Three Ways to Measure an Angle

Introduction :

When two rays intersect at a point they form an angle. The point at which they intersect is known as the vertex of the angle. The amount of rotation needed to rotate one ray about the vertex to coincide with the other ray is known as the magnitude or measure of an angle. The magnitude or measure of an angle is also equal to ratio of arc length to the length of the radius of the circle.

Explanation on three Ways to Measure an Angle:

What are the Three ways to measure an angle?

There are many ways to measure an angle. One way to measure angle is using degrees. The degree is further divided using decimal numbers and hence can express angles to any precision like hundredths, thousandths etc. Ex. 28.456 degrees represents 28 degrees and 456 thousandths. In this systme the degrees can also be subdivided into 60 parts. Each 1/60 of a degree is known as minute. The minute is again subdivided into 60 parts. Each 1/60 of a minute is known as second. Ex. 43 degrees 24 minutes 10 seconds. In this format the angle is usually written as 43° 24' 10''.

The second way to measure an angle is using radians. 1 radian is equal to the angle when arc length is equal to the radius of the circle. 1 radian is equal to 180º/∏ degrees or 57.296 degrees.

The third way to measure an angle is using gradien, gon or grad. In this system a circle is divided into 400 parts. Each part is equivalent to 1 grad.

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Three Ways to Measure an Angle: - Conversion from One Form to Another:

Lets see how one form is converted to the other forsm and viceversa - Three ways to measure an angle:

1) Convert Degree to radians and viceversa :

Degree to Radians: To convert angle in degrees to radians multipy the angle by `(pi)/(180 ^0)` .

Example: Convert 60 degrees to radians.

60° * `(pi)/(180 ^0)`  = `(pi)/(3)` radians

Radians to Degrees: To convert angle in radians to degrees multiply the angle by `(180)/(Pi)`

Example: Convert `(pi)/(2)` radian to degrees

`(pi)/(2)`  radian * `(180)/(Pi)` = 90°

2) Convert degree to Grad and viceversa :

Degree to Grads: To convert angle in degrees to grads multiply the angle by `(10)/(9)` .

Example: Convert 270 degrees to grads

270° * `(10)/(9)` = 300 grads

Grads to degrees: To convert angle in grads to degrees multiply the angle by `(9)/(10)`

Example: Convert 200 grads to degrees

200 * `(9)/(10)`  = 180°

Convert Radians to Grad and viceversa :-

Radians to Grads: To convert angle in radians to grads multipy the angle by `(200)/(pi)`

Example: Convert `(pi)/(4)`  radians to grads.

`(pi)/(4)`  * `(200)/(pi)`  = 50 grads

Grads to Radians: To convert angle in grads to radian multipy the angle by `(pi)/(200)`

Example: Convert 400 grads to radian

400 * `(pi)/(200)`  = 2∏ radians

Number Theory and Inequalities

Introduction for inequalities and number theory:

           Number theory is the branch of the pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into the several fields, according to the methods used and the type of questions investigated. These are the somewhat older terms, which are no longer as popular as they once were. Inequalities are the statement about the relative size or order of two objects or about whether they are the same or not.


Number Theory and Inequalities:-examples for Number Theory:

Ex 1:

Let a sequence can be defined by a1 = 1, a2 = 1, an = an–1 + an–2 for n > 2. Find the sequence.

Sol:

a1 = 1, a2 = 1an = an–1 + an–2 for n > 2

a3 = a2 + a1 = 1 + 1 = 2

a4 = a3 + a2 = 2 + 1 = 3

a5 = a4 + a3 = 3 + 2 = 5

? The sequence is: 1, 1, 2, 3, 5,...

Ex 2:

 Find the common difference and the next three terms of the A.P. 1, 4, 7, ...

Sol:

The common difference = 4 – 1 = 3

The next three terms are 7 + 3 = 10, 10 + 3 = 13

13 + 3 = 16

Integers:

Ex1:

The numbers 0, 1, -1, 2, -2 … are called integers

 1, 2, 3 … are called positive integers

 -1, -2, -3… are called negative integers. The collection of all integers is denoted by the letter Z. Thus Z = {…, -3, -2, -1, 0, 1, 2, 3…}.

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Number Theory and Inequalities:-examples for Inequalities:

 Solve the inequalities - x2 + 3x - 2 > 0

Sol:

- x2 + 3x - 2 > 0 ? - (x2 - 3x + 2) > 0

? x2 - 3x + 2 < 0

? (x - 1) (x - 2) < 0

On equating the factors to zero, we get x = 1, x = 2 are the roots of the quadratic equation. Plotting these roots on number line and making positive and negative alternatively from the right most part we get the corresponding number line as given below.

The three intervals are (- 8, 1), (1, 2) and (2, 8). Since the sign of (x - 1) (x - 2) is negative, select the interval in which (x - 1) (x - 2) is negative. ? x ? (1, 2)

Practice Problem for Inequalities Theory:

Solve : 4x2 - 25 = 0

Answer: Thus the solution set is (- 8,- 5/2 ] ? [5/2 , 8)

Inverse fraction

Introduction to Inverse fraction: 

          In mathematics, an inverse fraction or reciprocal for a number x, denoted by 1/x or x -1, is an element which when inverse by x yields the inverse identity, 1. the inverse of a fraction a/b is b/a. For the inverse fraction of a real number, divide 1 by the number. For example, the inverse fraction of 5 is one fifth (1/5 or 0.2), and the inverse fraction of 0.25 is 1 divided by 0.25, or 4.
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Inverse Fraction-example 1:

Find the inverse fraction of 5/2

Solution:

Inverse fraction is 5/2 = 2/5 

The solution of inverse fraction is 2/5

Inverse fraction-Example 2:

Find the inverse fraction of 8/4

Solution:

Inverse fraction is 8/4 = 4/8 

The solution of inverse fraction is 4/8

Inverse fraction-Example 3:

Find the inverse fraction of 4/2

Solution:

Inverse fraction is 4/2 = 2/4 

The solution of inverse fraction is 2/4

Inverse fraction-Example 4:

Find the inverse fraction of 3/2

Solution:

Inverse fraction is 3/2 = 2/3

The solution of inverse fraction is 2/3

Inverse fraction-Example 5:

Find the inverse fraction of 7/2

Solution:

Inverse fraction is 7/2 = 2/7 

The solution of inverse fraction is 2/7

Inverse fraction-Example 6:

Find the inverse fraction of 5/9

Solution:

Inverse fraction is 5/9 = 9/5 

The solution of inverse fraction is 9/5.

Inverse Fraction-example 7:

Find the inverse fraction of 5/1

Solution:

Inverse fraction is 5/1 = 1/5 

The solution of inverse fraction is 1/5

Inverse fraction-Example 8:

Find the inverse fraction of 9/2

Solution:

Inverse fraction is 9/2 = 2/9 

The solution of inverse fraction is 2/9

Inverse fraction-Example 9:

Find the inverse fraction of 11/2

Solution:

Inverse fraction is 11/2 = 2/11 

The solution of inverse fraction is 2/11

Inverse fraction-Example 10:

Find the inverse fraction of 5/12

Solution:

Inverse fraction is 5/12 = 12/5 

The solution of inverse fraction is 12/5

Inverse fraction-Example 11:

Find the inverse fraction of 5/23

Solution:

Inverse fraction is 5/23 = 23/5 

The solution of inverse fraction is 23/5

Inverse fraction-Example 12:

Find the inverse fraction of 5/21

Solution:

Inverse fraction is 5/21 = 21/5 

The solution of inverse fraction is 21/5

Inverse fraction-Example 13:

Find the inverse fraction of 15/2

Solution:

Inverse fraction is 15/2 = 2/15 

The solution of inverse fraction is 2/15

Inverse fraction-Example 14:

Find the inverse fraction of 5/17

Solution:

Inverse fraction is 5/17 = 17/5 

The solution of inverse fraction is 17/5

Inverse fraction-Example 15:

Find the inverse fraction of 13/2

Solution:

Inverse fraction is 13/2 = 2/13 

The solution of inverse fraction is 2/13.

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Introduction to equivalent fractions worksheet

Introduction to equivalent fractions worksheet:

            Before the introduction of  the decimal system children needed to learn a lot more about fractions,  as this was the only way to show a part of a whole number. In the past, using fraction such as 5 / 2 and 3 / 5 to describe shares of objects or groups of objects was common. Today these fractions have been replaced by decimals and the calculation are often done and writing fraction is done differently to whole numbers.

             A fraction is made up of a numerator and a denominator. This area of mathematics has often caused problems for both teachers and students alike, this concern however, is unnecessary if the correct grounding is given and basic concepts are understood.

Definition of Equivalent Fraction and Examples Worksheet:

Equivalent fractions:

              The equivalent fractions are fractions that are equal to the each other. We can use cross multiplication to decide to whether two fractions are equivalent. The fractions that show the same amount are called equivalent fractions.

              The equivalent fractions of the same value or equivalent means equal in value. Fraction can look different but be equivalent. These fractions are really the same,

            Example: 1/2 = 2/4 = 4/8

            The rule for equivalent fraction multiplying numerator and denominator of a fraction by the same number or a whole fraction, the results of fraction is said to be equivalent to the original fraction. The equivalent fraction that two fractions values have the same value and they retain of the same integrity and proportion.

             The common denominator is add and subtract fraction each fraction must have a common denominator they must be same thing. In fraction we must find a number that all the denominators will divide evenly into, Example look at the fraction 1 / 2 and 1 / 3.

             The denominators for these fractions are 2 and 3. A number that 2 and 3 will divide into evenly is 6. We can express both of these fraction as sixths, and so give them both a common denominator.                                                                                      

Equivalent Fraction Worksheet:

1.      3 / 4 = 6 / 8
2.      1 / 2 = 2 / 4
3.      2 / 3 = 4 / 6
4.      2 / 5 = 4 / 10
5.      2 / 9 = 4 / 18

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Simplify the Equivalent Fraction and Examples Worksheet:

Simplify the equivalent fraction:

                 The equivalent fraction simplify a fraction we find a number which will divide into both the  numerator and the denominator evenly, leaving no remainder .Example, to simplify the fraction  6 / 10 we divide the numerator and denominator by 2. So, 3 / 5 is the simplified fraction for 6 / 10.

Sample problem:

 8 / 4 = 4 / 2.
2 / 4 = 1 / 2
3 / 9 = 1 / 3
6 / 3 = 2 / 1 = 2
4 / 6 = 2 / 3.

Introduction for practice ratio problems

Introduction for practice ratio problems:

              In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient. Example: For every Spoon of sugar, you need 2 spoons of flour (1:2)
             In our daily life, by learning ratio and proportion many a times we compare two quantities of the same type. Thus, in convinced situations, comparison by division makes better sense than comparison by taking the difference. The comparison by division is the Ratio. We denote ratio-using symbol ‘:’ . If two ratios are equal, we state that they are in proportion and use the symbol ‘:’ or ‘=’ to equate the two ratios.

Practice Examples for Ratio Problems:

Problem 1:

             In a bag of red and green sweets, the ratio of red sweets to green sweets is 3:4. suppose  the bag contains  totally 120 green sweets then  how many red sweets are there?

Solution:

Step 1: Assign variables:

               Let x = red sweets

            Write the items in the ratio as a fraction.

                Red / green = 4 / 5 = x / 120

Step 2: Solve the equation 

                   Cross Multiply

                      4 × 120 = 5 × x
                             480 = 5x

                  Isolate variable x
                          x = 480 / 5 = 96

Answer: There are 96 red sweets.

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Problem 2:

There are 280 students in a certain school. If the number of boys in a school are 150. Find the ratio between

            i) Number of girls to number of boys

            ii) Number of boys to total number of students

Solution:

           Number of students = 280

           Number of boys = 150

Number of girls = Total number of students – Total number of boys

                             = 280 – 150

                             = 130

i) Number of girls to number of boys

             Number of girls: number of boys

                             = 130: 150

             Dividing each term by 10

                             = 13: 15
             So, ratio between girls to the boys is 13: 12

ii) Number of boys to total number of students

                Number of boys: number of students

                                   = 150: 280

                Dividing each term by 10

                                   = 15: 28
                So, ratio between boys to the students is 15 : 28

Problem 3:

Length and breadth of a rectangular field are 80 m and 20 m respectively. Find the ratio of the length to the breadth of the field.

Solution:

  Length of the rectangular field = 80 m

              Breadth of the rectangular field = 20 m

              The ratio of the length to the breadth is 80: 20

              The ratio can be written as

              = 80 / 20 = 4:1

Thus, the required ratio is 4:1

Practice Ratio Problems:

Practice problem 1:

              Find the ratio of 24 yards to 36 yards.

              Answer: 3: 4

Practice problem 2:

              Find the ratio of 39 days to 30 days.

              Answer: 13: 10

Practice problem 3:

              Mrs. Alice earns $ 160 per month. She spends $ 8 and saves the rest. Find the ratio of her salary to savings?

               Answer: 20: 1

Introduction to factorization in algebra

Introduction to factorization in algebra

                      In Algebra, factorization is the process of simplifying an algebric equation by factoring out the common factors from all terms. Thus, the expression can be expressed in terms of the product of two or more simple expressions. In some cases, the factorization of algebric equations is done by splitting some terms into two or more terms. For example the factorization of a quadratic equation in algebra is done by splitting the middle term in to two terms.

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                      For example, factorization of the polynomial, x² − 16 factors as (x − 4) (x + 4). In all cases, a product of simpler Expressions is obtained. Here x² – 16 is a second degree polynomial.Those factors  (x + 4) and (x – 4) are first degree polynomials. This factorization is useful in simplifying expressions.

Methods of Factorization:

1) Factorization in algebra by removing common factors:

When the terms of an algebraic expression X  have a common factor Y, we divide each term of X by Y and get an expression Z. Now, X is factored as Y × Z.

Ex:Factorize: 6x⁴y³ – 4x²y² + 10xy³.

Solution : We observe that 2xy² is a common factor .

6x⁴y³ – 4x²y² + 10xy³ =2xy²[ (6x⁴y³)/2xy² )-( 4x²y²/2xy² )+( 10xy³/2xy² ) ]=2xy²(3x³y – 2x + 5y).

2)Factorization in algebra  by grouping method:

Solution:Grouping method is used for the given expression contains three or more terms.

Steps:

In the first step, if the terms having common factors, then those terms to be grouped.

In the second step, the greatest common factor (GCF) is taken out.

Finally, in the third step, the distributive law is used to find the factors.

The distributive law is,

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

Factoring in algebra by grouping: x³+3x²−6x−18

Solution:In the first step, if the terms have common factors, then those terms to be grouped.

(x³+3x²) + (−6x−18)

Factor x² out of the first two terms, and factor −6 out of the second two terms.

x²(x+3)-6(x+3)

Note that there is a common factor, x+3. So, take (x+3) as common.

(x+3) (x²-6) is the final factorization.

x³+3x²−6x−18=(x+3) (x²-6).

Few more Methods of Factorization:

3) Factorization of quadratic expression in algebra:

Consider the quadratic expression like 15 – 2x – x².   

Solution: Writing in the standard form,

15 – 2x – x² = –x² – 2x + 15

                      = (–1) (x² + 2x – 15).

Using the splitting method, If we can find two numbers p and q such that

p + q = 2 and p q = -15, then we can get the factors.

So , p + q = 5+(-3)=2 and p*q =5*(-3)= -15

Hence, we get 15 – 2x – x² = (–1) (x² + 2x – 15).

                                                 = (–1) [(x+5) {x + (–3)}]

                                                 = (–1) (x +5)(x – 3)

                                                 = (x + 5) ( 3 – x).

                                                 = (3x + 1) (2x + 5)

4) Factoring Difference of Two Squares:

Factorization of the difference of the two squares in the form x² - y², we should remember the formula

(x²-y²)=(x + y) (x - y).This is the special cases of factorization.

Example: a²-4 = (a + 2) (a - 2)