At this lesson, the addition and subtraction of algebraic fractions with different denominators will be considered. We already know how to fold and subtract ordinary fractions with different denominators. For this, the fractions must be brought to a common denominator. It turns out that algebraic fractions obey the same rules. At the same time, we already know how to bring algebraic fractions to the overall denominator. The addition and subtraction of fractions with different denominators is one of the most important and complex topics in the course of grade 8. At the same time, this topic will meet in many themes of the algebra, which you will study in the future. As part of the lesson, we will study the rules for the addition and subtraction of algebraic fractions with different denominators, and also look whole line Typical examples.

Consider the simplest example For ordinary fractions.

Example 1.Fold the fractions :.

Decision:

Recall the rule of embedding frains. To begin with, the fraction must be brought to a common denominator. In the role of a common denominator for ordinary fractions stands the smallest common pain (NOC) source denominators.

Definition

The smallest natural number, which is divided simultaneously in numbers and.

To find the NOC, it is necessary to decompose the denominators for simple factors, and then choose all the simple factors that are included in the decomposition of both denominators.

; . Then in the NOC numbers should include two twos and two three :.

After finding a common denominator, it is necessary for each of the frains to find an additional multiplier (in fact, to divide the general denominator to the denominator of the corresponding fraction).

Then each fraction is multiplied by the optional factor. The fractions are obtained with the same denominators, fold and subtract which we learned at last lessons.

We get: .

Answer:.

We now consider the addition of algebraic fractions with different denominators. First, consider the fractions, whose denominators are numbers.

Example 2.Fold the fractions :.

Decision:

The solution algorithm is absolutely similar to the previous example. Easily choose a common denominator denominator: and additional faults for each of them.

.

Answer:.

So, formulate algorithm for addition and subtraction of algebraic fractions with different denominators:

1. Find the smallest common denominator fractions.

2. Find additional faults for each of the fractions (sharing a common denominator to the denominator of this fraction).

3. Draw the numerators to the corresponding additional faults.

4. Fold or subtract fraction, using the rules for addition and subtract fractions with the same denominators.

We now consider an example with fractions, in the denominator of which there are alphabetic expressions.

Example 3.Fold the fractions :.

Decision:

Since alphabetic expressions in both denominator are the same, then you should find a general denominator for numbers. The final general denominator will look at :. Thus, the solution of this example has the form:.

Answer:.

Example 4.Subtract fractions :.

Decision:

If you do not manage to "snatch" during the selection of a common denominator (it is impossible to decompose on multiplies or use the formulas of abbreviated multiplication), then as a common denominator, you have to take the product of the denominers of both fractions.

Answer:.

In general, when solving such examples, the most difficult task is to find a common denominator.

Consider a more complex example.

Example 5.Simplify :.

Decision:

When finding a common denominator, you must first try to decompose the denominators of the initial fractions on multipliers (to simplify the overall denominator).

In this case:

Then it is easy to define a common denominator: .

We define additional factors and solve this example:

Answer:.

Now fasten the rules for addition and subtract fractions with different denominators.

Example 6.Simplify :.

Decision:

Answer:.

Example 7.Simplify :.

Decision:

.

Answer:.

Consider now an example in which there are not two, but three fractions (after all, the rules for addition and subtraction for more fractions remain the same).

Example 8.Simplify :.

Calculator fractions Designed to quickly calculate operations with fractions, it will be easy to fold, multiply, divide or deduct.

Modern schoolchildren begin to study fractions already in grade 5, each year the exercises are complicated with them. Mathematical terms and quantities that we learn at school are rarely useful to us in adulthood. However, the fractions, unlike logarithms and degrees, are found in everyday life quite often (distance measurement, weighing goods, etc.). Our calculator is designed for quick operations with fractions.

To begin with, we define what fractions are and what they happen. We call the ratio of one number to another, this is a number consisting of a whole number of units.

Varieties of fractions:

• Ordinary
• Decimal
• Mixed

Example ordinary fractions:

The top value is the numerator, the lower denominator. Dash shows us that the upper number is divided into the lower. Instead of a similar writing format, when the packer is horizontally, you can write differently. You can put an inclined line, for example:

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

Decimal fractions They are the most popular variety of fractions. They consist of a whole part and fractional separated by the comma.

An example of decimal fractions:

0.2, or 6.71 or 0.125

Consist of an integer and fractional part. To find out the meaning of this fraction, you need to fold an integer and fraction.

An example of mixed fractions:

The fraction calculator on our site is able to quickly in the online mode of performing any mathematical transactions:

• Addition
• Subtraction
• Multiplication
• Division

To make the calculation, you need to enter numbers into the field and select the action. The fractions need to fill the numerator and denominator, an integer may not be written (if the fraction is ordinary). Do not forget to click on the "Equal" button.

It is convenient that the calculator immediately provides the process of solving an example with fractions, and not just a ready-made answer. It is thanks to the deployed solution that you can use this material When solving school challenges and for better development of the material passed.

You need to calculate the example:

After entering the indicators in the field form, we get:

To make an independent calculation, enter the data into the form.

Calculator fractions

Related sections.

At this lesson, the addition and subtraction of algebraic fractions with different denominators will be considered. We already know how to fold and subtract ordinary fractions with different denominators. For this, the fractions must be brought to a common denominator. It turns out that algebraic fractions obey the same rules. At the same time, we already know how to bring algebraic fractions to the overall denominator. The addition and subtraction of fractions with different denominators is one of the most important and complex topics in the course of grade 8. At the same time, this topic will meet in many themes of the algebra, which you will study in the future. Within the framework of the lesson, we will study the rules for the addition and subtraction of algebraic fractions with different denominators, and we will also analyze a number of typical examples.

Consider the simplest example for ordinary fractions.

Example 1.Fold the fractions :.

Decision:

Recall the rule of embedding frains. To begin with, the fraction must be brought to a common denominator. In the role of a common denominator for ordinary fractions stands the smallest common pain (NOC) source denominators.

Definition

The smallest natural number, which is divided simultaneously in numbers and.

To find the NOC, it is necessary to decompose the denominators for simple factors, and then choose all the simple factors that are included in the decomposition of both denominators.

; . Then in the NOC numbers should include two twos and two three :.

After finding a common denominator, it is necessary for each of the frains to find an additional multiplier (in fact, to divide the general denominator to the denominator of the corresponding fraction).

Then each fraction is multiplied by the optional factor. The fractions are obtained with the same denominators, fold and subtract which we learned at last lessons.

We get: .

Answer:.

We now consider the addition of algebraic fractions with different denominators. First, consider the fractions, whose denominators are numbers.

Example 2.Fold the fractions :.

Decision:

The solution algorithm is absolutely similar to the previous example. Easily choose a common denominator denominator: and additional faults for each of them.

.

Answer:.

So, formulate algorithm for addition and subtraction of algebraic fractions with different denominators:

1. Find the smallest common denominator fractions.

2. Find additional faults for each of the fractions (sharing a common denominator to the denominator of this fraction).

3. Draw the numerators to the corresponding additional faults.

4. Fold or subtract fraction, using the rules for addition and subtract fractions with the same denominators.

We now consider an example with fractions, in the denominator of which there are alphabetic expressions.

Example 3.Fold the fractions :.

Decision:

Since alphabetic expressions in both denominator are the same, then you should find a general denominator for numbers. The final general denominator will look at :. Thus, the solution of this example has the form:.

Answer:.

Example 4.Subtract fractions :.

Decision:

If you do not manage to "snatch" during the selection of a common denominator (it is impossible to decompose on multiplies or use the formulas of abbreviated multiplication), then as a common denominator, you have to take the product of the denominers of both fractions.

Answer:.

In general, when solving such examples, the most difficult task is to find a common denominator.

Consider a more complex example.

Example 5.Simplify :.

Decision:

When finding a common denominator, you must first try to decompose the denominators of the initial fractions on multipliers (to simplify the overall denominator).

In this case:

Then it is easy to define a common denominator: .

We define additional factors and solve this example:

Answer:.

Now fasten the rules for addition and subtract fractions with different denominators.

Example 6.Simplify :.

Decision:

Answer:.

Example 7.Simplify :.

Decision:

.

Answer:.

Consider now the example in which there are not two, but three fractions (after all, the rules of addition and subtraction for more fractions remain the same).

Example 8.Simplify :.

To the fraction to add an integer, it is enough to perform a number of actions, and rather counting.

For example, you have 7 - an integer, it must be added to the fraction 1/2.

We act as follows:

• 7 Multiply to the denominator (2), it turns out 14,
• to 14 add the upper part (1), 15,
• and we substitute the denominator.
• as a result, it turns out 15/2.

In such a simple way, you can add integers to fractional.

And in order to allocate an integer from the fraction, it is necessary to divide the numerator to the denominator, and the residue - and there will be a fraction.

The addition operation to the correct ordinary fraction of an integer is not difficult and sometimes consists of simply in the formation of a mixed fraction in which whole part It is put to the left of the fractional part, for example, such a fraction will be mixed:

However, more likely, when adding a fraction of an integer, an incorrect fraction is obtained, which the numerator turns out to be greater than the denominator. This operation is performed like this: an integer is represented as not proper crushed With the same denominator as the adaptable fraction and then simply fold the numerals of both fractions. For example, it will look like this:

5+1/8 = 5*8/8+1/8 = 40/8+1/8 = 41/8

In my opinion it is very simple.

For example, we have fraction 1/4 (this is the same as 0.25, that is, a quarter of an integer).

And to this quarter, you can add any integer, for example. three with a quarter:

3.25. Or in the fraraty it is expressed as: 3 1/4

Here, according to the sample of this example, any fractions with any integers can be folded.

You need to build an integer in the fraction with the denominator 10 (6/10). Next, led to the existing fraction to the general denominator 10 (35 \u003d 610). Well, and perform the operation as with conventional fractions 610 + 610 \u003d 1210 TOTAL 12.

It can be done in two ways.

one). The fraction can be translated into an integer and make addition. For example, 1/2 is 0.5; 1/4 equals 0.25; 2/5 is 0.4 and so on.

We take an integer 5, to which you need to add a fraction 4/5. We transform the fraction: 4/5 is 4 divided by 5 and get 0.8. Adds 0.8 to 5 and we get 5.8 or 5 4/5.

2). The second method: 5 + 4/5 \u003d 29/5 \u003d 5 4/5.

Addition of fractions Simple mathematical action, example, you need to fold an integer 3 and fraction 1/7. To fold these two numbers, you must have one denominator, so you must multiply on seven and divide on this figure, then you get 21/7 + 1/7, the denominator is one, fold 21 and 1, the answer is 22/7 .

Just take and add an integer to this fraction. It is necessary to 6 + 1/2 \u003d 6 1/2. Well, if it is a decimal fraction, it can be for example so 6 + 1.2 \u003d 7.2.

To fold the fraction and an integer, you need to add fractional and write them to an integer, in the form of a complex number, for example, with the addition of an ordinary fraction with an integer, we obtain: 1/2 +3 \u003d 3 1/2; When adding decimal fractions: 0,5 +3 =3,5.

The fraction itself is not an integer, by the fact that it does not reach it, but therefore it is not necessary to translate an integer in this fraction. Therefore, the whole number remains as a whole and fully demonstrates the full denomination, and the fraction of it plots, and shows how much this whole number is not enough before adding the next full score.

Academic example.

10 + 7/3 \u003d 10 integers and 7/3.

Unless of course there are whole, then they are summed up with integers.

12 + 5 7/9 \u003d 17 and 7/9.

It depends on what an integer and what fraction.

If a both allegations positive, You should assign this fraction to an integer. It turns out a mixed number. Moreover, there may be 2 cases.

Case 1.

• The crushing is correct, i.e. numerator less denominator. Then the mixed number received after the attribution and will be the answer.

4/9 + 10 \u003d 10 4/9 (ten whole four ninth).



Case 2.

• The fraction is wrong, i.e. Numerator more denominator. Then a small transformation is required. Incorrect fraction should be turned into a mixed number, in other words, allocate the whole part. This is done like this:



After that, to an integer, you need to add a whole part of the incorrect fraction and to the resulting amount to attribute its fractional part. In the same way, an integer is added to the mixed number.

1) 11/4 + 5 \u003d 2 3/4 + 5 \u003d 7 3/4 (7 three fourth).

2) 5 1/2 + 6 \u003d 11 1/2 (11 integer one second).

If one of the components or both negative, add additions by the rules of addition of numbers with different or identical signs. An integer is represented as the ratio of this number and 1, and then the numerator, and the denominator is multiplied by the number equal to the denominator of the fraction, to which an integer is added.

3) 1/5 + (-2) \u003d 1/5 + -2/1 \u003d 1/5 + -10/5 \u003d -9/5 \u003d -1 4/5 (minus 1 whole four fifters).

4) -13/3 + (-4) \u003d -13/3 + -4/1 \u003d -13/3 + -12/3 \u003d -25/3 \u003d -8 1/3 (minus 8 of integer one third).

Comment.

After dating S. negative numbers, when studying actions with them, grades grade 6 should understand that the negative fraction add a positive integer the same thing that deduct from natural Number fraction. This action is known to be done like this:



In fact, in order to make a fraction and a whole number, you just need to simply bring the existing integer to fractional, and it is simpler simple. You just need to take a denomoter (existing in the example) and make it a number of integer, multiplying it on this denominator and dividing it, here is an example:

2+2/3 = 2*3/3+2/3 = 6/3+2/3 = 8/3

Find the numerator and denominator. The fraction includes two numbers: the number that is located above the feature is called a numerator, and the number below is denominator. The denominator denotes the total number of parts on which some integer is broken, and the numerator is the number of such parts.

• For example, in the fraction ½ the numerator is 1, and the denominator 2.

Determine the denominator. If two or more fracts have a common denominator, in such fractions under the line there is the same number, that is, in this case, some integer is divided into the same number of parts. Folding a fraction with a common denominator is very simple, since the denominator of the total fraction will be the same as the fractions folded. For example:

• Droes 3/5 and 2/5 common denominator 5.
• Drinks 3/8, 5/8, 17/8 General denominator 8.