From the course of algebra in the school curriculum, we turn to specifics. In this article, we will explore in detail a special kind of rational expressions - rational fractions, and also analyze what characteristic identical transformations of rational fractions take place.

We note right away that rational fractions in the sense in which we define them below are called algebraic fractions in some algebra textbooks. That is, in this article we will mean the same thing as rational and algebraic fractions.

As usual, let's start with a definition and examples. Next, let's talk about reducing a rational fraction to a new denominator and about changing the signs of the members of the fraction. After that, we will analyze how the reduction of fractions is performed. Finally, let's dwell on the representation of a rational fraction as a sum of several fractions. We will provide all the information with examples with detailed descriptions of solutions.

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Definition and examples of rational fractions

Rational fractions are taught in 8th grade algebra lessons. We will use the definition of a rational fraction, which is given in the algebra textbook for 8 classes by Yu.N. Makarychev et al.

This definition does not specify whether the polynomials in the numerator and denominator of a rational fraction must be polynomials of the standard form or not. Therefore, we will assume that the notation of rational fractions can contain both standard and non-standard polynomials.

Here are a few examples of rational fractions... So, x / 8 and - rational fractions. And fractions and do not fit the voiced definition of a rational fraction, since in the first of them there is not a polynomial in the numerator, and in the second, both in the numerator and in the denominator there are expressions that are not polynomials.

Converting the numerator and denominator of a rational fraction

The numerator and denominator of any fraction are self-sufficient mathematical expressions, in the case of rational fractions, these are polynomials, in the particular case, monomials and numbers. Therefore, with the numerator and denominator of a rational fraction, as with any expression, it is possible to carry out identical transformations. In other words, the expression in the numerator of a rational fraction can be replaced by an expression identical to it, like the denominator.

Identical transformations can be performed in the numerator and denominator of a rational fraction. For example, in the numerator, you can group and bring similar terms, and in the denominator - the product of several numbers, replace it with its value. And since the numerator and denominator of a rational fraction are polynomials, it is possible to perform transformations characteristic of polynomials with them, for example, reduction to the standard form or representation in the form of a product.

For clarity, consider the solutions of several examples.

Example.

Convert rational fraction so that the numerator contains a polynomial of the standard form, and the denominator contains the product of polynomials.

Solution.

Reducing rational fractions to a new denominator is mainly used when adding and subtracting rational fractions.

Changing signs in front of a fraction, as well as in its numerator and denominator

The basic property of a fraction can be used to change the signs of the members of a fraction. Indeed, multiplying the numerator and denominator of a rational fraction by -1 is equivalent to changing their signs, and the result is a fraction that is identically equal to the given one. This transformation has to be addressed quite often when working with rational fractions.

Thus, if you simultaneously change the signs of the numerator and denominator of the fraction, you get a fraction equal to the original one. Equality corresponds to this statement.

Let's give an example. A rational fraction can be replaced with an identically equal fraction with changed signs of the numerator and denominator of the form.

One more identical transformation can be carried out with fractions, in which the sign changes either in the numerator or in the denominator. We will announce the corresponding rule. If you replace the sign of the fraction together with the sign of the numerator or denominator, you get a fraction that is identically equal to the original one. The written statement corresponds to the equalities and.

It is not difficult to prove these equalities. The proof is based on the properties of multiplication of numbers. Let us prove the first of them:. Equality is proved with the help of similar transformations.

For example, a fraction can be replaced with or.

To conclude this subsection, we present two more useful equalities and. That is, if you change the sign of only the numerator or only the denominator, then the fraction will change its sign. For example, and .

The considered transformations, which make it possible to change the sign of the members of a fraction, are often used when transforming fractionally rational expressions.

Reducing rational fractions

The next transformation of rational fractions, which is called cancellation of rational fractions, is based on the same basic property of a fraction. This transformation corresponds to equality, where a, b and c are some polynomials, and b and c are nonzero.

From the above equality, it becomes clear that reducing the rational fraction implies getting rid of the common factor in its numerator and denominator.

Example.

Reduce rational fraction.

Solution.

The common factor 2 is immediately visible, we will perform a reduction by it (when writing down the common factors, by which it is convenient to cross out). We have ... Since x 2 = x x and y 7 = y 3 y 4 (see if necessary), it is clear that x is the common factor of the numerator and denominator of the resulting fraction, like y 3. Let's reduce by these factors: ... This completes the reduction.

Above, we performed the reduction of the rational fraction sequentially. And it was possible to perform the reduction in one step, immediately reducing the fraction by 2 · x · y 3. In this case, the solution would look like this: .

Answer:

.

When canceling rational fractions, the main problem is that the common factor of the numerator and denominator is not always visible. Moreover, it does not always exist. In order to find the common factor or make sure that it is absent, you need to factor the numerator and denominator of the rational fraction. If there is no common factor, then the original rational fraction does not need to be canceled, otherwise, the cancellation is carried out.

In the process of reducing rational fractions, various nuances can arise. The main subtleties with examples and in detail are discussed in the article the reduction of algebraic fractions.

Concluding the conversation about cancellation of rational fractions, we note that this transformation is identical, and the main difficulty in its implementation lies in the factorization of the polynomials in the numerator and denominator.

Representation of a rational fraction as a sum of fractions

Quite specific, but in some cases very useful, is the transformation of a rational fraction, which consists in its representation as the sum of several fractions, or the sum of an integer expression and a fraction.

A rational fraction, in the numerator of which there is a polynomial, which is the sum of several monomials, can always be written as the sum of fractions with the same denominators, in the numerators of which the corresponding monomials are located. For example, ... This representation is explained by the rule of addition and subtraction of algebraic fractions with the same denominators.

In general, any rational fraction can be represented as a sum of fractions in many different ways. For example, the fraction a / b can be represented as the sum of two fractions - an arbitrary fraction c / d and a fraction equal to the difference between the fractions a / b and c / d. This statement is true, since the equality ... For example, a rational fraction can be represented as a sum of fractions in various ways: Let's represent the original fraction as the sum of an integer expression and a fraction. By dividing the numerator by the denominator in a column, we get the equality ... The value of the expression n 3 +4 for any integer n is an integer. And the value of a fraction is an integer if and only if its denominator is 1, −1, 3, or −3. These values ​​correspond to the values ​​n = 3, n = 1, n = 5, and n = −1, respectively.

Answer:

−1 , 1 , 3 , 5 .

Bibliography.

• Algebra: study. for 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2008 .-- 271 p. : ill. - ISBN 978-5-09-019243-9.
• A. G. Mordkovich Algebra. 7th grade. At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich. - 13th ed., Rev. - M .: Mnemosina, 2009 .-- 160 p.: Ill. ISBN 978-5-346-01198-9.
• A. G. Mordkovich Algebra. 8th grade. At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., Erased. - M .: Mnemozina, 2009 .-- 215 p.: Ill. ISBN 978-5-346-01155-2.
• Gusev V.A., Mordkovich A.G. Mathematics (manual for applicants to technical schools): Textbook. manual. - M .; Higher. shk., 1984.-351 p., ill.

Fractions of a unit and is represented as \ frac (a) (b).

Fraction numerator (a)- the number above the line of the fraction and showing the number of fractions by which the unit has been divided.

Fraction denominator (b)- the number below the line of the fraction and showing by how many fractions the unit was divided into.

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Basic property of a fraction

If ad = bc, then two fractions \ frac (a) (b) and \ frac (c) (d) are considered equal. For example, the fractions will be equal \ frac35 and \ frac (9) (15), since 3 \ cdot 15 = 15 \ cdot 9, \ frac (12) (7) and \ frac (24) (14) since 12 \ cdot 14 = 7 \ cdot 24.

It follows from the definition of equality of fractions that the fractions \ frac (a) (b) and \ frac (am) (bm), since a (bm) = b (am) is a clear example of the application of the combinational and displacement properties of multiplication of natural numbers in action.

Means \ frac (a) (b) = \ frac (am) (bm)- it looks like this basic property of a fraction.

In other words, we get a fraction equal to the given one by multiplying or dividing the numerator and denominator of the original fraction by the same natural number.

Fraction reduction Is a process of replacing a fraction, in which a new fraction is obtained equal to the original, but with a smaller numerator and denominator.

It is customary to reduce fractions based on the basic property of a fraction.

For example, \ frac (45) (60) = \ frac (15) (20)(the numerator and denominator are divisible by the number 3); the resulting fraction can again be reduced by dividing by 5, that is, \ frac (15) (20) = \ frac 34.

Irreducible fraction Is a fraction of the form \ frac 34 where the numerator and denominator are coprime numbers. The main purpose of reducing a fraction is to make the fraction irreducible.

Common denominator of fractions

Let's take two fractions as an example: \ frac (2) (3) and \ frac (5) (8) with different denominators 3 and 8. In order to bring these fractions to a common denominator and first multiply the numerator and denominator of the fraction \ frac (2) (3) at 8. We get the following result: \ frac (2 \ cdot 8) (3 \ cdot 8) = \ frac (16) (24)... Then we multiply the numerator and denominator of the fraction \ frac (5) (8) by 3. As a result, we get: \ frac (5 \ cdot 3) (8 \ cdot 3) = \ frac (15) (24)... So, the original fractions are reduced to a common denominator of 24.

Arithmetic operations on ordinary fractions

Adding ordinary fractions

a) With the same denominators, the numerator of the first fraction is added to the numerator of the second fraction, leaving the denominator the same. As you can see in the example:

\ frac (a) (b) + \ frac (c) (b) = \ frac (a + c) (b);

b) For different denominators, the fractions first lead to a common denominator, and then add the numerators according to rule a):

\ frac (7) (3) + \ frac (1) (4) = \ frac (7 \ cdot 4) (3) + \ frac (1 \ cdot 3) (4) = \ frac (28) (12) + \ frac (3) (12) = \ frac (31) (12).

Subtraction of common fractions

a) With the same denominators, the numerator of the second fraction is subtracted from the numerator of the first fraction, leaving the denominator the same:

\ frac (a) (b) - \ frac (c) (b) = \ frac (a-c) (b);

b) If the denominators of the fractions are different, then first the fractions lead to a common denominator, and then repeat the steps as in point a).

Multiplication of common fractions

Multiplication of fractions obeys the following rule:

\ frac (a) (b) \ cdot \ frac (c) (d) = \ frac (a \ cdot c) (b \ cdot d),

that is, the numerators and denominators are multiplied separately.

For example:

\ frac (3) (5) \ cdot \ frac (4) (8) = \ frac (3 \ cdot 4) (5 \ cdot 8) = \ frac (12) (40).

Division of ordinary fractions

Division of fractions is performed in the following way:

\ frac (a) (b): \ frac (c) (d) = \ frac (ad) (bc),

that is a fraction \ frac (a) (b) multiplied by a fraction \ frac (d) (c).

Example: \ frac (7) (2): \ frac (1) (8) = \ frac (7) (2) \ cdot \ frac (8) (1) = \ frac (7 \ cdot 8) (2 \ cdot 1 ) = \ frac (56) (2).

Reciprocal numbers

If ab = 1, then the number b is backward for the number a.

Example: for the number 9, the inverse is \ frac (1) (9), because 9 \ cdot \ frac (1) (9) = 1, for number 5 - \ frac (1) (5), because 5 \ cdot \ frac (1) (5) = 1.

Decimal fractions

Decimal a regular fraction is called, the denominator of which is 10, 1000, 10 \, 000, ..., 10 ^ n.

For example: \ frac (6) (10) = 0.6; \ enspace \ frac (44) (1000) = 0.044.

Incorrect numbers with the denominator 10 ^ n or mixed numbers are written in the same way.

For example: 5 \ frac (1) (10) = 5.1; \ enspace \ frac (763) (100) = 7 \ frac (63) (100) = 7.63.

Any ordinary fraction with a denominator that is a divisor of some power of 10 is represented as a decimal fraction.

Example: 5 is a divisor of 100, so the fraction \ frac (1) (5) = \ frac (1 \ cdot 20) (5 \ cdot 20) = \ frac (20) (100) = 0.2.

Arithmetic operations on decimal fractions

Adding Decimals

To add two decimal fractions, you need to arrange them so that the same digits and a comma under the comma are below each other, and then add the fractions as ordinary numbers.

Subtracting decimal fractions

It is performed in the same way as for addition.

Decimal multiplication

When multiplying decimal numbers, it is enough to multiply the given numbers, ignoring the commas (like natural numbers), and in the received answer, the comma on the right separates as many digits as they are after the comma in both factors in total.

Let's multiply 2.7 times 1.3. We have 27 \ cdot 13 = 351. Separate two digits on the right with a comma (the first and second numbers have one digit after the decimal point; 1 + 1 = 2). As a result, we get 2.7 \ cdot 1.3 = 3.51.

If in the result obtained there are fewer digits than must be separated by a comma, then the missing zeros are written in front, for example:

To multiply by 10, 100, 1000, it is necessary to transfer the comma in decimal fraction by 1, 2, 3 digits to the right (if necessary, a certain number of zeros are assigned to the right).

For example: 1.47 \ cdot 10 \, 000 = 14,700.

Division of decimal fractions

Dividing a decimal fraction by a natural number is done in the same way as dividing a natural number by a natural number. The comma in the quotient is placed after the division of the whole part is finished.

If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:

Consider dividing a decimal fraction by a decimal. Let's divide 2.576 by 1.12. First of all, we multiply the dividend and the divisor of the fraction by 100, that is, we move the comma to the right in the dividend and the divisor by as many digits as there are in the divisor after the comma (in this example, by two). Then you need to divide the fraction 257.6 by the natural number 112, that is, the problem is reduced to the case already considered:

It so happens that the final decimal fraction is not always obtained when dividing one number by another. The result is an infinite decimal. In such cases, they switch to ordinary fractions.

2.8: 0.09 = \ frac (28) (10): \ frac (9) (100) = \ frac (28 \ cdot 100) (10 \ cdot 9) = \ frac (280) (9) = 31 \ frac (1) (9).

In mathematics, a fraction is a number made up of one or more parts (fractions) of a unit. According to the form of notation, fractions are divided into ordinary (for example \ frac (5) (8)) and decimal (for example 123.45).

Definition. Common fraction (or simple fraction)

Ordinary (simple) fraction is a number of the form \ pm \ frac (m) (n) where m and n are natural numbers. The number m is called numerator of this fraction, and the number n is its denominator.

A horizontal or forward slash denotes a division sign, that is, \ frac (m) (n) = () ^ m / n = m: n

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

Definition. Correct and Incorrect Fractions

Correct a fraction with the modulus of the numerator less than the modulus of the denominator is called. For example, \ frac (9) (11), because 9

Wrong is a fraction in which the modulus of the numerator is greater than or equal to the modulus of the denominator. Such a fraction is a rational number, modulo greater than or equal to one. An example would be the fractions \ frac (11) (2), \ frac (2) (1), - \ frac (7) (5), \ frac (1) (1)

Along with an improper fraction, there is another notation for a number, which is called a mixed fraction (mixed number). This fraction is not ordinary.

Definition. Mixed fraction (mixed number)

Mixed shot is called a fraction written as an integer and a regular fraction and is understood as the sum of this number and a fraction. For example, 2 \ frac (5) (7)

(written as a mixed number) 2 \ frac (5) (7) = 2 + \ frac (5) (7) = \ frac (14) (7) + \ frac (5) (7) = \ frac (19 ) (7) (not written as an improper fraction)

A fraction is just a notation of a number. The same number can correspond to different fractions, both ordinary and decimal. Let's form a sign of equality of two ordinary fractions.

Definition. Equality of fractions

The two fractions \ frac (a) (b) and \ frac (c) (d) are equal if a \ cdot d = b \ cdot c. For example, \ frac (2) (3) = \ frac (8) (12) since 2 \ cdot12 = 3 \ cdot8

The main property of the fraction follows from the indicated sign.

Property. Basic property of a fraction

If the numerator and denominator of a given fraction are multiplied or divided by the same number, which is not equal to zero, then you get a fraction equal to the given one.

\ frac (A) (B) = \ frac (A \ cdot C) (B \ cdot C) = \ frac (A: K) (B: ​​K); \ quad C \ ne 0, \ quad K \ ne 0

Using the basic property of a fraction, you can replace a given fraction with another fraction equal to this one, but with a lower numerator and denominator. This replacement is called fraction reduction. For example, \ frac (12) (16) = \ frac (6) (8) = \ frac (3) (4) (here the numerator and denominator were divided first by 2, and then by another 2). The reduction of a fraction can be done if and only if its numerator and denominator are not mutually prime numbers. If the numerator and denominator of a given fraction are coprime, then the fraction cannot be canceled, for example, \ frac (3) (4) is an irreducible fraction.

Rules for positive fractions:

Of two fractions with the same denominators the greater is the fraction, the numerator of which is greater. For example, \ frac (3) (15)

Of two fractions with the same numerators the larger is the fraction, the denominator of which is smaller. For example, \ frac (4) (11)> \ frac (4) (13).

To compare two fractions with different numerators and denominators, you need to transform both fractions so that their denominators become the same. This is called common denominator conversion.

Possess the main property of the fraction:

Remark 1

If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then the result is a fraction equal to the original one:

$ \ frac (a \ cdot n) (b \ cdot n) = \ frac (a) (b) $

$ \ frac (a \ div n) (b \ div n) = \ frac (a) (b) $

Example 1

Let a square be given, divided into $ 4 $ equal parts. If we fill in $ 2 $ of $ 4 $ pieces, we get the filled $ \ frac (2) (4) $ of the whole square. If you look at this square, it is obvious that exactly half of it is painted over, i.e. $ (1) (2) $. Thus, we get $ \ frac (2) (4) = \ frac (1) (2) $. Factor the numbers $ 2 $ and $ 4 $:

Substitute these expansions into the equality:

$ \ frac (1) (2) = \ frac (2) (4) $,

$ \ frac (1) (2) = \ frac (1 \ cdot 2) (2 \ cdot 2) $,

$ \ frac (1) (2) = \ frac (2 \ div 2) (4 \ div 2) $.

Example 2

Is it possible to get an equal fraction if both the numerator and denominator of the given fraction are multiplied by $ 18 $ and then divided by $ 3 $?

Solution.

Let some ordinary fraction $ \ frac (a) (b) $ be given. By condition, the numerator and denominator of this fraction were multiplied by $ 18 $, and we got:

$ \ frac (a \ cdot 18) (b \ cdot 18) $

$ \ frac (a \ cdot 18) (b \ cdot 18) = \ frac (a) (b) $

$ \ frac (a \ div 3) (b \ div 3) $

According to the basic property of the fraction:

$ \ frac (a \ div 3) (b \ div 3) = \ frac (a) (b) $

Thus, the result was a fraction equal to the original.

Answer: You can get a fraction equal to the original.

Applying the basic property of a fraction

The main property of a fraction is most often used for:

• reducing fractions to a new denominator:
• reduction of fractions.

Reducing a fraction to a new denominator- replacing a given fraction with such a fraction that will be equal to it, but have a larger numerator and a larger denominator. To do this, the numerator and denominator of the fraction are multiplied by the same natural number, as a result of which, according to the basic property of the fraction, a fraction is obtained that is equal to the original, but with a large numerator and denominator.

Fraction reduction- replacing a given fraction with such a fraction that will be equal to it, but have a smaller numerator and a smaller denominator. To do this, the numerator and denominator of the fraction are divided by a positive common divisor of the numerator and denominator, other than zero, as a result of which, according to the basic property of the fraction, a fraction is obtained that is equal to the original, but with a smaller numerator and denominator.

If we divide (reduce) the numerator and denominator by their GCD, then the result is irreducible original fraction.

Reducing fractions

As you know, ordinary fractions are divided by contractible and irreducible.

To reduce the fraction, you need to divide both the numerator and the denominator of the fraction by their positive common divisor, which is not equal to zero. When reducing fractions, a new fraction is obtained with a smaller numerator and denominator, which, in terms of the basic property of the fraction, is equal to the original.

Example 3

Reduce the fraction $ \ frac (15) (25) $.

Solution.

Reduce the fraction by $ 5 $ (divide its numerator and denominator by $ 5 $):

$ \ frac (15) (25) = \ frac (15 \ div 5) (25 \ div 5) = \ frac (3) (5) $

Answer: $ \ frac (15) (25) = \ frac (3) (5) $.

Obtaining an irreducible fraction

Most often, a fraction is reduced to obtain an irreducible fraction equal to the original cancellable fraction. This result can be achieved by dividing both the numerator and the denominator of the original fraction by their GCD.

$ \ frac (a \ div gcd (a, b)) (b \ div gcd (a, b)) $ is an irreducible fraction, because according to the properties of the GCD, the numerator and denominator of this fraction are coprime numbers.

GCD (a, b) is the largest number by which both the numerator and denominator of $ \ frac (a) (b) $ can be divided. Thus, to reduce the fraction to an irreducible form, it is necessary to divide its numerator and denominator by their GCD.

Remark 2

Fraction reduction rule: 1. Find the GCD of two numbers that are in the numerator and denominator of the fraction. 2. Perform division of the numerator and denominator of the fraction by the found GCD.

Example 4

Reduce the fraction $ 6/36 $ to an irreducible form.

Solution.

Let us reduce this fraction by GCD $ (6.36) = 6 $, since $ 36 \ div 6 = 6 $. We get:

$ \ frac (6) (36) = \ frac (6 \ div 6) (36 \ div 6) = \ frac (1) (6) $

Answer: $ \ frac (6) (36) = \ frac (1) (6) $.

In practice, the phrase "reduce a fraction" implies that you need to reduce the fraction to an irreducible form.

Fractions

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very ..."
And for those who "very much ...")

Fractions in high school are not very annoying. For the time being. Until you come across powers with rational exponents and logarithms. But there…. You press, you press the calculator, and it shows a full display of some numbers. I have to think with my head like in the third grade.

Let's deal with fractions already, finally! Well, how much can you get confused in them !? Moreover, it's all simple and logical. So, what fractions are there?

Types of fractions. Transformations.

Fractions are of three types.

1. Ordinary fractions , for example:

Sometimes a slash is used instead of a horizontal line: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, bottom - the denominator. If you constantly confuse these names (it happens ...), tell yourself with the expression the phrase: " Zzzzz remember! Zzzzz referencing - behold zzzzz"You look, everything will be remembered.)

A dash, which is horizontal, which is oblique, means division the upper number (numerator) to the lower one (denominator). And that's it! Instead of a dash, it is quite possible to put a division sign - two dots.

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

32/8 = 32: 8 = 4

I'm not even talking about the fraction "4/1". Which is also just "4". And if it is not divided entirely, we leave it in the form of a fraction. Sometimes you have to do the reverse operation. Make a fraction of an integer. But more on that later.

2. Decimal fractions , for example:

It is in this form that you will need to write down the answers to the tasks "B".

3. Mixed numbers , for example:

Mixed numbers are hardly used in high school. In order to work with them, they must be translated into ordinary fractions in every way. But you definitely need to be able to do it! And then you get such a number in the puzzle and freeze ... From scratch. But we will remember this procedure! Below.

Most versatile common fractions... Let's start with them. By the way, if the fraction contains all sorts of logarithms, sines and other letters, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!

The main property of a fraction.

So let's go! For starters, I'll surprise you. The whole variety of transformations of fractions is provided by one and only property! It's called that, basic property of a fraction... Remember: if the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction will not change. Those:

It is clear that you can write further, until you turn blue. Do not let the sines and logarithms confuse you, we will deal with them further. The main thing is to understand that all these various expressions are the same fraction . 2/3.

Do we need it, all these transformations? And how! Now you will see for yourself. First, we use the basic property of the fraction for reduction of fractions... It would seem that the thing is elementary. Divide the numerator and denominator by the same number and all the cases! It is impossible to be mistaken! But ... man is a creative being. Mistakes can be everywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.

How to reduce fractions correctly and quickly, without doing unnecessary work, can be read in a special Section 555.

A normal student does not bother dividing the numerator and denominator by the same number (or expression)! It just crosses out everything that is the same above and below! This is where a typical mistake lurks, a blooper, if you like.

For example, you need to simplify the expression:

There is nothing to think about, we cross out the letter "a" above and two below! We get:

Everything is correct. But really you shared the whole numerator and the whole the denominator is "a". If you are used to just crossing out, then, in a hurry, you can cross out the "a" in the expression

and get it again

Which will be categorically wrong. Because here the whole the numerator on "a" is already does not share! This fraction cannot be canceled. By the way, such a reduction is, um ... a serious challenge to the teacher. This is not forgiven! Remember? When shortening, you need to divide the whole numerator and the whole denominator!

Reducing fractions makes life a lot easier. You get a fraction somewhere, for example 375/1000. And how to work with her now? Without a calculator? Multiply, say, add, square !? And if you are not too lazy, but neatly reduce it by five, and even by five, and even ... while it is being reduced, in short. We get 3/8! Much nicer, right?

The main property of a fraction allows you to convert ordinary fractions to decimal and vice versa. without calculator! This is important on the exam, right?

How to convert fractions from one type to another.

Decimal fractions are simple. As it is heard, it is written! Let's say 0.25. This is zero point, twenty-five hundredths. So we write: 25/100. Reducing (dividing the numerator and denominator by 25), we get the usual fraction: 1/4. Everything. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.

And if the integers are not zero? It's OK. We write down the entire fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three points, seventeen hundredths. We write 317 in the numerator and 100 in the denominator. We get 317/100. Nothing is reduced, everything means. This is the answer. Elementary Watson! From all that has been said, a useful conclusion: any decimal fraction can be turned into an ordinary one .

But the reverse conversion, ordinary to decimal, some cannot do without a calculator. And it is necessary! How will you write down your answer on the exam !? We carefully read and master this process.

What is the characteristic of the decimal fraction? She has in the denominator always costs 10, or 100, or 1000, or 10000, and so on. If your regular fraction has this denominator, no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. And if the answer to the task in section "B" is 1/2? What will we write in response? There decimals are required ...

Remembering basic property of a fraction ! Mathematics favorably allows the numerator and denominator to be multiplied by the same number. Anything, by the way! Except zero, of course. So we will apply this property to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course ...)? At 5, obviously. We boldly multiply the denominator (this is US it is necessary) by 5. But, then the numerator must also be multiplied by 5. This is already maths requires! We get 1/2 = 1x5 / 2x5 = 5/10 = 0.5. That's all.

However, all sorts of denominators come across. Will come across, for example, the fraction 3/16. Try, figure out here what to multiply 16 to make 100, or 1000 ... Not working? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide by a corner, on a piece of paper, as taught in elementary grades. We get 0.1875.

And there are also very nasty denominators. For example, you can't turn a fraction 1/3 into a good decimal. Both on a calculator and on a piece of paper, we get 0.3333333 ... This means that 1/3 is an exact decimal does not translate... The same as 1/7, 5/6, and so on. There are many untranslatable ones. Hence another useful conclusion. Not every fraction is converted to decimal !

By the way, this is useful information for self-testing. In section "B", the decimal fraction must be written in the answer. And you got, for example, 4/3. This fraction is not converted to decimal. This means that somewhere you went wrong along the way! Come back check the solution.

So, we figured out the common and decimal fractions. It remains to deal with the mixed numbers. To work with them, they all need to be converted into ordinary fractions. How to do it? You can catch a sixth grader and ask him. But the sixth grader will not always be at hand ... We will have to do it ourselves. This is not difficult. It is necessary to multiply the denominator of the fractional part by the whole part and add the numerator of the fractional part. This will be the numerator of the regular fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but in reality everything is elementary. Let's see an example.

Suppose you saw with horror the number in the puzzle:

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 the denominator of the ordinary fraction. We count the numerator. 7 multiply by 1 (whole part) and add 3 (fractional numerator). We get 10. This will be the numerator of the common fraction. That's all. It looks even simpler in mathematical notation:

Is it clear? Then consolidate your success! Convert to fractions. You should have 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction to a mixed number - is rarely required in high school. Well, if ... And if you are not in high school, you can look into the special Section 555. In the same place, by the way, you will learn about incorrect fractions.

Well, practically that's it. You remembered the types of fractions and realized how transfer them from one type to another. The question remains: why do it? Where and when to apply this deep knowledge?

I answer. Any example itself suggests the necessary actions. If in the example common fractions, decimals, and even mixed numbers are mixed together, we translate everything into common fractions. This can always be done... Well, if it is written, something like 0.8 + 0.3, then we think so, without any translation. Why do we need extra work? We choose the solution that is convenient US !

If the task is full of decimal fractions, but um ... some evil ones, go to the ordinary ones, try it! You look, everything will work out. For example, you have to square the number 0.125. It’s not so easy if you’re not out of the habit of calculator! Not only do you need to multiply the numbers in a column, so also think about where to insert the comma! It will definitely not work in the mind! And if we go to an ordinary fraction?

0.125 = 125/1000. Reduce it by 5 (this is for a start). We get 25/200. Once again by 5. We get 5/40. Oh, still shrinking! Back at 5! We get 1/8. We easily square it (in the mind!) And get 1/64. Everything!

Let's summarize this lesson.

1. Fractions are of three types. Ordinary, decimal and mixed numbers.

2. Decimal fractions and mixed numbers always can be converted to fractions. Reverse translation not always available.

3. The choice of the type of fractions for working with the task depends on this very task. If there are different types of fractions in one task, the safest thing is to switch to ordinary fractions.

Now you can practice. First, convert these decimal fractions to common ones:

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

You should get the following answers (in a mess!):

This concludes. In this lesson, we have refreshed key points on fractions. It happens, however, that there is nothing special to refresh ...) If someone has completely forgotten, or has not yet mastered ... Those can go to a special Section 555. All the basics are described in detail there. Many suddenly understand everything start. And the fractions decide on the fly).

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