rule checks for the divisibility of 5.
Rule 4: If the sum of the odd digits of the number when subtracted from the even digits of the number gives a number divisible by 11 or 0 then the number is divisible by 11 and is not a prime number. Exception is number 11. Number 11 is a prime number.
Ex: Consider number 14641.

Sol: Step 1: The sum of the numbers at odd places is 1+6+1 = 8

Step 2: Sum of the numbers at the even places is 4+4 =8

Step 3: Difference of the sum of the digits at odd and even places is 8-8 =0

Step 4: Hence, the number 14641 is divisible by 11 and is not a prime number.

Let us try some examples

Ex 1: Identify the prime numbers form the list

42, 57,83,75,98

Sol:   Step 1: 42 and 98 are even numbers hence are not prime

Step 2: Sum of the digits of 57 is 5+7=12 is divisible by 3.

Hence, the number is divisible by 3 and is not a prime number

Step 3: 75 – Ends with 5 and is divisible by 5 and hence not a prime number

Step 4: 83 – is the prime number in the list

Ex 2: Find the first prime number greater than 109?

Sol: This example can only be solved with a few trial and errors and with intelligent guesses. Let us check some of the numbers after 109. We can eliminate all odd numbers as they cannot be prime numbers.

Step 1: 111- This is divisible by 3 as the sum of the digits is 3

Step 2: 113 – This could be a prime number

Step 3: 115 –  This number ends with 5 and not a prime number

Step 4: The maximum possible divisor of a number is half the number. This means that a number can never be divided by a number greater than half its value. So for 113, we need to check whether any number less than 66 can divide the number.

Step 5: 113 cannot be divided by any number divisible by 2, 3 or 5 as 113 itself is not divisible by these numbers. This means that we should check whether 113 is divisible by other prime numbers like 7, 11,13,17,19,23,29,31, 37,41,43,47,53,59 and 61.Since, it is not divisible by any of these numbers, it is a prime number.

Checking whether a number is prime or not goes by the method of elimination. If the divisibility rules are used and if any of the divisors are found then it is definitely not a prime number. But if the number is big and you are not able to identify any divisors using the divisibility rules we CANNOT conclude that it is a prime number. It can have any higher prime number as divisor

For example 588139 is not divisible by 2,3, 5 or 11 etc. But we can write it as below

588139 = 839 x 701 and hence 588139 is NOT a prime number.

Here, 839 and 701 are prime numbers and hence identifying them as divisors manually will be difficult. However, the process can be carried out easily using soft wares.

Between, if you have problem on these topics adding rational expressions with different denominators, please browse expert math related websites for more help on help with math online.

Exercises on Prime Numbers.
Pro 1: Write the first five prime numbers greater than 100.

Ans: 101, 103, 107, 109 and 113

Pro 2: identify the prime number from the following

568, 965, 983, 1353

Ans: 983

Pro 3: Find the first prime number greater than 856

Ans: 857

Equilateral Triangle Prism

Introduction 

A triangular prism or three-sided prism is a form of prism; it is a polyhedron made of a triangular base, a translated copy, and three faces joining corresponding sides in geometry. Semi regular if the base faces are equilateral triangles, and the additional three faces are squares is called right triangular prism.3-sided bi pyramid form is dual of a equilateral prism. (Source: Wikipedia)

In this article we discuss how to find the volume and surface area of the equilateral triangle prism.

Examples Problem for Equilateral Triangle Prism:
1. Find the surface area and volume of an equilateral triangle prism. Bases of equilateral triangle prism are 5, and then the length of the equilateral triangle prism is 6.

Solution:

Given values,

Equilateral triangle prism base in a side a = 5

Equilateral triangle prism height in a side h = 6

Formula for Equilateral triangle prism,

Surface area = 2a2 `sqrt(3)`/4 + 3ah

Plug the given values,

= 2* 52 `sqrt(3)` /4 + 3*5*6

= 50*1.732/4 + 90

Calculate the values,

= 21.650+90

And then we get the answer is,

Equilateral triangle prism of surface area is = 111.650

Formula for Equilateral triangle prism,

Volume = a2 `sqrt(3)` h/ 4

Plug the given values,

= 52 `sqrt(3)`* 6/ 4

Calculate the values,

= 25*1.732*6/ 4

= 259.8/ 4

And then we get the answer is,

Equilateral triangle prism of volume is = 64.95.

Between, if you have problem on these topics math word problems for 5th grade, please browse expert math related websites for more help on adding and subtracting rational expressions with unlike denominators.

More Examples Problem for Equilateral Triangle Prism:

2. Find the surface area and volume of an equilateral triangle prism. Bases of equilateral triangle prism are 4, and then the length of the equilateral triangle prism is 3.

Solution:

Given values,

Equilateral triangle prism base in a side a = 4

Equilateral triangle prism height in a side h = 3

Formula for Equilateral triangle prism,

Surface area = 2a2`sqrt(3)` /4 + 3ah

Plug the given values,

= 2*42 `sqrt(3)` / 4 + 3*4*3

= 32*1.732/ 4 + 36

Calculate the values,

= 13.85+36

And then we get the answer is,

Equilateral triangle prism of surface area is = 49.85

Formula for Equilateral triangle prism,

Volume = a2 `sqrt(3)` h/ 4

Plug the given values,

= 42`sqrt(3)` 3/ 4

Calculate the values,

= 16*1.732*3/ 4

= 83.13/ 4

And then we get the answer is,

Equilateral triangle prism of volume is = 20.78.

Different Sets of Real Numbers

Introduction to different set of real numbers
Different set of real numbers are those which comprise of both, rational numbers like 34, -23/149 and irrational numbers like v4, 2.45655. This means the different set of real numbers consist of integers both positive and negative, fractions and decimal expansions that may or may not replicate.

A set can be defined as a collection of objects. The sets we are talking about in this pamphlet are based on real numbers. Any object that is present in the set can be termed as an element/member of the set of real numbers. Sets are represented by Capital letters like A,B,C etc or also by brackets like {…..} hemming in symbols for the elements that are present in the set.

I.e.  {1, 2, 3, 4}. This means the elements of the set are 1,2,3,4.

Keep in mind that two sets are known to be equal only if the elements in each of the sets are the same.



Real numbers can be marked on the number line, making it much easier to picture and understand.

I.e.

On careful observation of the number line you will notice that the numbers that are greater in value are seen to be on the right side of the number line. I.e. if m
A set of real numbers can be depicted on the number line as mentioned earlier, by shading or coloring the points whose coordinates are elements of the sets.

I like to share this solve by the substitution method with you all through my article.

Example on Different Set of Real Numbers:

let us study different set of real numbers with examples.

The natural numbers, also called as counting numbers or positive integers, are the numbers 1,2,3,4 etc, achieved by adding 1 repeatedly. The set 1,2,3,4 of all natural numbers are represented by the symbol N.


The integer comprises of all the natural numbers, the negatives of the natural numbers, and zero. The set of all integers  I.e. {-3,-2,-1, 0, 1, 2, 3} are given by the symbol N.


The rational numbers are numbers that are in the form a/b, such that a and b are integers; b? 0. Since b may be equal to 1, every integer is a rational number. Other illustrations of rational numbers are 9/5, 4/5 and -34/3. The set of all rational numbers is represented by the symbol Q (keep in mind this is to show that rational numbers are quotients of integers). Rational numbers that are in decimal form either lapse or begin to replicate a similar pattern imprecisely.


Irrational number are decimal representation that is non- lapsing or non- replicating.  Examples are v3= 1.7320508, and p= 3.1415926.


The amalgamation of  different rational and irrational numbers is the set of Real Numbers.

Adjacent Supplementary Angles

Introduction of supplementary angle:

1) If the sum of the measures of two angles is 180, the angles are auxiliary.

2) In other words, if the outer rays of two adjacent angle form a straight angle then the sum of their measures is 180

Note: to remember which word goes with which total, remember that they are in both alphabetical and numerical order: Complementary comes before Supplementary angle alphabetically, and 90 comes before 180 numerically.

Adjacent Angle:

Adjacent angles are pairs of angles that have the same vertex, share one side, but do not have any interior points in common.

More about Adjacent Angle:
Two angles with a common vertex and a common side, but no common interior points, are called adjacent angles.

In the above figure, `angle`DAC and `angle`BAC are adjacent angles; `angle`DAB and `angle`BAC are not adjacent angles.

If the two non-common sides of adjacent angles form opposite rays, then the angles are called a linear pair. Note that x and y are supplementary.

More about Supplementary Angles:
Two angles are supplementary and each is a supplement of the other if they are respectively congruent to the angles of a linear pair. Similarly, we define three angles to be supplementary if they are respectively congruent to the angles of a linear triple.

It easily follows, using the supplement postulate, that two angles are supplementary if and only if the sum of their measures in 180.

Principles for supplementary angles:

1) If two supplementary angles contain a and b, then a+b=108

Thus if angle of a and b are supplementary and a=140, then b=40

2) Adjacent angles are supplementary if their exterior sides lie in the same straight line.

3) If supplementary angles are congruent, each of them is a right angle (Equal supplementary angles are right angles).