Median Even Number

Introduction for Median of numbers:

Median is the center value of the given numbers or allocation in their ascending order. Median is the average value of the two center elements when the size of the allocation is even. Half the numbers in the list are fewer, and half the numbers are greater. To locate the Median, place the numbers you are given in value order and find the center number. But there are two center numbers (as happens when there is an even amount of numbers) then average those two numbers.

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Median of Numbers:

Odd number:

Example:

Find the Median of {8, 2 and 7}.

Put them in order: {2, 7, and 8},

The center number is 7,

So the median is 7.

Even numbers:

Example:

Find the Median of {8, 2, 7 and 3}.

Put them in order: {2, 3, 7, and 8},

The center numbers are 3 and 7;

The average of 3 and 4 is 3.5,

So the median is 3.5.

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Examples for Median of Numbers:

Example 1:

To find the median of numbers 5, 8, 7, 3, 4?

Solution:

Step 1:
Count the total numbers given.
There are 5 elements or numbers in the allocation.

Step 2:
Arrange the numbers in ascending order.
3, 4, 5, 7, 8

Step 3:
The total element in the allocation (5) is odd.
The middle position can be calculated using the formula. (n + 1) / 2.
So the middle position is (5 + 1) / 2 = 6 / 2 = 3.
The number at 3rd position is = Median = 5.

Example 2:

To find the median of numbers 6, 4, 7, 2, 1, 8?

Solution:

Step 1:
Count the total numbers given.

There are 6 elements or numbers in the allocation.

Step 2:
Arrange the numbers in ascending order.

1, 2, 4, 6, 7, 8

Step 3:
The total element in the allocation (6) is even.

As the whole is even, we have to take average of number at n / 2 and (n / 2) + 1

So the position are n / 2 = 6 / 2 = 3 and 4

The number at 3rd and 4th position are 4, 6

Step 4:
Find the median.

The average is (4 + 6) / 2 = Median = 5

Interest Growth Formula

Introduction to Interest growth formula:

Interest growth can calculated in two ways, they are

Simple interest growth

Compound interest growth

Simple interest growth:

The interest that is paid for only the borrowed amount not to interest calculated on the original amount.

Compound interest growth:

The interest that is paid not only for the borrowed amount but also to interest calculated on the original amount.

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Formula to Find Simple and Compound Interest Growth

Formula for calculating simple interest growth:

Interest= P*I*N

Here p is the original amount

I is the rate of interest

N is the number of years.

Formula for calculating compound interest growth:

A = P (1 + `r/q` ) nq  

Here (here the interest is compounded q times a year)

P is amount borrowed

r is the rate of interest

n is the number of year

q is the number time the interest is compounded.

A= P (1 + r) n 

Here the interest is compounded only one time  in a year

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Model Problems on Interest Growth Formula

Example: 1

Find the simple interest growth for the amount of 500$ and the rate of interest is 5% for 3 years

Solution:

Formula for calculating simple interest growth:

Interest= p*I*N

Here p is the original amount = $500

I is the rate of interest = 5%

N is the number of years = 3

Interest = 500*5%*3

Interest = 500*(5/100)*3

Interest = 5*5*3

Interest = $75

So the simple interest growth is $75

Example: 2

Find the amount using compound interest growth for the amount of 100$ and the rate of interest is 3.5% for 5 years if it is compounded 4 times?

Solution:

Formula for calculating compound interest growth:

A = P (1 + `r/q` ) nq

Here

P is amount borrowed = $1000

r is the rate of interest = 3.5%

n is the number of year =5

q is the number time the interest is compounded, q=4

Interest = 1000(1+ `(.035)/4` )20

=$1190.34

Amount after the 5years is $1190.34

Definition of Triangular Pyramid

Introduction to the definition of triangular pyramid:
In this article we see about definition of triangular pyramid. Pyramid is a polyhedron 3 – dimensional geometric shape. The pyramid has 4 – vertices. Out of them 3 are base of the pyramid and one is top of the pyramid.

Types of pyramid:

Square pyramid
Rectangular pyramid
Triangular pyramid
Pentagonal pyramid
These are some of the types of pyramid. In this section we see about definition of triangular pyramid.

Definition of Triangular Pyramid:
A pyramid which has a triangular base is said to be triangular pyramid.

Types of triangular pyramid:

Equilateral triangular pyramid – base of this pyramid is equilateral triangle
Isosceles triangular pyramid – base of this pyramid is isosceles triangle
Scalene triangular pyramid – base of this pyramid is scalene triangle
Volume of a triangular pyramid = `1/3` area of the triangle x length

Volume = `1/3` x `1/2 ` x base x height x length

That is volume = `1/6 xx b xx h xx l`

Let example we see some of the example problems for triangular pyramid.

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Example Problems – Definition of Triangular Pyramid:

Example problems 1 – definition of triangular pyramid:

Calculate the volume of triangular pyramid where the base is 6.5 m, height is 14.6 m and length is 15.4m.

Solution:

Given:

Base b = 6.5 m

Height h = 14.6 m

Length l = 15.4 m

Volume of triangular pyramid = `1/6` x b x h x l

= `1/6` x 6.5 x 14.6 x 15.4

= `1/6 ` 1461.46

= 243.57

Answer: volume of a triangular pyramid = 243.57 cubic meter.

Example problem 2 – definition of triangular pyramid:

Calculate the volume of triangular pyramid where the base is 12 m, height is 20 m and length is 15m.

Solution:

Given:

Base b = 12 m

Height h = 20 m

Length l = 15 m

Volume of triangular pyramid =` 1/6` x b x h x l

= `1/6` x 12 x 20 x 15

= `1/6` 3600

= 600

Answer: volume of a triangular pyramid = 600 cubic meter.

Add Improper Fractions

Introduction about fraction:
If two real numbers are in a / b form, then it said to be a fraction notation,

Here,   a (numerator)/ b (denominator) --- > both ‘a’ and ‘b’ not equal to zero.

Types of fraction:

1. Proper Fraction.

2. Improper Fraction.

3. Mixed Fraction.

Proper Fraction:

In fraction the numerator is lesser than the denominator means, such a fraction is called proper fraction.    

Example:

1/3, 2/5, 5/9.

Improper Fraction:

In fraction the numerator is greater than the denominator means, such a fraction is called improper fraction

Example:

5/3, 7/2, 9/5.

Mixed Fraction:

Mixed Fraction is the mixture of whole number and proper fraction.

Example 2 3/5, 1 ¾

Rules for Adding Improper Fraction:
Step 1: Initial step is the check whether the denominator is same or different.

Step 2: If the denominators are same add the numerators

Step 3: If the denominators are different then we have to make the common denominator by taking the Least common denominator

Step 4: Convert the denominator as a least common denominator

Step 5: Finally add the numerators

Note: When we add two improper fractions the result is also an improper fraction

Problems on add improper fraction

7/5+11/7                

Solution:

Step 1: Here the denominators are different

Step 2: Take the LCD for 5,7

LCD of 5,7 = 35

Step 3: To make the common denominator as 35

Multiply and divide by 7 for the first term and 5 for the second term

7/5*7/7 + 11/7 * 5/5

49/35+55/35

Step 4: Add the numerator,

49+55/35 = 104/35

Hence the answer for 7/5+11/7 =104/35

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Problems on Add Improper Fraction

9/7+14/11

Solution:

Step 1: Here the denominators are different

Step 2: Take the LCD for 7,11

LCD of 7,11=77

Step 3: To make the common denominator as 77

Multiply and divide by 11 for the first term and 7 for the second term

9/7*11/11 + 14/11 * 7/7

99/77+98/77

Step 4: Add the numerator,

98+99/77 = 197/77

Hence the answer for 9/7+14/11 =197/77

Problems on add improper fraction


6/5+12/10

Solution:

Step 1: Here the denominators are different

Step 2: Take the LCD for 5,10

LCD of 5,10 = 10

Step 3: To make the common denominator as 10

Multiply and divide by 1 for the first term

6/5*2/2 + 12/10

12/10+12/10

Step 4: Add the numerator,

12+12/10 = 24/10

Hence the answer for 6/5+12/10 =24/10

Tangent Angle Circle

Introduction to tangent angle circle:

The tangents to a circle are the lines that are from a point outside the circle and intersect the circle at only one point on the circle. When two tangents of a circle are from a single point and angle is formed between the two tangents of the circle. In the following article we will see in detail about the topic tangent angle circle.

More about Tangent Angle Circle:
The tangent of a circle are the lines that intersect the circle at only one point on the circle. If two tangents of a circle arises from a single point outside the circle then there is an angle formed between the tangents of the circle. This angle between the tangents of the circle can be related to the angle covered by the arcs that are formed by the intersection of the tangents with the circle.

Angle between the two tangents from a single point to the circle:

The angle x is the measure between the two tangents is half of the difference between the major arc and the minor arc.

Angle x = (Angle covered by the major arc – Angle covered by the minor arc)/2

Angle between a tangent and a secant from a single point to the circle:

In some cases there can be a tangent and a secant arising from a common point outside the circle in those cases the angle between a tangent and a secant from a point to the circle is half of the difference between the major arc angle and the minor arc angle.

Angle x = (Angle covered by major arc – Angle covered by the minor arc)/2

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Example Problem on Tangent Angle Circle:

1. Find the angle between the two tangents in the given diagram.

Solution:

The angle x = (Angle covered by major arc – angle covered by minor arc)/2

`= (245-115)/2`

`= 130/2`

`= 65` `degrees`

2. Find the angle between the two tangents to the circle in the given diagram.

Solution:

The angle x = (Angle covered by major arc – angle covered by minor arc)/2

`= (275-85)/2`

`= 190/2`

`= 95 degrees`

Practice problem on tangent angle circle:

1. Find the angle x between the tangent and the secant to the circle in the diagram.

Answer: 50 degrees

Real Numbers Integers Rational

Introduction to real numbers integers rational:

A real numbers integers rational number is a number that can be represented as a/b, where an a and b are integers and b ≠ 0. From the definition of a rational number, we see that fraction such as -1/2, 3/4 and 2/5 are rational numbers. If we replace b in a/b by an integer 1, we have a/b = a/1 =a. Hence the integers b = b/1, -8 = -8/1, 0 = 0/1 and so on are real number integers rational.

Properties of the Real Numbers Integers Rational:

In real numbers integers rational the division of two integers does not necessarily result in an integer. Hence, there is a need to include fractions to form a bigger set of numbers known as set of rational numbers. The rational numbers and the irrational numbers completely fill the number line and form the set of real numbers. We can summarise the properties of the rational number as given below. Let a, b and c be any rational numbers.

1) Commutative:

a + b = b + a.

a * b = b * a.

2) Associative:

(a + b)+c = a + (b + c)

(a * b)*c = a * (b * c)

3) Identity:

a + 0 = 0 + a = a

a * 1= 1 * a = a

4) Distributive:

Multiplication can be distributed over addition.

a * (b + c) = (a * b) + (a * c)          

5) Closure:

The system of rational numbers is closed under addition, subtraction, multiplication and division (except by 0).

a + b, a – b, a * b, a / b, (b ≠ 0) are all rational numbers.

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Examples for Real Numbers Integers Rational:

1) Find three real numbers integers rational between 1/5 and 1/3.

Solution:

Let q1, q2, q3, be the three required the rational numbers. Then

q1 = 1/2 (1/5 + 1/3) = 1/2 (3 + 5/ 15) = 1/2 * 8/15 = 4/15

q2 = 1/2 (1/2 + 1/3) = 1/2 (4 + 5/15) = 1/2 * 9/15 = 3/10

q3 = 1/2 (3/10 + 1/3) = 1/2 (9 + 10/30) = 1/2 * 19/30 = 19/60

1/5 < 4/15 < 3/10 < 19/60 < 1/3. 

Hence, three rational numbers between 1/5 and 1/3 are 4/15, 3/10 and 19/60.

2) Find nine real numbers integers rational numbers between 2/5 and 1/2.

Solution:

2/5 = 0.4, 1/2 = 0.5, clearly, we have

0.4 < 0.41 < 0.42 < 0.43 < 0.44 < 0.45 < 0.46 < 0.47 < 0.48 < 0.49 < 0.5

Therefore, nine rational numbers between 2/5 and 1/2 are

41/100, 42/100, 43/100, 44 /100, 45/100, 46/100, 47/100, 48/100, 49/100.