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The online fraction calculator allows you to perform simple arithmetic operations with fractions: adding fractions, subtracting fractions, multiplying fractions, dividing fractions. To make calculations, fill in the fields corresponding to the numerators and denominators of the two fractions.

Fractions in mathematics is a number representing a part of a unit or several parts of it.

A common fraction is written as two numbers, usually separated by a horizontal line indicating the division sign. The number above the line is called the numerator. The number below the line is called the denominator. The denominator of a fraction shows the number of equal parts into which the whole is divided, and the numerator of the fraction shows the number of these parts of the whole taken.

Fractions can be regular or improper.

• A fraction whose numerator is less than its denominator is called a proper fraction.
• An improper fraction is when the numerator of a fraction is greater than the denominator.

A mixed fraction is a fraction written as an integer and a proper fraction, and is understood as the sum of this number and the fractional part. Accordingly, a fraction that does not have an integer part is called a simple fraction. Any mixed fraction can be converted to an improper fraction.

In order to convert a mixed fraction into a common fraction, you need to add the product of the whole part and the denominator to the numerator of the fraction:

How to convert a common fraction to a mixed fraction

In order to convert an ordinary fraction to a mixed fraction, you must:

1. Divide the numerator of a fraction by its denominator
2. The result of division will be the whole part
3. The balance of the department will be the numerator

How to convert a fraction to a decimal

In order to convert a fraction to a decimal, you need to divide its numerator by its denominator.

In order to convert a decimal fraction to an ordinary fraction, you must:

How to convert a fraction to a percentage

In order to convert a common or mixed fraction to a percentage, you need to convert it to a decimal fraction and multiply by 100.

How to convert percentages to fractions

In order to convert percentages into fractions, you need to obtain a decimal fraction from the percentage (dividing by 100), then convert the resulting decimal fraction into an ordinary fraction.

Adding Fractions

The algorithm for adding two fractions is as follows:

1. Perform addition of fractions by adding their numerators.

Subtracting Fractions

Algorithm for subtracting two fractions:

1. Convert mixed fractions to ordinary fractions (get rid of the whole part).
2. Reduce fractions to a common denominator. To do this, you need to multiply the numerator and denominator of the first fraction by the denominator of the second fraction, and multiply the numerator and denominator of the second fraction by the denominator of the first fraction.
3. Subtract one fraction from another by subtracting the numerator of the second fraction from the numerator of the first.
4. Find the greatest common divisor (GCD) of the numerator and denominator and reduce the fraction by dividing the numerator and denominator by GCD.
5. If the numerator of the final fraction is greater than the denominator, then select the whole part.

Multiplying fractions

Algorithm for multiplying two fractions:

1. Convert mixed fractions to ordinary fractions (get rid of the whole part).
2. Find the greatest common divisor (GCD) of the numerator and denominator and reduce the fraction by dividing the numerator and denominator by GCD.
3. If the numerator of the final fraction is greater than the denominator, then select the whole part.

Division of fractions

Algorithm for dividing two fractions:

1. Convert mixed fractions to ordinary fractions (get rid of the whole part).
2. To divide fractions, you need to transform the second fraction by swapping its numerator and denominator, and then multiply the fractions.
3. Multiply the numerator of the first fraction by the numerator of the second fraction and the denominator of the first fraction by the denominator of the second.
4. Find the greatest common divisor (GCD) of the numerator and denominator and reduce the fraction by dividing the numerator and denominator by GCD.
5. If the numerator of the final fraction is greater than the denominator, then select the whole part.

Online calculators and converters:

Let us analyze two types of solutions to systems of equations:

1. Solving the system using the substitution method.
2. Solving the system by term-by-term addition (subtraction) of the system equations.

In order to solve the system of equations by substitution method you need to follow a simple algorithm:
1. Express. From any equation we express one variable.
2. Substitute. We substitute the resulting value into another equation instead of the expressed variable.
3. Solve the resulting equation with one variable. We find a solution to the system.

To solve system by term-by-term addition (subtraction) method need to:
1. Select a variable for which we will make identical coefficients.
2. We add or subtract equations, resulting in an equation with one variable.
3. Solve the resulting linear equation. We find a solution to the system.

The solution to the system is the intersection points of the function graphs.

Let us consider in detail the solution of systems using examples.

Example #1:

Let's solve by substitution method

Solving a system of equations using the substitution method

2x+5y=1 (1 equation)
x-10y=3 (2nd equation)

1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, which means that it is easiest to express the variable x from the second equation.
x=3+10y

2.After we have expressed it, we substitute 3+10y into the first equation instead of the variable x.
2(3+10y)+5y=1

3. Solve the resulting equation with one variable.
2(3+10y)+5y=1 (open the brackets)
6+20y+5y=1
25y=1-6
25y=-5 |: (25)
y=-5:25
y=-0.2

The solution to the equation system is the intersection points of the graphs, therefore we need to find x and y, because the intersection point consists of x and y. Let's find x, in the first point where we expressed it we substitute y.
x=3+10y
x=3+10*(-0.2)=1

It is customary to write points in the first place we write the variable x, and in the second place the variable y.
Answer: (1; -0.2)

Example #2:

Let's solve using the term-by-term addition (subtraction) method.

Solving a system of equations using the addition method

3x-2y=1 (1 equation)
2x-3y=-10 (2nd equation)

1. We choose a variable, let’s say we choose x. In the first equation, the variable x has a coefficient of 3, in the second - 2. We need to make the coefficients the same, for this we have the right to multiply the equations or divide by any number. We multiply the first equation by 2, and the second by 3 and get a total coefficient of 6.

3x-2y=1 |*2
6x-4y=2

2x-3y=-10 |*3
6x-9y=-30

2. Subtract the second from the first equation to get rid of the variable x. Solve the linear equation.
__6x-4y=2

5y=32 | :5
y=6.4

3. Find x. We substitute the found y into any of the equations, let’s say into the first equation.
3x-2y=1
3x-2*6.4=1
3x-12.8=1
3x=1+12.8
3x=13.8 |:3
x=4.6

The intersection point will be x=4.6; y=6.4
Answer: (4.6; 6.4)

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What are irrational equations and how to solve them

Equations in which the variable is contained under the radical sign or under the sign of raising to a fractional power are called irrational. When we deal with fractional powers, we deprive ourselves of many mathematical operations to solve the equation, so irrational equations are solved in a special way.

Irrational equations are usually solved by raising both sides of the equation to the same power. In this case, raising both sides of the equation to the same odd power is an equivalent transformation of the equation, and raising it to an even power is a unequal transformation. This difference is obtained due to such features of raising to a power, such as if raised to an even power, then negative values ​​are “lost”.

The point of raising both sides of an irrational equation to a power is the desire to get rid of “irrationality.” Thus, we need to raise both sides of the irrational equation to such a degree that all fractional powers of both sides of the equation turn into integers. After which you can look for a solution to this equation, which will coincide with the solutions to the irrational equation, with the difference that in the case of raising to an even power, the sign is lost and the final solutions will require verification and not all will be suitable.

Thus, the main difficulty is associated with raising both sides of the equation to the same even power - due to the inequality of the transformation, extraneous roots may appear. Therefore, it is necessary to check all found roots. Those who solve an irrational equation most often forget to check the found roots. It is also not always clear to what degree an irrational equation must be raised in order to get rid of irrationality and solve it. Our smart calculator was created specifically to solve irrational equations and automatically check all the roots, which will save you from forgetfulness.

Free online irrational equations calculator

Our free solver will allow you to solve an irrational equation online of any complexity in a matter of seconds. All you need to do is simply enter your data into the calculator. You can also find out how to solve the equation on our website. And if you still have questions, you can ask them in our VKontakte group.

Purpose of the service. The matrix calculator is designed to solve systems of linear equations using a matrix method (see example of solving similar problems).

Instructions. To solve online, you need to select the type of equation and set the dimension of the corresponding matrices. where A, B, C are the specified matrices, X is the desired matrix. Matrix equations of the form (1), (2) and (3) are solved through the inverse matrix A -1. If the expression A·X - B = C is given, then it is necessary to first add the matrices C + B and find a solution for the expression A·X = D, where D = C + B. If the expression A*X = B 2 is given, then the matrix B must first be squared.

It is also recommended to familiarize yourself with the basic operations on matrices.

Example No. 1. Exercise. Find the solution to the matrix equation
Solution. Let's denote:
Then the matrix equation will be written in the form: A·X·B = C.
The determinant of matrix A is equal to detA=-1
Since A is a non-singular matrix, there is an inverse matrix A -1 . Multiply both sides of the equation on the left by A -1: Multiply both sides of this equation on the left by A -1 and on the right by B -1: A -1 ·A·X·B·B -1 = A -1 ·C·B -1 . Since A A -1 = B B -1 = E and E X = X E = X, then X = A -1 C B -1

Inverse matrix A -1:
Let's find the inverse matrix B -1.
Transposed matrix B T:
Inverse matrix B -1:
We look for matrix X using the formula: X = A -1 ·C·B -1

Answer:

Example No. 2. Exercise. Solve matrix equation
Solution. Let's denote:
Then the matrix equation will be written in the form: A·X = B.
The determinant of matrix A is detA=0
Since A is a singular matrix (the determinant is 0), therefore the equation has no solution.

Example No. 3. Exercise. Find the solution to the matrix equation
Solution. Let's denote:
Then the matrix equation will be written in the form: X A = B.
The determinant of matrix A is detA=-60
Since A is a non-singular matrix, there is an inverse matrix A -1 . Let's multiply both sides of the equation on the right by A -1: X A A -1 = B A -1, from where we find that X = B A -1
Let's find the inverse matrix A -1 .
Transposed matrix A T:
Inverse matrix A -1:
We look for matrix X using the formula: X = B A -1


Answer: >

Instructions

Note:π is written as pi; square root as sqrt().

Step 1. Enter a given example consisting of fractions.

Step 2. Click the “Solve” button.

Step 3. Get detailed results.

To ensure that the calculator calculates fractions correctly, enter the fraction separated by the “/” sign. For example: . The calculator will calculate the equation and even show on the graph why this result was obtained.

What is an equation with fractions

A fractional equation is an equation in which the coefficients are fractional numbers. Linear equations with fractions are solved according to the standard scheme: the unknowns are transferred to one side, and the known ones to the other.

Let's look at an example:

Fractions with unknowns are transferred to the left, and other fractions are transferred to the right. When numbers are transferred beyond the equal sign, then the sign of the numbers changes to the opposite:

Now you only need to perform the actions of both sides of the equality:

The result is an ordinary linear equation. Now you need to divide the left and right sides by the coefficient of the variable.

Solve equations with fractions online updated: October 7, 2018 by: Scientific Articles.Ru