Subtracting Fractions

Subtraction is one of the basic operations in math. It is a process of finding the difference of two numbers. In case of whole numbers the process is simple but in case of fractions, the process may involve a number of steps.

How to Subtract Fractions
As mentioned earlier, subtraction of fractions require a number of steps. Of course, in the simplest cases, where the denominators of the fractions are same, the process is just like subtracting whole numbers. All that need to be done is just subtract the numerators over the given common denominator.

But in case of fractions with different denominators, first task is to find the equivalents of the given fractions with same denominator.  Such a equivalent common denominator is also called as Lowest Common Divisor. It is same as the lowest common multiple of the denominator.

How do you Subtract Fractions
Let us concentrate on the case of subtractions of fractions with different denominators, as the case of subtraction of fractions with same denominators is as simple as subtraction of whole numbers.
The easiest method for students to understand is the method of equivalent fractions. The first step here is, determine the lowest common multiple of the given denominators.

Then find ‘equivalent fractions’ of the given ones with the lowest common multiple as the denominator. That is rewrite the given fractions as if their denominators are changed to the lowest common multiple found. For example, 1/2 can be rewritten in the equivalent form as 2/4, 4/8,8/16 etc. depending on the need.
The subtraction of given fractions is same as the subtraction of their equivalent fractions and now the process is simple because the denominators are made same.

Subtract Fractions
Let us discuss a specific example to illustrate how to subtract fractions with different denominators. Let is consider that 2/5 is subtracted from 3/7.

The denominators are 5 and 7 and the lowest common multiple is 35. The equivalent fraction of 2/5 with denominator is 14/35 and that of 3/7 with the same denominator is 15/35.

Now subtraction of 2/5 from 3/7 is same as subtraction of 14/35 from 15/35 which is (15 – 14)/35 = 1/35. Thus the answer is 1/35.

Subtracting Fractions with Whole Numbers
For subtracting fractions with whole numbers, first subtract the fraction parts and then the whole number parts and combine them For example, 5(1/2) – 3 (1/4) is done by (1/2) – (1/4) 1/4 and 5 – 3 = 2, which gives the final answer as 2(1/4).

However trouble arises when the fraction part of the first mixed number is less than that of the second mixed number. In such cases, convert the first fraction part to an improper fraction by borrowing 1 from the whole number. The rest of the procedure is same as explained but remember the whole number part of the first one is reduced by 1!

How to Multiply Two Digit Numbers

If we have practiced and hopefully memorized the multiplication tables and Multiplication Rules, then we can solve any multiplication problem. We just have to understand the system to how to do it. We know that multiplication is just repeated addition but if we have large numbers, we cannot add them repeatedly to get the solution. To multiply one digit number and two digit number with a one digit number is an easy process but multiplying two double digits numbers uses a different process that needs to be followed.

How to Multiply Two Digit Numbers - We will learn here how Double Digit Multiplication works. Let us start with an example of the multiplication problem of two double digits numbers. Let us do 16 times 19; we can break down the Two Digit Multiplication into series of steps which are given as follows: -

• In 2 Digit Multiplication, firstly we take the numbers present in ones place and multiply them together; in this case we multiply 9 times 6 which equal 54.
• With the product of the digits at ones that is 54, we only write the 4 down while multiplication and 5 is carried forward on the tenth place just like when we add two numbers.
• Then we multiply 9 times 1 which is 9 and then add 5, and which is 14 so the solution to 9 times 16 is 144. Thus 16 times 9 is 144.
• Then we take the digit on the tens place on one number that is one in the number 19 and multiply it with 16, which gives us 16.
Now we have two solutions, one is 144 and other is 16, to find the final answer we add a zero to the second solution and then add the solutions together. That means 16 will become 160 and then we add it to 144 which equals 304. So the solution of 16 times 19 is 304.

 Only by using these breaking down method we have solved this big problem.  To multiply two digit numbers we can either break the one double digit number into two digits. For example, in the previous example, we can break 19 into 1 and 9. 9 is at ones place so we can write it as 9 but 1 is at tens place so we can use its face value which is 10. Then we can multiply 9 by 16 which give us 144 and then we multiply 10 by 16 which equals 160. We can add both the solutions to get the solution 304.

Definition of an Acute Angle

The different types of angle have different types of names. Where the angle is less than 90 radiant is called as acute angle. The acute angle is smaller angle compared with other type of angle such as right angle, obtuse angle, straight angle, reflex angle, and full angle.  The angle which is larger is called as reflex angle.

Definition of a Acute Angle
Definition acute angle, an angle which is less then 90 radiant is called as acute angle. Compared to the other types of angle, an acute angle is very smaller that is in between 0-90 radiant. Also that acute triangle are those where all the interior angle are acute.

Definition of an Acute Angle
An angle of less then 90˚ is called as acute angle and it has three angles. An angle which is exactly 90˚ is a right angle but it is not an acute angle, since it is less than 90˚ or between 0˚-90˚.

Acute Angle Definition Math
An angle which measures less than 90˚ or in between 0˚-90˚ is called as acute angle. In other words, acute angle is a positive angle that measures less than 90˚. Simply it is defined as, an angle of less than 90˚.

Geometry Acute Angle
In geometry, an angle is line segments that intersect in the same plane or it is the amount of bend between two lines. The angles are formed by two intersecting rays at the point of intersection or sharing of same end point. The angles are used to measure the turns in between the two arms. The unit of an angle is degree or radiant. In order to measure an angle from a circle that is a two dimensional plane, if the angle measures greater than 0˚ and less than 90˚ is called as acute angle. The acute angle is referred by the alphabets like ABC or DEF.

In above figure angle A is acute which measures greater than 0˚ and less than 90˚. It is formed by the intersection of two rays like AB and AC, which are called as sides of angle or arms. Where the angle is formed is called as vertex. In figure, the angle 1 which is, marked by square sign is called acute angle. Acute angle is smaller than right angle which is exactly equal to 90˚. Similarly the angle 2 is complementary of an angle 1. Complementary angle of an acute angle is also acute as it measures greater than 0˚ and less than 90˚.

Exterior Angles in Polygon and Triangle

Exterior Angle
Consider a shape such as square ABCD. Extend the horizontal bottom of the square at the point D up to F. Now we have a vertical line BD and a horizontal line DF joining at the common vertex D. The angle BDF formed between the original side BD of the mathematical shape i.e. square and the extended side DF is termed to be the exterior angle. The angle formed on the inner side of BD is the interior angle BDC.  The sum of the interior angle and the exterior angle formed using the side BD is 180 degrees. This forms a horizontal straight line CF measuring 180 degrees.

Polygon
A mathematical shape which has straight sides and flat shape is termed to be polygon.

Exterior Angles Polygon
Exterior Angles Polygon definition states that: With one of the angles of the polygon, a linear pair is formed by an angle which is termed as an exterior angle of the polygon.

At every vertex of a polygon, two exterior angles can be formed at the maximum.  Each exterior angle of a polygon is formed in between a side of the polygon and the line extended from its adjacent side.

Finding Exterior Angles of Polygons
We can find the exterior angles formula for polygons. The important point to be noted here is that at each vertex of the polygon, two equal exterior angles can be drawn but the formula for finding the exterior angles of polygon uses only one exterior angle per vertex.

Polygon Exterior Angles Formula:
The polygon exterior angles formula states that the sum of Polygon Exterior Angles is 360 degrees, irrespective of the type of the polygon. In other words, we can say that the sum of all the exterior angles of a polygon is equal to one full revolution.

Sum of one exterior angle of all vertex of any polygon = 360 degrees.

In case of a regular polygon, the formula to find any exterior angle is obtained by dividing the sum of the exterior angles i.e. 360 by the number of angles, say “n”.

The value of an angle of a regular polygon = 360/n

Here are few examples of Hexagon:

Example 1:
Hexagon has 6 sides. Therefore every exterior angle in hexagon = 360/6 = 60 degrees.

Example 2:
The value of an angle of a regular polygon is 30 degrees. How many sides are there in the polygon?
360/n = 30 which can be rewritten as 360 = 30n.
Thus, n is given by dividing 360 by 30 which results in 12.

Exterior Angles Triangle
In case of triangle, exterior angle lies in between one side of the given triangle and the extension of the other side of the same triangle. Exterior Angles of Triangle can be obtained by adding the measures of the two of the non-adjacent interior angles.

Math Homework Help

As many parents will know, math homework is something kids avoid like the plague. Math may not be everyone's favorite subject but there is no denying it's importance in day to day life. Math skills are necessary and students who take an interest in the subject early on, will have a much easier time studying it in high school.

Much of the lack of interest in math boils down to the way it is delivered in class. To ensure that every student understands what is being taught, teachers need to employ different techniques to explain the concepts. This does not happen very often and students vary between having a vague idea and being completely clueless. When they have to finish their homework, in most cases students simply don't know enough to do their work. Some parents help out but in many households parents are either too pressed for time or not well-versed enough in math themselves to help their kids.

Getting help with math is the best solutions for students and parents who find themselves in this situation. Math tutoring has become very popular over the past decade with several students signing up for them, starting as early as elementary school. Math tutors wok with students on an individual basis, giving them ample time to learn at their own pace and clear every single doubt. Hundreds of students in schools across the country use math tutoring and as a result, have aced their tests and exams, getting As and Bs where they were previously failing.

Many math helpers feature help with math homework as a regular part of their services that students can make use of everyday. Unlike online calculators and programs which calculate the answer for the questions students input, getting professional help ensures that students really learn the concept or theory and how to put it into practice. Students can also practice with math tutors which allows them to really explore the topic as there is someone who can correct them, if need be. Homework help has helped students keep up with their schedules and submit assignments on time, all the while learning more about the topic.

All about acute angles

A very common question for a 5th grader who has just started to learn geometry would be: what is an acute angle? Or What is a acute angle? (Considering that they are not really sure what article to use before the word ‘acute’). Let us now describe an acute angle in simple terms.

A acute angle:
Look at the hands of a clock on your bedroom wall when the time is say 2:00 pm. The hour and the minute hand of the clock make an angle with each other at the centre of the clock. That angle would be an acute angle. From 12:00 noon to 3:00 pm, the angles made by the hands of the clock are acute angles. At 3:00 pm the hands of the clock are at right angles to each other. After 3:00 pm the angle between the hands of the clock are obtuse angles.

If you have a pine tree around your home, look at the top of the tree. The angle made by the two lines in the form of an inverted V at the top is an acute angle.

Acute angle definition:
An angle whose measure is more than 0 degrees and less than 90 degrees is called an acute angle. If the angle measure is in radians, then the angle whose measure is more than 0 radians and less than pi/2 radians is called an acute angle.
Mathematically it is written like this:
If angle A is such that measure of angle A = m0 < = (mIn geometry an acute angle can be generally sketched as follows:

Acute angles in geometry:

An equilateral triangle has all three angles as acute angles. Each of the angles in an equilateral triangle is 60 degrees or pi/3 radians. See picture below:


In a right triangle, one of the angles is 90 degrees (or pi/2 radians), but the other two angles are acute angles. For example see the picture below:

The angle subtended by the chord of a circle at any point in the major segment of the circle would always be a right angle, except if the chord is a diameter of the circle. In such a case, the angle subtended would be a right angle. See the following figure to understand that better:

In the above figure, BC is the chord of a circle and the angle subtended by BC at A = α. Α would be an acute angle.

Basics about Circles

Definition: A circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). Properties of a circle encompass use of terms such as chord, segment, sector, diameter etc of a circle. Now let us try to understand some other terms related to properties of circles.

Properties of circle:
A straight line that intersects a circle in two distinct points is called a secant to that circle. In the picture below, we have a circle with centre at C. A line l intersects this circle in two points, A and B. This line is a secant to the circle.

A straight line that intersects (or touches) a circle in just one point is called a tangent to the circle at that point. For a circle at a given point, there can be only one tangent. The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide. See picture below.

Circle theorems:

1. The tangent at any point of a circle is perpendicular to the radius through the point of contact. This we can see in the above picture. The tangent is perpendicular to the radius that joins the point of contact with the centre of the circle.

2. The lengths of tangents drawn from an external point to a circle are equal.

Circle formulas geometry:

Area of a circle: Area of a circle is given by the formula:
A = pi r^2
Area of semi circle: Area of a semi circle of radius r is given by the formula:
A = (pi/2)r^2

Segment of a circle:

The portion (or part) of the circular region enclosed between a chord and the corresponding arc of the circle is called a segment of the circle.

In the picture above, the orange portion is called the minor segment of the circle and the yellow portion is the major segment of the circle. The minor segment corresponds to minor arc and the major segment corresponds to the major arc of a circle.

Area of a segment of a circle is found using the formula below:

Where, theta is the angle subtended by the chord at the centre of the circle and r is the radius of the circle.

Calculate Time - Fourth Grade Math

In grade four, new concepts are introduced in math. Some new concepts are time, multiples and factors, addition and subtraction of three digit numbers, unitary method, measures of length, mass and capacity, fractional numbers, addition and subtraction of fractions, decimals, addition and subtraction of decimals, Introduction to angles.

Introduction to grade four math:

In grade four, the topic time contains the following sub units.

- Measurement of time

-Calender

- Time in second

- Addition and subtraction of time

In this article let us learn about 24-hour clock time.

In the present day world business houses, airlines, railways are busy round the clock. Hence it is convenient to use 24 - hour time representation instead of a.m. and p.m.

1. 12 O' clock midnight is expressed as 00 00 or 24 00

2. 12 O' clock is expressed as 12 00

3. The time between 12 O' clock noon and 12 O' clock midnight is expressed by adding 12 hours to the given hours period.

For example:

25 minutes past 6 in the evening is expressed as 18 25

45 minutes past 11 midnight is written as 23 45

Rules for Writing 24-hour Clock Time:

A day begins at 12 midnight (00:00 hours) and hence at 12 midnight the following day.

Thus 1 day = 24 hours

Rule 1: For any time in a.m. we simply put down the time by writing hours and minutes in two digits numbers.

Rule 2: For any time written in p.m. we simply add 12 hours to the number of hours period and write minutes without separating them.

6:25 a.m is written as 06 25 hours

10:45 a.m is written as 10 45 hours

3:10 p.m. is written as 15 10 hours (3 + 12 = 15)

10:50 p.m. is written as 22 50 hours (10 + 12 = 22)

Example Problems on Grade Fourth Math:

Ex 1: Express 11:25 p.m. in the 24 hours system.

Sol:

Step 1: See which rule can be used.

Step 2: Since the time given is in p.m., add 12 to 11

Step 3: So, 11:25 p.m. = (11 + 12 hours) : 25 min

= 23 25 hours

Ex 2: Express 18 30 hours in terms of a.m. or p.m.

Sol:

18 30 hours means (12 + 6 hours) 30 minutes

= 6:30 p.m.

Ex 3: Express 07:45 p.m. in the 24 hours system.

Sol:

Step 1: See which rule can be used.

Step 2: Since the time given is in p.m., add 12 to 7

Step 3: So, 07:45 p.m. = (7 + 12 hours) : 45 min

= 19 : 45 hours

Trigonometric Integrals

Trigonometry is a fundamental concept of mathematics. It is used in calculus functions and vectors. In this topic we have to use trigonometry as integral function. That means how to integrate trigonometric functions. For this we also have to know what is integration?  Integration means to calculate area of a given curve, and the curve is a closed curve made by x axis and y axis.

Trigonometric integrals mean integration of trigonometric functions. As we know these trigonometric functions are basic formulas for solving trigonometric integral. To more simplify this term, let’s take an example like sin2X. This is a trigonometric function. And we integrate this function for this first we have to expand this term by using formula of trigonometry. After expanding we carry out the constant term then by using product rule of integral, we can integrate this trigonometric function.

Above example is simple it has only one trigonometric function but trigonometric function may be combine with other function also. It can be algebraic function with trigonometry, logarithmic function with trigonometry and exponential function with trigonometry. These are also called integrals of trigonometric functions. To solve this type of problem either we can use integration by substitution method or integration by parts method.

Inverse trigonometric integrals such as sin^-1X and cos ^-1X etc. now to integrate this type of functions we have to use basics of calculus. We need  to take this function equal to any constant like Y. means we have to write Y= sin ^-1X. now we transfer sin function to other site the we get. X=sin Y. Now we can simply integrate this term.

Trigonometric substitution integrals, here we also integrate trigonometric functions and calculus functions, but procedure is different. To integrate this type of function we have to substitute and equal trigonometric term in place of other trigonometric term. The first from of integrals is integration of [f’(x)/f(x)] dx=logf(x) . In this form integral of a function whose numerator is the exact derivative of its denominator and equal to the logarithmic of its denominator? The second form is, in the integrand consist of the product of a constant power of a function f(x) and the derivative of f(x), to obtain the integral we increase the index by unity and then divide by increase index. This procedure is known as power formula. Lets take an example suppose we have to integrate (4x^3/1+x^4) dx= ln (1+x^4). By using this method we substitute 1+x^4 = any constant term like (t), and after that we integrate this function.

Exponential Function an Introduction

An Exponential Function is a function which involves exponent which is the variable part rather than the base as in any normal function. For instance f(x)= x^3 is a function and an exponential function is something like g(x)= 3^x, here the exponent or the power is a variable (x) and the fixed value is the base (3). So, the definition of Exponential functions can be given as a function whose base is a fixed value and the exponent a variable. Example: f(x) = 5^x, here the base 5 is fixed value and the exponent ‘x’ is the variable.
In general, we can define Exponential Functions as a function which is written in the form ‘a^x’ in which ‘a’ is the base which is a fixed value or constant (‘a ‘not equal to 1) and ‘x’ the variable which is any real number. The most common exponential function we come across in math is e^x which is known as the Euler’s number.
Let us now take a quick look at the Exponential Function Properties. Consider the Exponential function f(x) = b^x for which the properties are as follows:
• The domain of the exponential function consist of all real numbers
• The range is the collection of all positive real numbers
• When b is greater than 1 then the function is an increasing function also called exponential growth function and when b is less than 1 then the function is a decreasing function also called exponential decay function
• The other properties that an exponential function satisfy are,
1. b^x.b^y = b^(x+y) [when bases are same and a multiplication operation then we can add the powers]
2. b^x/b^y = b^(x-y)[when bases are same and a division operation then we can subtract the powers]
3. (b^x)^y = b^(xy) [when a base is raised to a power x and raised to whole power y then we can multiply the powers]
4. a^x.b^x= (a.b)^x [when bases are different with the same power and a multiplication operation then we can multiply the bases whole raised to power]

We come across a function called an Inverse Exponential Function; this is nothing but a logarithm function.  We know that the exponential function is written in the form f(x) = b^x, to find the inverse of a given function we need to interchange x and y and solve for y. By interchanging we get x = b^y  and then solving for y gives us y = log x (base b) which is a logarithm function.

Points and lines tutoring

Tutor is the person who teaches the kids and this teaching section is the tutoring. Tutoring is an open source for the students to gain knowledge that is in online a point is nothing but the dot , it has no dimension or no width, it’s only a simple black dot. In geometry co ordinates of a point which shows the particular place in a segment for representation.Line has two end points is called segment. Line segment is denoted with a connected piece of line.line segments names  has two endpoints and it is named by its endpoints.

Points and Lines Tutoring:

Tutoring about the geometric points and lines we have to know the classification of a points and lines. points and lines classification are as follows.

Collinear points:
When three or more points lies on the same line is said to be collinear points.

Midpoint:
A halfway point where line segment divides into two equal parts are called midpoint.

Equidistant point:
A point which is said to be equidistant in a line segment where point is equal length from other points which are in congruent then the point is equidistant point.

Parallel line segment:
Two lines which does not touch each other are called parallel lines.

Perpendicular line segment:
Two line segment  that form a L shape are called perpendicular lines.

Problems in Points and Lines Tutoring:

Example 1:
Find the distance between the points A(6,3) and B (2,1).

Solution:
Let assume "d" be the distance between A and B.           (x1,y1)= (6,3), (x2,y2)= (2,1).

Then d (A, B) =`sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

= `sqrt((2-6)^2 +(1-3))^2)`

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

= `sqrt(16+4)`

=`sqrt20`
=2`sqrt5`

Example 2:
Find co-ordinate of the mid point of the line segment joining given points A(-5,3) and B(2,1)

Solution:
The required mid point is
Formul a   `((x_1+x_2)/2 ,(y_1+y_2)/2)` here,  (x1, y1) = (-5,3),(x2, y2) = (2,1)

=  `((-5+2)/(2))``((3+1)/(2)) `

=   `(-3/2) ` , ` (4/2)`

=    `(-3/2, 2)`

Example 3:
Find the slope of the lines given (8,-5) and (4,2)

Solution:
(x1,y1)= (8,-5), (x2,y2)= (4,2).
We know to find slope of line,m=` (y_2-y_1) /(x_2-x_1)`

=`(2+5)/(4-8)`

m =`7/-4`

Example 4:
Find the equation of the line having slope  3 and y-intercept 5.

Solution:
Applying the equation of the line is y = mx + c
Given,       m =3 ,c = 5
y =  3x +5

or  y = 3x+5
or  -3x+y-5 = 0
3x-y+5 = 0.

Introduction for division math facts

Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient is greater than one; otherwise it is less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × 1⁄b. When written as a product, it obeys all the properties of multiplication. (Source: Wikipedia)

Definition and Steps for Division Math Facts:

Definition for division:

Division is defined as an arithmetic function, which is the opposed process of multiplication. From the process of division, the proportion or ratio of two numbers be capable of be calculated.

Otherwise, the process of decision how many periods of one number is included in a further one. Symbol of division is ‘/’ or ‘÷’.

If we divide a number by another number, then

Dividend = (Divisor * Quotient) + Remainder

Steps for division math facts:

Step1. Division of two integers by the related signs resolve be positive sign

a) Positive integer ÷ positive integer = positive integer

b) Negative integer ÷ negative integer = positive integer

Step2. Division of two integers by the unlike signs will be negative

a) Positive integer ÷ negative integer = negative integer

b) Negative integer ÷ positive integer = negative integer.              

Division Math Facts Example Problems and Practice Problem:

Example problem for division math facts:

1. Solve the following division

36 ÷ 6

Solution:

36 ÷ 6

= (6 * 6) ÷ 6

Answer: 6

2. Solve the following division,

32 ÷ 4

Solution:

32 ÷ 4

= (4 * 8) ÷ 4

Answer: 8

3. Solve the following division,

48 ÷ 8

Solution:

48 ÷ 8

= (6 * 8) ÷ 8

Answer: 6

Practice problem for division math facts:

1.      - 49 ÷ 7 = -7 (unlike sign)

2.      56 ÷ 7 = 8 (like sign)

3.      48 ÷ 12 = 4 (like sign)

4.      81 ÷ 9 = 9(like sign).

Learning subtraction of square roots

A square of a number a is a number x. therefore x2=a .A number x whose square is a. Every positive real number a has a unique positive square root, called the principal square root. Square root denoted by a radical sign as sqrt of a. For positive ax, the principal square root can also be written in exponent notation, as a1/2. We can undo a exponent with a radical, and a radical can undo a power. The “`sqrt(a)` “symbol is called the "radical “symbol..The line across the top is called the vinculum.

Subtracting Square Root Terms

Subtracting square roots

Subtracting square roots is combining like terms when we need to do that with algebraic expressions. The induces (a square roots index is 2 `root(2)(a)` , a cube roots index is 3  `root(3)(a)` , a 4th roots index is 4 `root(4)(a)` ,a 5th roots index is 5 `root(5)(a)` etc.) or the radicands (enclosed by parentheses after SQRT or the expression under the root sign) are the same.

Just as with "regular" numbers, square roots can be subtracted together. But you might not be able to simplify the subtraction all the way down to one number. Just as "you can't subtract apples and oranges", so also you cannot combine "unlike" radicals. To subtract radical terms together, they have to have the same radical part.

Simplifying Square-Root Terms

Simplify a square root, we take out anything that is a perfect square; that is, we take out front anything that has two copies of the same factor.

We can raise numbers to powers other than just 2; we can cube things, raise them to the fourth power, raise them to the 100th power, and so forth.

(ab)^2=a^2b^2  and`sqrt(ab)`   = `sqrt(a)``sqrt(b)` but we can’t write this subtraction of square root  `sqrt(a-b)`  = `sqrt(a)` - `sqrt(b)`

Example Problems on Subtracting Square Root

Example Problems

1. (4 * `sqrt(2)` ) - (5 * `sqrt(2)` ) + (12 * `sqrt(2)` )
Solution: Combine like
= (4 - 5 + 12) * `sqrt(2)`
Answer is 11 * `sqrt(2)`

2. (53 * `sqrt(5)` ) - (5 * `sqrt(5)` )
Solution: Combine like terms
(53 - 5) * `sqrt(5)`
= 48 * `sqrt(5)`

3. (40 * `sqrt(5)` ) - (48* `sqrt(5)` )
Solution: Combine like terms by subtracting the numerical coefficients.
(40 - 48) * `sqrt(5)`
= -8 *`sqrt(5)`
`sqrt(3x+1)`-( -`sqrt(2x-1)` )= 1    subtracting 2 square roots with variables
`sqrt(3x+1)` = 1 - `sqrt(2x-1)`
Take square both sides
3x + 1 = 1 - 2 `sqrt(2x-1)` + 2x-1
3x + 1 - 1 -2x + 1 = -2 `sqrt(2x-1)`
x+1 = -2 `sqrt(2x-1)`
take Square both sides again
x^2 + 2x + 1 = 4(2x -1)
x^2 + 2x + 1 = 8x -4
x^2 -6x + 5 = 0
(x-5)(x-1) = 0
x1 = 5, x^2 = 1

Ogive – The cumulative line graph

In statistics, a frequency chart displays the given data, in which the frequency of each data item is found.  What does frequency mean? Frequency as we use in case of the frequency of the flight from one place to another means the number of times the particular flight travels from one place to another.  In statistics, frequency is used to display the number of times the data item occurs in a data set.  Tally marks or tallies are used to record and show the frequency of an item in a data. Now, let us learn about cumulative frequency. It is the total of the frequency and all the frequencies below it in a frequency distribution.  In simple words, it is the running total of frequencies. Given frequency of a set of data, the Ogive chart looks something similar to the chart given below:

Age Frequency      cumulative frequency
8      4 4
9      6 4+6 =10
10    15 10+15=25
11     9 25+9=34
12    18 34+18 = 52
13    10 52+10 =62

The Ogive Definition can be given as a distribution curve in which the frequencies are cumulative
Now that we have the cumulative frequencies, we shall now plot the graph. To plot the graph we take the ages on the x-axis and the cumulative frequencies on the y-axis as we plot a normal line graph. Once all the points are plotted, we now join the points. The curve we get is the cumulative frequency curve, also called the Ogive

We can define Ogive as a cumulative frequency graph which is a curve or graph showing the cumulative frequency for a given set of data. When the given data is an un-grouped data, to get Ogive, we find the cumulative frequency of the data and plot that on the y-axis and the given data to which cumulative frequency is calculated is taken on the x-axis. The graph we get is the Ogive of ungrouped data. When the data given is a grouped data, we divide the group into classes with upper and lower boundary which is taken on the x-axis and the cumulative frequency of the data on the y-axis. The graph we get here is the Ogive of a grouped data.

Ogive Example
For example, let us assume the amount of savings for the months of January and March as $200 and the savings of $125 for the months February, April and May.  For the given data, the ogive displays a running total of the savings with the amount saved in dollars on the y-axis and the months on the x-axis.

Trigonometric Identities | Theorems Based on Trigonometric Identities

Trigonometric Identies are some identies used in Trigonometry in order to make the calculations easier.
Trigonometry is a word consisting of three Greek words " Tri" means three, "Gon" means side, and "Metron" means measure. Thus, trigonometry is a study related to the measures of sides and angles of a triangle. Trigonometry is mainly used by captains of ships to find the direction and distance of islands and light houses from sea. Trigonometry is also used in astronomy, geography and engineering.
Trigonometric Ratios
In any right-angled triangle ABC,
let  angle B = 90 o  and angle C = T.                                                      

Line segment AC is the hypotenuse.
With reference to angle C, we can say that,
Line segment AB is the opposite side of Line segment BC is the adjacent side of Therefore, trigonometric ratios are given as,


Trigonometric Identities
Basic trigonometric identities are:
sin^2 T + cos^2 T = 1.
tan^2 T + 1 = sec^2 T
1 + cot 2T  = cosec^2 T
Theorems Based on Trigonometric Identities

Theorem 1: sin^2 T + cos^2 T =1
In right-angled triangle ABC, let angle< B = 90, angle< C = T.
Let AB = a, BC = b, and AC = c
By Pythagorean theorem we can say,
(hypotenuse)^2 = ( side)^2 + (side)^2
From figure we can say,
(AC)^2 =  (AB)^2 + (BC)^2
c^2 =  a^2 + b^2
divide throughout by c^2, we get,
(c^2 ) / c^2 =  ( a^2 + b^2 ) / c^2
1  =  a^2 / c^2 + b^2 / c^2
=  ( AB )^2 / ( AC)^2 + ( BC)^2 / (AC)^2
=   (AB / AC)^2 + ( BC / AC)^2
=   ( sin T )^2 +  ( cos T )^2
Therefore,
1 = sin^2 T + cos^2 T

Theorem 2: tan2 T + 1 = sec2 T
We have sin^2 T + cos^2 T = 1
Divide on both sides by cos^2 T,
( sin^2 T + cos^2 T ) / cos^2 T  =  1 / cos^2 T
(sin^2 T / cos^2 T) + (cos^2 T / cos^2 T)  =  1 / cos^2 T
By using trigonometric ratios,
sin T/ cos T  =  tan T
1 / cos T  =  sec T
substitute the values we get,
( sin T / cos T )^2 + 1  =  ( 1 / cos T)^2
(tan T)^2 + 1  =  ( sec T )^2
tan^2 T + 1  =  sec^2 T
Theorem 3: 1 + cot2 T  = cosec^2 T
We have sin^2 T + cos^2 T = 1
Divide on both sides by sin^2 T,
( sin^2 T + cos^2 T ) / sin^2 T  =  1 / sin^2 T
(sin^2 T / sin^2 T) + (cos^2 T / sin^2 T)  =  1 / sin^2 T
By using trigonometric ratios,
cos T/ sin T  =  cot T
1 / sin T  =  cosec T
substitute the values we get,
1 +  ( cos T / sin T )^2   =  ( 1 / sin T)^2
1 +  (cot T)^2  =  ( cosec T )^2
1 +  cot 2T  =  cosec^2 T

Standard Deviation of Mean in a nutshell

Standard deviation of Mean is the measure of the spread of the data about the mean value. If the standard deviation is low it shows that the values of the data are not spread out much and if the standard deviation is high it shows that the values of the data are spread out. At times we come across data which has the same mean but different range; to compare the sets of data standard deviation is very useful.  The average squared deviation from the mean is called the Variance. The square root of variance is the Standard Deviation of Mean. It is a statistical measure to know how the data is spread in the distribution, in simple words statistical measure of dispersion. Standard Deviation Means is also called the Mean of the Means.

In a population Variance is given by the formula: sigma^2 =summation[x – mu]^2/n
Where, x is each value in the data, mu is the mean of the data, n is the total number of values in the data.  Usually variance is estimated from a sample in a population. Variance calculated from a sample is given by the formula: sigma^2 = summation[x – x bar] ^2/ (n-1), here, x is each value from the sample, x bar is the mean of the values in the sample; n-1 is one less than the total number of values in the sample.  One Standard Deviation of the Mean is given by sigma= sqrt [summation[x – x bar] ^2/ (n-1)]

Standard Deviation of the Mean Equation
The equation or the formula to be used to calculate the standard deviation depends on whether the data is grouped or non-grouped. For example, given data, 42, 35, 48, 53, 47 is a non-grouped data.

In such a case, the standard deviation of the mean is calculated using the equation:
sigma = sqrt [summation (x- x bar) ^2/ (n-1)] where sigma is the standard deviation, (x-x bar) ^2 is the square of the deviations of the data values and n is the total number of values. Let us consider the data given below
Hours of components Frequency
300-400                   13
400-500                   25
500-600                   66
600-700                            58
700-800                   38
Understanding statistics problems is always challenging for me but thanks to all math help websites to help me out.
The data is a grouped data, here the standard deviation of the mean is estimated using the equation given by, sigma = sqrt [summation f(x-x bar) ^2/summation (f)] where sigma is the standard deviation, f is the frequency, x is each value of the data, x bar is the mean of the data values, summation is the sum of.

Introduction to indefinite integrals

In derivatives we learn about the differentiability of a function on some interval I and if it is differentiable, how to find its unique derivative f’ at each point of I. In application of derivatives we learn that using derivative we can find the slope of the tangent at any point on the curve, we can find the rate of change of one variable with respect to the other. Now let us look at an operation that is inverse to differentiation. For example we know that the derivative of x^5 with respect to x is 5x^4. Suppose the question is like this: derivative of which function is 5x^4∫ Then it may not be that easy to find the answer. It is a question of inverse operation to differentiation.

The answer to the question: " Whose derivative is a given function f ∫ " is provided by an operation called anti derivation. It is possible that we may not get an answer to this question or we may have more than one answer. For example, (d/dx) (x^4) = 4x^3, (d/dx)(x^4 + 3) = 4x^3 and in general (d/dx)(x^4+C) = 4x^3, where c is some constant.

Definition of integration (integrals): If we can find a function g defined on the interval I such that (d/dx)(g(x)) = f(x), for all x belonging to I, then g(x) is called a primitive or anti derivative or indefinite integral of f(x). It is denoted by ∫ f(x) dx and is called indefinite integral of f(x) with respect to x. The process (operation) of finding g(x), given f(x) is called indefinite integration.

Thus the question when can we find the integral of f cannot be easily answered. There are some sufficient conditions such as, continuous functions and monotonic functions have integrals. Sin x/x is continuous, hence ∫(sin x/x)dx is defined but cannot be expressed as any known elementary function. Similarly, ∫ v(x^3+1) dx and ∫v(csc x) are defined but canoe be expressed as known elementary functions. If anti derivative of f exists, then it is called integrable function.
(1) ∫ f(x) dx means, integral of f(x) with respect to x.
(2) In ) ∫ f(x) dx, f(x) is called the integrand.
(3) In ) ∫ f(x) dx, ∫ …. dx indicates the process of integration with respect to x.

For evaluating indefinite integrals we use the following standard table of indefinite integrals:

Calculus - Limits and Continuity

The concept of Limitlays the foundation for the popular branch of Mathematics called Calculus. Calculusinvolves the analysis of functions and their behaviour. To study the behaviour of functions one needs to have a good hold on the fundamental concepts of Limit and Continuity. Calculus Limits and Continuity is the language of science and engineering.Limits and Continuity in Calculus have led to the development of ideas like derivative, integration etc.

Consider a function f(x):R->R and ‘a’ is any point in the domain of the function, limit of the function f(x) exists at x = a, if f(x) = f(a), where x is arbitrarily close to a. The idea is that as we approach the point x = a on the real line, f(x) approaches f (a). The limiting value of the function when x is very close to ‘a’ is ‘f (a)’. In the language of mathematics it is commented as “lim x->a, f(x) =f(a)”.

Continuity is another important concept which is based on the concept of limits. The function f(x) is said to be continuous at x = a, if the limit of the function at x = a on approaching the point x = a from both sides is equal to the value of the function at x = a. Precisely this can be stated as – if  lim h->0, f(a+h) = f(a-h)=f(a), then f(x) is continuous at x = a. Mathematically this can be stated as – a function is said to be continuous at a point if the right hand limit and the left hand limit of the function at that point is equal to the value of the function at that point. Geometrically speaking, a continuousfunction is the one which can be sketched on paper without lifting the pen even once.


 Properties of Continuous Functions:There are two important results for continuous functions which are stated in the form of theorems-
• Intermediate value theorem: The intermediate value theoremstates that if f is a real valued continuousfunctionon the closed interval [a, b] andt is any number between f(a) and f(b), then there exists a numberc in [a, b] such that f(c) = t.Example: The height of a child increases from 1 m to 1.7 m between the ages of eight and sixteen years, then, at some point of timebetween eight and sixteen years of age, the child must have had a height of 1.5 m.
• Extreme value theorem: The extreme value theorem states that if f is continuousreal valued function on [a,b], then f has a maximumvalue in [a,b], i.e. there exists some c in [a,b] such that f(c) >= f(x)  for all x in  [a,b]. The same holds for the minimum of f(x). For example, consider the function sin(x) where x lies between [0,2*pi]. This function attains at maximum at x = pi/2 and a minimum at x = 3*pi/2.

Derivatives of Exponential Functions with Trignometric function power

What are exponential functions? A mathematical function which is in the form, f(x) = a^x is called an exponential function, here x is a variable and a is a constant which is the base of the function (a is greater than zero but does not equal one). The most commonly used exponential function is e which is the natural exponential function which is denoted as e^x. Some definitions of e are
• e=lim(n?inifinity)[1+1/n]^n
• Lim(h?0)[e^h-1]/h=1, where e is a unique positive number
• e=summation(n=0 to infinity)[1/n(factorial)]
We know that derivative is defined as, f’(x) = lim(h?0)[f(x+h) – f(x)]/h
Using the above, we can find the derivative of the natural exponential function f(x) = e^x
d[e^x]/dx = lim(h?0)[e^(h+x) – e^x]/h
    =lim(h?0)[e^h.e^x – e^x]/h  [using rule of exponents]
    =lim(h?0)e^x[e^h-1]/h            [taking e^x common]
    =e^x lim(h?0)[e^h-1]/h
    =e^x .1
    = e^x
As we observe, e^x is its own derivative. Let us find the derivative of the exponential function with trigonometric function as the power.

E-sinx
We have, derivative of e^u is given by e^u.du/dx. So, the derivative of E-sinx written as e^-sinx would be d[e^-sinx]/dx. Taking –sinx = u which gives du/dx = -cosx, we get, e^u. du/dx which equals e^-sinx.(-cosx)
Finally we get the derivative of E-sinx as –cosx.e^-sinx

E –sinx
We have, y= e^-sinx
     y= e^u where u= -sinx, derivative of u= du/dx = -cosx
taking derivative, we get,
dy/dx= dy/du. du/dx
          = e^-sinx. [-cosx]
          = -cosx.e^-sinx

E^-sinx
Let us find the derivative of the exponential function with the trigonometric function –sinx as the power.
Using the chain rule, we get,
d[e^(-sinx)]/dx = d[e^(-sinx)/dx. d[-sinx]/dx
             = e^(-sinx). (-cosx)
             = -cosx.e^(-sinx)

Basics of exponential functions

An exponential function is a function of the form y = a^x where a belongs to positive real numbers and x is any real number. We shall first try to understand such numbers with the help of graphs.

Exponential function graph:
Let us try to graph exponential function y = 2^x. So here we see that a = 2 (which is a positive real number and x is any real number). To obtain certain number of points on the graph we construct the following table:

Note that both the above functions are not inverse of each other. With this understanding, let us now define an exponential function.

Definition: Let a belongs to R+. Then the function f: R ->R+, f(x) = a^x is called an exponential function. a is the base of the function. The corresponding exponential function equation would be: y = a^x
For example:
f(x) = 3^x, g(x) = (1/4)^x (where x belongs to R), h(x) = 1^x, (where x belongs to R), are all exponential functions.

Natural exponential function:
We are familiar with the irrational number pi which we come across in connection with the area and circumference of a circle. There is another important irrational number, which is denoted by e and which lies between 2 and 3. Its approximate value is 2.71828. Exact value of e is given by the sequence: e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ....An exponential function to the base e, f:R->R+, f(x) = e^x is called the natural exponential function. This function is very often used in study of various branches of science and math.

Inverse of exponential function:
Inverse of an exponential function is a logarithmic function. In other words, the exponential function and the logarithmic function are inverses of each other. Thus, if f:R->R+, f(x) = a^x, a belongs to R+ -{1}, then f^(-1):R+->R, f^(-1)(x) = log(a)x [read that as log of x base a]

Derivative of exponential function:
Based on limit definition of derivatives, the derivative of an exponential function can be shown as follows:
If y= a^x, then dy/dx = (a^x)Ln(a)

Trigonometry Made Simple

Introduction to Trigonometry:
Trigonometry is a derived from a Greed word ‘tri’ (meaning three) and ‘gon’ (meaning sides and ‘metron’ (meaning measure). Trigonometry is the study of relationships between the sides and angles of a triangle. The earliest known work on Trigonometry was recorded in Egypt and Babylon. Early Astronomers used it to find out the distances of the stars and planets from the Earth. Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on Trigonometric concepts.

The Trigonometric Ratios of the angle A in right triangle ABC are defined as:
Sine of angle A = (side opposite to angle A)/hypotenuse; cosec = 1/Sine
Cosine of angle A = (side adjacent to angle A)/hypotenuse; sec = 1/cosine
Tangent of angle A = (side opposite to angle A)/(side opposite to angle A); cot =1/tan
The trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the lengths of its sides

Trigonometry Problems and Answers:
Problem: Given angle A=51 degrees, Adjacent side length = x and opposite side length = 10. Find x and H, hypotenuse of the triangle
Answer: tan(A) = opposite side/adjacent side
tan(51)= 10/x
X = 10/(tan 51) = 8.1 (two significant digits)
Sin(A) = opposite side/hypotenuse
Sin(51) = 10/H
H = 10/ Sin(51) = 13 (two significant digits)
x=8.1 and H=13

Problem: If sin 3A = cos (A- 26 degrees), where 3A is an acute angle, find the value of A
Answer:  sin 3A can be written as cos (90-3A)
      So, we get            cos (90-3A) = cos(A-26degrees)
Since both 90-3A and A-26 are both acute angles,
90 – 3A = A- 26
4A = 116
A = 29 degrees

Problem:  An observer 1.5m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from his eyes is 45 degrees. Calculate the height of the chimney.
Answer: Let us draw a rough triangle ABC with the right angle at B. let us draw a line DE parallel to BC such that AB (AE+EB) will be the height of the chimney,  CD (equal to BE) the observer and Angle ADE, the angle of elevation.  Here, ADE is the right triangle, right-angled at E
We have, AB = AE +EB = AE +1.5 DE = CB = 28.5 (distance from the chimney)
Let us use the tangent of the angle of elevation
tan(45degrees) = AE/DE
1 = AE/28.5
AE= 28.5
Height of the chimney = AE +EB =28.5 +1.5 = 30 m

Parallelogram Definition

Let us learn about the Parallelogram Definition general, the diagonals of a parallelogram are having different lengths. The two diagonals in the figure which intersects at a particular point and lie in the interior part of parallelogram. When two pairs of the sides are opposite and they are parallel to each other.Then it is called as parallelogram .Now let us see about the parallelograms sides introduction.In parallelograms introduction, we can draw a pair of parallel lines. Draw another pair of parallel lines intersecting the former.Thus the parallelogram can be formed.Thus we can say that the pair of opposite sides of parallelogram is of equal length. Similarly we can also learn about other topics such as types of lines.Hope you like the above example of Parallelogram Definition.Please leave your comments, if you have any doubts.

Absolute Measures of Dispersion

One can define dispersion and measure it in different ways. The common dispersion definition is: “way to measure variation of data”. In general, if the dispersion is large then the variation of variable values will be huge. If the dispersion is small, then the variation is close.

Different Measures of Dispersion
The measures of dispersion can be classified into two types namely Absolute measures of dispersion and Relative measures of dispersion. Rest of the article covers these two types in detail.

Absolute Measures of Dispersion
The statistical data observations will generally be specified in some units. If the dispersion values have to be specified in the same units as that of the statistical data, then you have to choose an absolute measure of dispersion. For example, if the statistical data is represented in grams then the measure of dispersion must also be in grams.

There are various absolute measures of dispersion. Commonly used absolute measures are:
• Range
• Mean Deviation
• Variance and Standard Deviation

Here is an overview and example of one absolute measure called Range:

The Range is the simplest measure of dispersion. It is the difference between the top most value and the lowest value in the statistical set of data provided. For example, Jackson took 5 Mathematics tests in one period of time. The test marks are: 70, 60, 50, 80, and 90. What is the range of the marks? The range is calculated by subtracting the lowest value 50 from the highest value 90 and the result will be 40. Thus, the range is 40.

Relative Measures of Dispersion
If you have to compare dispersion of two or more statistical data sets which are of different units, then you have to opt for relative measures of dispersion. These dispersion measures are dimensionless and they are used to establish dispersion relation between any data sets. Common relative measures of dispersion are:
• Coefficient of Range
• Coefficient of Mean Deviation
• Coefficient of Variation
• Coefficient of Standard Deviation

Here is an overview and example of one absolute measure called Coefficient of Variation:

The coefficient of variation is the percentage of variation obtained by dividing standard deviation by mean. For example, a school has two sections for X standard. The average score of students in first section are 85 and the second section average scores are 80. The standard deviations of the two sections are 8 and 7 respectively. Which X standard section has large variation in its scores?

In this problem, the average score denotes the mean and the values of standard deviation are also specified. Now to find the variation of data by calculating coefficient of variation:

Coefficient of variance of section A is 9.4%, which is obtained by dividing standard deviation value 8 by the mean 85 and multiplying the result 0.2 by 100.

Coefficient of variance of section B is 8.75%, which is obtained by dividing 7 by 80 and multiplying the result by 100.

Comparing both the coefficients, section A is consistent when compared to section B. Hence, section B has large variation in its scores.

Harmonic Mean

Harmonic Mean

Mean is an important concept in measures of central tendency. Measures of central tendency give the measure for the center of the data. We have different types of measures of central tendency those are Mean, Median and Mode. Mean is classified into three parts those are Arithmetic Mean, Geometric Mean and Harmonic Mean. Arithmetic Mean is simply the ratio of sum of observations and the number of observations. Geometric Mean is the nth root of the product of the observations, where ‘n’ is the number of observations. What is Harmonic Mean? Harmonic Mean Definition is the reciprocal of arithmetic mean of the reciprocal of the observations. Depends on the situation we have to know which mean is correct. We have relation among these three means, that is, Arithmetic Mean = Geometric Mean = Harmonic Mean. Harmonic Mean Formula can be described as

Weighted Mean is useful in some cases where each
observation do not have an equal importance. In general arithmetic mean we are giving equal importance to each observation but this is not always the case. When all the observations are not equally important then we have to use weighted arithmetic mean. In weighted arithmetic mean we do not take sum of the observation we multiply the observations with respect to their corresponding importance. In Weighted Harmonic Mean is the reciprocal of weighted arithmetic mean of the reciprocal of observations.

Harmonic Means may not applicable in all cases if zero value present in the observations then reciprocal of zero do not exist, hence harmonic mean also do not exist. Harmonic Mean is useful when there are extreme values present in the data then it gives true picture of the average of the data. The Harmonic Mean is better average when the numbers are defined in a relation to some unit.

 For example in case of averaging speed Harmonic Mean is better measure than Arithmetic Mean. Suppose we have to find the average speed of a person travelling from place A to place B. If the person travelling with 10kmph in first hour and 15kmph in second hour then the average speed is the arithmetic mean. If the person travels first half distance with the speed 10kmph and remaining distance with the speed 15kmph then Harmonic Mean is better measure then arithmetic mean. Harmonic Mean is also called as sub- contrary mean. Harmonic Mean is useful in case of finding averages involving rates and ratios.  

Metric units and conversions

Unit conversion is converting from one unit to another of the same quantity. Unit conversion is important as it uses the metric system. Unit conversion is used to convert from one given unit to the other desired unit. Unit conversion is done for length, weight, volume, temperature, energy etc.


Unit of weight
The standard unit of mass in the metric system is gram. As per the International system of units, the S.I unit of weight is same as that of force and that is Newton but the basic unit of weight is kilogram which is usually denoted as ‘kg’. The other units for weight are kilogram (kg), milligram (mg), centigram, gram (g), decagram, hectogram, ounce and pound etc. These units are used in weight conversion that is to convert one unit from another. For example:- 1 kg = 1000g or 1 g = 1000mg.

Unit of Length – Different units of length are used in different parts of the world. As per the International system of units, the S.I unit of length is metre which is usually denoted by ‘m’. For example 10 metres is written as 10 m. There are many other units of length that are used in length conversion, some of them are millimeter (mm) , centimeter (cm), kilometer (km), decameter (dm) etc. These units can be converted to other units such as 1 mm = 1/1000 m and 1 km = 1000 m or 1 cm = 1/100 m.

Unit of Volume – There are many units which are used for volume. As per the International system of units, the S.I unit of volume is cubic metre which is usually denoted as m^3. The other units that are used to represent volume are cubic centimetre , cubic metre. Volume conversion can be done by converting one unit from another like 1 cubic metre is equal to 1000 litres or 1 millilitre is equal to 1 cubic centimetre

The unit conversions chart is very important as it helps in converting one unit from another. The chart helps in converting any quantity in the given units to the desired units. The online volume conversion chart helps in converting the difficult units.

How to convert units?
Unit conversions can be done by using the following steps: -
1. Write the terms which are given in the given unit.
2. Write the conversion factor for the given and the desired units.
3. Write it as a fraction with the given units as a denominator or in the opposite direction.
4. Cancel the ‘like’ units.
5. Multiply odd units.

Matrix- Cofactor and Adjoint

Adjoint matrix
Definition: Transpose of a cofactor matrix is called Adjoint of matrix.
 Adjoint is defined only for square matrices.
It involves two operations, which can be done in either order. We can take transpose of the given matrix and then replace the individual elements by their corresponding cofactors or the other way.
Transpose of a matrix:
Matrix obtained on interchanging the elements of rows and columns is called transpose of a matrix.

Transpose of  is  Cofactor:
The value of the determinant obtained by eliminating the row and column in which the element is present is called cofactor of the element. The cofactor is preceded with + or – sign depending upon the position of the element.
Adjoint of a 3 X 3 Matrix

Cofactor of 1 is obtained as follows:
Step 1: Identify the position of the element. [(Row, column) = (i, j)]
Step 2: remove the row and column in which 1 is present.
Step 3: Write all the other elements in the determinant form
Step 4: Evaluate the determinant append with










































For finding the Adjoint of 2 X 2 matrix, though we adopt the same technique, yet there is simple way to evaluate.

Adjoint of a 2 X 2 matrix:

Matrix Adjoint:
Adjoint of a matrix is used in evaluating the multiplicative inverse of a matrix. Multiplicative inverse is defined only non-singular matrices.

Scientific notation :word problem and expanded form

We come across large numbers in our day  to day life .we can express large number conveniently expressed using exponents.every large number can be expressed as k x 10n , where k is some natural number .however , for the sake of uniformity , we write the number in the form k x 10n  where k is terminating decimal number greater than or equal to 1 and less than 10 and n is a natural number .To  express large number into standard form is scientific notation.
Scientific notation rules
1) For the positive number , move the decimal  point to the left to bring non-zero digit to the left of the decimal point .
Example of scientific notation  :To express 41460. 5 in the standard form of decimal , we have to move decimal point 4 places on the left to get 4.14605 , so 41460.5 in the standard form of decimal is 4.14605 x 104.
2) If the given number is less  than one , then move the decimal point to the right to obtain one non –zero  digit  to the left  of the decimal point. If the decimal point is moved p places to the right ,then multiply the new number by 10^-n to express the given number in the standard form of decimal
Example of scientific notation  : To Express 0.03453 in the standard form of decimal we will have to move the decimal poit 2 places on the right to get 3.453 .so , 0.03453 in the standard form of decimal  is 3.453 x 10^-2
Scientific notation word problems
Example 1: A lake covers an area of about 3.5*10^5 square feet and its average depth of lake  is about 16 feet.   Calculate  the cubic feet of water in the lake.
Solution1) Volume  of lake =  area x depth
                                             =  3.5 *10^5 x 16 cubic feet
                                            = 56 *10^5 cubicfeet
                                           = 5.6 x 10^6 cubic feet
Expanded form : The expanded  form of a number can also be expressed  in term of powers of 10 by using
10^0 = 1 , 10^1 = 10 , 10^2 = 100
Example of expanded form :
1) 6467 = 6 x1000+ 4 x 100+ 6 x10 + 7
2) 7465267= 7 x 1000000+ 4 x100000+6x10000+5x1000+2x100+6x10+7
                               = 7x10^6+4x10^5+6x10^4+x10^3+2x10^2+6x10^1+7x10^0

Solving Complex Fractions

Complex fractions are those fractions which have a fraction in numerator or denominator or in both. For example, 1/ (¾), ½ / 3, (¾ )/ (½ ) are some of the complex fractions

Let us now learn to simplify complex fractions. While simplifying complex fraction, let us consider the following example for better understanding,

Complex Fractions

Simplify the complex fraction 4a²b/(8/ab)
Here the numerator is 4a²b and
        the denominator is 8/ab
It can be written as 4a²b  ÷ (8/ab)
to simplify, we need to flip the fraction in the denominator and multiply it with the terms in the numerator as follows,
             4a²b × ab/8 (division of fractions)
Now, we can simplify the terms
           4/8 × a²b × ab
which gives us ½ a³b²


Complex fraction solver
In solving complex fractions, we can use one more method, which is the LCD method. Let us learn how to solve  complex fractions with some example problems

1.Simplify (4/5)/(2/15)
Solution: Numerator = 4/5
Denominator = 2/15
  LCD of 5 and 15 (denominators of the two fractions) is 15
Multiply LCD with each of the fractions of the numerator and the denominator
4/5 x 15 = 4 x 3 = 12
2/15 x 15 = 2 x 1 = 2
The simplified fraction, 12/2 = 6

2.Simplify (1/a + 1/b) ÷ (1/a – 1/b)
Solution: (1/a + 1/b) = (b + a)/ab
(1/a – 1/b) = (b – a)/ab
LCD in this case would be ‘ab’
(b+a)/ab x ab = (b+a)
(b-a)/ab x ab = (b-a)
the simplified fraction is (b+a)/(b-a)

3.Simplify (x²/4)/(y/x)
Solution: x²/4 .  x/y  
= x³/4y
4.How to simplify complex fraction given below
               (a+b)/(x-y) / (a² - b²)/(x² - y²)
Solution: We re-write the given complex fraction,
          (a+b)/ (x-y)  ×  (x² - y²)/(a² - b²)
we have, (a² - b²) = (a+b)(a-b)
        (a+b)/ (x-y) × (x+y)(x-y)/(a+b)(a-b)
on simplification, we get   (x+y)/(a-b)
5.simplify the complex fraction given,
4 ¾ /3 ½
Solution: Numerator = 4 ¾ = 19/4
               Denominator = 3 ½ = 7/2
LCD of 4 and 2 is 4
19/4 x 4 = 19
7/2 x 4 = 14
the simplified fraction is 19/14

Algebraic Numbers

In today's post i will help you in learning the concept of algebraic numbers.

Algebraic number is a number that exist in a polynomial equations having integers, co efficient and so on. There are different types of algebraic numbers and these are as follows namely:

Natural numbers
Rational numbers
Irrational numbers
Complex numbers

Next time i will help you with some other concept such as algebraic manipulation.

You can also avail your help from online tutoring. Not just in algebra but in other concepts such as calculus tutoring and so on.

Do post your comments.

Sample Statistics

Let's learn about mean in sample statistics in today's session of learning.

Sample statistics is nothing but making use of numerical data in order to do some data research and data analysis. It is also the technique to summarize numerical data in statistics. Mean is used to summarize this numerical data in statistics and therefore, it is such an important concept in sample statistics.

Next time i will help you with the concept of errors in sampling in statistics. One can also connect with an online tutor for more help. Not just in statistics but one can avail to the help of algebra tutors as well.

Do post your comments.

Mixed fractions

Let's learn about mixed fractions and subtracting mixed fractions in today's learning.

Mixed fractions are those fractions which are formed of a whole number and a fraction. In order to subtract mixed fractions, we need to first convert the mixed fractions to improper fractions and then use the same method of subtract fractions. Below are the steps of subtracting mixed fractions:

Step 1) Convert mixed to improper fraction.
Step 2) If denominators are different, find the LCM and take a common denominator.
Step 3) Subtract the numerators.

For more help avail to an online tutor and get your required help. Not just fractions but you can avail to free algebra tutoring and so on as well.

Do post your comments.