What is the largest number that can be used to reduce a fraction? Online calculator. Reducing fractions (irregular, mixed)
In this article we will look at basic operations with algebraic fractions:
- reducing fractions
- multiplying fractions
- dividing fractions
Let's start with reductions algebraic fractions .
It would seem that, algorithm obvious.
To reduce algebraic fractions, need to
1. Factor the numerator and denominator of the fraction.
2. Reduce equal factors.
However, schoolchildren often make the mistake of “reducing” not the factors, but the terms. For example, there are amateurs who “reduce” fractions by and get as a result , which, of course, is not true.
Let's look at examples:
1.
Reduce fraction: 
1. Let us factorize the numerator using the formula of the square of the sum, and the denominator using the formula of the difference of squares
2. Divide the numerator and denominator by
2.
Reduce fraction: 
1. Let's factorize the numerator. Since the numerator contains four terms, we use grouping.
2. Let's factorize the denominator. We can also use grouping.
3. Let's write down the fraction that we got and reduce the same factors:
Multiplying algebraic fractions.
When multiplying algebraic fractions, we multiply the numerator by the numerator, and multiply the denominator by the denominator.
Important! There is no need to rush to multiply the numerator and denominator of a fraction. After we have written down the product of the numerators of the fractions in the numerator, and the product of the denominators in the denominator, we need to factor each factor and reduce the fraction.
Let's look at examples:
3. Simplify the expression:
1. Let’s write the product of fractions: in the numerator the product of the numerators, and in the denominator the product of the denominators:
2. Let's factorize each bracket:
Now we need to reduce the same factors. Note that the expressions and differ only in sign: 
and as a result of dividing the first expression by the second we get -1.
So, 
We divide algebraic fractions according to the following rule:
That is To divide by a fraction, you need to multiply by the "inverted" one.
We see that dividing fractions comes down to multiplying, and multiplication ultimately comes down to reducing fractions.
Let's look at an example:
4. Simplify the expression:
This article continues the topic of converting algebraic fractions: consider such an action as reducing algebraic fractions. Let's define the term itself, formulate a reduction rule and analyze practical examples.
Yandex.RTB R-A-339285-1
The meaning of reducing an algebraic fraction
In materials about common fractions, we looked at its reduction. We defined reducing a fraction as dividing its numerator and denominator by a common factor.
Reducing an algebraic fraction is a similar operation.
Definition 1
Reducing an algebraic fraction is the division of its numerator and denominator by a common factor. In this case, in contrast to the reduction of an ordinary fraction (the common denominator can only be a number), the common factor of the numerator and denominator of an algebraic fraction can be a polynomial, in particular, a monomial 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 the number 3, resulting in: x 2 + 2 x y 6 x 3 · y + 12 · x 2 · y 2 . We can reduce the same fraction by the variable x, and this will give us the expression 3 x + 6 y 6 x 2 y + 12 x y 2. It is also possible to reduce a given fraction by a monomial 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 reducing an algebraic fraction is a fraction greater than simple type, at best, is an irreducible fraction.
Are all algebraic fractions subject to reduction?
Again, from materials on ordinary fractions, we know that there are reducible and irreducible fractions. Irreducible fractions are fractions that do not have common factors in the numerator and denominator other than 1.
It’s the same with algebraic fractions: they may have common factors in the numerator and denominator, or they may not. The presence of common factors allows you to simplify the original fraction through reduction. When there are no common factors, it is impossible to optimize a given fraction using the reduction method.
IN general cases For a given type of fraction, it is quite difficult to understand whether it can be reduced. Of course, in some cases the presence of a common factor between the numerator and denominator is obvious. For example, in the algebraic fraction 3 x 2 3 y it is quite clear that the common factor is the number 3.
In the fraction - x · y 5 · x · y · z 3 we also immediately understand that it can be reduced by x, or y, or x · y. And yet, much more often there are examples of algebraic fractions, when the common factor of the numerator and denominator is not so easy to see, and even more often, it is simply absent.
For example, we can reduce the fraction x 3 - 1 x 2 - 1 by x - 1, while the specified common factor is not present in the entry. But the fraction x 3 - x 2 + x - 1 x 3 + x 2 + 4 · x + 4 cannot be reduced, since the numerator and denominator do not have a common factor.
Thus, the question of determining the reducibility of an algebraic fraction is not so simple, and it is often easier to work with a fraction of a given form than to try to find out whether it is reducible. In this case, such transformations take place that in particular cases make it possible to determine the common factor of the numerator and denominator or to draw a conclusion about the irreducibility of a fraction. We will examine this issue in detail in the next paragraph of the article.
Rule for reducing algebraic fractions
Rule for reducing algebraic fractions consists of two sequential actions:
- finding common factors of the numerator and denominator;
- if any are found, the action of reducing the fraction is carried out directly.
The most convenient method for finding common denominators is to factor the polynomials present in the numerator and denominator of a given algebraic fraction. This allows you to immediately clearly see the presence or absence of common factors.
The very action of reducing an algebraic fraction is based on the main property of an algebraic fraction, expressed by the equality undefined, where a, b, c are some polynomials, and b and c are non-zero. The first step is to reduce the fraction to the form a · c b · c, in which we immediately notice the common factor c. The second step is to perform a reduction, i.e. transition to a fraction of the form a b .
Typical examples
Despite some obviousness, let us clarify the special case when the numerator and denominator of an algebraic fraction are equal. Similar fractions are identically equal to 1 on the entire ODZ of the variables of this fraction:
5 5 = 1 ; - 2 3 - 2 3 = 1 ; x x = 1 ; - 3, 2 x 3 - 3, 2 x 3 = 1; 1 2 · x - x 2 · y 1 2 · x - x 2 · y ;
Since ordinary fractions are a special case of algebraic fractions, let us recall how they are reduced. The natural numbers written in the numerator and denominator are factored into prime factors, then the common factors are canceled (if any).
For example, 24 1260 = 2 2 2 3 2 2 3 3 5 7 = 2 3 5 7 = 2 105
The product of simple identical factors can be written as powers, and in the process of reducing a fraction, use the property of dividing powers with identical bases. Then the above solution would be:
24 1260 = 2 3 3 2 2 3 2 5 7 = 2 3 - 2 3 2 - 1 5 7 = 2 105
(numerator and denominator divided by a common factor 2 2 3). Or for clarity, based on the properties of multiplication and division, we give the solution the following form:
24 1260 = 2 3 3 2 2 3 2 5 7 = 2 3 2 2 3 3 2 1 5 7 = 2 1 1 3 1 35 = 2 105
By analogy, the reduction of algebraic fractions is carried out, in which the numerator and denominator have monomials 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 needs to be reduced.
Solution
It is possible to write the numerator and denominator of a given fraction as a product of simple factors and variables, and then carry out the reduction:
27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z = - 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 = = - 3 · 3 · a · a · a 2 · c · c · c · c · c · c = - 9 a 3 2 c 6
However, a more rational way would be to write the solution as an expression with powers:
27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z = - 3 3 · a 5 · b 2 · c · z 2 · 3 · a 2 · b 2 · c 7 · z = - 3 3 2 · 3 · a 5 a 2 · b 2 b 2 · c c 7 · z z = = - 3 3 - 1 2 · a 5 - 2 1 · 1 · 1 c 7 - 1 · 1 = · - 3 2 · a 3 2 · c 6 = · - 9 · a 3 2 · c 6 .
Answer:- 27 a 5 b 2 c z 6 a 2 b 2 c 7 z = - 9 a 3 2 c 6
When the numerator and denominator of an algebraic fraction contain fractional numerical coefficients, there are two possible ways of further action: either divide these fractional coefficients separately, or first get rid of the fractional coefficients by multiplying the numerator and denominator by a certain natural number. The last transformation is carried out due to the basic property of an algebraic fraction (you can read about it in the article “Reducing an algebraic fraction to a new denominator”).
Example 2
The given fraction is 2 5 x 0, 3 x 3. It needs to be reduced.
Solution
It is possible to reduce the fraction this way:
2 5 x 0, 3 x 3 = 2 5 3 10 x x 3 = 4 3 1 x 2 = 4 3 x 2
Let's try to solve the problem differently, having first gotten rid of fractional coefficients - multiply the numerator and denominator by the least common multiple of the denominators of these coefficients, i.e. on LCM (5, 10) = 10. Then we get:
2 5 x 0, 3 x 3 = 10 2 5 x 10 0, 3 x 3 = 4 x 3 x 3 = 4 3 x 2.
Answer: 2 5 x 0, 3 x 3 = 4 3 x 2
When we reduce algebraic fractions general view, in which the numerators and denominators can be either monomials or polynomials, there may be a problem when the common factor is not always immediately visible. Or moreover, it simply does not exist. Then, to determine the common factor or record the fact of its absence, the numerator and denominator of the algebraic fraction are factored.
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 needs to be reduced.
Solution
Let's factor the polynomials in the numerator and denominator. Let's put it out of brackets:
2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49)
We see that the expression in parentheses can be converted using abbreviated multiplication formulas:
2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49) = 2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7)
It is clearly seen that it is possible to reduce a fraction by a common factor b 2 (a + 7). Let's make a reduction:
2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7) = 2 (a + 7) b (a - 7) = 2 a + 14 a b - 7 b
Let us write a short solution without explanation as a chain of equalities:
2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49) = = 2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7) = 2 (a + 7) b (a - 7) = 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 = 2 a + 14 a b - 7 b.
It happens that common factors are hidden by numerical coefficients. Then, when reducing fractions, it is optimal to put the numerical factors at higher powers of the numerator and denominator out of brackets.
Example 4
Given the algebraic fraction 1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2 . It is necessary to reduce it if possible.
Solution
At first glance, the numerator and denominator do not exist common denominator. However, let's try to convert the given fraction. Let's take out the factor x in the numerator:
1 5 x - 2 7 x 3 y 5 x 2 y - 3 1 2 = x 1 5 - 2 7 x 2 y 5 x 2 y - 3 1 2
Now you can see some similarity between the expression in brackets and the expression in the denominator due to x 2 y . Let us take out the numerical coefficients of the higher powers of these polynomials:
x 1 5 - 2 7 x 2 y 5 x 2 y - 3 1 2 = x - 2 7 - 7 2 1 5 + x 2 y 5 x 2 y - 1 5 3 1 2 = = - 2 7 x - 7 10 + x 2 y 5 x 2 y - 7 10
Now the common factor becomes visible, we carry out the reduction:
2 7 x - 7 10 + x 2 y 5 x 2 y - 7 10 = - 2 7 x 5 = - 2 35 x
Answer: 1 5 x - 2 7 x 3 y 5 x 2 y - 3 1 2 = - 2 35 x .
Let us emphasize that the skill of reducing rational fractions depends on the ability to factor polynomials.
If you notice an error in the text, please highlight it and press Ctrl+Enter
So we got to the reduction. The basic property of a fraction is applied here. BUT! Not so simple. With many fractions (including those from the school course), it is quite possible to get by with them. What if we take fractions that are “more abrupt”? Let's take a closer look! I recommend looking at materials with fractions.So, we already know that the numerator and denominator of a fraction can be multiplied and divided by the same number, the fraction will not change. Let's consider three approaches:
Approach one.
To reduce, divide the numerator and denominator by a common divisor. Let's look at examples:
Let's shorten:
In the examples given, we immediately see which divisors to take for reduction. The process is simple - we go through 2,3,4,5 and so on. In most school course examples, this is quite enough. But if it’s a fraction:
Here the process of selecting divisors can take a long time;). Of course, such examples are outside the school curriculum, but you need to be able to cope with them. Below we will look at how this is done. For now, let's get back to the downsizing process.
As discussed above, in order to reduce a fraction, we divided by the common divisor(s) we determined. Everything is correct! One has only to add signs of divisibility of numbers:
- if the number is even, then it is divisible by 2.
- if a number from the last two digits is divisible by 4, then the number itself is divisible by 4.
— if the sum of the digits that make up the number is divisible by 3, then the number itself is divisible by 3. For example, 125031, 1+2+5+0+3+1=12. Twelve is divisible by 3, so 123031 is divisible by 3.
- if the end of a number is 5 or 0, then the number is divisible by 5.
— if the sum of the digits that make up the number is divisible by 9, then the number itself is divisible by 9. For example, 625032 =.> 6+2+5+0+3+2=18. Eighteen is divisible by 9, which means 623032 is divisible by 9.
Second approach.
To put it briefly, in fact, the whole action comes down to factoring the numerator and denominator and then reducing equal factors in the numerator and denominator (this approach is a consequence of the first approach):
Visually, in order to avoid confusion and mistakes, equal factors are simply crossed out. Question - how to factor a number? It is necessary to determine all divisors by searching. This is a separate topic, it is not complicated, look up the information in a textbook or on the Internet. You won't encounter any big problems with factoring numbers that are present in school fractions.
Formally, the reduction principle can be written as follows:
Approach three.
Here is the most interesting thing for the advanced and those who want to become one. Let's reduce the fraction 143/273. Try it yourself! Well, how did it happen quickly? Now look!
We turn it over (we change places of the numerator and denominator). Divide the resulting fraction with a corner and convert it to mixed number, that is, we select the whole part:
It's already easier. We see that the numerator and denominator can be reduced by 13:
Now don’t forget to turn the fraction back over again, let’s write down the whole chain:
Checked - it takes less time than searching through and checking divisors. Let's return to our two examples:
First. Divide with a corner (not on a calculator), we get:
This fraction is simpler, of course, but the reduction is again a problem. Now we separately analyze the fraction 1273/1463 and turn it over:
It's easier here. We can consider a divisor such as 19. The rest are not suitable, this is clear: 190:19 = 10, 1273:19 = 67. Hurray! Let's write down:
Next example. Let's shorten 88179/2717.
Divide, we get:
Separately, we analyze the fraction 1235/2717 and turn it over:
We can consider a divisor such as 13 (up to 13 is not suitable):
Numerator 247:13=19 Denominator 1235:13=95
*During the process, we saw another divisor equal to 19. It turns out that:
Now we write down the original number:
And it doesn’t matter what is larger in the fraction - the numerator or the denominator, if it is the denominator, then we turn it over and proceed as described. This way we can reduce any fraction; the third approach can be called universal.
Of course, the two examples discussed above are not simple examples. Let's try this technology on the “simple” fractions we have already discussed:
Two quarters.
Seventy-two sixties. The numerator is greater than the denominator; there is no need to reverse it:
Of course, the third approach was applied to such simple examples just as an alternative. The method, as already said, is universal, but not convenient and correct for all fractions, especially for simple ones.
The variety of fractions is great. It is important that you understand the principles. Strict rules there is simply no way to work with fractions. We looked, figured out how it would be more convenient to act, and moved forward. With practice, skill will come and you will crack them like seeds.
Conclusion:
If you see a common divisor(s) for the numerator and denominator, use them to reduce.
If you know how to quickly factor a number, then factor the numerator and denominator, then reduce.
If you can’t determine the common divisor, then use the third approach.
*To reduce fractions, it is important to master the principles of reduction, understand the basic property of a fraction, know approaches to solving, and be extremely careful when making calculations.
And remember! It is customary to reduce a fraction until it stops, that is, reduce it as long as there is a common divisor.
Sincerely, Alexander Krutitskikh.
Fractions and their reduction is another topic that begins in 5th grade. Here the basis of this action is formed, and then these skills are drawn by a thread into higher mathematics. If the student does not understand, then he may have problems in algebra. Therefore, it is better to understand a few rules once and for all. And also remember one prohibition and never break it.
Fraction and its reduction
Every student knows what it is. Any two digits located between a horizontal line are immediately perceived as a fraction. However, not everyone understands that any number can become it. If it is an integer, then it can always be divided by one, and then you get an improper fraction. But more on that later.
The beginning is always simple. First you need to figure out how to reduce a proper fraction. That is, one whose numerator is less than its denominator. To do this, you will need to remember the basic property of a fraction. It states that when multiplying (as well as dividing) its numerator and denominator at the same time same number it turns out to be an equivalent fraction to the original one.
Division actions that are performed in this property and result in reduction. That is, to simplify it as much as possible. A fraction can be reduced as long as there are common factors above and below the line. When they are no longer there, reduction is impossible. And they say that this fraction is irreducible.
Two ways
1.Step by step reduction. It uses an estimation method where both numbers are divided by the minimum common factor that the student notices. If after the first contraction it is clear that this is not the end, then the division continues. Until the fraction becomes irreducible.
2. Finding the largest common divisor at the numerator and denominator. This is the most rational way to reduce fractions. It involves factoring the numerator and denominator into prime factors. Among them, you then need to choose all the same ones. Their product will give the greatest common factor by which the fraction is reduced.
Both of these methods are equivalent. The student is encouraged to master them and use the one he likes best.
What if there are letters and addition and subtraction operations?
The first part of the question is more or less clear. Letters can be abbreviated just like numbers. The main thing is that they act as multipliers. But many people have problems with the second one.
Important to remember! You can only reduce numbers that are factors. If they are summands, it is impossible.
In order to understand how to reduce fractions of the form algebraic expression, you need to learn the rule. First, express the numerator and denominator as a product. Then you can reduce if common factors appear. To represent it in the form of multipliers, the following techniques are useful:
- grouping;
- bracketing;
- application of abbreviated multiplication identities.
Moreover, the latter method makes it possible to immediately obtain the terms in the form of multipliers. Therefore, it should always be used if a known pattern is visible.
But this is not scary yet, then tasks with degrees and roots appear. That's when you need to gain courage and learn a couple of new rules.
Expression with degree
Fraction. The numerator and denominator are the product. There are letters and numbers. And they are also raised to a power, which also consists of terms or factors. There is something to be afraid of.
In order to understand how to reduce fractions with powers, you will need to learn two things:
- if the exponent contains a sum, then it can be decomposed into factors, the powers of which will be the original terms;
- if the difference, then the dividend and the divisor, the first will have the minuend to the power, the second will have the subtrahend.
After completing these steps, the total multipliers become visible. In such examples there is no need to calculate all powers. It is enough to simply reduce degrees with the same exponents and bases.
In order to finally master how to reduce fractions with powers, you need a lot of practice. After several similar examples, actions will be performed automatically.
What if the expression contains a root?
It can also be shortened. Only again, following the rules. Moreover, all those described above are true. In general, if the question is how to reduce a fraction with roots, then you need to divide.
It can also be divided into irrational expressions. That is, if the numerator and denominator contain identical factors, enclosed under the sign of the root, then they can be safely reduced. This will simplify the expression and complete the task.
If, after the reduction, irrationality remains under the fraction line, then you need to get rid of it. In other words, multiply the numerator and denominator by it. If common factors appear after this operation, they will need to be reduced again.
That's probably all about how to reduce fractions. There are few rules, but only one prohibition. Never shorten terms!
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 detailed solution, which 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
Help for website development of the project
Dear Site Visitor.
If you were unable to find what you were looking for, be sure to write about it in the comments, what is currently missing on the site. This will help us understand in which direction we need to move further, and other visitors will soon be able to receive the necessary material.
If the site turned out to be useful to you, donate the site to the project only 2 ₽ and we will know that we are moving in the right direction.
Thank you for stopping by!
I. Procedure for reducing an algebraic fraction using an online calculator:
- 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.
- To set negative fraction, put a minus sign on the whole part of the fraction.
- 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:
