Solving equations with a variable in the denominator of a fraction. How to solve equations with fractions

The lowest common denominator is used to simplify this equation. This method is used when you cannot write a given equation with one rational expression on each side of the equation (and use the crisscross method of multiplication). This method is used when you are given a rational equation with 3 or more fractions (in the case of two fractions, it is better to use criss-cross multiplication).

Fraction calculator designed for quickly calculating operations with fractions, it will help you easily add, multiply, divide or subtract fractions.

Modern schoolchildren begin studying fractions already in the 5th grade, and exercises with them become more complicated every year. The mathematical terms and quantities that we learn in school can rarely be useful to us in adult life. However, fractions, unlike logarithms and powers, are found quite often in everyday life (measuring distances, weighing goods, etc.). Our calculator is designed for quick operations with fractions.

First, let's define what fractions are and what they are. Fractions are the ratio of one number to another; it is a number consisting of an integer number of fractions of a unit.

Types of fractions:

• Ordinary
• Decimal
• Mixed

Example ordinary fractions:

The top value is the numerator, the bottom is the denominator. The dash shows us that the top number is divisible by the bottom number. Instead of this writing format, when the dash is horizontal, you can write differently. You can put an inclined line, for example:

1/2, 3/7, 19/5, 32/8, 10/100, 4/1

Decimals are the most popular type of fractions. They consist of an integer part and a fractional part, separated by a comma.

Example of decimal fractions:

0.2 or 6.71 or 0.125

Consist of a whole number and a fractional part. To find out the value of this fraction, you need to add the whole number and the fraction.

Example of mixed fractions:

The fraction calculator on our website is able to quickly perform any mathematical operations with fractions online:

• Addition
• Subtraction
• Multiplication
• Division

To carry out the calculation, you need to enter numbers in the fields and select an action. For fractions, you need to fill in the numerator and denominator; the whole number may not be written (if the fraction is ordinary). Don't forget to click on the "equal" button.

It’s convenient that the calculator immediately provides the process for solving an example with fractions, and not just a ready-made answer. It is thanks to the detailed solution that you can use this material to solve school problems and to better master the material covered.

You need to perform the example calculation:

After entering the indicators into the form fields, we get:

To make your own calculation, enter the data in the form.

Fraction calculator

Related sections.

Actions with fractions. In this article we will look at examples, everything in detail with explanations. We will consider ordinary fractions. We'll look at decimals later. I recommend watching the whole thing and studying it sequentially.

1. Sum of fractions, difference of fractions.

Rule: when adding fractions with equal denominators, the result is a fraction - the denominator of which remains the same, and its numerator will be equal to the sum of the numerators of the fractions.

Rule: when calculating the difference between fractions with the same denominators, we obtain a fraction - the denominator remains the same, and the numerator of the second is subtracted from the numerator of the first fraction.

Formal notation for the sum and difference of fractions with equal denominators:

Examples (1):

It is clear that when ordinary fractions are given, then everything is simple, but what if they are mixed? Nothing complicated...

Option 1– you can convert them into ordinary ones and then calculate them.

Option 2– you can “work” separately with the integer and fractional parts.

Examples (2):

More:

What if the difference of two mixed fractions is given and the numerator of the first fraction is less than the numerator of the second? You can also act in two ways.

Examples (3):

*Converted to ordinary fractions, calculated the difference, converted the resulting improper fraction to a mixed fraction.

*We broke it down into integer and fractional parts, got a three, then presented 3 as the sum of 2 and 1, with one represented as 11/11, then found the difference between 11/11 and 7/11 and calculated the result. The meaning of the above transformations is to take (select) a unit and present it in the form of a fraction with the denominator we need, then we can subtract another from this fraction.

Another example:

Conclusion: there is a universal approach - in order to calculate the sum (difference) of mixed fractions with equal denominators, they can always be converted to improper ones, then perform the necessary action. After this, if the result is an improper fraction, we convert it to a mixed fraction.

Above we looked at examples with fractions that have equal denominators. What if the denominators are different? In this case, the fractions are reduced to the same denominator and the specified action is performed. To change (transform) a fraction, the basic property of the fraction is used.

Let's look at simple examples:

In these examples, we immediately see how one of the fractions can be transformed to get equal denominators.

If we designate ways to reduce fractions to the same denominator, then we will call this one METHOD ONE.

That is, immediately when “evaluating” a fraction, you need to figure out whether this approach will work - we check whether the larger denominator is divisible by the smaller one. And if it is divisible, then we carry out the transformation - we multiply the numerator and denominator so that the denominators of both fractions become equal.

Now look at these examples:

This approach is not applicable to them. There are also ways to reduce fractions to a common denominator; let’s consider them.

Method TWO.

We multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first:

*In fact, we reduce fractions to the form when the denominators become equal. Next, we use the rule for adding fractions with equal denominators.

Example:

*This method can be called universal, and it always works. The only downside is that after the calculations you may end up with a fraction that will need to be further reduced.

Let's look at an example:

It can be seen that the numerator and denominator are divisible by 5:

Method THREE.

You need to find the least common multiple (LCM) of the denominators. This will be the common denominator. What kind of number is this? This is the smallest natural number that is divisible by each of the numbers.

Look, here are two numbers: 3 and 4, there are many numbers that are divisible by them - these are 12, 24, 36, ... The smallest of them is 12. Or 6 and 15, 30, 60, 90 are divisible by them.... The least is 30. The question is - how to determine this least common multiple?

There is a clear algorithm, but often this can be done immediately without calculations. For example, according to the above examples (3 and 4, 6 and 15) no algorithm is needed, we took large numbers (4 and 15), doubled them and saw that they are divisible by the second number, but pairs of numbers can be others, for example 51 and 119.

Algorithm. In order to determine the least common multiple of several numbers, you must:

- decompose each number into SIMPLE factors

— write down the decomposition of the BIGGER of them

- multiply it by the MISSING factors of other numbers

Let's look at examples:

50 and 60 => 50 = 2∙5∙5 60 = 2∙2∙3∙5

in the expansion of a larger number one five is missing

=> LCM(50,60) = 2∙2∙3∙5∙5 = 300

48 and 72 => 48 = 2∙2∙2∙2∙3 72 = 2∙2∙2∙3∙3

in the expansion of a larger number two and three are missing

=> LCM(48.72) = 2∙2∙2∙2∙3∙3 = 144

* The least common multiple of two prime numbers is their product

Question! Why is finding the least common multiple useful, since you can use the second method and simply reduce the resulting fraction? Yes, it is possible, but it is not always convenient. Look at the denominator for the numbers 48 and 72 if you simply multiply them 48∙72 = 3456. You will agree that it is more pleasant to work with smaller numbers.

Let's look at examples:

*51 = 3∙17 119 = 7∙17

the expansion of a larger number is missing a triple

=> NOC(51,119) = 3∙7∙17

Now let's use the first method:

*Look at the difference in the calculations, in the first case there are a minimum of them, but in the second you need to work separately on a piece of paper, and even the fraction that you received needs to be reduced. Finding the LOC simplifies the work significantly.

More examples:

*In the second example it is clear that the smallest number that is divisible by 40 and 60 is 120.

RESULT! GENERAL COMPUTING ALGORITHM!

— we reduce fractions to ordinary ones if there is an integer part.

- we bring fractions to a common denominator (first we look at whether one denominator is divisible by another; if it is divisible, then we multiply the numerator and denominator of this other fraction; if it is not divisible, we act using the other methods indicated above).

- Having received fractions with equal denominators, we perform operations (addition, subtraction).

- if necessary, we reduce the result.

- if necessary, then select the whole part.

2. Product of fractions.

The rule is simple. When multiplying fractions, their numerators and denominators are multiplied:

Examples:

Instructions

Reduction to a common denominator.

Let the fractions a/b and c/d be given.

The numerator and denominator of the first fraction are multiplied by LCM/b

The numerator and denominator of the second fraction are multiplied by LCM/d

An example is shown in the figure.

To compare fractions, you need to add them to a common denominator, then compare the numerators. For example, 3/4< 4/5, см. .

Adding and subtracting fractions.

To find the sum of two ordinary fractions, they need to be brought to a common denominator, then add the numerators, leaving the denominator unchanged. An example of adding fractions 1/2 and 1/3 is shown in the figure.

The difference of fractions is found in a similar way; after finding the common denominator, the numerators of the fractions are subtracted, see the figure.

When multiplying ordinary fractions, the numerators and denominators are multiplied together.

In order to divide two fractions, a fraction of the second fraction is necessary, i.e. change its numerator and denominator, and then multiply the resulting fractions.

Video on the topic

Sources:

• fractions grade 5 using an example
• Basic fraction problems

Module represents the absolute value of the expression. Straight brackets are used to denote a module. The values ​​contained in them are considered modulo. Solving the module consists of opening parentheses according to certain rules and finding the set of expression values. In most cases, the module is expanded in such a way that the submodular expression receives a number of positive and negative values, including a zero value. Based on these properties of the module, further equations and inequalities of the original expression are compiled and solved.

Instructions

Write the original equation with . To do this, open the module. Consider each submodular expression. Determine at what value of the unknown quantities included in it the expression in modular brackets becomes zero.

To do this, equate the submodular expression to zero and find the resulting equation. Write down the values ​​you find. In the same way, determine the values ​​of the unknown variable for each module in the given equation.

Draw a number line and plot the resulting values ​​on it. The values ​​of the variable in the zero module will serve as constraints when solving the modular equation.

In the original equation, you need to expand the modular ones, changing the sign so that the values ​​of the variable correspond to those displayed on the number line. Solve the resulting equation. Check the found value of the variable against the constraint specified by the module. If the solution satisfies the condition, it is true. Roots that do not satisfy the restrictions must be discarded.

In the same way, expand the modules of the original expression, taking into account the sign, and calculate the roots of the resulting equation. Write down all the resulting roots that satisfy the constraint inequalities.

Fractional numbers allow you to express the exact value of a quantity in different forms. You can do the same math operations with fractions as you can with whole numbers: subtraction, addition, multiplication, and division. To learn to decide fractions, we must remember some of their features. They depend on the type fractions, the presence of an integer part, a common denominator. Some arithmetic operations require the fractional part of the result to be reduced after execution.

You will need

• - calculator

Instructions

Take a close look at the numbers. If among the fractions there are decimals and irregular ones, sometimes it is more convenient to first perform operations with decimals, and then convert them to the irregular form. Can you translate fractions in this form initially, writing the value after the decimal point in the numerator and putting 10 in the denominator. If necessary, reduce the fraction by dividing the numbers above and below by one divisor. Fractions in which an integer part is isolated must be converted to the wrong form by multiplying it by the denominator and adding the numerator to the result. This value will become the new numerator fractions. To select a whole part from an initially incorrect one fractions, you need to divide the numerator by the denominator. Write the whole result from fractions. And the remainder of the division will become the new numerator, denominator fractions it does not change. For fractions with an integer part, it is possible to perform actions separately, first for the integer and then for the fractional parts. For example, the sum of 1 2/3 and 2 ¾ can be calculated:
- Converting fractions to the wrong form:
- 1 2/3 + 2 ¾ = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12;
- Summation of separately integer and fractional parts of terms:
- 1 2/3 + 2 ¾ = (1+2) + (2/3 + ¾) = 3 +(8/12 + 9/12) = 3 + 12/17 = 3 + 1 5/12 = 4 5 /12.

For with values ​​below the line, find the common denominator. For example, for 5/9 and 7/12 the common denominator will be 36. For this, the numerator and denominator of the first fractions you need to multiply by 4 (you get 28/36), and the second one - by 3 (you get 15/36). Now you can perform the calculations.

If you are going to calculate the sum or difference of fractions, first write the found common denominator under the line. Perform the necessary actions between the numerators, and write the result above the new line fractions. Thus, the new numerator will be the difference or sum of the numerators of the original fractions.

To calculate the product of fractions, multiply the numerators of the fractions and write the result in place of the numerator of the final fractions. Do the same for the denominators. When dividing one fractions write down one fraction on the other, and then multiply its numerator by the denominator of the second. In this case, the denominator of the first fractions multiplied accordingly by the second numerator. In this case, a kind of revolution occurs fractions(divisor). The final fraction will be the result of multiplying the numerators and denominators of both fractions. It's not hard to learn fractions, written in the condition in the form of “four-story” fractions. If it separates two fractions, rewrite them using the “:” separator and continue with normal division.

To obtain the final result, reduce the resulting fraction by dividing the numerator and denominator by one whole number, the largest possible in this case. In this case, there must be integers above and below the line.

Please note

Do not perform arithmetic with fractions whose denominators are different. Choose a number such that when you multiply the numerator and denominator of each fraction by it, the result is that the denominators of both fractions are equal.

Useful advice

When writing fractional numbers, the dividend is written above the line. This quantity is designated as the numerator of the fraction. The divisor, or denominator, of the fraction is written below the line. For example, one and a half kilograms of rice as a fraction will be written as follows: 1 ½ kg of rice. If the denominator of a fraction is 10, the fraction is called a decimal. In this case, the numerator (dividend) is written to the right of the whole part, separated by a comma: 1.5 kg of rice. For ease of calculation, such a fraction can always be written in the wrong form: 1 2/10 kg of potatoes. To simplify, you can reduce the numerator and denominator values ​​by dividing them by one integer. In this example, you can divide by 2. The result will be 1 1/5 kg of potatoes. Make sure that the numbers you are going to perform arithmetic with are presented in the same form.

Instructions

Click once on the “Insert” menu item, then select “Symbol”. This is one of the easiest ways to insert fractions into the text. It consists in the following. The set of ready-made symbols includes fractions. Their number, as a rule, is small, but if you need to write ½ in the text rather than 1/2, then this option will be the most optimal for you. In addition, the number of fraction characters may depend on the font. For example, for the Times New Roman font there are slightly fewer fractions than for the same Arial. Vary fonts to find the best option when it comes to simple expressions.

Click on the “Insert” menu item and select the “Object” sub-item. A window will appear in front of you with a list of possible objects to insert. Choose among them Microsoft Equation 3.0. This app will help you type fractions. And not only fractions, but also complex mathematical expressions containing various trigonometric functions and other elements. Double-click on this object with the left mouse button. A window will appear in front of you containing many symbols.

To print a fraction, select the symbol representing a fraction with an empty numerator and denominator. Click on it once with the left mouse button. An additional menu will appear, clarifying the scheme itself. fractions. There may be several options. Select the one that suits you best and click on it once with the left mouse button.

Solving equations with fractions Let's look at examples. The examples are simple and illustrative. With their help, you will be able to understand in the most understandable way.
For example, you need to solve the simple equation x/b + c = d.

An equation of this type is called linear, because The denominator contains only numbers.

The solution is performed by multiplying both sides of the equation by b, then the equation takes the form x = b*(d – c), i.e. the denominator of the fraction on the left side cancels.

For example, how to solve a fractional equation:
x/5+4=9
We multiply both sides by 5. We get:
x+20=45
x=45-20=25

Another example when the unknown is in the denominator:

Equations of this type are called fractional-rational or simply fractional.

We would solve a fractional equation by getting rid of fractions, after which this equation, most often, turns into a linear or quadratic equation, which is solved in the usual way. You just need to consider the following points:

• the value of a variable that turns the denominator to 0 cannot be a root;
• You cannot divide or multiply an equation by the expression =0.

This is where the concept of the region of permissible values ​​(ADV) comes into force - these are the values ​​of the roots of the equation for which the equation makes sense.

Thus, when solving the equation, it is necessary to find the roots, and then check them for compliance with the ODZ. Those roots that do not correspond to our ODZ are excluded from the answer.

For example, you need to solve a fractional equation:

Based on the above rule, x cannot be = 0, i.e. ODZ in this case: x – any value other than zero.

We get rid of the denominator by multiplying all terms of the equation by x

And we solve the usual equation

5x – 2x = 1
3x = 1
x = 1/3

Answer: x = 1/3

Let's solve a more complicated equation:

ODZ is also present here: x -2.

When solving this equation, we will not move everything to one side and bring the fractions to a common denominator. We will immediately multiply both sides of the equation by an expression that will cancel out all the denominators at once.

To reduce the denominators, you need to multiply the left side by x+2, and the right side by 2. This means that both sides of the equation must be multiplied by 2(x+2):

This is the most common multiplication of fractions, which we have already discussed above.

Let's write the same equation, but slightly differently

The left side is reduced by (x+2), and the right by 2. After the reduction, we obtain the usual linear equation:

x = 4 – 2 = 2, which corresponds to our ODZ

Answer: x = 2.

Solving equations with fractions not as difficult as it might seem. In this article we have shown this with examples. If you have any difficulties with how to solve equations with fractions, then unsubscribe in the comments.