In mathematics, the fraction is a number consisting of one or more parts (fractions) of a unit. In the form of recording, the fractions are divided into ordinary (example \\ FRAC (5) (8)) and decimal (for example 123.45).

Definition. Ordinary fraction (or simple fraction)

Ordinary (simple) fraction The number of species \\ pm \\ frac (m) (n) is called where M and N are natural numbers. The number M is called numerator of this fraction, and the number n is her denominator.

Horizontal or oblique feature indicates a fission sign, that is, \\ Frac (M) (n) \u003d () ^ m / n \u003d m: n

Ordinary fractions are divided into two types: the correct and incorrect.

Definition. Proper and incorrect fraction

Right It is called a fraction in which the numerator module is less than the denominator module. For example, \\ FRAC (9) (11), because 9

Wrong The fraction in which the numerator module is greater than or equal to the denominator module. Such a fraction is a rational number, the module is greater or equal to one. An example will be fractions \\ FRAC (11) (2), \\ FRAC (2) (1), - \\ FRAC (7) (5), \\ FRAC (1) (1)

Along with the wrong fraction there is a different recording of a number called a mixed fraction (mixed number). Such a fraction is not ordinary.

Definition. Mixed fraction (mixed number)

Mixed fraction It is called a fraction recorded in the form of an integer and proper fraction and is understood as the amount of this number and fraction. For example, 2 \\ FRAC (5) (7)

(recording in the form of a mixed number) 2 \\ FRAC (5) (7) \u003d 2 + \\ FRAC (5) (7) \u003d \\ FRAC (14) (7) + \\ FRAC (5) (7) \u003d \\ FRAC (19 ) (7) (write in the form of incorrect fraction)

The fraction is just a record of the number. The same number can correspond to different fractions, both ordinary and decimal. We form a sign of the equality of two ordinary fractions.

Definition. Sign of fraction equality

Two fractions \\ FRAC (A) (B) and \\ FRAC (C) (D) are equalif a \\ cdot d \u003d b \\ cdot c. For example, \\ FRAC (2) (3) \u003d \\ FRAC (8) (12) as 2 \\ Cdot12 \u003d 3 \\ CDOT8

From the specified feature follows the main property of the fraction.

Property. The main property of the fraci

If the numerator and denominator of this fraction is multiplied or divided into one and the same number, unequal zero, then it turns out the fraction equal to this.

\\ FRAC (A) (B) \u003d \\ FRAC (A \\ CDOT C) (B \\ Cdot C) \u003d \\ FRAC (A: K) (B: \u200b\u200bK); \\ quad c \\ ne 0, \\ quad k \\ ne 0

Using the basic properties of the fraction, it is possible to replace this fraction by another fraction equal to this, but with a smaller numerator and denominator. Such a replacement is called the cutting of the fraction. For example, \\ FRAC (12) (16) \u003d \\ FRAC (6) (8) \u003d \\ FRAC (3) (4) (here the numerator and the denominator were divided first by 2, and then another 2). Reducing the fraction can be carried out if and only if its numerator and denominator are not mutually simple numbers. If the numerator and the denominator of this fraction are mutually simple, it is impossible to cut it, for example, \\ FRAC (3) (4) is an unstable fraction.

Rules for positive fractions:

Of two fractions with the same denominators More that fraction whose numerator is greater. For example, \\ FRAC (3) (15)

Of two fractions with identical numerals More that fraction, the denominator is less. For example, \\ FRAC (4) (11)\u003e \\ FRAC (4) (13).

To compare two fractions with different numerals and denominators, you need to convert both fractions so that their denominators become the same. Such a transformation is called fractions to a common denominator.

This lesson will consider the basic property of algebraic fraction. The ability to correctly and without errors to apply this property is one of the most important basic skills in the whole course of school mathematics and will meet not only throughout the study of this topic, but also in almost all mathematics sections. Previously, the reduction of ordinary fractions was already studied, and at this lesson the reduction of rational fractions will be considered. Despite the rather large external difference, which exists between rational and ordinary fractions, they have a lot of common, namely, and ordinary, and rational fractions are inherent in the same basic property and general rules for performing arithmetic action. Within the framework of the lesson, we will face concepts: cutting the fractions, multiplication and division of the numerator and denominator for the same expression - and consider examples.

Recall the main thing property of ordinary fraci: The fraction value will not change if its numerator and the denominator are simultaneously multiplied or divided into one and the same different number from zero. Recall that the division of the numerator and denominator of the fraction on the same number is different from zero abbreviation.

For example:, while the value of fractions does not change. However, often with the use of this property, many allow standard errors:

1) - The example of dividing only one term from the numerator to 2 is allowed in the above example, and not the entire number. The correct sequence of actions looks like this: or .

2) - Here we see a similar mistake, however, in addition, 0, not as a result of the division, and not 1, which is even more frequent and coarse error.

Now it is necessary to go to consideration algebraic fraci. Recall this concept from the previous lesson.

Definition.Rational (algebraic) fraction - fractional expression of the view, where - polynomials. - numerator denominator.

Algebraic fractions are, in a sense, the generalization of ordinary fractions and above them can be carried out the same operations as over ordinary fractions.

And the numerator, and the denoter denoter can be multiplied and divided into one and the same polynomial (single) or a number other than zero. It will be the identical conversion of algebraic fraction. Recall that as before, the division of the numerator and denominator of the fraction on the same expression is called from zero abbreviation.

The main property of algebraic fraction Allows you to cut the fractions and bring them to the smallest common denominator.

To reduce ordinary fractions, we resorted to the main theorem of arithmetic, decomposed the numerator, and the denominator for simple factors.

Definition.Prime number - a natural number, which is divided only by one and itself. All other natural numbers are called composite. 1 is neither simple or constant number.

Example 1.a), where the multipliers on which the numerals are decomposed and the denominators of these frains are simple numbers.

Answer.; .

Consequently for reducing fractions It is necessary to pre-decompose the numerator and denominator of the fraction, and then divide them into common multipliers. Those. It should be carried out by the methods of decomposition of polynomials to multipliers.

Example 2. Reduce fraction a) , b), c).

Decision. but). It should be noted that in the numerator there is a full square, and in the denominator the difference of squares. After cut, it is necessary to indicate that, in order to avoid division on zero.

b) . In the denominator, a common numerical factor is made, which is useful to do in almost any case when it is possible. Similarly, with the previous example, we indicate that.

in) . In the denominator, we endure the minus (or formally formally). Do not forget that when cutting.

Answer.;; .

Now we will give an example for bringing to a common denominator, it is done similarly to ordinary fractions.

Example 3.

Decision.To find the smallest common denominator it is necessary to find the smallest common pain (Nok.) Two denominators, i.e. NOK (3; 5). In other words, find the smallest number that is divided by 3 and 5 at the same time. Obviously, this number 15, it can be written in this way: NOK (3; 5) \u003d 15 is a common denominator of these fractions.

To convert the denominator 3 to 15, it must be multiplied by 5, and to convert 5 to 15, it must be multiplied by 3. According to the main property of the algebraic fraction, it is necessary to multiply by the same numbers and the corresponding numerators of the specified fractions.

Answer.; .

Example 4.Lead to a general denomoter and.

Decision. We will conduct similar to the previous example of action. The smallest overall multiple NOC denominators (12; 18) \u003d 36. We give to this denomoter both fractions:

and .

Answer.; .

Now consider the examples demonstrating the use of fraction reduction techniques to simplify them in more complex cases.

Example 5.Calculate the value of the fraction: a), b), B).

but) . With a reduction, we use the rule of degree division.

After we repeated use the main property of ordinary fractionYou can go to the consideration of algebraic fractions.

Example 6.Simplify the fraction and calculate with the specified values \u200b\u200bof the variables: a); b);

Decision.When approaching the solution, the following option is possible - immediately substitute the values \u200b\u200bof the variables and start calculating the fraction, but in this case the decision is greatly complicated and the time it is necessary for its decision, not to mention the danger to be mistaken in complex computing. Therefore, it is convenient to first simplify the expression in the letter form, and then substitute the values \u200b\u200bof the variables.

but) . When a multiplier is reduced, it is necessary to check if it does not turn to zero in the specified values \u200b\u200bof variables. When substitution, we obtain that it makes it possible to reduce this multiplier.

b). In the denominator, we take a minus, as we have already done it in example 2. When reducing on again, we check if we do not divide to zero :.

Answer.; .

Example 7.Lead to the general denominator of the fraction a) and, b) and, c) and.

Decision. a) In this case, we will approach the solution as follows: we will not use the concept of NOC, as in the second example, and simply multiply the denominator of the first fraction on the denominator of the second and on the contrary - this will allow you to bring the fraction to the same denominator. Of course, we do not forget to multiply the numerals of fractions on the same expressions.

. In the numerator opened brackets, and in the denominator used the color difference formula.

. Similar actions.

It can be seen that this method allows you to multiply the denominator and the numerator of one fraction on the element from the channel of the second fraction, which is not enough. With another, the fraction is carried out similar actions, and the denominators are given to the total.

b) We do similar to the previous point of action:

. Multiply the numerator and the denominator on that element of the channel of the second fraction, which was lacking (in this case, to the entire denominator).

. Similarly.

in) . In this case, we were multiplied by 3 (the multiplier which is present in the denominator of the second fraction and is absent in the first).

.

Answer. but) ; b); , in) ; .

In this lesson, we studied the main property of algebraic fraction and considered the basic tasks with its use. In the next lesson, we will describe in more detail bringing fractions to a common denominator using the formulas of abbreviated multiplication and the grouping method when expanding on multipliers.

Bibliography

1. Bashmakov M.I. Algebra Grade 8. - M.: Enlightenment, 2004.
2. Dorofeyev G.V., Suvorova S.B., Baynovich E.A. and others. Algebra 8. - 5th ed. - M.: Enlightenment, 2010.
3. Nikolsky S.M., Potapov MA, Reshetnikov N.N., Shevkin A.V. Algebra Grade 8. Textbook for general education institutions. - M.: Education, 2006.

1. Ege in mathematics ().
2. Festival of pedagogical ideas "Open Lesson" ().
3. Mathematics at school: Purchasing plans ().

Homework

Speaking of mathematics, it is impossible not to recall the fraraty. Their study pays a lot of attention and time. Remember how many examples you had to decide to assimilate certain rules for working with fractions, as you remembered and applied the main property of the fraction. How many nerves were spent for finding a common denominator, especially if there were more than two terms in the examples!

Let's remember what it is, and a little refreshing in memory basic information and rules for working with fractions.

Definition of fractions

Let's start, perhaps, from the most important thing - the definition. The fraction is a number that consists of one or more units. The fractional number is written in the form of two numbers, separated by horizontal or scytheless feature. At the same time, the upper (or first) is called the numerator, and the lower (second) - denominator.

It is worth noting that the denominator shows how many parts are divided, and the numerator is the number taken by the share or parts. Often the fraction if they are correct, less than one.

Now let's consider the properties of these numbers and the basic rules that are used when working with them. But before we disassemble such a thing as the "basic property of rational fraction," we will talk about the types of fractions and their features.

What are the frauds

There are several types of such numbers. First of all, these are ordinary and decimal. The first represents the appearance of the recording by us using the horizontal or oblique feature. The second type of fractions is denoted by the so-called positional recording, when the integer follows first, and then, after the comma, the fractional part is indicated.

It is worth noting that both decimal and ordinary fractions are equally used in mathematics. The main property of the fraction is valid only for the second option. In addition, in ordinary fractions, the correct and incorrect numbers are distinguished. The first numerator is always less denominator. We also note that such a fraction is less than one. In the wrong fraction, the opposite is a numerator more denominator, and she herself is more than one. In this case, an integer can be distinguished from it. In this article we will consider only ordinary fractions.

Properties of fractions

Any phenomenon, chemical, physical or mathematical, has its own characteristics and properties. No exception and fractional numbers. They have one important feature, with the help of which these or other operations can be carried out. What is the main property of the fraction? The rule says that if its numerator and denominator multiplied or divided into one and the same rational number, we get a new fraction, the value of which will be equal to the value of the original. That is, multiplying two parts of a fractional number 3/6 to 2, we will get a new shot 6/12, while they will be equal.

Based on this property, you can cut the fraction, as well as select common denominators for a particular pair of numbers.

Operations

Despite the fact that the fractions seem to be more complex to us, compared with them, you can also perform basic mathematical operations, such as addition and subtraction, multiplication and division. In addition, there is such a specific effect as a reduction of fractions. Naturally, each of these actions is made according to a specific rules. Knowledge of these laws facilitates work with fractions, makes it easier and interesting. That is why then we will consider the basic rules and algorithms of action when working with such numbers.

But before talking about such mathematical operations as addition and subtraction, we will analyze such an operation as bringing to a common denominator. Here we just do and use the knowledge of what the main property of the fraction exists.

Common denominator

In order for the number to lead to a common denominator, first need to find the smallest common multiple for two denominators. That is the smallest number, which is simultaneously divided into both denominator without a residue. The easiest way to choose the NOC (the smallest common multiple) is to write it into a line for one denominator, then for the second and find the coincident number among them. In the event that the NOC is not found, that is, these numbers do not have a common multiple number, multiply them, and the value obtained is considered for the NOC.

So, we found NOC, now you should find an additional factor. To do this, it is necessary to divide the NOC to the shooters of the frains and write down the resulting number above each of them. Next, you should multiply the numerator and the denominator to the received additional factor and write the results in the form of a new fraction. If you doubt that the number you received is still, remember the main property of the fraction.

Addition

We now turn directly to mathematical operations over fractional numbers. Let's start with the simplest. There are several embodiments of fractions. In the first case, both numbers have the same denominator. In this case, it remains only to fold the numbers among themselves. But the denominator does not change. For example, 1/5 + 3/5 \u003d 4/5.

If fractions have different denominators, you should bring them to general and only then perform addition. How to do this, we disassembled a little higher. In this situation, you just use the main property of the fraction. The rule will cause the number to the general denominator. In this case, the value will not change in any way.

Alternatively, it may happen that the fraction is mixed. Then you should first fold with the whole parts, and then fractional.

Multiplication

Does not require any tricks, and to fulfill this action, it is not necessary to know the main property of the fraction. It is enough to first multiply the numerals and denominators. In this case, the product of numerals will become a new numerator, and the denominers are a new denominator. As you can see, nothing complicated.

The only thing about you is required is knowledge of multiplication tables, as well as attentiveness. In addition, after receiving the result, it is necessary to check whether this number can be reduced or not. How to cut the fraction, we will tell a little later.

Subtraction

Performing should be guided by the same rules as when adding. Thus, in numbers with the same denominator, it is sufficient from the numerator of the decrease in the numerator subtractable. In the event that frains have different denominators, you should bring them to general and then perform this operation. As in the same case with addiction, you will need to use the basic property of algebraic fraction, as well as skills in finding NOC and common divisors for fractions.

Division

And the latter, the most interesting operation when working with such numbers - division. It is quite simple and does not cause special difficulties even in those who are poorly dealt with how to work with fractions, especially perform addition and subtraction operations. During the division, this rule acts as multiplication on the back shot. The main property of the fraction, as in the case of multiplication, will be involved for this operation. We will analyze more.

When dividing numbers, divisible remains unchanged. The fraction of the divider turns into a reverse, that is, the numerator with the denominator changes in places. After that, the number is multiplied with each other.

Abbreviation

So, we have already disassembled the definition and structure of fractions, their types, rules for operations on these numbers, found out the main property of algebraic fraction. Now let's talk about such an operation as a reduction. The cutting of the fraction is the process of its conversion - dividing the numerator and denominator to the same number. Thus, the fraction is reduced without changing its properties.

Usually, when making a mathematical operation, it is necessary to carefully look at the result obtained in the end and find out whether it is possible to reduce the resulting fraction or not. Remember that in the final result, a fractional short-circuit is always recorded.

Other operations

Finally, we note that we have listed far from all transactions on fractional numbers, mentioning only the most famous and necessary. The fraction can also be compared, convert to decimal and vice versa. But in this article we did not consider these operations, since in mathematics they are carried out much less often than those that have been given above.

conclusions

We talked about fractional numbers and operations with them. We also disassemble the basic property but we note that all these issues we considered casual. We only led the most famous and used rules, gave the most important, in our opinion, tips.

This article is designed to quickly refresh you forgotten information about the frauds, rather than give new information and "score" the head of endless rules and formulas, which, most likely, you will not use it.

We hope that the material presented in the article simply and concisely became useful for you.

Drobi.

Attention!
This topic has additional
Materials in a special section 555.
For those who are strongly "not very ..."
And for those who are "very ...")

The fractions in high schools are not very annoyed. For the time being. So far, do not come up with degrees with rational indicators and logarithms. And here .... You give, you give a calculator, and he all the complete scoreboard does not seem to. I have to think about thinking as in the third grade.

Let's figure out with the fractions finally! Well, how much can you get confused!? Moreover, it's simple and logical. So, what are the fractions?

Types of fractions. Conversion.

The fraraty is three species.

1. Ordinary fractions , eg:

Sometimes instead of horizontal screenshots, they put an inclined line: 1/2, 3/4, 19/5, well, and so on. Here we will often be this writing to use. The upper number is called numerator, Lower - denominator. If you constantly confuse these names (happens ...), tell me with the phrase expression: " ZZZZapumnney! ZZZZnamer - Vni zZZZy! "You look, everything and zzzzozomnikh.)

Chertochka that is horizontal that inclined means division top number (numerator) to the bottom (denominator). And that's all! Instead of a screw, it is quite possible to put a fission sign - two points.

When the division is possible, it should be done. So, instead of fractions "32/8", it is much more pleasant to write the number "4". Those. 32 Just divide by 8.

32/8 = 32: 8 = 4

I'm not talking about the fraction "4/1". Which is also just "4". And if it is not divided by a lot, we leave, in the form of a fraction. Sometimes there is a reverse operation to do. Make an integer fraction. But about this below.

2. Decimal fractions , eg:

It is in this form that will need to record the answers to the tasks "B".

3. Mixed numbers , eg:

Mixed numbers are practically not used in high school. In order to work with them, they must be translated into ordinary fractions. But it's necessary to be able to do! And then there will be such a number in a task and hang ... in an empty place. But we will remember this procedure! Slightly lower.

The most universal Ordinary fractions. With them and start. By the way, if there are all sorts of logarithms, sinuses and other beaks, it does not change anything. In the sense that all actions with fractional expressions are no different from action with ordinary fractions!

The main property of the fraction.

So let's go! To begin with, I will surprise you. All fraction transformation varieties is provided by one-sole property! It is called the main property of the fraci. Remember: if the numerator and denominator of the fraci multiply (divided) per and the same number, the fraction will not change. Those:

It is clear that you can write further before the formation. Sinuses and logarithms let you do not embarrass, we'll figure it out with them. The main thing is to understand that all these diverse expressions are one and the same fraction . 2/3.

And we need it, all these transformations? And how! Now you will see. To begin with, we will use the main property of the fraction for reducing fractions. It would seem that the thing is elementary. We divide the numerator and denominator for the same number and all things! It is impossible to make a mistake! But ... a person is creative creature. Make a mistake everywhere! Especially if you have to reduce the fraction of type 5/10, but a fractional expression with all sorts of beaks.

As properly and quickly cut the fraction, without making any extra work, you can read in a special section 555.

The normal student is not bothering the division of the numerator and the denominator on the same number (or expression)! He simply jumps all the same on top and bottom! Here is a typical mistake, a lap, if you want.

For example, you need to simplify the expression:

There is nothing to think here, you jump up the letter "A" from above and a twice from below! We get:

That's right. But really you divided all Numerator I. all danger on "A". If you are used to simply cross, then, you need, you can cross "A" in expression

and get again

What will be categorically incorrect. Because here all Numerator on "A" already not divide! It is impossible to cut this fraction. By the way, such a reduction is, GM ... a serious challenge to the teacher. This is not forgiven! Remember? When cutting, we need to share all Numerator I. all denominator!

Reducing fractions greatly facilitates life. It turns out somewhere you have fraction, for example 375/1000. And how now to work with her? Without a calculator? Multiply, say, fold, in a square to erect!? And if you don't be lazy, yes, it is accurate enough to cut five, and even five, and even ... while it is reduced, in short. We get 3/8! Much more pleasant, right?

The main property of the fraction allows us to translate ordinary fractions to decimal and vice versa without calculator! This is important to the exam, right?

How to translate fractions from one species to another.

With decimal fractions, everything is simple. As heard, it is written! Let's say 0.25. This is a zero whole, twenty-five hundredths. Yes, we write: 25/100. We reduce (divide the numerator and denominator on 25), we get the usual fraction: 1/4. Everything. It happens, and nothing is reduced. Type 0.3. These are three tenths, i.e. 3/10.

And if integers - not zero? Nothing wrong. We write down the entire fraction without any commas In the numerator, and in the denominator is what hearse. For example: 3.17. These are three integers, seventeen hundredths. We write in the numerator 317, and in the denominator 100. We get 317/100. Nothing is reduced, it means everything. This is the answer. Elementary Watson! Of all the told useful conclusion: any decimal fraction can be turned into an ordinary .

But the inverse transformation, ordinary to decimal, some without a calculator cannot do. But you must! How do you write to write on the exam!? Carefully read and master this process.

Decimal fraction than characteristic? She has in the denominator always It costs 10, or 100, or 1000, or 10,000 and so on. If your usual fraction has such a denominator, there are no problems. For example, 4/10 \u003d 0.4. Or 7/100 \u003d 0.07. Or 12/10 \u003d 1.2. And if in response to the task section "in" turned out 1/2? What will we write in response? There are decimal required ...

Remember the main property of the fraci ! Mathematics favorably allows you to multiply the numerator and denominator for the same number. For any, by the way! In addition to zero, of course. So applies this property to yourself! What can be multiplied by the denominator, i.e. 2 So that it become 10, or 100, or 1000 (smaller better, of course ...)? 5, obviously. Boldly multiply the denominator (this us it is necessary) by 5. But, then the numerator must be multiplied, too, for 5. This is already mathematics Requires! We obtain 1/2 \u003d 1x5 / 2x5 \u003d 5/10 \u003d 0.5. That's all.

However, the denominators are all sorts. Will come, for example, the fraction 3/16. Try, figure out here, on which 16 multiply, so that 100 it happens, or 1000 ... does not work? Then you can simply separate 3 to 16. Behind the lack of a calculator, you will have to divide the corner, on a piece of paper, as in junior grades. We get 0.1875.

And there are completely bad denominants. For example, a fraction of 1/3, well, do not turn into a good decimal. And on the calculator, and on a piece of paper, we will get 0,3333333 ... This means that 1/3 in an exact decimal fraction does not translate. Just as 1/7, 5/6 and so on. Many of them undeveloped. From here another useful conclusion. Not every ordinary fraction is translated into decimal !

By the way, this is useful information for self-test. In the section "B" in response, you need a decimal fraction to record. And you have it, for example, 4/3. This fraction is not translated into decimal. This means that somewhere you made a mistake on the road! Return, check the solution.

So, with ordinary and decimal fractions figured out. It remains to deal with mixed numbers. To work with them, they must be translated into ordinary fractions. How to do it? You can catch a sixth grader and ask him. But not always the sixth grader will be at hand ... you have to. It's not hard. It is necessary a denominator of a fractional part to multiply by a whole part and add a fractional part numerator. It will be a numerator of the usual fraction. And the denominator? The denominator will remain the same. It sounds difficult, but in fact everything is elementary. We look an example.

Let in a challenge you with horror saw a number:

Calmly, without panic, we think. The whole part is 1. one. Fractional part - 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be a denominator of an ordinary fraction. We consider the numerator. 7 Multiply with 1 (whole part) and add 3 (numerator of the fractional part). We get 10. It will be a numerator of an ordinary fraction. That's all. Even easier, it looks in a mathematical record:

Clear? Then secure success! Translate into ordinary fractions. You should work 10/7, 7/2, 23/10 and 21/4.

Reverse operation - Translation of incorrect fraction in a mixed number - in high schools is rarely required. Well, if so ... and if you are not in high schools - you can look into a special section 555. There, by the way, and about the wrong fraraty will learn.

Well, almost everything. You remembered the types of fractions and understood as Translate them from one species to another. The question remains: what for do it? Where and when to apply these deep knowledge?

I answer. Any example itself suggests the necessary actions. If an example was mixed into a bunch of ordinary fractions, decimal, and even mixed numbers, we translate everything into ordinary fractions. It can always be done. Well, if it is written, something like 0.8 + 0.3, then I believe without any translation. Why do we need extra work? We choose that path the solution that is convenient us !

If the task is complete decimal fractions, but um ... angry some, go to ordinary, try! You look, everything will work. For example, it will be in a square to erect a number 0.125. Not so easy if you did not pay off from the calculator! Not only you need to multiply the column, so think, where to insert comma! In mind it will not be exactly! And if you go to an ordinary fraction?

0.125 \u003d 125/1000. Reducing on 5 (this is for starters). We get 25/200. Once again by 5. We get 5/40. Oh, still cuts! Again on 5! We get 1/8. Easily erected into a square (in the mind!) And we get 1/64. Everything!

Let's summarize this lesson.

1. Fruit is three species. Ordinary, decimal and mixed numbers.

2. Decimal fractions and mixed numbers always You can translate into ordinary fractions. Reverse translation not always available.

3. Selecting the type of fractions to work with the task depends on this very task. If there are different types of fractions in one task, the most reliable - go to ordinary fractions.

Now you can take care. To begin with, translate these decimal fractions to ordinary:

3,8; 0,75; 0,15; 1,4; 0,725; 0,012

There must be such answers (in disorder!):

On this and end. In this lesson, we refreed in memory key moments for fractions. It happens, however, it is especially nothing to refreshing ...) If someone who has completely forgotten, or has not mastered ... the one can go into a special section 555. There all the foundations are detailed. Many suddenly understand everything Start. And decide the fraraty with the lea).

If you like this site ...

By the way, I have another couple of interesting sites for you.)

It can be accessed in solving examples and find out your level. Testing with instant check. Learn - with interest!)

You can get acquainted with features and derivatives.

Disassembled in detail the main property of the fraciThe wording is given, the proof and explaining example is given. The use of the main property of the fraction with the reduction of fractions and bringing fractions to a new denominator is also considered.

Navigating page.

The main property of the fraction - the wording, proof and explanatory examples

Let's consider an example illustrating the main property of the fraction. Let we have a square divided into 9 "big" squares, and each of these "large" squares is divided into 4 "small" squares. Thus, it is also possible to say that the original square is divided into 4 · 9 \u003d 36 "small" squares. Form 5 "big" squares. At the same time, 4 · 5 \u003d 20 "small" squares will be painted. We give a drawing that meets our example.

The painted part is 5/9 of the original square, or that the same, 20/36 of the original square, that is, the fractions 5/9 and 20/36 are equal to: or. Of these equalities, as well as from equalities 20 \u003d 5 · 4, 36 \u003d 9 · 4, 20: 4 \u003d 5 and 36: 4 \u003d 9, it follows that.

To secure disassembled material, consider the solution of the example.

Example.

The numerator and the denominator of some ordinary fraction was multiplied by 62, after which the numerator and denominator of the resulting fraci was divided into 2. Is it equal to the resulting fraction of the original?

Decision.

The multiplication of the numerator and denominator of the fraction on any natural number, in particular by 62, gives a fraction that, due to the basic properties of the fraction, is equal to the initial one. The main property of the fraction allows the fact that after dividing the numerator and denominator of the resulting fraction on 2 will turn out to be a fraction that will be equal to the initial fraction.

Answer:

Yes, the resulting fraction is equal to the source.

Application of the main properties of the fraction

The main property of the fraction is mainly applied in two cases: first, when we bring fractions to a new denominator, and, secondly, when cutting fractions.

The main property of the fraction makes it possible to reduce fractions, and as a result, turn from the initial fraction to the fraction equal to it, but with a smaller numerator and denominator. The cutting of the fraction lies in the division of the numerator and the denominator of the original fraction on any different number of the positive number and the denominator (if there are no such common divisors, then the initial fraction is inconsistent, that is, it is not subject to reduction). In particular, dividing on will lead the initial fraction to an incomprehensive form.

Bibliography.

• Vilekin N.Ya., Zhokhov V.I., Chesnokov A.S., Schwarzburg S.I. Mathematics: Tutorial for 5 CL. general educational institutions.
• Vilenkin N.Ya. and others. mathematics. Grade 6: Textbook for general educational institutions.

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