Thursday, December 27, 2012

Polynomials Chart

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminate) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number. (Source Wikipedia)

In this article polynomial chart we see about basic concepts of polynomial, its types of polynomial ,some example problems

Polynomial Types:


Basic concepts of polynomials:
Polynomial is nothing but algebraic expression and also concept of algebras. More types of polynomials are available in the algebra depends on the number of terms. Based on the number of terms polynomial was classified four types

Different types of polynomial:

Polynomial chart

Types of polynomial

Monomial:

If the expression having one term mean it was called as monomial

Example: 7x ,8x2

Binomial:

If the expression having two terms mean it was called as binomial

Example: 6x+4x

Trinomial

If the expression having three terms mean it was called as trinomial

Example: 3x+8x2+9

Polynomial:

If the expression having more than three terms mean it was called as polynomial

Example: 5x2+12x3+9x+10

Polynomial operations are addition of polynomial, subtraction of polynomial, multiplication of polynomial, division of polynomial.

Example Problems in Polynomial:

Example problems in Polynomial degree chart:

Polynomial addition chart:

Example 1:

Add the polynomial 3x2+5x+2 and 5x+6

Given polynomials: 3x2+5x+2,5x+6

Now we have to arrange the  terms for addition

After than add the terms one by one.

This is a polynomial addition chart

Polynomial addition

Example 2:

Polynomial multiplication chart:

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

Now we have to multiply the one terms with another terms

And then add the terms

Polynomial multiplication

Example 3:

Degree of polynomial:

(9z9 +8 z4 − 6z5 + 8) Find the degree of polynomial for each term?

Degree of polynomial for first term=9

Degree of polynomial for second term =4

Degree of polynomial for third term=5

Degree of polynomial for fourth term=0

Highest degree of polynomial is 9


Tuesday, December 25, 2012

Ratio to Fraction Converter

Ratio :

In mathematics, The ratio can be used to relate two quantities by using the symbol : Also it can be expressed as follows,

  • x is to y
  • the ratio of x to y
  • x : y

Fraction :

In mathematics , Part of the whole can be expressed as fraction. There are three kinds of fraction
  • Proper fraction
  • Improper fraction
  • Mixed fraction

In this article we are going to see about how to simplify the ration as fraction by using the ratio to fraction converter.

Ratio to Fraction Converter :

Converter:

The electronic or software device that can perform the operations Quickly. The ratio to fraction converter can be used to convert fraction for the given ratio.

Ratio to fraction converter

Fig(i) Ratio to fraction converter

Let us see some problems on ratio to fraction convertor.

Problems on Ratio to Fraction Converter :

Problem 1:

Convert the ratio 45 : 180 into simplified fraction

Solution:

Given,The ratio 45 : 180

We need to convert the given ratio into fraction .

we know that 45 : 180 = ` 45/ 180`

Divided by  45 on both numerator and denominator,

`45/180` = `( 45 / 45 ) / ( 180 / 180 )`

= `1 / 4`

Answer: The simplified fraction of the given fraction is  `1/4` .


Problem 2:

In a bag, there is Blue  and Green balls, the ratio of Blue balls to Green balls is 5:6. If the bag contains 180 Blue balls, how many green balls are there?

Solution:

Given The ratio of the Blue and green balls = 5 : 6

Number of blue balls = 180

Let us take x = green balls

To find the green balls we need to convert the given ratio into fraction,

Write the items in the ratio as a fraction.

`(blue) / (green)` = `5/6` = `x / 180`

`5/6` = `x / 180`

Multiply by 6 on both sides,

5 = `x / 180 `

5 = `x / 30`

Now multiply by 30 on both sides,

5 * 30 = x

150 = x

x = 150

Total number of black balls = 150

Answer: Green balls = 150

Sunday, December 23, 2012

Square Root of Complex Number

A square root of an x is a numeral r such to r2 = x, or a numeral r whose square is x. each non-negative real number x have a exclusive non-negative square root, recognized the principal square root, signify by a essential symbol as . For optimistic x, the main square root knows how to as well be written in exponent information, as x1/2.  (Source: Wikipedia)

Square Root of Complex Number:


A complex number is a number regarding of a real also imaginary part. It know how to be written in the structure a + bi, where a and b are real numbers, also i is the ordinary imaginary unit among the property i 2 = −1. The complex numbers include the ordinary factual numbers, however expand them by adding in extra numbers and equally increasing the considerate of addition also multiplication.

Every positive numbers x contain two square roots. One of them is` sqrt(x)` , to be constructive, with the other `-sqrt(x)` that is negative. Collectively, these two roots are signifying `+-sqrt(x)` . Square roots of unconstructive numbers know how to be converse in the structure of complex numbers. More commonly, square roots know how to be thinks in any context in that a notion of square of several arithmetical objects is definite.

Example for Square Root of Complex Number:


Example 1:

Solve square root of compel numbers `sqrt(3) - isqrt(6)`

Solution:

Step 1: the given square root complex number is `sqrt(3) - isqrt(6)`

Step 2: `x = sqrt(3) and y = -sqrt(6)`



Step 3:    `r =sqrt( x^2+y^2)`

Step 4: substitute x and y values

`r =sqrt( (sqrt(3)^2)+(-sqrt(6)^2))`

Step 5:   r = 3

Example 2:

Solve square root of comple numbers 4+13i

Solution:


Step 1: the given square root complex number is   4+13i

Step 2: the given problem of the form is a+bi

Step 3: `x = 4 and y = 13`

Step 4:    `r =sqrt( x^2+y^2)`

Step 5: substitute x and y values

`r =sqrt( (4^2)+(13^2))`

Step 6:    `sqrt(377)`

Step 7:   r = 19

Step 8: finding b value

` b =sqrt((r-x)/2)) `

Step 9:    ` b =sqrt(((19)-4)/2) `

Step 10:    ` b =sqrt((15)/2) `

Step 11:  b = 2

substitute y and b in a

Step 12:  a = `y/(2b)`

a = `13/(2(2))`

a = 3

Step 13: find square root of  r1 and r2

r1 =a + bi = 3 + 2i

r2 = -a-bi = -3-2i

so the square root of complex numbers are 3+2i and -3-2i

Wednesday, December 19, 2012

Simplifying Radicals Activity

Simplifying radicals activity involves the process of solving radicals equation with step by step solution. Activity is the process of solving equations with radicals symbol. The square root symbol is also represented as radicals. Simplifying radicals is easily carried out by performing squaring operations on the given equation is known as simplifying radicals activity. The following are the example problems which explain the radicals activity.

In logarithm, the radical pictogram is represented by √. The symbol n√x , n, x and √ are represented by index , radical and and radical symbol. Steps to multiplying radicals is the similar way of ordinary multiplications, additionally it has the following conditions. When multiplying two or more radicals, we must multiply the numbers exterior radicals and then multiply the numbers in the interior radicals.

Steps to multiplying radicals:

The following steps to multiplying radicals, when the radicals have the indistinguishable key, n:

Step 1:  Utilize the multiplying radicals rule for nth roots to find the product of the radicals.

Step 2:  Simplify the product by factoring and taking the nth root of the factors that are ideal nth powers.

Rules on multiplying radicals:

The following rules for steps to multiplying radicals helps the easy way of understanding and simplifying the radical expression.

Rule 1:

n√x . n√y  =  n√(x . y)

Rule 2:

n√( x + y) . n√(x – y) = n√(x2 – y2)

Rule 3:

(x. n √y)n  =  xn . y

Simplifying Radicals Activity Example Problems:

Ex:1 Solve the radicals.


`sqrt(u^2-5u+7) = 1`

Sol:

Given equation is
`sqrt(u^2-5u+7) = 1`

To solve the radical symbol perform squaring operation on both sides
`[sqrt(u^2-5u+7)]^2 = (1)^2`

And simplify.
u 2 – 5u+7= 1

Make the above equation in factor form.
u 2 - 5 u + 6 = 0

The above equation is in quadratic equation form with two solutions
u = 3 and u = 2 is the answer.

Ex:2 Solve the radicals.

`sqrt(3u-5) = u - 1`

Sol:


Given equation is
`sqrt(3u-5) = u - 1`

To solve the radical symbol perform squaring operation on both sides

`[sqrt( 3u-5)]^2 = (u-1)^2`

Simplify the above equation
3 u - 5 = u 2 - 2 u + 1

Change the above equation in factor form.
u 2 - 5 u + 6 = 0

The above equation is in quadratic equation form with two solutions
u = 2 and u = 3 is the answer.

Ex:3 Solve the radicals.

`sqrt(u^2-16u+37) = 3`

Sol:

Given equation is
`sqrt(u^2-16u+37) = 3`

To solve the radical symbol perform squaring operation on both sides
`[sqrt(u^2-16u+37)]^2 = (3)^2`

And simplify.
u 2 – 16u+37= 9

Make the above equation in factor form.
u 2 - 16 u + 28 = 0

The above equation is in quadratic equation form with two solutions
u = 2 and u = 14 is the answer.

Practice Problems on Simplifying Radicals:

Q:1 Solve the radicals.

` sqrt(u^2-12u+29) = 3`

Answer: u = 2 and u = 10.

Q:2 Solve the radicals.

`sqrt(4u-3) = u -2`

Answer: u = 7 and u = 1

Multiplying Radicals:


Ex:1 (a)     Steps to multiplying radicals √2 . √36.

Sol:

1.   Multiply the i nside radical numbers  :          = √(2 .36)   = √72

2.   Simplify radicals if possible             :          =  √(2 . 62 )

3.   Answer                                           :          = 6 √2

(b) Steps to multiplying radicals  √5 . √7

Sol:

1.    Multiply the inside radical numbers = √(5 . 7)

2.    Simplify radicals if possible     =  √35

(c)   Steps to multiplying radicals                                      4√7 . 7√5

Sol:

1.   Multiply the outside of radical numbers first      (4 . 7) = 28

2.   Multiply the inside radical  numbers              √(7 . 5)  = √35

3. . Put steps 1 and 2 together                                   28√35

4.  Answer                                                                 28√35

Ex:2 (a) Steps to multiplying radicals √(4 + 6) . √(4 - 6)

Sol:

√(4 + 6) . √(4 - 6)       (given)

= √(42 – 62)

= √(16 – 36)

= √-20                  (Simplify radicals if possible )

= 2√-5                           (Answer)

(b) Steps to multiplying radicals √(3 + 2) . √(3 - 2)

Sol:

√(3 + 2) . √(3 - 2)           (given)

= √(32 – 22)

= √(9 -4)                  (Simplify radicals if possible )

= √5                                  (Answer)

Examples on rule:

(a) Steps to multiplying radicals  (7√3)2

Sol:

(7√3)2          (given)

= 72 . 3

= 49 . 3

= 147                  (Answer)

(b) Steps to multiplying radicals  (3√2)2

Sol:

(3√2)2            (given)

=  32 . 2

=  9 . 2

=  18               (Answer)

Sunday, December 16, 2012

Standard Deviation Percentage

 The standard deviation is  most commonly used term in statistics. The relative standard deviations to consider the accuracy of compute the standard deviation of given analytical data. The Standard deviation is the square root of average squared deviation from the mean.

Standard Deviation  :

Standard  deviation is the arithmetic mean of all the deviation of observations taken about their mean

Standard equation When each of the given terms has frequency 1  
Let x1, x2, …, xn be the n given observations and let M be their mean. Then, the variance σ2 is given by
σ2 =[ (x1  M)2 + (x2  M)2 + … + (xn – M)2 ] / n = Σ d2i /n,
where the deviation from the mean, di = (xi  M)
And, therefore, the standard deviation σ is given by
σ = + √{Σ(xi  M)2/n} = √Σ di2/n, where di = (xi – M)

Formula to Calculate Percentage of Standard Deviation:


When frequencies of the variable are given
In this case, the variance is given by
σ2 = (Σ fi di2 fi) & S.D.= σ = √(Σ fi di2)/n
we proceed in same way  as we have done earlier
but here each di is multiplied by correponding fi
and apply the above formula
and we get standard deviation for frequency distribution

Ex :  Find the variance and standard deviation from the following frequency distribution table:

Variable (xi)246810121416
Frequency (fi)445158545

Sol :  We have

Variable
xi
Frequency
fi
fi xi_ 
di = (xi – M)
di2fi di2
248– 749196
44`16–     525100
6530–  3945
815120–  1115
10880118
125603945
14456525100
16580749245
Σ fi = 50Σ fi xi = 450Σ fi di2 = 754

... M =
450/50
= 9

... Variance, σ2 = Σ fi di2/ Σ fi = 754/50 = 15.08
And, standard deviation, σ = √15.08 = 3.88

Percentage of standard deviation:

Percentage of standard deviation or relative standard deviation = (standard deviation / mean)  x 100.

Calculating Percentage of Standard Deviation:

Calculate the variance as well as the standard deviation percentage of the given table of the data:

xi7101216182528
fi251310641

Solution:
Presenting the data in tabular form, we get

xififixi(xi - mean)(xi - mean)2fi(xi - mean)2
7214-864128
10550-525125
1213156-39117
16101601110
1861083954
25410010100400
2812813169169
416161003


Here, N= 41 and `sum_(i=1)^7` fixi  = 616.

Therefore, mean = (`sum_(i=1)^7` fixi ) `-:` N = (1/41) x 616   =  15



and                `sum_(i=1)^7`fi(xi - mean)2   = 1003.

Hence, Variance(σ2)  =  (1/ N) `sum_(i=1)^7` fi(xi - mean)2  =  (1/41) x 1003   =  24.46

and

Standard deviation (σ) = `sqrt(24.46)`   =  4.94

Relative standard deviation or standard deviation percentage = (σ / mean) x 100  =  (4.94 / 15 ) x 100     =   32.9%

Wednesday, December 12, 2012

Division of Decimal Numbers

There are some rules of dividing the decimal numbers. Different methods of division are applied in dividing the decimal numbers.

(1)   Division of a decimal by a whole number:

Rules: (a) Divide as in division of numbers.

(b) When you reach the tenth digit, place the decimal in the quotient.

When the number of digits in the dividend is less and the division is not complete, keep adding zero at every step till the division is complete.

Example:  (1) 100.4 ÷ 25

25)100.4(4.016

100_
40-------------- Put zero

25_
150------------ Put zero

150
X

(2)   1.2 ÷ 25                                    ‌‌‌

25)1.20(0.048

100_
200               Since 12 is not multiply of 25, so 0 is added.

200_
Xx

(2)   Division by decimal numbers by 10, 100, 1000 etc.



Rules: (a) While dividing a decimal by 10, 100 or 1000 etc, multiples of 10 the decimal shifts to the left by as many places as there are zeros in the divisor.

(b) If the number of places in the integral part is less, then put the required number of zeros to the left of the integral part, then shift the decimal point.



Example: (a) 71.6 ÷ 10



=       716    ÷10
          10

           716      x    1_
=        10           10



=      716
       100

= 7.16 Ans.



(b) 923.07 ÷ 100



92307 ÷ 100

100

=      92307      x     1__
         100              100

=         92307
           10000

=        9.2307 Ans.

Rules of Division of Decimal Numbers Continued

(3)   Division of decimal numbers by multiples of 10, 100, 1000 etc.

Example: 245.1 ÷30

3)245.1(81.7

24___
x  5

3__
21

21_
X                  so, 245.1 ÷ 30 =8.17



245.1

30           = 245.1 =    245.1     x     1_

3x10        3               10

=   81.7 x    1

10       =   8.17 Ans.

(4)   Division of a decimal by a decimal:

Example: 14.7 ÷ 2.1

147 ÷ 21

10    10

=   147   x   10

10         21

= 7 Ans.

Alternative method: (a) count the number of decimal digits in the divisor.

(b) Move the decimal in dividend that many places to right.

(c) Write the divisor without the decimal.

(d) Now divide the numbers as usual.

(5)   Division of a whole number by a decimal:

Rules: (a) Count the number of decimal digits in the divisor.

(b) Add as many zeros to the dividend.

(c) Remove the decimal in the divisor.

(d) Divide as usual.

Example: 42 ÷ 0.7

42.0 ÷ 0.7

= 420 ÷ 7 = 60 Ans.

Exercise of Decimal Number Fractions

Divide: (1) 234.65 by 25

(2) 193.92 by 800

(3) 8.16 by 0.24

(4) 68 by 4.25

(5) 44 by 176

Answer:  (1) 9.386 (2) 0.2424 (3) 34 (4) 16 (5) 0.25

Sunday, December 9, 2012

Trig Equation Solving Examples


Trigonometric equations is shortly called as trig equations.  An equation involving trigonometrical function is called a trigonometrical equation.

cosθ =`1/2` , tanθ = 0, cos2θ − 2sinθ =`1/2 `

There are some examples for trigonometrical equations. To solve these equations we find all replacements for the variable θ that make the equations true. A solution of a trigonometrical equation is the value of the unknown angle that satisfies the equation. A trigonometrical equation may have infinite number of solutions. The solution in which the absolute value of the angle is the least is called principal solution. In this article let us study trig equations solving examples.

Trig Equation Solving Examples:

Let us see sample problems for trig equation solving examples.

General solutions of sin θ = 0 ; cosθ = 0 ; tan θ = 0

Find the principal value of the following:

(i) cosx =`sqrt3/2`

Solution: (i) cosx =`sqrt3/2` > 0

∴ x lies in the first or fourth quadrant. Principal value of x must be in[0, π]. Since cosx is positive the principal value is in the first quadrant

cosx =`sqrt3/2` = cos`pi/6` and `pi/6`

∈ [0, π]∴ The principal value of x is `pi/6` .

(ii) cosθ = −`sqrt3/2` < 0

Since cos θ is negative, θ lies in the second or third quadrant. But the  principal value must be in [0, π] i.e.  Within  1st  or  2nd  quadrant. The principal value is in the 2nd quadrant.

cosθ = −`sqrt3/2` = cos (180° − 30°) = cos150°.

The principal value is θ = 150° =`(5pi)/6` .

Trig Equation Solving Examples:

Projection formula

In any triangle ABC  a = b cos C + c cosB

True with usual notations and it is called projection formula.

Proof:

In triangle ABC, draw AD perpendicular to BC.From the right angled triangles ABD and ADC,

cosB =BD

AB ⇒ BD = AB × cosB

cosC =DC

AC ⇒ DC = AC × cosC

But BC = BD + DC = AB cosB + AC cosC

a = c cosB + b cosC

or a = b cosC + c cosB


Solve : sin2x + sin6x + sin4x = 0

Solution:

sin2x + sin6x + sin4x = 0 or (sin6x + sin2x) + sin4x = 0 or 2sin4x. cos2x + sin4x = 0

sin4x (2 cos2x + 1) = 0

when sin4x = 0 ⇒ 4x = nπ or x =`(npi)/4` ; n ∈ Z

When 2 cos2x + 1 = 0 ⇒ cos 2x =− `1/2`

= − cos`pi/3` = cos (π −π3)= cos`(2pi/3)`

∴ 2x = 2nπ ±`(2pi)/3` or x = n π ±`pi/3`

Hence x =`(npi)/4` or x = nπ ±`pi/3`

; n ∈ Z

Please visit this website and Know more on Derivative of Cosine .


Wednesday, December 5, 2012

The Number -‘One followed by Six zeros’



The number system started with the natural number system .Now we have huge numbers
being represented. They can be either positive or negative. They can be fractions. They can
be rational or irrational. The natural numbers start with 1 and extend up to infinity. One
million is also natural number. One million is nothing but 10 lakhs. It is the number of zeros
that follow ‘1’ that is important. The number of zeros decides the value of the number.
In ten lakhs, we have six zeros. In ten we have one zero. In hundred two zeros, thousand
three zeros, ten thousand four zeros, lakh five zeros and finally in ten lakhs six zeros. So,
the number of zeros is very important. Now we will try to write 1 million in numbers to
understand this. It comes after the number 999999 in the natural number system. It is also
the number which comes before the number 1000001 in the natural number system. So,
we now understand how to write 1 million after coming to know its position in the natural
number system. Expressing  million in numbers is quite easy and counting up to a million can
be a difficult task.

In the modern era we use the term a million rather than saying ten lakhs or thousand
thousands to represent the same number. Using this term has become very easy. So, we
need to know how to write 1 million in numbers as this term is very commonly used. The
word million is derived is one of the most beautiful languages in the world, Greek. This
shows how to write one million in numbers and can be clearer if we understand it to be just
a number on the natural number system number line.

Just studying a million is not enough we need to study its multiples too. If we do that
we can be clearer with the concepts. So, how to write millions in numbers if there are
numbers greater than a million. We now come to the concept of a billion. In a billion there
are nine zeros. One is followed by nine zeros to form a billion. This shows that there are
1000 millions in a billion. A billion is also known as a milliard. The natural number system
further increases to trillion, quadrillion and so on. A trillion consists of a one followed by 12
zeros.These are very large numbers.

Thursday, November 29, 2012

Derivative Math Problems


In math, differentiation is the process of finding the derivative which means measuring how a function changes with respect to its input. The derivative of y with respect to x is given by`(dy)/(dx)` . The reverse process of derivative is antiderivative. Following is the example and practice problems of math derivative which helps you for learning derivative in math.

Example Problems of Math Derivative:

Learn math derivative with Example problem 1:

Find the derivative of the function y = 7x8 + 2x6 + 5x5

Solution:

Step 1: Given function

y = 7x8 + 2x6 + 5x5

Step 2: Differentiate the given function y = 7x8 + 2x6 + 5x5 with respect to ' x ', to get `(dy)/dx`

`(dy)/dx` = 56x7 + 12x5 + 25x4

Learn math derivative with example problem 2:

Find the derivative of the function y = sin (7x4 + 1)

Solution:

Step 1: Given function

y = sin (7x4 + 1)

Step 2: Differentiate the given function y = sin (7x4 + 1) with respect to ' x ', to get `(dy)/dx`

`(dy)/dx` = [cos(7x4 + 1)] (28x3)

= 28x3 cos (7x4 + 1)

Learn math derivative with example problem 3:

Find the derivative of the function y = ecos(9x - 14)

Solution:

Step 1: Given function

y = ecos (9x - 14)

Step 2: Differentiate the given function y =  ecos (9x - 14) with respect to ' x ', to get `(dy)/dx`

`(dy)/dx` = ecos (9x - 14) (- 9sin (9x - 14))

= - 9 ecos (9x - 14) sin (9x - 14)

Practice Problems of Math Derivative:

A few practice problems are given below with solutions which gives you idea how to differentiate the function.

1) Find the derivative of the function y = 16x2 + 11x + 5

2) Find the derivative of the function y = - 5tan (7x2 + 2)

3) Find the derivative of the function y = `6/5` e sin (5x - 9)

Solutions:

1) 32x + 11

2) - 70x sec2 (7x2 + 2)

3) 6e sin (5x - 9) cos (5x - 9)

Thursday, November 22, 2012

Solving Equations with Exponents

Introduction:

Exponent equations are the equations in which variable appear as an exponent.

To solve these equations rules and laws of exponents are used. Exponent equations are of two types
(1) Exponent equations in which bases are same
(2) Exponent equations in which bases are different.

Steps to Solve Equations with Exponent

Solving Exponential Equations of the same base

1) Ignore the bases, and simply set the exponents equal to each other
2) Solve for the variable

 When the bases of the terms are different

1) Ignore the exponents; rewrite both of the bases as powers of same number.
For example if there are 2 and 4 in the bases, then convert base 4, in to base 2
by writing it again as (2)^2
2) once the bases are same , ignore them
3) Equalize the exponents
4) Solve for variable

Simple Problems of Equations with Exponents

  Solve for variable Answer
1. 3m  =  35 Since the bases are the same, set the exponents equal to one another:
m = 5
2. 5t   = 125 125can be expressed as a power of 5:
5= 53
t = 3
3.  493y=343 49 and 343 can be expressed as a power of 7:

[(7)2]3y = 73

76y = 73
6y = 3
y = 1/2

More Problems of Equations with Exponents

  Solve for x. Answer
1.  52x+1  =  53x-2 Since the bases are the same, set the exponents equal to one another:
2x + 1 = 3x - 2
3 = x
2.  32x-1  = 27x 27 can be expressed as a power of 3:
32x-1  = 33x

2x - 1 = 3x

-1 = x
3.   43x-8  = 162x 16 can be expressed as a power of 4:

43x-8= [(4)2]2x

3x - 8 = 4x
 
-8 = x


Monday, November 19, 2012

Units of Meters


In this article discuss about standard units (international system) of meters. The meter (or meter), symbol m, is the base unit of length in the International System of Units (SI).The meters is the length of path travelled by light in vacuum during a time interval of 1/299792458 of a seconds.

The Basic Units of Length is Meters (m):

Linear measures     Short form of writing

Millimeter                         mm

Centimeter                         cm

Decimeter                          dm

Meter                                m

Decameter                         dam

Hectometer                         hm

Kilometer                           km

Some examples of units values:

1 mm = 0.001 m

1 cm = 0.01 m

1 dc = 0.1 m

1 km = 1000 m

Units of Meters - Examples:

Units of meters - Example 1:

Convert 378.6 cm into meter

Solution:

Here the conversion is from centimeter to meter. i.e. from lower unit to higher unit

100cm = 1m.

Hence 378.6cm = 378.6/100 m

= 3.786 m (shifting the decimal two digits to the left)

Units of meters - Example 2:

Convert 40.1735 km into meter.

Solution:

Here the conversion is from higher to lower. Hence we have to shift the decimal point to the right.

1km = 1000 m

40.1735 km = 40.1735 × 1000 m

= 40173.5 m

Units of meters - Example 3:

Convert 6m into millimeter.

Solution:

Note: Here we are converting the unit meter into millimeter. i.e. from higher unit to lower unit. So the operation should be multiplication.

1 m = 1000 mm

Hence 6 m = 6 × 1000 mm

= 6000 mm

Units of meters - Example 4:

Convert 778.6 cm into meter

Solution:

Here the conversion is from centimeter to meter. i.e. from lower unit to higher unit

100cm = 1m.

Hence 778.6cm = 778.6/100 m

= 7.786 m (shifting the decimal two digits to the left)

Units of meters - Example 5:

Convert 90.1736 km into meter.

Solution:

Here the conversion is from higher to lower. Hence we have to shift the decimal point to the right.

1km = 1000 m

90.1736 km = 90.1736 × 1000 m

= 90173.6 m

Units of meters - Example 6:

Convert 9m into millimeter.

Solution:

Note: Here we are converting the unit meter into millimeter. i.e. from higher unit to lower unit. So the operation should be multiplication.

1 m = 1000 mm

Hence 9 m = 9 × 1000 mm

= 9000 mm.

Sunday, October 28, 2012

Convert Fractions to Decimals


How to convert fractions to decimals with calculator ?

In this article, let us learn what is fraction and what is decimal and how to convert fractions to decimals using calculator.

Fractions: Let us divide a circle into four parts and take a part away and ask yourself a question,"How many parts are taken away from the whole ?" It is one part from 4. we could represent this as 1/4. So, fraction is a number composed of two parts say top part which is called as numerator and the bottom part is called as denominator.So, numerator gives the required part whereas denominator part gives the number of parts in the whole.

Example: `(15)/(17)`

Decimals: Decimals are numbers which has a dot which we say it as decimal point.

Example: 0.234, 67.45

Convert Fractions to Decimal( Relation)


We could observe that 0.5 is in 1/10th position. It means that 0.5 = 5/10

We could observe 9 in 0.59 is in 1/100th position. It means that 0.59 = 59/100

We could observe 1 in 0.591  is in 1/1000th position. It means that 0.591= 591/1000

If any fraction has the denominator has 10, 100, 1000or any multiples of 10  then the fraction is called as decimal fraction.

Convert Fractions to Decimal( Examples)

Consider a fraction `(1)/(4)`

Let us see how to convert the above fraction to decimal using calculator.

Choose a number such that the number when multiplied with denominator becomes a number which is a multiple of 10.So, i am choosing the number 25 since 25 x 4 = 100

Multiply the number 25 with the numerator and denomiantor.

So, 1 x 25 = 25 and 4 x 25 = 100

1/4 = 25/100

Since 100 has two zeros keep decimal point after counting 2 numbers from the right.

So, 25/100 = 0.25

In some case, we could not make the denominator as multiples of 10. For eample, 2/3. We could not find a number such that when denominator multiplied by the number gives denominator as multiples of 10.But 3 x 333 = 999

Since 999 is nearest to 1000, multiply the numerator and denomiantor with 333. So,

2/3 = 2 x333 / 3 x 333 = 666/999 = 0.666 (approximately)

Otherwise, to convert fractions to decimals with calculator, enter the numerator in the calculator and press the division sign and then the denominator. The resulting number in the calculator is the decimal.

Tuesday, October 23, 2012

Solve Rational Equation Problems


In algebra solving rational equation is very simple,we have few rules to solve any type of equations. Whatever we do on  one side  of the equation, we must do to the other side also. If you have fractions, we can try to eliminate them by multiplying by the common denominator. If there are quadratics involved in our equations, we must get all the  terms to one side with zero on the other.The basic rational expression is in the form of fraction.where there is at least one variable in the denominator.

Solved Example Based on Rational Equation :

Ex 1:Solve `3/x+6=2/(4x)`

Sol:

Step 1: The given equation is

`3/x+6=2/(4x)`

subtracting both sides by `2/(4x)`

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

Step 2: Rearrange the equation.

`3/x-2/(4x)+6=0`

subtracting both sides by 6.

`3/x-2/(4x)+6-6=0-6`

`3/x-2/(4x)=-6`

Step 3: Find the l.c.d(least common denominator) x and 4x

L.C.D=4x

`(12-2)/(4x)=-6`

`10/(4x)=-6`

Step 4:Multiply both sides by 4x.

`10/(4x)xx4x=-6xx4x`

10=-24x

Step 5:Divide both sides by -24

`10/(-24)=(-24x)/(-24)`

`x=-5/12`

The solution is  [x=-5/12]

Example Based on Rational Equation:

Ex 2: Solve `x/(x-2)+1/(x-4)=2/(x^2-6x+8)`

Sol:

Step 1:first factor the `x^2-6x+8`

factors=(x-4)(x-2)

Step 2:convert common denominator to all

`(x/(x-2))((x-4)/(x-4))+(1/(x-4))((x-2)/(x-2))=2/((x-2)(x-4))`

`(x^2-4x)/((x-2)(x-4))+(x-2)/((x-2)(x-4))=2/((x-2)(x-4))`

`(x^(2)-4x)+(x-2)=2`

`x^(2)-4x+x-2=2`

`x^(2)-3x-4=0`

`(x-4)(x+1)=0`

`x=4 or x=-1`

If we submit the x=4 in the denominator,it is division by zero,so x=4 is not considerable.

x=-1   is the answer

Friday, October 19, 2012

Positive Integers Tutoring



Tutoring means a tutor taking an interactive session to individual student or a group of student in a class through online. The learning integer is defined the equal to the whole number. The negative numbers are including in the learning integer tutor. The integers do not used the fraction number. Each positive integer contains same negative integers. Let us discuss about the positive integers tutoring.

Positive Integer Tutoring

The general structure of integer is {…, -4, -3, -2, -1, 0, 1, 2, 3, 4…}. The integer is commonly representing the three types of numbers.

The first type is positive counting number. The second type is negative counting number. The zero value is third type.

The positive integer is equal to the whole number. The general structure of the positive integer is {1, 2, 3, 4, 5…}. The positive integer is also called as the positive number counting. This number is specifying positive value only. The + is the sign of the positive integers.

The value +86 and 86 is representing the same value. The positive integers do not necessary for the sign representation. The value 55 is automatically representing the positive integer.

The non negative integer is equal representation of the positive integer. The main difference of the positive integer and non negative integer is the positive integer starts with 1 but the non negative integer starts with 0. Many operations are performing the positive integer tutoring.

Example Problem of Positive Integer Tutoring

Problem 1:

Calculate the following positive integer.

2415 + 1564 =?

Solution

2415
1564(+)
--------
3979
---------

Problem 2:

Calculate the following positive integer.

4698 - 2456 =?

Solution

4698
2456 (-)
--------
2242
---------

Practice problem of positive integer tutoring

1. Add the positive integers 568 + 637.

2. Subtract the integers 634 – 542.

Answer

1. 1205

2. 92

Wednesday, October 3, 2012

Theory of Proportions

Introduction to theory of proportions:
       The theory of proportion is one of the basic topic in mathematic. In our usual life, there are a lot of occasions as we compare two quantities by means of their measurements. As soon as we compare two quantities of the same type by division, we contain a ratio of those two quantities.
Definition of Ratio: Ratio means similarity of two like quantities by division.

Definition of Theory Proportions:

      Proportions is a correspondence of two ratios.
      Consider the proportions
                a: b = c: d
      The first and fourth terms (a and d) are knoen as the extreme terms or extremes.
      The second and third terms (b and c) are known as the middle terms or means.
Important property:
      Product of extremes = Product of means.

Examples of Theory Proportions:

Let us see some examples of theory of proportions.
Example 1:
      Verify 5: 6 = 10: 12 is a proportion or not.
Solution:
      Product of extremes = 5*12 = 60
      Product of means = 6*10 = 60.
             `:.`   60 = 60
       These two products are equal.
            `:.` 5:6 = 8: 6 is a proportion.
Example 2:
       Verify 6: 7 = 12: 15 is a proportion or not.
Solution:
       Product of extremes = 6*15 = 90
       Product of means = 7*12 = 84
                          `:.`    90 = 84
       These two products are not equal.
         So, 6: 7 =12: 15 is not a proportion.
Example 3:
       If 2: 3= 6:_ is a proportion, find the missing term.
Solution:
       Let us assume the missing value is x
       Product of extremes = 2*x
       Product of means = 3*6 =18
       Since it is a proportion, 2*x =18
                                          2x = 18
       Divide both sides by 2 on both sides we get,
                                        2x/2 = 18/2
                                             x = 18/2 = 9
       `:. ` The missing term is 9
       So the proportion is 2:3 =6: 9

Example 4:
       The income and Savings of a family are into the ratio 8: 3 If the income of the family is Rs. 3,300.Find how much is being saved.
Solution:
       Let us savings be Rs. x.
          `:.` The proportion is 8: 3 = 3300: x
                     (Income: saving) = (Income: saving)
                                       11x = 9900
                                   11x/11= 9900/11
                                           x= 900
      Therefore, the savings = Rs. 900.

Tuesday, October 2, 2012

Step by Step Adding Fractions

Introduction to step by step adding fractions:
                  A fraction 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, ⅝, ¾, etc.) and which consist of a numerator and a denominator.

                                                                                                                                                 

Step by Step Adding Fractions:

1. Add the two fractions `20/3` and `22/4`
         Solution:
                                   The  two fractions are  `20/3` and `22/4`
                 Step1:             Add two fraction
                                              =`20/3` +`22/4`
                 Step2:          There the denominator are different  so we need to take lcm  to preform addition
                                                           =`(80+66)/12`
                  Step3:           By adding 80 and 66 the answer is 146
                                                         =`146/12`
                   Step4:                This can be simplified has
                                                           =12.1
 2. Add the two fractions `40/4` and `50/5`
            Solution:
                                  The  two fractions are `40/4` and `50/5`
                   Step1:                 Add two fraction
                                                     =`40/4` +`50/5`
                   Step2:     There the denominator is sameso we need to take lcm  to preform addition
                                                            =`(200+200)/20`
                   Step3:       By adding 200 and 200 the answer is 400
                                                                =`400/20`
                   Step4:           This can be simplified has
                                                       = 1
 3. Add the two fractions `60/5` and `70/6`
          Solution:
                                  The  two fractions are `60/5` and` 70/6`
                    Step1:          Add two fraction
                                                  =`60/5` +`70/6`
                    Step2:   There the denominator is same so we need to take lcm  to preform addition
                                                        =`(360+350)/30`
                    Step3:     By adding 360 and350 the answer is 710
                                                              =`710/30`
                    Step4:        This can be simplified has
                                                              = 23.66

Step by Step Adding Fractions:

4. Add the fractions `5/6` and `25/30`
      Solution:
                                        The two fractions are `5/6` and `25/30`
                 Step1:         The given Two fractions are equivalent fractions
                                         By simplifying `25/30` we get 5/6
                  Step2:
                                          Now we need to find the sum of `5/6` and `5/6`
                                                                   = `5/6` +`5/6`
                   Step3 :            There the denominator are same so add the numerators in the fractions together
                                                                   =`(5+5)/6`
                                              The sum of 5 and 5 is 10
                                               So `(5+5)/6` can be written as `10/6`
                  Step4:
                                             This can be reduced further as `5/3`
5. Add the two fractions `80/6` and `90/6`
        Solution:
                                     The  two fractions are `80/6` and `90/6`
                  Step1:             Add two fraction
                                                =`80/6` +`90/6`
                   Step2:       There the denominator is same so add the numerators in the fractions together
                                                            =`(80+90)/6`
                      Step3:        By adding 80 and 90 the answer is 170
                                                          =`170/6`
                      Step4:          This can be simplified has
                                                        = 28.33


 6. Add the two fractions `1000/3` and `120/4`
         Solution:
                                  The  two fractions are `100/3` and `120/4`
                Step1:           Add two fraction
                                         =`100/3` +`120/4`
                Step2:     There the denominator is same so we need to take lcm  to preform addition
                                           =`(400+3690)/12`
                Step3:      By adding 400 and 360 the answer is 760
                                                  =`760/12`
                Step4:        This can be simplified has
                                                  = 63.33