In this article, we will analyze what the main property of a fraction is, formulate it, give a proof and a good example. Then we will consider how to apply the basic property of a fraction when performing the actions of reducing fractions and bringing fractions to a new denominator.
All common fractions possess the most important property, which we call the main property of the fraction, and it sounds like this:
Definition 1
If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then the result will be a fraction equal to the given one.
Let's represent the main property of a fraction in the form of equality. For natural numbers a , b and m the equalities will be valid:
a m b m = a b and a: m b: m = a b
Consider the proof of the main property of a fraction. Based on the properties of multiplication of natural numbers and the properties of division of natural numbers, we write the equalities: (a · m) · b = (b · m) · a and (a: m) · b = (b: m) · a. So the fractions a m b m and a b , as well as a: m b: m and a b are equal by the definition of equality of fractions.
Let's look at an example that graphically illustrates the main property of a fraction.
Example 1
Let's say we have a square divided into 9 "big" parts-squares. Each "big" square is divided into 4 smaller ones. It is possible to say that the given square is divided into 4 9 = 36 "small" squares. Highlight 5 "large" squares with color. In this case, 4 · 5 = 20 "small" squares will be colored. Let's show a picture demonstrating our actions:
The colored part is 59 of the original figure or 2036 which is the same. Thus, the fractions 5 9 and 20 36 are equal: 5 9 = 20 36 or 20 36 = 5 9 .
These equalities, as well as the equalities 20 = 4 5, 36 = 4 9, 20: 4 = 5 and 36: 4 = 9, make it possible to conclude that 5 9 = 5 4 9 4 and 20 36 = 20 4 36 4 .
To consolidate the theory, we will analyze the solution of an example.
Example 2
It is given that the numerator and denominator of some ordinary fraction were multiplied by 47, after which these numerator and denominator were divided by 3. Is the resulting fraction equal to the given one?
Solution
Based on the basic property of a fraction, we can say that multiplying the numerator and denominator of a given fraction by a natural number 47 will result in a fraction equal to the original one. We can assert the same thing by dividing further by 3. Ultimately, we will get a fraction equal to the given one.
Answer: Yes, the resulting fraction will be equal to the original.
Application of the basic property of a fraction
The main property is used when you need to bring fractions to a new denominator and when reducing fractions.
Reducing a fraction to a new denominator is the act of replacing a given fraction with a fraction equal to it, but with a larger numerator and denominator. To bring a fraction to a new denominator, you need to multiply the numerator and denominator of the fraction by the required natural number. Operations with ordinary fractions would be impossible without a way to bring fractions to a new denominator.
Definition 2
Fraction reduction- the action of the transition to a new fraction equal to the given one, but with a smaller numerator and denominator. To reduce a fraction, you need to divide the numerator and denominator of the fraction by the same necessary natural number, which will be called common divisor.
There are cases when there is no such common divisor, then they say that the original fraction is irreducible or cannot be reduced. In particular, reducing a fraction by using the greatest common factor will make the fraction irreducible.
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Fractions
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
Fractions in high school are not very annoying. For the time being. Until you come across exponents with rational exponents and logarithms. And there…. You press, you press the calculator, and it shows all the full scoreboard of some numbers. You have to think with your head, like in the third grade.
Let's deal with fractions, finally! Well, how much can you get confused in them!? Moreover, it is all simple and logical. So, what are fractions?
Types of fractions. Transformations.
Fractions are of three types.
1. Common fractions , For example:
Sometimes, instead of a horizontal line, they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens ...), tell yourself the phrase with the expression: " Zzzzz remember! Zzzzz denominator - out zzzz u!" Look, everything will be remembered.)
A dash, which is horizontal, which is oblique, means division top number (numerator) to bottom number (denominator). And that's it! Instead of a dash, it is quite possible to put a division sign - two dots.
When the division is possible entirely, it must be done. So, instead of the fraction "32/8" it is much more pleasant to write the number "4". Those. 32 is simply divided by 8.
32/8 = 32: 8 = 4
I'm not talking about the fraction "4/1". Which is also just "4". And if it doesn’t divide completely, we leave it as a fraction. Sometimes you have to do the reverse. Make a fraction from a whole number. But more on that later.
2. Decimals , For example:
It is in this form that it will be necessary to write down the answers to tasks "B".
3. mixed numbers , For example:
Mixed numbers are practically not used in high school. In order to work with them, they must be converted to ordinary fractions. But you definitely need to know how to do it! And then such a number will come across in the puzzle and hang ... From scratch. But we remember this procedure! A little lower.
Most versatile common fractions. Let's start with them. By the way, if there are all sorts of logarithms, sines and other letters in the fraction, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!
Basic property of a fraction.
So let's go! First of all, I will surprise you. The whole variety of fraction transformations is provided by a single property! That's what it's called basic property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction will not change. Those:
It is clear that you can write further, until you are blue in the face. Do not let sines and logarithms confuse you, we will deal with them further. The main thing to understand is that all these various expressions are the same fraction . 2/3.
And we need it, all these transformations? And how! Now you will see for yourself. First, let's use the basic property of a fraction for fraction abbreviations. It would seem that the thing is elementary. We divide the numerator and denominator by the same number and that's it! It's impossible to go wrong! But... man is a creative being. You can make mistakes everywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.
How to reduce fractions correctly and quickly without doing unnecessary work can be found in special Section 555.
A normal student does not bother dividing the numerator and denominator by the same number (or expression)! He just crosses out everything the same from above and below! This is where it hides typical mistake, blooper if you want.
For example, you need to simplify the expression:
There is nothing to think about, we cross out the letter "a" from above and the deuce from below! We get:
Everything is correct. But really you shared the whole numerator and the whole denominator "a". If you are used to just cross out, then, in a hurry, you can cross out the "a" in the expression
and get again
Which would be categorically wrong. Because here the whole numerator on "a" already not shared! This fraction cannot be reduced. By the way, such an abbreviation is, um ... a serious challenge to the teacher. This is not forgiven! Remember? When reducing, it is necessary to divide the whole numerator and the whole denominator!
Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. And how to work with her now? Without a calculator? Multiply, say, add, square!? And if you are not too lazy, but carefully reduce by five, and even by five, and even ... while it is being reduced, in short. We get 3/8! Much nicer, right?
The basic property of a fraction allows you to convert ordinary fractions to decimals and vice versa without calculator! This is important for the exam, right?
How to convert fractions from one form to another.
It's easy with decimals. As it is heard, so it is written! Let's say 0.25. It's zero point, twenty-five hundredths. So we write: 25/100. We reduce (divide the numerator and denominator by 25), we get the usual fraction: 1/4. All. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.
What if integers are non-zero? It's OK. Write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three whole, seventeen hundredths. We write 317 in the numerator, and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary Watson! From all of the above, a useful conclusion: any decimal fraction can be converted to a common fraction .
But the reverse conversion, ordinary to decimal, some cannot do without a calculator. And it is necessary! How will you write down the answer on the exam!? We carefully read and master this process.
What is a decimal fraction? She has in the denominator Always is worth 10 or 100 or 1000 or 10000 and so on. If your usual fraction has such a denominator, there is no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. And if in the answer to the task of section "B" it turned out 1/2? What will we write in response? Decimals are required...
We remember basic property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. For anyone, by the way! Except zero, of course. Let's use this feature to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But, then the numerator must also be multiplied by 5. This is already mathematics demands! We get 1/2 \u003d 1x5 / 2x5 \u003d 5/10 \u003d 0.5. That's all.
However, all sorts of denominators come across. For example, the fraction 3/16 will fall. Try it, figure out what to multiply 16 by to get 100, or 1000... Doesn't work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide in a corner, on a piece of paper, as they taught in elementary grades. We get 0.1875.
And there are some very bad denominators. For example, the fraction 1/3 cannot be turned into a good decimal. Both on a calculator and on a piece of paper, we get 0.3333333 ... This means that 1/3 into an exact decimal fraction does not translate. Just like 1/7, 5/6 and so on. Many of them are untranslatable. Hence another useful conclusion. Not every common fraction converts to a decimal. !
By the way, this helpful information for self-test. In section "B" in response, you need to write down a decimal fraction. And you got, for example, 4/3. This fraction is not converted to decimal. This means that somewhere along the way you made a mistake! Come back, check the solution.
So, with ordinary and decimal fractions sorted out. It remains to deal with mixed numbers. To work with them, they all need to be converted to ordinary fractions. How to do it? You can catch a sixth grader and ask him. But not always a sixth grader will be at hand ... We will have to do it ourselves. It is not difficult. Multiply the denominator of the fractional part by the integer part and add the numerator of the fractional part. This will be the numerator of a common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but it's actually quite simple. Let's see an example.
Let in the problem you saw with horror the number:
Calmly, without panic, we understand. The whole part is 1. One. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of an ordinary fraction. That's all. It looks even simpler in mathematical notation:
Clearly? Then secure your success! Convert to common fractions. You should get 10/7, 7/2, 23/10 and 21/4.
The reverse operation - converting an improper fraction into a mixed number - is rarely required in high school. Well, if... And if you - not in high school - you can look into the special Section 555. In the same place, by the way, you will learn about improper fractions.
Well, almost everything. You remembered the types of fractions and understood How convert them from one type to another. The question remains: For what do it? Where and when to apply this deep knowledge?
I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers, we convert everything into ordinary fractions. It can always be done. Well, if something like 0.8 + 0.3 is written, then we think so, without any translation. Why do we need extra work? We choose the solution that is convenient us !
If the task is entirely decimals, but um... some evil ones, go to the ordinary ones, try it! Look, everything will be fine. For example, you have to square the number 0.125. Not so easy if you have not lost the habit of the calculator! Not only do you need to multiply the numbers in a column, but also think about where to insert the comma! It certainly doesn't work in my mind! And if you go to an ordinary fraction?
0.125 = 125/1000. We reduce by 5 (this is for starters). We get 25/200. Once again on 5. We get 5/40. Oh, it's shrinking! Back to 5! We get 1/8. Easily square (in your mind!) and get 1/64. All!
Let's summarize this lesson.
1. There are three types of fractions. Ordinary, decimal and mixed numbers.
2. Decimals and mixed numbers Always can be converted to common fractions. Reverse Translation not always available.
3. The choice of the type of fractions for working with the task depends on this very task. In the presence of different types fractions in one task, the most reliable thing is to switch to ordinary fractions.
Now you can practice. First, convert these decimal fractions to ordinary ones:
3,8; 0,75; 0,15; 1,4; 0,725; 0,012
You should get answers like this (in a mess!):
On this we will finish. In this lesson, we brushed up on the key points on fractions. It happens, however, that there is nothing special to refresh ...) If someone has completely forgotten, or has not mastered it yet ... Those can go to a special Section 555. All the basics are detailed there. Many suddenly understand everything are starting. And they solve fractions on the fly).
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)
you can get acquainted with functions and derivatives.
Speaking of mathematics, one cannot help but remember fractions. Their study is given a lot of attention and time. Remember how many examples you had to solve in order to learn certain rules for working with fractions, how you memorized and applied the main property of a fraction. How many nerves were spent to find a common denominator, especially if there were more than two terms in the examples!
Let's remember what it is, and refresh our memory a little about the basic information and rules for working with fractions.
Definition of fractions
Let's start with the most important thing - definitions. A fraction is a number that consists of one or more unit parts. A fractional number is written as two numbers separated by a horizontal or slash. In this case, the upper (or first) is called the numerator, and the lower (second) is called the denominator.
It is worth noting that the denominator shows how many parts the unit is divided into, and the numerator shows the number of shares or parts taken. Often fractions, if they are correct, are less than one.
Now let's look at the properties of these numbers and the basic rules that are used when working with them. But before we analyze such a concept as "the main property of a rational fraction", let's talk about the types of fractions and their features.
What are fractions
There are several types of such numbers. First of all, these are ordinary and decimal. The first are the type of record already indicated by us using a horizontal or slash. The second type of fractions is indicated using the so-called positional notation, when the integer part of the number is indicated first, and then, after the decimal point, the fractional part is indicated.
It is worth noting here that in mathematics both decimal and ordinary fractions are used equally. The main property of the fraction is valid only for the second option. In addition, in ordinary fractions, right and wrong numbers are distinguished. For the former, the numerator is always less than the denominator. Note also that such a fraction is less than unity. In an improper fraction, on the contrary, the numerator is greater than the denominator, and it itself is greater than one. In this case, an integer can be extracted from it. In this article, we will consider only ordinary fractions.
Fraction properties
Any phenomenon, chemical, physical or mathematical, has its own characteristics and properties. Fractional numbers are no exception. They have one important feature, with the help of which it is possible to carry out certain operations on them. What is the main property of a fraction? The rule says that if its numerator and denominator are multiplied or divided by the same rational number, we get a new fraction, the value of which will be equal to the value of the original. That is, multiplying the two parts of the fractional number 3/6 by 2, we get a new fraction 6/12, while they will be equal.
Based on this property, you can reduce fractions, as well as select common denominators for a particular pair of numbers.
Operations
Although fractions seem more complex to us, they can also perform basic mathematical operations, such as addition and subtraction, multiplication and division. In addition, there is such a specific action as the reduction of fractions. Naturally, each of these actions is performed according to certain rules. Knowing these laws makes it easier to work with fractions, making it easier and more interesting. That is why further we will consider the basic rules and the algorithm of actions when working with such numbers.
But before we talk about such mathematical operations as addition and subtraction, we will analyze such an operation as reduction to a common denominator. This is where the knowledge of what basic property of a fraction exists will come in handy.
Common denominator
In order to reduce a number to a common denominator, you first need to find the least common multiple of the two denominators. That is smallest number, which is simultaneously divisible by both denominators without a remainder. The easiest way to find the LCM (least common multiple) is to write in a line for one denominator, then for the second and find a matching number among them. In the event that the LCM is not found, that is, these numbers do not have a common multiple, they should be multiplied, and the resulting value should be considered as the LCM.
So, we have found the LCM, now we need to find an additional multiplier. To do this, you need to alternately divide the LCM into denominators of fractions and write down the resulting number over each of them. Next, multiply the numerator and denominator by the resulting additional factor and write the results as a new fraction. If you doubt that the number you received is equal to the previous one, remember the main property of the fraction.
Addition
Now let's go directly to mathematical operations on fractional numbers. Let's start with the simplest. There are several options for adding fractions. In the first case, both numbers have the same denominator. In this case, it remains only to add the numerators together. But the denominator does not change. For example, 1/5 + 3/5 = 4/5.
If the fractions have different denominators, they should be reduced to a common one and only then the addition should be performed. How to do this, we have discussed with you a little higher. In this situation, the main property of the fraction will come in handy. The rule will allow you to bring the numbers to a common denominator. The value will not change in any way.
Alternatively, it may happen that the fraction is mixed. Then you should first add together the whole parts, and then the fractional ones.
Multiplication
It does not require any tricks, and in order to perform this action, it is not necessary to know the basic property of the fraction. It is enough to first multiply the numerators and denominators together. In this case, the product of the numerators will become the new numerator, and the product of the denominators will become the new denominator. As you can see, nothing complicated.
The only thing that is required of you is knowledge of the multiplication table, as well as attentiveness. In addition, after receiving the result, you should definitely check whether this number can be reduced or not. We will talk about how to reduce fractions a little later.
Subtraction
Performing should be guided by the same rules as when adding. So, in numbers with the same denominator, it is enough to subtract the numerator of the subtrahend from the numerator of the minuend. In the event that the fractions have different denominators, you should bring them to a common one and then perform this operation. As with the analogous addition case, you will need to use the basic property of an algebraic fraction, as well as skills in finding the LCM and common factors for fractions.
Division
And the last, most interesting operation when working with such numbers is division. It is quite simple and does not cause any particular difficulties even for those who do not understand how to work with fractions, especially to perform addition and subtraction operations. When dividing, such a rule applies as multiplication by a reciprocal fraction. The main property of a fraction, as in the case of multiplication, will not be used for this operation. Let's take a closer look.
When dividing numbers, the dividend remains unchanged. The divisor is reversed, i.e. the numerator and denominator are reversed. After that, the numbers are multiplied with each other.
Reduction
So, we have already examined the definition and structure of fractions, their types, the rules of operations on given numbers, and found out the main property of an algebraic fraction. Now let's talk about such an operation as reduction. Reducing a fraction is the process of converting it - dividing the numerator and denominator by the same number. Thus, the fraction is reduced without changing its properties.
Usually, when performing a mathematical operation, you should carefully look at the result obtained in the end and find out whether it is possible to reduce the resulting fraction or not. Remember that in final result a fractional number that does not require reduction is always written.
Other operations
Finally, we note that we have listed far from all operations on fractional numbers, mentioning only the most famous and necessary. Fractions can also be compared, converted to decimals, and vice versa. But in this article we did not consider these operations, since in mathematics they are carried out much less frequently than those that we have given above.
conclusions
We talked about fractional numbers and operations with them. We also analyzed the main property. But we note that all these issues were considered by us in passing. We have given only the most well-known and used rules, we have given the most important, in our opinion, advice.
This article is intended to refresh the information you have forgotten about fractions, rather than give new information and "fill" your head with endless rules and formulas, which, most likely, will not be useful to you.
We hope that the material presented in the article simply and concisely has become useful to you.
When studying ordinary fractions, we encounter the concepts of the main property of a fraction. A simplified form is necessary for solving examples with ordinary fractions. This article involves the consideration of algebraic fractions and the application to them of the main property, which will be formulated with examples of the scope of its application.
Formulation and rationale
The main property of a fraction has a formulation of the form:
Definition 1
When simultaneously multiplying or dividing the numerator and denominator by the same number, the value of the fraction remains unchanged.
That is, we get that a · m b · m = a b and a: m b: m = a b are equivalent, where a b = a · m b · m and a b = a: m b: m are considered valid. The values a , b , m are some natural numbers.
Dividing the numerator and denominator by a number can be represented as a · m b · m = a b . This is similar to solving example 8 12 = 8: 4 12: 4 = 2 3 . When dividing, an equality of the form a is used: m b: m \u003d a b, then 8 12 \u003d 2 4 2 4 \u003d 2 3. It can also be represented as a m b m \u003d a b, that is, 8 12 \u003d 2 4 3 4 \u003d 2 3.
That is, the main property of the fraction a · m b · m = a b and a b = a · m b · m will be considered in detail in contrast to a: m b: m = a b and a b = a: m b: m .
If the numerator and denominator contain real numbers, then the property is applicable. We must first prove the validity of the written inequality for all numbers. That is, prove the existence of a · m b · m = a b for all real a , b , m , where b and m are non-zero values to avoid division by zero.
Proof 1
Let a fraction of the form a b be considered part of the record z, in other words, a b = z, then it is necessary to prove that a · m b · m corresponds to z, that is, to prove a · m b · m = z. Then this will allow us to prove the existence of the equality a · m b · m = a b .
The fraction bar means the division sign. Applying the relationship with multiplication and division, we get that from a b = z after transformation we get a = b · z . According to the properties of numerical inequalities, both parts of the inequality should be multiplied by a number other than zero. Then we multiply by the number m, we get that a · m = (b · z) · m . By property, we have the right to write the expression in the form a · m = (b · m) · z . Hence, it follows from the definition that a b = z . That's all the proof of the expression a · m b · m = a b .
Equalities of the form a · m b · m = a b and a b = a · m b · m make sense when instead of a , b , m there are polynomials, and instead of b and m they are non-zero.
The main property of an algebraic fraction: when you simultaneously multiply the numerator and denominator by the same number, we get an identically equal to the original expression.
The property is considered fair, since operations with polynomials correspond to operations with numbers.
Example 1
Consider the example of the fraction 3 · x x 2 - x y + 4 · y 3 . It is possible to convert to the form 3 x (x 2 + 2 x y) (x 2 - x y + 4 y 3) (x 2 + 2 x y).
Multiplication by the polynomial x 2 + 2 · x · y was performed. In the same way, the main property helps to get rid of x 2, which is present in the fraction of the form 5 x 2 (x + 1) x 2 (x 3 + 3) given by the condition, to the form 5 x + 5 x 3 + 3. This is called simplification.
The main property can be written as expressions a · m b · m = a b and a b = a · m b · m , when a , b , m are polynomials or ordinary variables, and b and m must be non-zero.
Scope of application of the main property of an algebraic fraction
The use of the main property is relevant for reduction to a new denominator or when reducing a fraction.
Definition 2
Reduction to a common denominator is the multiplication of the numerator and denominator by a similar polynomial to obtain a new one. The resulting fraction is equal to the original.
That is, a fraction of the form x + y x 2 + 1 (x + 1) x 2 + 1 when multiplied by x 2 + 1 and reduced to a common denominator (x + 1) (x 2 + 1) will get the form x 3 + x + x 2 y + y x 3 + x + x 2 + 1 .
After performing operations with polynomials, we get that the algebraic fraction is converted to x 3 + x + x 2 y + y x 3 + x + x 2 + 1.
Reduction to a common denominator is also performed when adding or subtracting fractions. If fractional coefficients are given, then it is first necessary to make a simplification, which will simplify the form and the very finding of the common denominator. For example, 2 5 x y - 2 x + 1 2 = 10 2 5 x y - 2 10 x + 1 2 = 4 x y - 20 10 x + 5.
The application of the property when reducing fractions is performed in 2 stages: decomposing the numerator and denominator into factors to find the common m, then making the transition to the form of the fraction a b , based on the equality of the form a · m b · m = a b .
If a fraction of the form 4 x 3 - x y 16 x 4 - y 2 after decomposition is converted to x (4 x 2 - y) 4 x 2 - y 4 x 2 + y, it is obvious that the general the multiplier is the polynomial 4 · x 2 − y . Then it will be possible to reduce the fraction according to its main property. We get that
x (4 x 2 - y) 4 x 2 - y 4 x 2 + y = x 4 x 2 + y. The fraction is simplified, then when substituting the values, it will be necessary to perform much less actions than when substituting into the original one.
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Possess basic property of a fraction:
Remark 1
If the numerator and denominator of a fraction are multiplied or divided by the same natural number, then as a result we get a fraction equal to the original one:
$\frac(a\cdot n)(b\cdot n)=\frac(a)(b)$
$\frac(a\div n)(b\div n)=\frac(a)(b)$
Example 1
Let a square divided into $4$ equal parts be given. If $2$ of $4$ parts are shaded, we get the shaded $\frac(2)(4)$ of the entire square. If you look at this square, it is obvious that exactly half of it is shaded, i.e. $(1)(2)$. Thus, we get $\frac(2)(4)=\frac(1)(2)$. Let's factorize the numbers $2$ and $4$:
Substitute these expansions into equality:
$\frac(1)(2)=\frac(2)(4)$,
$\frac(1)(2)=\frac(1\cdot 2)(2\cdot 2)$,
$\frac(1)(2)=\frac(2\div 2)(4\div 2)$.
Example 2
Is it possible to get an equal fraction if both the numerator and denominator of the given fraction are multiplied by $18$ and then divided by $3$?
Solution.
Let some ordinary fraction $\frac(a)(b)$ be given. By condition, the numerator and denominator of this fraction were multiplied by $ 18 $, we got:
$\frac(a\cdot 18)(b\cdot 18)$
$\frac(a\cdot 18)(b\cdot 18)=\frac(a)(b)$
$\frac(a\div 3)(b\div 3)$
According to the basic property of a fraction:
$\frac(a\div 3)(b\div 3)=\frac(a)(b)$
Thus, the resulting fraction is equal to the original.
Answer: You can get a fraction equal to the original.
Application of the basic property of a fraction
The main property of a fraction is most often used for:
- converting fractions to a new denominator:
- fraction abbreviations.
Bringing a fraction to a new denominator- replacement of a given fraction with a fraction that will be equal to it, but have a larger numerator and a larger denominator. To do this, the numerator and denominator of the fraction are multiplied by the same natural number, as a result of which, according to the main property of the fraction, a fraction is obtained that is equal to the original one, but with a larger numerator and denominator.
Fraction reduction- replacement of a given fraction with a fraction that will be equal to it, but have a smaller numerator and a smaller denominator. To do this, the numerator and denominator of the fraction are divided by a positive common divisor numerator and denominator other than zero, as a result of which, according to the main property, the fractions receive a fraction equal to the original one, but with a smaller numerator and denominator.
If we divide (reduce) the numerator and denominator by their GCD, then the result is irreducible form of the original fraction.
Fraction reduction
As you know, ordinary fractions are divisible by contractible And irreducible.
To reduce a fraction, you need to divide both the numerator and denominator of the fraction by their positive common divisor, not zero. When reducing the fraction, a new fraction is obtained with a smaller numerator and denominator, which, according to the main property of the fraction, is equal to the original one.
Example 3
Reduce the fraction $\frac(15)(25)$.
Solution.
Reduce the fraction by $5$ (divide its numerator and denominator by $5$):
$\frac(15)(25)=\frac(15\div 5)(25\div 5)=\frac(3)(5)$
Answer: $\frac(15)(25)=\frac(3)(5)$.
Getting an irreducible fraction
Most often, a fraction is reduced to obtain an irreducible fraction equal to the original reducible fraction. This result can be achieved by dividing both the numerator and denominator of the original fraction by their GCD.
$\frac(a\div gcd (a,b))(b\div gcd (a,b))$ is an irreducible fraction, because according to the properties of GCD, the numerator and denominator of a given fraction are coprime numbers.
GCD(a,b) is the largest number by which both the numerator and denominator of the fraction $\frac(a)(b)$ can be divided. Thus, to reduce a fraction to an irreducible form, it is necessary to divide its numerator and denominator by their gcd.
Remark 2
Fraction reduction rule: 1. Find the GCD of two numbers that are in the numerator and denominator of the fraction. 2. Perform the division of the numerator and denominator of the fraction by the found GCD.
Example 4
Reduce the fraction $6/36$ to an irreducible form.
Solution.
Let's reduce this fraction by GCD$(6,36)=6$, because $36\div 6=6$. We get:
$\frac(6)(36)=\frac(6\div 6)(36\div 6)=\frac(1)(6)$
Answer: $\frac(6)(36)=\frac(1)(6)$.
In practice, the phrase "reduce a fraction" implies that you need to reduce the fraction to an irreducible form.
