In this lesson, we will study the main property of the fraci, learn what fractions are equal to each other. We will learn to reduce the fractions, to determine whether the fraction is reduced or not, we will practic a reduction in the frains and find out when it is worth using a reduction, and when not.

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The main property of the fraci

Imagine such a situation.

At the table 3 man I. 5 Apples. Divide 5 Apples on three. Each is obtained by \\ (\\ mathbf (\\ FRAC (5) (3)) \\) of the apple.

And at the next table still 3 man and too 5 Apples. Each again by \\ (\\ Mathbf (\\ FRAC (5) (3)) \\)

At the same time 10 Apple I. 6 human. Each software \\ (\\ MathBF (\\ FRAC (10) (6)) \\)

But this is the same.

\\ (\\ MathBF (\\ FRAC (5) (3) \u003d \\ FRAC (10) (6)) \\)

These fractions are equivalent.

You can enlarge twice the number of people and twice the number of apples. The result will be the same.

In mathematics, this is formulated as:

If the numerator and denominator of the fraction are multiplied or divided into one and the same number (not equal to 0), then the new fraction will be equal to the original.

This property is sometimes called " the main property of the fraci ».

$$ \\ MathBF (\\ FRAC (A) (B) \u003d \\ FRAC (A \\ CDOT C) (B \\ CDOT C) \u003d \\ FRAC (A: D) (B: \u200b\u200bD)) $$

For example, the path from the city to the village 14 km.

We go on the road and determine the path traveled by kilometer columns. After passing six columns, six kilometers, we understand that we passed \\ (\\ mathbf (\\ FRAC (6) (14)) \\) paths.

But if we do not see the columns (maybe they were not installed), you can consider electric columns along the road. Them 40 pieces per kilometer. That is, everything 560 Along the way. Six kilometers- \\ (\\ mathbf (6 \\ cdot40 \u003d 240) \\) pillars. That is, we passed 240 of 560 Pillar- \\ (\\ MathBF (\\ FRAC (240) (560)) \\)

\\ (\\ MathBF (\\ FRAC (6) (14) \u003d \\ FRAC (240) (560)) \\)

Example 1.

Mark the point with coordinates ( 5; 7 ) on the coordinate plane XOY.. It will correspond to the fraction \\ (\\ mathbf (\\ FRAC (5) (7)) \\)

Connect the origin of the coordinate with the resulting point. Build another point that has coordinates of two times large previous ones. What fraction did you get? Will they be equal?

Decision

Destruction on the coordinate plane can be marked with a point. To portray the fraction \\ (\\ mathbf (\\ FRAC (5) (7)) \\), we note the point with the coordinate 5 along the axis Y. and 7 along the axis X.. We will spend direct from the beginning of the coordinates through our point.

On the same straight line will be the point corresponding to the fraction \\ (\\ mathbf (\\ FRAC (10) (14)) \\)

They are equivalent: \\ (\\ MathBF (\\ FRAC (5) (7) \u003d \\ FRAC (10) (14)) \\)

This article continues the topic of transformation of algebraic fractions: Consider such an action as a reduction in algebraic fractions. Let us give the definition of the term itself, we formulate the reduction rule and analyze practical examples.

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The meaning of the reduction of algebraic fraction

In materials by Ob. ordinary fraci We considered its reduction. We have determined the reduction of ordinary fraction as a division of its number and denominator for a common factor.

Reducing the algebraic fraction is a similar action.

Definition 1.

Reducing algebraic fractions - This is the division of its numerator and denominator for a general factor. At the same time, in contrast to the reduction of an ordinary fraction (the total denominator can only be a number), the total multiplier of the numerator and denominator of the algebraic fraction can serve as a polynomial, in particular, or a number.

For example, the algebraic fraction 3 · x 2 + 6 · x · y 6 · x 3 · y + 12 · x 2 · y 2 can be reduced by number 3, as a result, we obtain: x 2 + 2 · x · y 6 · x 3 · y + 12 · x 2 · y 2. We can cut the same fraction to the variable x, and it will give us the expression 3 · x + 6 · y 6 · x 2 · y + 12 · x · y 2. Also a given fraction can be reduced by one-sided 3 · X.or any of the polynomials X + 2 · Y, 3 · x + 6 · y, x 2 + 2 · x · y or 3 · x 2 + 6 · x · y.

The ultimate goal of the reduction of algebraic fraction is the fraction of a simpler view, in best case - Unstable fraction.

Are all algebraic fractions subject to reduction?

Again, from materials on ordinary fractions, we know that there are cuts and non-interpretable fractions. Unstable is a fraction who do not have common multipliers of the numerator and denominator different from 1.

With algebraic fractions, everything is the same: they may have common multipliers of the numerator and denominator, may not have. The presence of general factors makes it possible to simplify the initial fraction by reducing. When there are no general multipliers, it is impossible to optimize the specified fraction of the reduction.

In general cases set The fraction is quite difficult to understand whether it is subject to a reduction. Of course, in some cases, the presence of a common multiplier of the numerator and denominator is obvious. For example, in algebraic fractions 3 · x 2 3 · y, it is absolutely clear that the total factor is the number 3.

In the fraction - x · y 5 · x · y · z 3 We also immediately understand that it is possible to reduce it on x, or y, or on x · y. And yet, it is much more common examples of algebraic fractions, when the general multiplier of the numerator and the denominator is not so easy to see, and even more often - he is simply absent.

For example, the fraction of x 3 - 1 x 2 - 1 we can cut on x - 1, while the specified general multiplier in the record is missing. But the fraction x 3 - x 2 + x - 1 x 3 + x 2 + 4 · x + 4 is impossible to expose the reduction, since the numerator and the denominator do not have a common factor.

Thus, the question of finding out the reduction of algebraic fraction is not as simple, and it is often easier to work with the fraction of a given species than trying to figure out whether it is reduced. At the same time, there are such transformations that in particular cases allow you to determine the total multiplier of the numerator and the denominator or to conclude the fragility of the fraction. We will analyze in detail this question in the next paragraph of the article.

The rule of reduction of algebraic fractions

The rule of reduction of algebraic fractions consists of two consecutive actions:

• finding common multipliers of the numerator and denominator;
• if such, the implementation of the cutting effect of the fraction is directly.

The most convenient method of finding common denominators is the decomposition of polynomials existing in the numerator and denominator of a given algebraic fraction. This allows you to immediately see the presence or absence of general multipliers.

The effect of the reduction of algebraic fraction is based on the main property of an algebraic fraction expressed by the equality undefined, where a, b, C is some polynomials, and B and C - non-zero. The first step, the fraction is given to the form A · C B · C, in which we immediately notice the general factor c. The second step is to reduce, i.e. Transition to fraction of the form a b.

Characteristic examples

Despite some evidence, clarify about private caseWhen the numerator and denominator of algebraic fraction are equal. Similar fractions are identically equal to 1 throughout the odd variable of this fraction:

5 5 \u003d 1; - 2 3 - 2 3 \u003d 1; x x \u003d 1; - 3, 2 · x 3 - 3, 2 · x 3 \u003d 1; 1 2 · x - x 2 · y 1 2 · x - x 2 · y;

Since ordinary fractions are a special case of algebraic fractions, we will remind you how to reduce them. Natural numbers recorded in a numerator and denominator are laid out to simple multipliers, then general factors are reduced (if any).

For example, 24 1260 \u003d 2 · 2 · 2 · 3 2 · 2 · 3 · 3 · 5 · 7 \u003d 2 3 · 5 · 7 \u003d 2 105

The work of simple identical factors can be written as degrees, and in the process of reducing the fraction to use the property of degree in degrees with the same bases. Then the above decision would be:

24 1260 \u003d 2 3 · 3 2 2 · 3 2 · 5 · 7 \u003d 2 3 - 2 3 2 - 1 · 5 · 7 \u003d 2 105

(Numerator and denominator are divided into a common factor 2 2 · 3). Or for clarity, relying on the properties of multiplication and division, we will give this type of decision:

24 1260 \u003d 2 3 · 3 2 2 · 3 2 · 5 · 7 \u003d 2 3 2 2 · 3 3 2 · 1 5 · 7 \u003d 2 1 · 1 3 · 1 35 \u003d 2 105

By analogy, the algebraic fractions are reduced, in which the numeric and the denominator have universal with integer coefficients.

Example 1.

The algebraic fraction is given - 27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · C 7 · Z. It is necessary to make it reduced.

Decision

It is possible to write a numerator and denominator of a given fraction as a product of simple multipliers and variables, after which the reduction:

27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · C 7 · z \u003d - 3 · 3 · 3 · a · a · a · a · a · b · b · c · z 2 · 3 · A · A · b · b · C · C · C · C · C · C · C · Z \u003d \u003d - 3 · 3 · A · A · A 2 · C · C · C · C · C · C \u003d - 9 · a 3 2 · C 6

However, a more rational way will record a solution in the form of expressions with degrees:

27 · a 5 · b 2 · C · Z 6 · A 2 · B 2 · C 7 · Z \u003d - 3 3 · A 5 · B 2 · C · Z 2 · 3 · A 2 · B 2 · C 7 · z \u003d - 3 3 2 · 3 · a 5 a 2 · b 2 B 2 · Cc 7 · zz \u003d \u003d - 3 3 - 1 2 · a 5 - 2 1 · 1 · 1 C 7 - 1 · 1 \u003d · - 3 2 · a 3 2 · C 6 \u003d · - 9 · A 3 2 · C 6.

Answer: - 27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · C 7 · z \u003d - 9 · a 3 2 · C 6

When there are fractional numerical coefficients in a numerator and denominator of algebraic fraction, two ways of further action are possible: or separately divide these fractional coefficients, or to pre-get rid of fractional coefficients, multiplying the numerator and denominator for some kind natural number. The last transformation is carried out due to the basic properties of the algebraic fraction (it is possible to read about it in the article "Running an algebraic fraction for a new denominator").

Example 2.

The fraction 2 5 · x 0, 3 · x 3 is given. It is necessary to reduce it.

Decision

It is possible to reduce the fraction in this way:

2 5 · x 0, 3 · x 3 \u003d 2 5 3 10 · x x 3 \u003d 4 3 · 1 x 2 \u003d 4 3 · x 2

Let us try to solve the problem otherwise, pre-getting rid of fractional coefficients - multiply the numerator and denominator to the smallest general multiple denominators of these coefficients, i.e. on NOC (5, 10) \u003d 10. Then we get:

2 5 · x 0, 3 · x 3 \u003d 10 · 2 5 · x 10 · 0, 3 · x 3 \u003d 4 · x 3 · x 3 \u003d 4 3 · x 2.

Answer: 2 5 · x 0, 3 · x 3 \u003d 4 3 · x 2

When we reduce algebraic fractions general viewIn which the numerals and denominators can be both single-wing and polynomials, a problem is possible when the general factor is not always visible immediately. Or moreover, he simply does not exist. Then, to determine the general factor or fixing the fact about its absence, the numerator and the denominator of the algebraic fraction lay out on multipliers.

Example 3.

The rational fraction 2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 is given. It is necessary to cut it.

Decision

We will decompose polynomials in a numerator and denominator. Implement for braces:

2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 \u003d 2 · b 2 · (A 2 + 14 · A + 49) B 3 · (A 2 - 49)

We see that the expression in brackets can be converted using the formulas of abbreviated multiplication:

2 · b 2 · (A 2 + 14 · A + 49) B 3 · (A 2 - 49) \u003d 2 · B 2 · (A + 7) 2 B 3 · (A - 7) · (A + 7)

It is clearly noticeable that it is possible to reduce the fraction on the general factory B 2 · (A + 7). We will reduce:

2 · b 2 · (A + 7) 2 B 3 · (A - 7) · (A + 7) \u003d 2 · (A + 7) B · (A - 7) \u003d 2 · A + 14 A · B - 7 · B.

A brief decision without explanation we write as a chain of equalities:

2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 \u003d 2 · b 2 · (A 2 + 14 A + 49) B 3 · (A 2 - 49) \u003d \u003d 2 · b 2 · (A + 7) 2 B 3 · (A - 7) · (A + 7) \u003d 2 · (A + 7) B · (A - 7) \u003d 2 · A + 14 A · b - 7 · b

Answer: 2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 \u003d 2 · A + 14 A · B - 7 · b.

It happens that common factors are hidden by numeric coefficients. Then, when cutting fractions, the optimal numerical factors with the senior degrees of the numerator and the denominator to take place behind the brackets.

Example 4.

Dana algebraic fraction 1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2. It is necessary to carry out its reduction, if possible.

Decision

At first glance, the numerator and denominator does not exist general denominator. However, let's try to convert a given fraction. I will bring a multiplier x in a numerator:

1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2 \u003d x · 1 5 - 2 7 · x 2 · y 5 · x 2 · y - 3 1 2

Now a certain similarity of expressions in brackets and expressions in the denominator due to x 2 · y . I will bring numerical coefficients for the bracket with senior degrees of these polynomials:

x · 1 5 - 2 7 · x 2 · y 5 · x 2 · y - 3 1 2 \u003d x · - 2 7 · - 7 2 · 1 5 + x 2 · y 5 · x 2 · y - 1 5 · 3 1 2 \u003d - - 2 7 · x · - 7 10 + x 2 · y 5 · x 2 · y - 7 10

Now the general multiplier becomes visible, we carry out a reduction:

2 7 · x · - 7 10 + x 2 · y 5 · x 2 · y - 7 10 \u003d - 2 7 · x 5 \u003d - 2 35 · x

Answer: 1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2 \u003d - 2 35 · x.

Let the emphasis on the fact that the skill of the reduction of rational fractions depends on the ability to spread polynomials to multipliers.

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In this article we will look at basic actions with algebraic fractions:

• reducing fractions
• multiplication of fractions
• division of fractions

Let's start by S. reducing algebraic fractions.

It would seem that, algorithm Obvious.

To reduce algebraic fractions, need to

1. Ensure the numerator and denominator of the fractions on multipliers.

2. Reduce the same multipliers.

However, schoolchildren often make a mistake, "cutting" are not multipliers, but the components. For example, there are amateurs that in the fraction "are reduced" on and result in the result, which, of course, is incorrect.

Consider examples:

1. Reduce fraction:

1. Spreads the number of the sum of the square of the square on the multipliers, and the denominator according to the square difference formula

2. We divide the numerator and denominator on

2. Reduce fraction:

1. Spread the numerator on multipliers. Since the numerator contains four terms, we will apply a grouping.

2. Spread the denominator on multipliers. Also apply a grouping.

3. We write the fraction that we turned out and cut the same multipliers:

Multiplying algebraic fractions.

When multiplying algebraic fractions, we multiply the numerator to the numerator, and the denominator multiplies to the denominator.

Important! No need to hurry to perform multiplication in the numerator and denoter of the fraction. After we recorded the product of the fractions of the fractions in the numerator, and in the denominator, the product of the denominers, you need to decompose every multiplier to multipliers and cut the fraction.

Consider examples:

3. Simplify the expression:

1. We write the work of fractions: in the numerator, the product of numerals, and in the denominator, the product of the denominers:

2. Spread each bracket for multipliers:

Now we need to cut the same multipliers. Note that expressions and differ only in the sign: And as a result of division of the first expression on the second, we obtain -1.

So,

We carry out the division of algebraic fractions by such a rule:

I.e to divide the fraction, you need to multiply to the "inverted".

We see that the division of fractions is reduced to multiplication, and multiplication, ultimately, is reduced to the reduction of fractions.

Consider an example:

4. Simplify the expression:

Reducing fractions is needed in order to bring the fraction to more simplicityFor example, in response as a result of solving the expression.

Reducing fractions, definition and formula.

What is the reduction of fractions? What does shorten the fraction mean?

Definition:
Reducing fractions - This separation in the fraction of the numerator and the denominator for the same positive number is not equal to zero and one. As a result, the reduction turns out the fraction with a smaller numerator and the denominator equal to the previous fraction according to.

Formula Reducing fractions The main property of rational numbers.

\\ (\\ FRAC (P \\ Times N) (Q \\ Times N) \u003d \\ FRAC (P) (Q) \\)

Consider an example:
Reduce fraction \\ (\\ FRAC (9) (15) \\)

Decision:
We can decompose the fraction on simple multipliers and reduce the general factors.

\\ (\\ FRAC (9) (15) \u003d \\ FRAC (3 \\ Times 3) (5 \\ Times 3) \u003d \\ FRAC (3) (5) \\ Times \\ Color (Red) (\\ FRAC (3) (3) ) \u003d \\ FRAC (3) (5) \\ Times 1 \u003d \\ FRAC (3) (5) \\)

Answer: After the reduction, the fraction \\ (\\ FRAC (3) (5) \\) was obtained. According to the main property of rational numbers, the initial and the resulting fraction is equal.

\\ (\\ FRAC (9) (15) \u003d \\ FRAC (3) (5) \\)

How to cut the fraction? Reduction of fractions to an inocarabula.

To get as a result of an unstable fraction, you need find the highest general divisor (Node) For numerator and denominator.

There are several ways to find a node. We will use the example of the decomposition of numbers to simple factors.

Get an inconspicuous fraction \\ (\\ FRAC (48) (136) \\).

Decision:
We find a node (48, 136). Speak numbers 48 and 136 on simple multipliers.
48=2⋅2⋅2⋅2⋅3
136=2⋅2⋅2⋅17
Node (48, 136) \u003d 2⋅2⋅2 \u003d 6

\\ (\\ FRAC (48) (136) \u003d \\ FRAC (\\ COLOR (RED) (2 \\ Times 2 \\ Times 2) \\ Times 2 \\ Times 3) (\\ Color (Red) (2 \\ Times 2 \\ Times 2) \\ Times 17) \u003d \\ FRAC (\\ COLOR (RED) (6) \\ Times 2 \\ Times 3) (\\ Color (Red) (6) \\ Times 17) \u003d \\ FRAC (2 \\ Times 3) (17) \u003d \\ The reduction rule is a fraction before an in-law.

It is necessary to find the largest common divider for numerals and denominator.

1. It is necessary to divide the numerator and the denominator to the greatest common divisor as a result of the division to receive an unstable fraction.
2. Example:

Reduce the fraction \\ (\\ FRAC (152) (168) \\).

Decision:
We find a node (152, 168). Speak numbers 152 and 168 on simple multipliers.
152=2⋅2⋅2⋅19
168=2⋅2⋅2⋅3⋅7
Node (152, 168) \u003d 2⋅2⋅2 \u003d 6

\\ (\\ FRAC (152) (168) \u003d \\ FRAC (\\ COLOR (RED) (6) \\ Times 19) (\\ Color (Red) (6) \\ Times 21) \u003d \\ FRAC (19) (21) \\)

Answer: \\ (\\ FRAC (19) (21) \\) Unstable fraction.

Reducing improper fraction.

How to cut irregular fraction?
Rules for reducing fractions for the correct and incorrect fractions are the same.

Consider an example:
Reduce the wrong fraction \\ (\\ FRAC (44) (32) \\).

Decision:
Sick on simple multipliers numerator and denominator. And then general factors will reduce.

\\ (\\ FRAC (44) (32) \u003d \\ FRAC (\\ COLOR (RED) (2 \\ Times 2) \\ Times 11) (\\ Color (Red) (2 \\ Times 2) \\ Times 2 \\ Times 2 \\ Times 2 ) \u003d \\ FRAC (11) (2 \\ Times 2 \\ Times 2) \u003d \\ FRAC (11) (8) \\)

Reducing mixed fractions.

Mixed fractions on the same rules as ordinary fractions. The only difference is that we can the whole part does not touch, but cutting part or mixed fraction Translate to the wrong fraction, cut and translate back to the correct fraction.

Consider an example:
Reduce the mixed fraction \\ (2 \\ FRAC (30) (45) \\).

Decision:
By two ways:
The first way:
We have a fractional part to simple multipliers, and we will not touch the whole part.

\\ (2 \\ FRAC (30) (45) \u003d 2 \\ FRAC (2 \\ Times \\ COLOR (RED) (5 \\ Times 3)) (3 \\ Times \\ Color (Red) (5 \\ Times 3)) \u003d 2 \\ The second way:

We first translate into the wrong fraction, and then we cut to simple multipliers and reduce. The resulting incorrect fraction will be translated into the correct one.
\\ (2 \\ FRAC (30) (45) \u003d \\ FRAC (45 \\ Times 2 + 30) (45) \u003d \\ FRAC (120) (45) \u003d \\ FRAC (2 \\ Times \\ Color (Red) (5 \\ Times 3) \\ Times 2 \\ Times 2) (3 \\ Times \\ Color (Red) (3 \\ Times 5)) \u003d \\ FRAC (2 \\ Times 2 \\ Times 2) (3) \u003d \\ FRAC (8) (3) \u003d 2 \\ FRAC (2) (3) \\)

Questions on the topic:

Is it possible to cut the fractions when adding or subtracting?
Answer: No, you must first fold or subtract fractions according to the rules, but only then cut. Consider an example:
Calculate the expression \\ (\\ FRAC (50 + 20-10) (20) \\).

You often make an error reducing the same numbers in the numerator and the denominator in our case number 20, but it cannot be reduced until you add and subtract.

Decision:
\\ (\\ FRAC (50+ \\ Color (Red) (20) -10) (\\ Color (RED) (20)) \u003d \\ FRAC (60) (20) \u003d \\ FRAC (3 \\ Times 20) (20) \u003d \\ FRAC (3) (1) \u003d 3 \\)

What numbers can you cut a fraction?

Answer: You can cut the fraction to the largest common divider or the usual divider of the numerator and the denominator. For example, fraction \\ (\\ FRAC (100) (150) \\).
We write to simple multipliers of the number 100 and 150.

The greatest common divider will be the number of nodes (100, 150) \u003d 2⋅5⋅5 \u003d 50
100=2⋅2⋅5⋅5
150=2⋅5⋅5⋅3
\\ (\\ FRAC (100) (150) \u003d \\ FRAC (2 \\ Times 50) (3 \\ Times 50) \u003d \\ FRAC (2) (3) \\)

Received an incomprehensible fraction \\ (\\ FRAC (2) (3) \\).

But it is not necessary to always be divided into nodes not always need an unstable fraction, you can reduce the fraction on a simple divider of the numerator and the denominator. For example, in the number 100 and 150, the total divider 2. Spel the fraction \\ (\\ FRAC (100) (150) \\) by 2.

\\ (\\ FRAC (100) (150) \u003d \\ FRAC (2 \\ Times 50) (2 \\ Times 75) \u003d \\ FRAC (50) (75) \\)

Received a reduction fraction \\ (\\ FRAC (50) (75) \\).

What fractions can be cut?
Answer: You can cut the fractions that the numerator and the denominator have a common divider. For example, the fraction \\ (\\ FRAC (4) (8) \\). In the number 4 and 8 there is a number for which they both share this number 2. Therefore, such a fraction can be reduced by the number 2.

Example:
Compare two fractions \\ (\\ FRAC (2) (3) \\) and \\ (\\ FRAC (8) (12) \\).

These two fractions are equal. Consider detailed fraction \\ (\\ FRAC (8) (12) \\):

\\ (\\ FRAC (8) (12) \u003d \\ FRAC (2 \\ Times 4) (3 \\ Times 4) \u003d \\ FRAC (2) (3) \\ Times \\ FRAC (4) (4) \u003d \\ FRAC (2) (3) \\ Times 1 \u003d \\ FRAC (2) (3) \\)

From here we get, \\ (\\ FRAC (8) (12) \u003d \\ FRAC (2) (3) \\)

Two fractions are equal then and only if one of them is obtained by reducing the other fraction on the general multiplier of the numerator and denominator.

Example:
Reduce if the following fractions are possible: a) \\ (\\ FRAC (90) (65) \\) b) \\ (\\ FRAC (27) (63) \\) B) \\ (\\ FRAC (17) (100) \\) d) \\ (\\ FRAC (100) (250) \\)

Decision:
a) \\ (\\ FRAC (90) (65) \u003d \\ FRAC (2 \\ Times \\ Color (Red) (5) \\ Times 3 \\ Times 3) (\\ Color (RED) (5) \\ Times 13) \u003d \\ FRAC (2 \\ Times 3 \\ Times 3) (13) \u003d \\ FRAC (18) (13) \\)
b) \\ (\\ FRAC (27) (63) \u003d \\ FRAC (\\ COLOR (RED) (3 \\ Times 3) \\ Times 3) (\\ Color (Red) (3 \\ Times 3) \\ Times 7) \u003d \\ FRAC (3) (7) \\)
c) \\ (\\ FRAC (17) (100) \\) Osturbable fraction
d) \\ (\\ FRAC (100) (250) \u003d \\ FRAC (\\ COLOR (RED) (2 \\ Times 5 \\ Times 5) \\ Times 2) (\\ Color (Red) (2 \\ Times 5 \\ Times 5) \\ In this article we will analyze in detail how it is held

Reducing fractions . First, we will discuss what the fraction is called a reduction. After that, let's talk about bringing a reduced fraction to an incomprehensive form. Further, we will get a rule of reduction of fractions and, finally, consider examples of applying this rule.Navigating page.

What does shorten the fraction mean?

We know that ordinary fractions are divided into reduced and non-constructed fractions. By names, it is possible to guess that the reduced fraction can be reduced, and non-conscript - it is impossible.

What does shorten the fraction mean?

Reduce fraction - It means splitting its numerator and a denominator on their positive and different from one. It is clear that as a result of the reduction of the fraction, a new fraction with a smaller number and denominator is obtained, and, by virtue of the basic properties of the fraction, the resulting fraction is equal to the source.

For example, we will reduce the ordinary fraction 8/24, separating its numerator and denominator to 2. In other words, we will reduce the fraction 8/24 to 2. Since 8: 2 \u003d 4 and 24: 2 \u003d 12, as a result of such a reduction, it turns out a fraction 4/12, which is equal to the initial fraction 8/24 (see equal and unequal fractions). In the end we have.

Bringing ordinary fractions to nonstorm

Usually the ultimate goal of the reduction of the fraction is to obtain a non-interpretable fraction, which is equal to the initial reduced fraction. This goal can be achieved if it is reduced by the initial reduced fraction on its numerator and denominator. As a result of such a reduction, an unstable fraction is always obtained. Indeed, fraction is non-worn, because from it is known that and -. Here, let's say that the greatest common divisor of the numerator and denominator of the fraction is the greatest number that can be reduced by this fraction.

So, bringing ordinary fractions to an incomprehensive form It is to divide the numerator and denominator of the initial reduced fraction on their node.

We will analyze an example, for which we will return to the fraction 8/24 and reduce it to the largest common divisor of numbers 8 and 24, which is 8. Since 8: 8 \u003d 1 and 24: 8 \u003d 3, then we arrive at the non-interpretable fraction 1/3. So, .

Note that under the phrase "cut a fraction" often implies the leading of the initial fraction precisely to an incomprehensive form. In other words, the cutting of the fraction is very often called the division of the numerator and the denominator on their greatest common divisor (and not on any of their common divisor).

How to cut a fraction? Rule and fraction reduction examples

It remains only to disassemble the shortage of fractions, which explains how to reduce this fraction.

The reduction rule of fractions Consists of two steps:

• first, there is a node of the numerator and denominator of the fraction;
• secondly, the division of the numerator and the denominator of the fraction on their nodes is carried out, which gives an incomprehensive fraction equal to the initial one.

We will understand an example of a reduction of the fraci According to the voiced rule.

Example.

Reduce fraction 182/195.

Decision.

We carry out both steps prescribed by the rules of the cutting of the fraction.

First we find Nod (182, 195). It is most convenient to use the Euclide algorithm (see): 195 \u003d 182 · 1 + 13, 182 \u003d 13 · 14, that is, node (182, 195) \u003d 13.

Now we divide the numerator and denominator of the fraction 182/195 by 13, while we get an incompreheral fraction 14/15, which is equal to the initial fraction. On this cutting of the fraction is completed.

Briefly the solution can be written like this :.

Answer:

On this with a reduction of fractions, it is possible to finish. But for the completeness of the picture, consider two more ways to reduce fractions, which are usually applied in easy cases.

Sometimes the numerator and denominator of the cutting fraction is easy. Reduce the fraction in this case is very simple: you only need to remove all common multipliers from the numerator and denominator.

It is worth noting that this method directly follows from the rule of reduction of fractions, since the product of all common simple multipliers of the numerator and the denominator is equal to their greatest general divisor.

We will analyze the solution of the example.

Example.

Reduce fraction 360/2 940.

Decision.

Spread the nipple and denominator for simple multipliers: 360 \u003d 2 · 2 · 2 · 3 · 3 · 5 and 2 940 \u003d 2 · 2 · 3 · 5 · 7 · 7. In this way, .

Now we get rid of general multipliers in the numerator and denominator, for convenience, they simply cry out: .

Finally, I turn out the remaining multipliers:, and the reduction of the fraction is completed.

Here is a brief record of the decision: .

Answer:

Consider another way to reduce the fraction, which consists in a consistent reduction. Here at each step there is a reduction in the fraction on some common divisor of the numerator and denominator, which is either obvious or easily determined by