Subtracting Fractions

The procedure of fraction addition is more or less the same as that used in the subtraction of fractions. This implies that if you have the know-how of adding fractions, you will have an easier time subtracting them. Subtraction of fractions is also one of the most common operations that we are likely to come across in many mathematical computations. All the same, if you have a problem subtracting fractions, there is absolutely no need to worry because what you need is only a few steps and procedure away.

To begin with, it is important to understand the structure of the fraction. A fraction can either be proper improper or mixed. In cases where you are required to perform subtraction of improper fractions, the first step should be to change it into an improper fraction first. A fraction has basically two numbers. One on top and the other at the bottom. The one on top is called the numerator, and the one at the bottom is the denominator. For example:

Given a/b, a is the numerator and b are the denominators.

Having understood the structure of a basic fraction, we shall consider two methods that can be used to subtract fractions basing on the number of fractions being subtracted. These methods require the understanding of the structure of a fraction.

Method of Common Denominator

Method of Least Common Denominator

  1. Method of Common Denominator

This method is well applicable when you are subtracting one fraction from another, meaning, you are dealing with two fractions only. Since by now we know the meaning of a denominator, it will also be-be helpful to understand the meaning of “Common Denominator.” What is a Common Denominator? When the denominators of the two fractions in question are the same or identical, we say that there exists a common denominator.

Therefore, the main aim of this method is to ensure that the denominators of the fractions are the same before performing the subtraction. When the denominators are not the same, subtraction cannot be done. How do we then ensure that the fractions have the same denominators? Multiply each fraction – the numerator and the denominator – by the denominator of the other fraction.

For illustration consider;

a/b – c/d = (a*d)/(b*d) – (c*b)/(d*b)

Once the denominators of the two fractions is now common, (b*d), the ‘new’ numerator are simply added.

Example; 1/2 – 1/3

(1*3)/(2*3) – (1*2)/(3*2) = 3/6 – 2/6 = 5/6

 

  1. Method of Least Common Denominator

This method is used when subtracting two or more fractions. The knowledge of the Least Common Multiple (LCM) is needed in this method. The LCM of the denominators is determined and used in the calculations. To find the LCM of the denominators, follow the steps below;

List the multiples of each denominator

Select the multiples that are common – appearing for all the denominators

Choose the smallest among the common list of the multiples. This is the LCM

After getting the LCM, divide the LCM by each denominator and multiply the result by the numerator of each fraction separately before you proceed to simply subtract the ‘new’ numerators and using the LCM as the denominator. Simplify the resultant fraction.

To clearly illustrate the long theory, we will need an example;

Question; 1/2 – 1/4 – 1/5

List the multiples of each denominator

2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22…

4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40…

5 = 5, 10, 15, 20, 25, 30, 35, 40…

Select the common multiples

20…

Choose the smallest = 20. This is the LCM.

[(1*10) – (1*5) – (1*4)]/20 = 9/20.

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When Quadratic Formula is Used

With elementary algebra, quadratic formula is used for the solution of a quadratic equation. You can find many ways that you can use to solve this problem without using the quadratic formula like graphing, completion of the square or factoring while using the quadratic formula maybe most convenient method. The quadratic equation looks like ax2+bx+c=0.  With this equation, x stands for unknown but a, b, with C should be constant, and A should not be zero. Someone may verify that the quadratic formula does satisfy quadratic equation through inserting the first number in the last number.

quadratic formula
quadratic formula

The solution found with the quadratic formula is known as the root for a quadratic equation.  In geometry, such roots can represent the value of x which is the parabola given will cross at the axis of X. The formula can yield zero of different parabola while quadratic formula can give axis to the symmetry of a parabola. This is normally used in order to determine the number of the zero which the quadratic equation does have.

The formula may get derived from the simple application with a technique application needed to complete a square. This is why; a derivation can be sometime left like the exercise of the students who wish to experience the rediscovery of the formula.

A quadratic equation can be in the form to complete a square when it is applied. You may add the constant on the two sides of an equation like the left-hand side which become the complete square.  The terms should be rearranged at a right-hand side so that we can obtain the common denominator. When the square had been completed, then the square root of the two sides have to be found and then isolating the x to get the quadratic formula.

The solution of a quadratic equation can be gotten by using different alternatives of the derivation with the minor differences, and this is through manipulating a. Some of the old sources may use the alternative of parameterization for the equation by using b with a magnitude of the half of a common number.  Even if the results can be different from the first solution, however, they are also equivalent.

Without the need to go in the parabolas like the geometrical objects of the cone, the parabola will be the curve which is being described as the second degree of the polynomial.

The early method to solve the quadratic equation was done in the geometry.   The Babylonian cuneiform tablet had the problems that are reducible in solving a quadratic equation. Egyptian Berlin Papyrus that dated back in the middle kingdom had some solution for a two-term equation.   In Greek, the mathematician Euclid was using the geometric method in order to solve the quadratic equation in the Elements Book 2.  The laws of the quadratic equation do appear in Nine Chapters on Mathematical Art circa from China. A Greek mathematician Diophantus in the work he did on the arthmetica, he solved some quadratic equations which are recognizable to the algebraic done by Euclid. The solution he used was able to give one root, regardless if positive roots were used.

What are the Basic and Right Rules for Dividing Fractions?

Fractions have its set of rules for multiplication, division, subtraction and also for addition. Multiplication of fractions is easy, but the division of fractions it little difficult. However, if you know the rule of dividing fractions, you can easily divide two fractions very easily. Fraction is one of the important concepts of the algebra, and you should know the operations on fractions like the multiplication and the division. It is important as you will come across with fractions while studying the algebraic equation and various problems of algebra. Here, you can get informed about the division as well as multiplication of fractions.

Before you study how to divide the fractions, you should know how to multiply fractions. It will help you in the division also. If you do not know how to multiply two fractions, you can’t divide the fractions. The rule of multiplication is very simple. Multiply the denominator of the first fraction with another fraction and just like multiply the numerators of both fractions. So you will get a new fraction after multiplication.

How to divide the fractions?

The basic rule for the division of fractions is simple. Dividing fractions is followed by the reciprocal and multiplication. It’s the single operation in which you have needed to reciprocal a fraction. You can better get it by reading the simple steps of division listed below.

Step 1: first invert the divisor fraction only and use the inverted divisor.

    Step 2: now, rewrite the problem and replace the multiplication sign “x” to the division sign “÷” and also replace the divisor fraction by its reciprocal.

    Step 3: multiply the numbers at the numerator place of both fractions.

    Step 4: multiply the numbers at the denominator place of both fractions.

Step 5: if possible, reduce the fraction obtained by multiplication into its lowest form.

Example:  

Math
Math

Important thing to consider

But keep in mind that you have only to reciprocal only a single fraction. So, do not reciprocal or invert both the fractions which you are using for the division. So, you have only required inverting the divisor fraction only. There is no need to invert the dividend according to the rule of fraction division. In simple words, only invert the second fraction and use the first fraction as it is to solve fraction division problems. It is the most important consideration for dividing fractions correctly.

The same rule will be followed in the problems of dividing a fraction by a whole number. You have to invert the whole number in this case as the whole number is the divisor. So, the whole number will automatically convert into a fraction. You will get a new fraction and then multiply the numerators and denominators.

In this case, convert the whole number into its reciprocal by putting 1 in the numerator and the number at denominator place. So five becomes 1/5, and it is the reciprocal of that number.

Math
Math

Thus, you can easily get the right solution of the dividing fractions problems. You can divide a fraction by another fraction by following the reciprocal and multiplication rule.

How to benefit from Online Equation Solver

If you are not able to do the mathematical questions faster, then the equation solver is here to help you out, while enhancing your skills to solve the problems even in the future.  When you use a broadband connection or a computer, the student may easily get rid of the math anxiety. When you get the help with the online environment, then it is a phenomenal way that you can solve a complex math problem. The student should take many sessions as he likes and he can use the computer at the place he desires. Using the solver is the easy way and fun way to understand the concepts while at the same time, it saves time and it is affordable.  Some of the students may find that math is difficult and they become anxious when it comes to taking math classes. However, while learning online, it can help them to get important steps in solving the math problems at once. The use of the solver is done one on one, and it helps to solve the problems that involve the use of math. The online solver will make the children to be more confident when they are face to face with tricky problems.

When you use the equation solver, you will easily understand and do the exercise in a more streamlined manner. Normally, the solver will take you through how to solve the problem, and it may be used by the students who want to share the homework and getting the correct solution within few minutes. Besides the whiteboard, there is a chat option which is available, and it makes the communication process easy with an available tutor.  When there is no face to face interaction, the student may overcome his fear, and he can ask the questions he wants to.

Equation Solver
Equation Solver

Equation solver makes the learning process easy since it is personalized to what the student wants and he can easily learn to complete the assignment when he is alone at home.

The ability to understanding math problem in the class room is difficult sometime for the students who are anxious about math. They may be afraid to ask the questions when they are in school that when they are doing it online, it will be easier to get help.

Math is viewed by many students as the most difficult subject to learn. The reason for this is because the students should learn how to solve the problems step by step.  There are some rules and formula to use if you want to solve the mathematical problems. Math involves geometry, statistics, calculus, algebra and pre-algebra. Trigonometry is also one of the topics which students around all grades may have to cover for their curriculum. Besides, the solver can be used in many areas of real life, starting from shopping, working on a recipe or making the budget.   To be able to learn math well, there is a need of doing the regular math.  The learning centers are mostly expensive, and the student will need to be present at a fixed time or date. With the online math solver, this is no longer a requirement.

How to Solve Equations using Quadratic Formula Calculator

Quadratic formula is the best and the easiest way of solving quadratic equations. Most mathematicians believe that this formula can solve even the complex quadratic equation you can think of. Then there is a quadratic formula calculator that uses the quadratic formula to solve equations. First of all a quadratic formula looks like the one below Math

Where x is the variable being solved, a, b and c being the numerical coefficients of the quadratic equation that you may come across. They literally represent any number that you can think of. The values of a, b and c could be 6, 11, and 28.

Learning the calculator through examples

If you do not know how to use the above formula, sit back and relax. But be very attentive. There is a website that you can use to learn these things. It is quickmath.com; the best online math tutoring site that anyone can access for free.

Let me show you how the site uses examples to solve quadratic equations using the quadratic formula calculator.

Example 1

Given an equation like this,  Math

It can be solved this way;

Math

Here, a equals to 10, b equals to -7, and c equals to -23

Therefore, we will get  Math

 

The final result will be Math

 

X= 1.8575 or

X=1.1575

The two answers should give you the same results. You can proof it by substituting x in the equation using any of its value we obtained here.

Example 2

Let us take a look at another example of a quadratic equation and try to solve it using the quadratic formula calculator. Remember, you can get to learn all these from http://www.quickmath.com.

An equation could be in the form of words. So, you will first have to deduce it there and form a quadratic equation then try to solve it.

Question: Solve for the value of x .if three times x squared plus two x minus eight equals to 10.

Okay, this is not a good example but was just trying to illustrate how you may come across such equations.

Forming an equation will be like Math

Using the quadratic formula of  Math

The equation will need to be simplified further since we still have some like terms that have not been grouped together. The final equation should be; Math

Therefore, an equals to 3, b equals to 2 and value of c will be -18

Math

Math

The values of x will be -2.805 and 2.14

You can use these values to ascertain that they are correct. If you did not get the above values for x, then you may have skipped some steps, and therefore the best thing you can do is to retrace your steps till the place you got lost. From there, you can start following the steps again. The good thing is that they are pretty straight forward steps.

Make sure you practice doing this regularly until you master it. Once that happens, nothing quadratic will prevent you from passing that math exam you are about to sit for.

 

Want to know how Factor Calculator Works?

Factor calculator tries to simplify the any given quadratic equation. The equation is normally in the form of . The calculator then walks it way down until it gets an answer. The values of a, b and c are always known while that of x has to be solved. In this scenario, x is the variable, while a, b and c are the numerical coefficients of the given quadratic equation. One more thing, “a” should never be equal to zero otherwise the equations ceases to be a quadratic equation but instead becomes a linear equation.

So, for factor calculator to work effectively using the factoring method, the equation should be factor able otherwise another method will have to be deployed in order for you to get an answer of the value of variable x. when I say factor able, I mean that there have to be two numbers that have to multiply to the equal constant value of “c” and at the same time these two numbers need, to sum up to the numerical coefficient value of “b” the numbers are called the factors of “c”.

Illustration by example

People always tend to remember how something works when they have been shown an example and have also tried another example by their selves. So for you not to forget, I will show you two examples but make sure you go and look for other examples on your own.

Example 1

You have been told to factor 

Which two numbers can I multiply them together and get -4 and at the same time sum them up to get 3? They should definitely be 4 and -1.

Therefore, when substituted, they should be

The answer is (x -1) (x + 4). You can double check the answer by expanding what you got. Always, it should lead you back to the initial quadratic equation.

Example 2

Try and factor 

a*c = 6*-6 = 36 and b = 5.

The positive factors of -36 are; 1, 2, 3, 4,6,9,12,18 and 36

But the product of the two numbers is -36 and the sum is 5. It, therefore, one of the numbers has to be negative in order to satisfy this equation.  The best combination to satisfy this will thus has to be

-4 and 9, such that -4 * 9 = -36 and -4 + 9 = 5

Substituting 5x will result in an equation like this

From there, we can then factor the first two and then the rest.

This is what you should get if you are doing it on your book,

2x (3x-2) +3(3x-2). Our common factor here is (3x-2), and thus the final result should be

(2x+3) (3x-2).

You can try to expand it on your own. The answer should be equal to our first or original equation. If not, then you may have missed something along the way. I urge you to go retrace your steps and find out where you went wrong.

You can always go online and look for tutoring sites. The best that most people visit is quickmath.com. It is really cool.

How to use a Factoring Calculator

What is a factoring method in mathematical aspects?

First, it is important for us to understand what a “factoring” means in mathematical point of view. It means that two numbers have to multiply to equal constant term “c” and yet again sum up to “b” the coefficient on the x term. This method is used to solve quadratic equations that are factor able. So, the factoring calculator tries to solve the equation of it as long as the right values of the coefficients have been fed to that calculator.

Normally, the quadratic equation takes the form of  whereby the values of a, b and c are known while that of x has to be solved or obtained from the equation after it has been solved. In other words, it is safe to say that a, b and c are the numerical coefficients of the given quadratic equations.

To understand the factoring formula better, it is best to illustrate it using an example.

Solving quadratic equations using factoring method

We can take an equation like  and have been told to factor it.

First, we have to find the factors of 6 that add up to 5. But because we can write 6 as the product of 2 and 3, and the two also add up to 5, i.e. 2+3=5. Then, I will have to use 2 and 3 instead of the initial 6.

The above polynomial can be simplified further into the form of (x + m) and (x + n)

Therefore, it is all right to rewrite is this way (x + 2) (x +3).

= (x + 2) (x +3). You can leave it there or continue to get the values of x, which will be

X = – 2 and x =-3 However, this really won’t change anything. So you can leave it at the (x + 2) (x +3) point.

We can take a look at one more example for you to understand it clearly.

Factor,

The best way to approach such the problem is to apply the differences of squares rule which is

=(x + y) (x – y)

Therefore, = ((x – 2)-3) ((x – 2) +3). Simplifying it further will land us on these solutions:

(x – 5)(x + 1)

These two examples have been clearly explained. You can look for many other examples online or on books. There is this website that will save you a lot of time since it has many examples that have been explained step wise. The site is www.quickmath.com. You can visit the page and look for the factoring page within the home page. When you click on us, you will be redirected to a web page that contains the information I am talking about.  You can also raise the alarm if you think the answer they gave is not correct. You will have to send them an email and clearly, tell them what you think is the right answer, and I am sure they will update it immediately.