Let's understand what reducing fractions is, why and how to reduce fractions, and give the rule for reducing fractions and examples of its use.

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What is "reducing fractions"

To reduce a fraction is to divide its numerator and denominator by a common factor that is positive and different from one.

As a result of this action, a fraction with a new numerator and denominator will be obtained, equal to the original fraction.

For example, let's take common fraction 6 24 and shorten it. Divide the numerator and denominator by 2, resulting in 6 24 = 6 ÷ 2 24 ÷ 2 = 3 12. In this example, we reduced the original fraction by 2.

Reducing fractions to irreducible form

In the previous example, we reduced the fraction 6 24 by 2, resulting in the fraction 3 12. It is easy to see that this fraction can be further reduced. Typically, the goal of reducing fractions is to end up with an irreducible fraction. How to reduce a fraction to its irreducible form?

This can be done by reducing the numerator and denominator by their greatest common factor (GCD). Then, by the property of the greatest common divisor, the numerator and denominator will have mutually prime numbers, and the fraction will be irreducible.

a b = a ÷ N O D (a , b) b ÷ N O D (a , b)

Reducing a fraction to an irreducible form

To reduce a fraction to an irreducible form, you need to divide its numerator and denominator by their gcd.

Let's return to the fraction 6 24 from the first example and bring it to its irreducible form. The greatest common divisor of the numbers 6 and 24 is 6. Let's reduce the fraction:

6 24 = 6 ÷ 6 24 ÷ 6 = 1 4

Reducing fractions is convenient to use so as not to work with large numbers. In general, there is an unspoken rule in mathematics: if you can simplify any expression, then you need to do it. Reducing a fraction most often means reducing it to an irreducible form, and not simply reducing it by the common divisor of the numerator and denominator.

Rule for reducing fractions

To reduce fractions, just remember the rule, which consists of two steps.

Rule for reducing fractions

To reduce a fraction you need:

1. Find the gcd of the numerator and denominator.
2. Divide the numerator and denominator by their gcd.

Let's look at practical examples.

Example 1. Let's reduce the fraction.

Given the fraction 182 195. Let's shorten it.

Let's find the gcd of the numerator and denominator. To do this, in this case it is most convenient to use the Euclidean algorithm.

195 = 182 1 + 13 182 = 13 14 N O D (182, 195) = 13

Divide the numerator and denominator by 13. We get:

182 195 = 182 ÷ 13 195 ÷ 13 = 14 15

Ready. We have obtained an irreducible fraction that is equal to the original fraction.

How else can you reduce fractions? In some cases, it is convenient to factor the numerator and denominator into prime factors, and then remove all common factors from the upper and lower parts of the fraction.

Example 2. Reduce the fraction

Given the fraction 360 2940. Let's shorten it.

To do this, imagine the original fraction in the form:

360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7

Let's get rid of the common factors in the numerator and denominator, resulting in:

360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7 = 2 3 7 7 = 6 49

Finally, let's look at another way to reduce fractions. This is the so-called sequential reduction. Using this method, the reduction is carried out in several stages, in each of which the fraction is reduced by some obvious common factor.

Example 3. Reduce the fraction

Let's reduce the fraction 2000 4400.

It is immediately clear that the numerator and denominator have a common factor of 100. We reduce the fraction by 100 and get:

2000 4400 = 2000 ÷ 100 4400 ÷ 100 = 20 44

20 44 = 20 ÷ 2 44 ÷ 2 = 10 22

We reduce the resulting result again by 2 and obtain an irreducible fraction:

10 22 = 10 ÷ 2 22 ÷ 2 = 5 11

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Children at school learn the rules of reducing fractions in 6th grade. In this article, we will first tell you what this action means, then we will explain how to convert a reducible fraction into an irreducible fraction. The next point will be the rules for reducing fractions, and then we will gradually get to the examples.

What does it mean to "reduce a fraction"?

So, we all know that ordinary fractions are divided into two groups: reducible and irreducible. Already by the names you can understand that those that are contractible are contracted, and those that are irreducible are not contracted.

• To reduce a fraction means to divide its denominator and numerator by their (other than one) positive divisor. The result, of course, is a new fraction with a smaller denominator and numerator. The resulting fraction will be equal to the original fraction.

It is worth noting that in mathematics books with the task “reduce a fraction,” this means that you need to reduce the original fraction to this irreducible form. If we talk in simple words, then dividing the denominator and numerator by their greatest common divisor is a reduction.

How to reduce a fraction. Rules for reducing fractions (grade 6)

So there are only two rules here.

1. The first rule of reducing fractions is to first find the greatest common factor of the denominator and numerator of your fraction.
2. The second rule: divide the denominator and numerator by the greatest common divisor, ultimately obtaining an irreducible fraction.

How to reduce an improper fraction?

The rules for reducing fractions are identical to the rules for reducing improper fractions.

In order to reduce improper fraction, first you will need to write the denominator and numerator into simple factors, and only then reduce the common factors.

Reducing mixed fractions

The rules for reducing fractions also apply to reducing mixed fractions. There is only a small difference: we can not touch the whole part, but reduce the fraction or convert the mixed fraction into an improper fraction, then reduce it and again convert it into a proper fraction.

Reduce mixed fractions possible in two ways.

First: write the fractional part into prime factors and then leave the whole part alone.

The second way: first convert it to an improper fraction, write it into ordinary factors, then reduce the fraction. Convert the already obtained improper fraction into a proper fraction.

Examples can be seen in the photo above.

We really hope that we were able to help you and your children. After all, they are often inattentive in class, so they have to study more intensively at home on their own.

First level

Converting Expressions. Detailed Theory (2019)

Converting Expressions

We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:

“It’s much simpler,” we say, but such an answer usually doesn’t work.

Now I will teach you not to be afraid of any such tasks. Moreover, at the end of the lesson, you yourself will simplify this example to (just!) an ordinary number (yes, to hell with these letters).

But before you start this lesson, you need to be able to handle fractions and factor polynomials. Therefore, first, if you have not done this before, be sure to master the topics “” and “”.

Have you read it? If yes, then you are now ready.

Basic simplification operations

Now let's look at the basic techniques that are used to simplify expressions.

The simplest one is

1. Bringing similar

What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics. Similar are terms (monomials) with the same letter part. For example, in sum, similar terms are and.

Do you remember?

To bring similar means to add several similar terms to each other and get one term.

How can we put the letters together? - you ask.

This is very easy to understand if you imagine that the letters are some kind of objects. For example, a letter is a chair. Then what is the expression equal to? Two chairs plus three chairs, how many will it be? That's right, chairs: .

Now try this expression: .

To avoid confusion, let different letters represent different objects. For example, - is (as usual) a chair, and - is a table. Then:

chairs tables chair tables chairs chairs tables

The numbers by which the letters in such terms are multiplied are called coefficients. For example, in a monomial the coefficient is equal. And in it is equal.

So, the rule for bringing similar ones is:

Examples:

Give similar ones:

Answers:

2. (and similar, since, therefore, these terms have the same letter part).

2. Factorization

This is usually the most important part in simplifying expressions. After you have given similar ones, most often the resulting expression needs to be factorized, that is, presented as a product. This is especially important in fractions: in order to be able to reduce a fraction, the numerator and denominator must be represented as a product.

You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned. To do this, decide a few examples(needs to be factorized):

Solutions:

3. Reducing a fraction.

Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?

That's the beauty of downsizing.

It's simple:

If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.

This rule follows from the basic property of a fraction:

That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).

To reduce a fraction you need:

1) numerator and denominator factorize

2) if the numerator and denominator contain common factors, they can be crossed out.

The principle, I think, is clear?

I would like to draw your attention to one thing typical mistake when contracting. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.

No abbreviations if the numerator or denominator is a sum.

For example: we need to simplify.

Some people do this: which is absolutely wrong.

Another example: reduce.

The “smartest” will do this: .

Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.

But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.

Here's another example: .

This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:

You can immediately divide it into:

To avoid such mistakes, remember easy way how to determine whether an expression is factorized:

The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation. That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized). If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).

To consolidate, solve a few yourself examples:

Answers:

1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:

The first step should be factorization:

4. Adding and subtracting fractions. Reducing fractions to a common denominator.

Adding and subtracting ordinary fractions is a familiar operation: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators. Let's remember:

Answers:

1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:

2. Here the common denominator is:

3. Here, first of all, we convert mixed fractions into improper ones, and then according to the usual scheme:

It's a completely different matter if the fractions contain letters, for example:

Let's start with something simple:

a) Denominators do not contain letters

Here everything is the same as with ordinary numerical fractions: we find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:

Now in the numerator you can give similar ones, if any, and factor them:

Try it yourself:

b) Denominators contain letters

Let's remember the principle of finding a common denominator without letters:

· first of all, we determine the common factors;

· then we write out all the common factors one at a time;

· and multiply them by all other non-common factors.

To determine the common factors of the denominators, we first factor them into prime factors:

Let us emphasize the common factors:

Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:

This is the common denominator.

Let's get back to the letters. The denominators are given in exactly the same way:

· factor the denominators;

· determine common (identical) factors;

· write out all common factors once;

· multiply them by all other non-common factors.

So, in order:

1) factor the denominators:

2) determine common (identical) factors:

3) write out all the common factors once and multiply them by all other (non-underlined) factors:

So there's a common denominator here. The first fraction must be multiplied by, the second - by:

By the way, there is one trick:

For example: .

We see the same factors in the denominators, only all with different indicators. The common denominator will be:

to a degree

to a degree

to a degree

to a degree.

Let's complicate the task:

How to make fractions have the same denominator?

Let's remember the basic property of a fraction:

Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!

See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?

So, another unshakable rule:

When you reduce fractions to a common denominator, use only the multiplication operation!

But what do you need to multiply by to get?

So multiply by. And multiply by:

We will call expressions that cannot be factorized “elementary factors.” For example, - this is an elementary factor. - Same. But no: it can be factorized.

What about the expression? Is it elementary?

No, because it can be factorized:

(you already read about factorization in the topic “”).

So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will deal with them in the same way.

We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).

The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:

Another example:

Solution:

Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:

Great! Then:

Another example:

Solution:

As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:

It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:

So let's write:

That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.

Now let's bring it to a common denominator:

Got it? Let's check it now.

Tasks for independent solution:

Answers:

Here we need to remember one more thing - the difference of cubes:

Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .

A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their double product. The partial square of the sum is one of the factors in the expansion of the difference of cubes:

What to do if there are already three fractions?

Yes, the same thing! First of all, let's make sure that maximum amount the factors in the denominators were the same:

Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.

We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:

Hmm... It’s clear what to do with fractions. But what about the two?

It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:

Exactly what is needed!

5. Multiplication and division of fractions.

Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:

Did you count?

It should work.

So, let me remind you.

The first step is to calculate the degree.

The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.

And finally, we perform addition and subtraction. Again, in any order.

But: the expression in brackets is evaluated out of turn!

If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.

What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):

Okay, it's all simple.

But this is not the same as an expression with letters?

No, it's the same! Only instead of arithmetic operations you need to do algebraic ones, that is, the actions described in previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.

Usually our goal is to represent the expression as a product or quotient.

For example:

Let's simplify the expression.

1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:

It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).

2) We get:

Multiplying fractions: what could be simpler.

3) Now you can shorten:

OK it's all over Now. Nothing complicated, right?

Another example:

Simplify the expression.

First, try to solve it yourself, and only then look at the solution.

First of all, let's determine the order of actions. First, let's add the fractions in parentheses, so instead of two fractions we get one. Then we will do division of fractions. Well, let's add the result with the last fraction. I will number the steps schematically:

Now I’ll show you the process, tinting the current action in red:

Finally, I will give you two useful tips:

1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.

2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.

Here are some tasks for you to solve on your own:

And what was promised at the very beginning:

Solutions (brief):

If you have coped with at least the first three examples, then you have mastered the topic.

Now on to learning!

CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS

Basic simplification operations:

• Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
• Factorization: putting the common factor out of brackets, applying it, etc.
• Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
  1) numerator and denominator factorize
  2) if the numerator and denominator have common factors, they can be crossed out.

  IMPORTANT: only multipliers can be reduced!

• Adding and subtracting fractions:
  ;
• Multiplying and dividing fractions:
  ;

In this article we will go into detail about reduction algebraic fractions . First, let's figure out what is meant by the term “reduction of an algebraic fraction” and find out whether an algebraic fraction is always reducible. Below we present a rule that allows this transformation to be carried out. Finally, we will consider solutions to typical examples that will allow us to understand all the intricacies of the process.

Page navigation.

What does it mean to reduce an algebraic fraction?

While studying, we talked about their reduction. we called dividing its numerator and denominator by a common factor. For example, the common fraction 30/54 can be reduced by 6 (that is, its numerator and denominator divided by 6), which leads us to the fraction 5/9.

By reducing an algebraic fraction we mean a similar action. Reduce an algebraic fraction- this means dividing its numerator and denominator by a common factor. But if the common factor of the numerator and denominator of an ordinary fraction can only be a number, then the common factor of the numerator and denominator of an algebraic fraction can be a polynomial, in particular, a monomial or number.

For example, an algebraic fraction can be reduced by the number 3, giving the fraction . It is also possible to perform a contraction to the variable x, resulting in the expression . The original algebraic fraction can be reduced by the monomial 3 x, as well as by 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 to obtain a fraction more simple type, V best case scenario– irreducible fraction.

Can any algebraic fraction be reduced?

We know that ordinary fractions are divided into . Irreducible fractions do not have common factors in the numerator and denominator other than one, and therefore cannot be reduced.

Algebraic fractions may or may not have common factors in the numerator and denominator. If there are common factors, it is possible to reduce an algebraic fraction. If there are no common factors, then simplifying an algebraic fraction by reducing it is impossible.

In general, according to appearance algebraic fraction, it is quite difficult to determine whether it can be reduced. Of course, in some cases the common factors of the numerator and denominator are obvious. For example, it is clearly seen that the numerator and denominator of an algebraic fraction have a common factor 3. It is also easy to notice that an algebraic fraction can be reduced by x, by y, or directly by x·y. But much more often, the common factor of the numerator and denominator of an algebraic fraction is not immediately visible, and even more often, it simply does not exist. For example, it is possible to reduce a fraction by x−1, but this common factor is not clearly present in the notation. And an algebraic fraction it is impossible to reduce, since its numerator and denominator do not have common factors.

In general, the question of the reducibility of an algebraic fraction is very difficult. And sometimes it is easier to solve a problem by working with an algebraic fraction in its original form than to find out whether this fraction can be reduced first. But there are still transformations that in some cases make it possible, with relatively little effort, to find the common factors of the numerator and denominator, if any, or to conclude that the original algebraic fraction is irreducible. This information will be disclosed in the next paragraph.

Rule for reducing algebraic fractions

The information from the previous paragraphs allows you to naturally perceive the following rule for reducing algebraic fractions, which consists of two steps:

• first, the common factors of the numerator and denominator of the original fraction are found;
• if there are any, then a reduction is made by these factors.

The indicated steps of the announced rule need clarification.

The most convenient way to find common ones is to factor the polynomials in the numerator and denominator of the original algebraic fraction. In this case, the common factors of the numerator and denominator immediately become visible, or it becomes clear that there are no common factors.

If there are no common factors, then we can conclude that the algebraic fraction is irreducible. If common factors are found, then in the second step they are reduced. The result is a new fraction of a simpler form.

The rule for reducing algebraic fractions is based on the basic property of an algebraic fraction, which is expressed by the equality, where a, b and c are some polynomials, and b and c are non-zero. At the first step, the original algebraic fraction is reduced to the form from which the common factor c becomes visible, and at the second step the reduction is performed - the transition to the fraction.

Let's move on to solving examples using this rule. On them we will analyze all the possible nuances that arise when factoring the numerator and denominator of an algebraic fraction into factors and subsequent reduction.

Typical examples

First, we need to talk about reducing algebraic fractions whose numerator and denominator are the same. Such fractions are identically equal to one on the entire ODZ of the variables included in it, for example,
and so on.

Now it doesn’t hurt to remember how to reduce ordinary fractions - after all, they are a special case of algebraic fractions. Natural numbers in the numerator and denominator of a common fraction, after which the common factors are canceled (if any). For example, . The product of identical prime factors can be written in the form of powers, and used when abbreviating. In this case, the solution would look like this: , here we divided the numerator and denominator by a common factor 2 2 3. Or, for greater clarity, based on the properties of multiplication and division, the solution is presented in the form.

Absolutely similar principles are used to reduce algebraic fractions, the numerator and denominator of which contain monomials with integer coefficients.

Example.

Cancel an algebraic fraction .

Solution.

You can represent the numerator and denominator of the original algebraic fraction as a product of prime factors and variables, and then carry out the reduction:

But it is more rational to write the solution in the form of an expression with powers:

Answer:

.

As for the reduction of algebraic fractions that have fractional numerical coefficients in the numerator and denominator, you can do two things: 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. We talked about the last transformation in the article bringing an algebraic fraction to a new denominator; it can be carried out due to the basic property of an algebraic fraction. Let's understand this with an example.

Example.

Perform fraction reduction.

Solution.

You can reduce the fraction as follows: .

And it was possible to first get rid of fractional coefficients by multiplying the numerator and denominator by the denominators of these coefficients, that is, by LCM(5, 10)=10. In this case we have .

Answer:

.

We can move on to algebraic fractions general view, in which the numerator and denominator can contain both numbers and monomials, as well as polynomials.

When reducing such 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 a common factor or verify its absence, you need to factor the numerator and denominator of an algebraic fraction.

Example.

Reduce a rational fraction .

Solution.

To do this, factor the polynomials in the numerator and denominator. Let's start by putting it out of brackets: . Obviously, the expressions in parentheses can be transformed using

Division and the numerator and denominator of the fraction on their common divisor, different from one, is called reducing a fraction.

To reduce a common fraction, you need to divide its numerator and denominator by the same natural number.

This number is the greatest common divisor of the numerator and denominator of the given fraction.

The following are possible decision recording forms Examples for reducing common fractions.

The student has the right to choose any form of recording.

Examples. Simplify fractions.

Reduce the fraction by 3 (divide the numerator by 3;

divide the denominator by 3).

Reduce the fraction by 7.

We perform the indicated actions in the numerator and denominator of the fraction.

The resulting fraction is reduced by 5.

Let's reduce this fraction 4) on 5·7³- the greatest common divisor (GCD) of the numerator and denominator, which consists of the common factors of the numerator and denominator, taken to the power with the smallest exponent.

Let's factor the numerator and denominator of this fraction into prime factors.

We get: 756=2²·3³·7 And 1176=2³·3·7².

Determine the GCD (greatest common divisor) of the numerator and denominator of the fraction 5) .

This is the product of common factors taken with the lowest exponents.

gcd(756, 1176)= 2²·3·7.

We divide the numerator and denominator of this fraction by their gcd, i.e. by 2²·3·7 we get an irreducible fraction 9/14 .

Or it was possible to write the decomposition of the numerator and denominator in the form of a product of prime factors, without using the concept of power, and then reduce the fraction by crossing out the same factors in the numerator and denominator. When there are no identical factors left, we multiply the remaining factors separately in the numerator and separately in the denominator and write out the resulting fraction 9/14 .

And finally, it was possible to reduce this fraction 5) gradually, applying signs of dividing numbers to both the numerator and denominator of the fraction. We reason like this: numbers 756 And 1176 end in an even number, which means both are divisible by 2 . We reduce the fraction by 2 . The numerator and denominator of the new fraction are numbers 378 And 588 also divided into 2 . We reduce the fraction by 2 . We notice that the number 294 - even, and 189 is odd, and reduction by 2 is no longer possible. Let's check the divisibility of numbers 189 And 294 on 3 .

(1+8+9)=18 is divisible by 3 and (2+9+4)=15 is divisible by 3, hence the numbers themselves 189 And 294 are divided into 3 . We reduce the fraction by 3 . Further, 63 is divisible by 3 and 98 - No. Let's look at other prime factors. Both numbers are divisible by 7 . We reduce the fraction by 7 and we get the irreducible fraction 9/14 .