How to reduce fractions with whole numbers. Online calculator for reducing algebraic fractions with a detailed solution allows you to reduce a fraction and convert an improper fraction to a proper fraction

Without knowing how to reduce a fraction and having a stable skill in solving such examples, it is very difficult to study algebra in school. The further you go, the more it interferes with your basic knowledge of reducing fractions. new information. First, powers appear, then factors, which later become polynomials.

How can you avoid getting confused here? Thoroughly consolidate skills in previous topics and gradually prepare for knowledge of how to reduce a fraction, which becomes more complex from year to year.

Basic knowledge

Without them, you will not be able to cope with tasks of any level. To understand, you need to understand two simple points. First: you can only reduce factors. This nuance turns out to be very important when polynomials appear in the numerator or denominator. Then you need to clearly distinguish where the multiplier is and where the addend is.

The second point says that any number can be represented in the form of factors. Moreover, the result of reduction is a fraction whose numerator and denominator can no longer be reduced.

Rules for reducing common fractions

First, you should check whether the numerator is divisible by the denominator or vice versa. Then it is precisely this number that needs to be reduced. This is the simplest option.

The second is analysis appearance numbers. If both end in one or more zeros, then they can be shortened by 10, 100 or a thousand. Here you can notice whether the numbers are even. If yes, then you can safely cut it by two.

The third rule for reducing a fraction is to factor it into prime factors numerator and denominator. At this time, you need to actively use all your knowledge about the signs of divisibility of numbers. After this decomposition, all that remains is to find all the repeating ones, multiply them and reduce them by the resulting number.

What if there is an algebraic expression in a fraction?

This is where the first difficulties appear. Because this is where terms appear that can be identical to factors. I really want to reduce them, but I can’t. Before you can reduce an algebraic fraction, it must be converted so that it has factors.

To do this, you will need to perform several steps. You may need to go through all of them, or maybe the first one will provide a suitable option.

Check whether the numerator and denominator or any expression in them differ by sign. In this case, you just need to put minus one out of brackets. This produces equal factors that can be reduced.

See if it is possible to remove the common factor from the polynomial out of brackets. Perhaps this will result in a parenthesis, which can also be shortened, or it will be a removed monomial.

Try to group the monomials in order to then add a common factor to them. After this, it may turn out that there will be factors that can be reduced, or again the bracketing of common elements will be repeated.

Try to consider abbreviated multiplication formulas in writing. With their help, you can easily convert polynomials into factors.

Sequence of operations with fractions with powers

In order to easily understand the question of how to reduce a fraction with powers, you need to firmly remember the basic operations with them. The first of these is related to the multiplication of powers. In this case, if the bases are the same, the indicators must be added.

The second is division. Again, for those that have the same reasons, the indicators will need to be subtracted. Moreover, you need to subtract from the number that is in the dividend, and not vice versa.

The third is exponentiation. In this situation, the indicators are multiplied.

Successful reduction will also require the ability to reduce degrees to on the same grounds. That is, to see that four is two squared. Or 27 - the cube of three. Because reducing 9 squared and 3 cubed is difficult. But if we transform the first expression as (3 2) 2, then the reduction will be successful.

In this article we will look in detail at how reducing fractions. First, let's discuss what is called reducing a fraction. After this, let's talk about reducing a reducible fraction to an irreducible form. Next we will obtain the rule for reducing fractions and, finally, consider examples of the application of this rule.

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What does it mean to reduce a fraction?

We know that ordinary fractions are divided into reducible and irreducible fractions. You can guess from the names that reducible fractions can be reduced, but irreducible fractions cannot.

What does it mean to reduce a fraction? Reduce fraction- this means dividing its numerator and denominator by their positive and different from unity. It is clear that as a result of reducing a fraction, a new fraction is obtained with a smaller numerator and denominator, and, due to the basic property of the fraction, the resulting fraction is equal to the original one.

For example, let's reduce the common fraction 8/24 by dividing its numerator and denominator by 2. In other words, let's reduce the fraction 8/24 by 2. Since 8:2=4 and 24:2=12, this reduction results in the fraction 4/12, which is equal to the original fraction 8/24 (see equal and unequal fractions). As a result, we have .

Reducing ordinary fractions to irreducible form

Typically, the ultimate goal of reducing a fraction is to obtain an irreducible fraction that is equal to the original reducible fraction. This goal can be achieved by reducing the original reducible fraction into its numerator and denominator. As a result of such a reduction, an irreducible fraction is always obtained. Indeed, a fraction is irreducible, since it is known that And - . Here we will say that the greatest common divisor of the numerator and denominator of a fraction is the largest number by which this fraction can be reduced.

So, reducing a common fraction to an irreducible form consists of dividing the numerator and denominator of the original reducible fraction by their gcd.

Let's look at an example, for which we return to the fraction 8/24 and reduce it by the greatest common divisor of the numbers 8 and 24, which is equal to 8. Since 8:8=1 and 24:8=3, we come to the irreducible fraction 1/3. So, .

Note that the phrase “reduce a fraction” often means reducing the original fraction to its irreducible form. In other words, reducing a fraction very often refers to dividing the numerator and denominator by their greatest common factor (rather than by any common factor).

How to reduce a fraction? Rules and examples of reducing fractions

All that remains is to look at the rule for reducing fractions, which explains how to reduce a given fraction.

Rule for reducing fractions consists of two steps:

• firstly, the gcd of the numerator and denominator of the fraction is found;
• secondly, the numerator and denominator of the fraction are divided by their gcd, which gives an irreducible fraction equal to the original one.

Let's sort it out example of reducing a fraction according to the stated rule.

Example.

Reduce the fraction 182/195.

Solution.

Let's carry out both steps prescribed by the rule for reducing a fraction.

First we find GCD(182, 195) . It is most convenient to use the Euclid algorithm (see): 195=182·1+13, 182=13·14, that is, GCD(182, 195)=13.

Now we divide the numerator and denominator of the fraction 182/195 by 13, and we get the irreducible fraction 14/15, which is equal to the original fraction. This completes the reduction of the fraction.

Briefly, the solution can be written as follows: .

Answer:

This is where we can finish reducing fractions. But to complete the picture, let's look at two more ways to reduce fractions, which are usually used in easy cases.

Sometimes the numerator and denominator of the fraction being reduced is not difficult. Reducing a fraction in this case is very simple: you just need to remove all common factors from the numerator and denominator.

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

Let's look at the solution to the example.

Example.

Reduce the fraction 360/2 940.

Solution.

Let's factor the numerator and denominator into simple factors: 360=2·2·2·3·3·5 and 2,940=2·2·3·5·7·7. Thus, .

Now we get rid of the common factors in the numerator and denominator; for convenience, we simply cross them out: .

Finally, we multiply the remaining factors: , and the reduction of the fraction is completed.

Here is a short summary of the solution: .

Answer:

Let's consider another way to reduce a fraction, which consists of sequential reduction. Here, at each step, the fraction is reduced by some common divisor of the numerator and denominator, which is either obvious or easily determined using

Reducing fractions is necessary in order to reduce the fraction to more simple view, for example, in the answer obtained as a result of solving an expression.

Reducing fractions, definition and formula.

What is reducing fractions? What does it mean to reduce a fraction?

Definition:
Reducing Fractions- this is the division of the numerator and denominator of a fraction into the same thing positive number not equal to zero and one. As a result of the reduction, a fraction with a smaller numerator and denominator is obtained, equal to the previous fraction according to.

Formula for reducing fractions main property rational numbers.

\(\frac(p \times n)(q \times n)=\frac(p)(q)\)

Let's look at an example:
Reduce the fraction \(\frac(9)(15)\)

Solution:
We can factor a fraction into prime factors and cancel common factors.

\(\frac(9)(15)=\frac(3 \times 3)(5 \times 3)=\frac(3)(5) \times \color(red) (\frac(3)(3) )=\frac(3)(5) \times 1=\frac(3)(5)\)

Answer: after reduction we got the fraction \(\frac(3)(5)\). According to the basic property of rational numbers, the original and resulting fractions are equal.

\(\frac(9)(15)=\frac(3)(5)\)

How to reduce fractions? Reducing a fraction to its irreducible form.

To get an irreducible fraction as a result, we need find the greatest common divisor (GCD) for the numerator and denominator of the fraction.

There are several ways to find GCD; in the example we will use the decomposition of numbers into prime factors.

Get the irreducible fraction \(\frac(48)(136)\).

Solution:
Let's find GCD(48, 136). Let's write the numbers 48 and 136 into prime factors.
48=2⋅2⋅2⋅2⋅3
136=2⋅2⋅2⋅17
GCD(48, 136)= 2⋅2⋅2=6

\(\frac(48)(136)=\frac(\color(red) (2 \times 2 \times 2) \times 2 \times 3)(\color(red) (2 \times 2 \times 2) \times 17)=\frac(\color(red) (6) \times 2 \times 3)(\color(red) (6) \times 17)=\frac(2 \times 3)(17)=\ frac(6)(17)\)

The rule for reducing a fraction to an irreducible form.

1. You need to find the greatest common divisor for the numerator and denominator.
2. You need to divide the numerator and denominator by the greatest common divisor to obtain an irreducible fraction as a result of division.

Example:
Reduce the fraction \(\frac(152)(168)\).

Solution:
Let's find GCD(152, 168). Let's write the numbers 152 and 168 into prime factors.
152=2⋅2⋅2⋅19
168=2⋅2⋅2⋅3⋅7
GCD(152, 168)= 2⋅2⋅2=6

\(\frac(152)(168)=\frac(\color(red) (6) \times 19)(\color(red) (6) \times 21)=\frac(19)(21)\)

Answer: \(\frac(19)(21)\) is an irreducible fraction.

Reducing improper fractions.

How to reduce an improper fraction?
The rules for reducing fractions are the same for proper and improper fractions.

Let's look at an example:
Reduce the improper fraction \(\frac(44)(32)\).

Solution:
Let's write the numerator and denominator into simple factors. And then we’ll reduce the common factors.

\(\frac(44)(32)=\frac(\color(red) (2 \times 2 ) \times 11)(\color(red) (2 \times 2 ) \times 2 \times 2 \times 2 )=\frac(11)(2 \times 2 \times 2)=\frac(11)(8)\)

Reducing mixed fractions.

Mixed fractions follow the same rules as ordinary fractions. The only difference is that we can do not touch the whole part, but reduce the fractional part or Convert a mixed fraction to an improper fraction, reduce it and convert it back to a proper fraction.

Let's look at an example:
Cancel the mixed fraction \(2\frac(30)(45)\).

Solution:
Let's solve it in two ways:
First way:
Let's write the fractional part into simple factors, but we won't touch the whole part.

\(2\frac(30)(45)=2\frac(2 \times \color(red) (5 \times 3))(3 \times \color(red) (5 \times 3))=2\ frac(2)(3)\)

Second way:
Let's first convert it to an improper fraction, and then write it into prime factors and reduce. Let's convert the resulting improper fraction into a proper fraction.

\(2\frac(30)(45)=\frac(45 \times 2 + 30)(45)=\frac(120)(45)=\frac(2 \times \color(red) (5 \times 3) \times 2 \times 2)(3 \times \color(red) (3 \times 5))=\frac(2 \times 2 \times 2)(3)=\frac(8)(3)= 2\frac(2)(3)\)

Related questions:
Can you reduce fractions when adding or subtracting?
Answer: no, you must first add or subtract fractions according to the rules, and only then reduce them. Let's look at an example:

Evaluate the expression \(\frac(50+20-10)(20)\) .

Solution:
They often make the mistake of abbreviating same numbers In our case, the numerator and denominator have the number 20, but they cannot be reduced until you have completed the addition and subtraction.

\(\frac(50+\color(red) (20)-10)(\color(red) (20))=\frac(60)(20)=\frac(3 \times 20)(20)= \frac(3)(1)=3\)

What numbers can you reduce a fraction by?
Answer: You can reduce a fraction by the greatest common factor or the common divisor of the numerator and denominator. For example, the fraction \(\frac(100)(150)\).

Let's write the numbers 100 and 150 into prime factors.
100=2⋅2⋅5⋅5
150=2⋅5⋅5⋅3
The greatest common divisor will be the number gcd(100, 150)= 2⋅5⋅5=50

\(\frac(100)(150)=\frac(2 \times 50)(3 \times 50)=\frac(2)(3)\)

We got the irreducible fraction \(\frac(2)(3)\).

But it is not necessary to always divide by gcd; an irreducible fraction is not always needed; you can reduce the fraction by a simple divisor of the numerator and denominator. For example, the number 100 and 150 have a common divisor of 2. Let's reduce the fraction \(\frac(100)(150)\) by 2.

\(\frac(100)(150)=\frac(2 \times 50)(2 \times 75)=\frac(50)(75)\)

We got the reducible fraction \(\frac(50)(75)\).

What fractions can be reduced?
Answer: You can reduce fractions in which the numerator and denominator have a common divisor. For example, the fraction \(\frac(4)(8)\). The number 4 and 8 have a number by which they are both divisible - the number 2. Therefore, such a fraction can be reduced by the number 2.

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

These two fractions are equal. Let's take a closer look at the fraction \(\frac(8)(12)\):

\(\frac(8)(12)=\frac(2 \times 4)(3 \times 4)=\frac(2)(3) \times \frac(4)(4)=\frac(2) (3)\times 1=\frac(2)(3)\)

From here we get, \(\frac(8)(12)=\frac(2)(3)\)

Two fractions are equal if and only if one of them is obtained by reducing the other fraction by the common factor of the numerator and denominator.

Example:
If possible, reduce the following fractions: a) \(\frac(90)(65)\) b) \(\frac(27)(63)\) c) \(\frac(17)(100)\) d) \(\frac(100)(250)\)

Solution:
a) \(\frac(90)(65)=\frac(2 \times \color(red) (5) \times 3 \times 3)(\color(red) (5) \times 13)=\frac (2 \times 3 \times 3)(13)=\frac(18)(13)\)
b) \(\frac(27)(63)=\frac(\color(red) (3 \times 3) \times 3)(\color(red) (3 \times 3) \times 7)=\frac (3)(7)\)
c) \(\frac(17)(100)\) irreducible fraction
d) \(\frac(100)(250)=\frac(\color(red) (2 \times 5 \times 5) \times 2)(\color(red) (2 \times 5 \times 5) \ times 5)=\frac(2)(5)\)

It is based on their basic property: if the numerator and denominator of a fraction are divided by the same non-zero polynomial, then an equal fraction will be obtained.

You can only reduce multipliers!

Members of polynomials cannot be abbreviated!

To reduce an algebraic fraction, the polynomials in the numerator and denominator must first be factorized.

Let's look at examples of reducing fractions.

The numerator and denominator of the fraction contain monomials. They represent work(numbers, variables and their powers), multipliers we can reduce.

We reduce the numbers by their greatest common divisor, that is, by the largest number by which each of these numbers is divided. For 24 and 36 this is 12. After reduction, 2 remains from 24, and 3 from 36.

We reduce the degrees by the degree with the lowest index. To reduce a fraction means to divide the numerator and denominator by the same divisor, and subtract the exponents.

a² and a⁷ are reduced to a². In this case, one remains in the numerator of a² (we write 1 only in the case when, after reduction, there are no other factors left. From 24, 2 remains, so we do not write 1 remaining from a²). From a⁷, after reduction, a⁵ remains.

b and b are reduced by b; the resulting units are not written.

c³º and c⁵ are shortened to c⁵. From c³º what remains is c²⁵, from c⁵ is one (we don’t write it). Thus,

The numerator and denominator of this algebraic fraction are polynomials. You cannot cancel terms of polynomials! (you cannot reduce, for example, 8x² and 2x!). To reduce this fraction, you need . The numerator has a common factor of 4x. Let's take it out of brackets:

Both the numerator and denominator have the same factor (2x-3). We reduce the fraction by this factor. In the numerator we got 4x, in the denominator - 1. For 1 property algebraic fractions, the fraction is 4x.

You can only reduce factors (you cannot reduce this fraction by 25x²!). Therefore, the polynomials in the numerator and denominator of the fraction must be factorized.

The numerator is the complete square of the sum, the denominator is the difference of squares. After decomposition using abbreviated multiplication formulas, we obtain:

We reduce the fraction by (5x+1) (to do this, cross out the two in the numerator as an exponent, leaving (5x+1)² (5x+1)):

The numerator has a common factor of 2, let's take it out of brackets. The denominator is the formula for the difference of cubes:

As a result of the expansion, the numerator and denominator received the same factor (9+3a+a²). We reduce the fraction by it:

The polynomial in the numerator consists of 4 terms. the first term with the second, the third with the fourth, and remove the common factor x² from the first brackets. We decompose the denominator using the sum of cubes formula:

In the numerator, let’s take the common factor (x+2) out of brackets:

Reduce the fraction by (x+2):

Online calculator performs reduction of algebraic fractions in accordance with the rule of reducing fractions: replacing the original fraction with an equal fraction, but with a smaller numerator and denominator, i.e. Simultaneously dividing the numerator and denominator of a fraction by their common greatest common factor (GCD). The calculator also displays a detailed solution that will help you understand the sequence of the reduction.

Given:

Solution:

Performing fraction reduction

checking the possibility of performing algebraic fraction reduction

1) Determination of the greatest common divisor (GCD) of the numerator and denominator of a fraction

determining the greatest common divisor (GCD) of the numerator and denominator of an algebraic fraction

2) Reducing the numerator and denominator of a fraction

reducing the numerator and denominator of an algebraic fraction

3) Selecting the whole part of a fraction

separating the whole part of an algebraic fraction

4) Converting an algebraic fraction to a decimal fraction

converting an algebraic fraction to decimal

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I. Procedure for reducing an algebraic fraction using an online calculator:

1. To reduce an algebraic fraction, enter the values ​​of the numerator and denominator of the fraction in the appropriate fields. If the fraction is mixed, then also fill in the field corresponding to the whole part of the fraction. If the fraction is simple, then leave the whole part field blank.
2. To specify a negative fraction, place a minus sign on the whole part of the fraction.
3. Depending on the specified algebraic fraction, the following sequence of actions is automatically performed:

• determining the greatest common divisor (GCD) of the numerator and denominator of a fraction;
• reducing the numerator and denominator of a fraction by gcd;
• highlighting the whole part of a fraction, if the numerator of the final fraction is greater than the denominator.
• converting the final algebraic fraction to a decimal fraction rounded to the nearest hundredth.

II. For reference:

A fraction is a number consisting of one or more parts (fractions) of a unit. Common fraction(simple fraction) is written as two numbers (the numerator of the fraction and the denominator of the fraction) separated by a horizontal bar (the fraction bar) indicating the division sign. The numerator of a fraction is the number above the fraction line. The numerator shows how many shares were taken from the whole. The denominator of a fraction is the number below the fraction line. The denominator shows how many equal parts the whole is divided into. A simple fraction is a fraction that does not have a whole part. A simple fraction can be proper or improper. A proper fraction is a fraction whose numerator is less than its denominator, so a proper fraction is always less than one. Example of proper fractions: 8/7, 11/19, 16/17. An improper fraction is a fraction in which the numerator is greater than or equal to the denominator, so an improper fraction is always greater than or equal to one. Example of improper fractions: 7/6, 8/7, 13/13. mixed fraction is a number that contains a whole number and a proper fraction, and denotes the sum of that whole number and the proper fraction. Any mixed fraction can be converted to an improper fraction. Example mixed fractions: 1¼, 2½, 4¾.

III. Note:

1. Source data block highlighted yellow , intermediate calculation block allocated blue , the solution block is highlighted in green.
2. To add, subtract, multiply and divide ordinary or mixed fractions, use the online fraction calculator with detailed solution.