In the article we will show how to solve the fraci On simple understandable examples. Tell me what fraction and consider decision fractions!

Concept drobi. It is introduced into the mathematics course since the 6th grade of high school.

The fractions are: ± x / y, where y is a denominator, it reports to how many parts divided the whole, and X is a numerator, it reports how many such parts took. For clarity, take an example with a cake:

In the first case, the cake was cut equally and took one half, i.e. 1/2. In the second case, the cake was cut into 7 parts, of which 4 parts took, i.e. 4/7.

If a part of the division of one number to another is not an integer, it is written in the form of a fraction.

For example, expression 4: 2 \u003d 2 gives an integer, but 4: 7 is not divided by a focus, so such an expression is recorded in the form of fractions 4/7.

In other words fraction - This is an expression that denotes the division of two numbers or expressions, and which is written with a fractional feature.

If the numerator is less than the denominator - the fraction is correct, if on the contrary - incorrect. The fraction may include an integer.

For example, 5 whole 3/4.

This record means that in order to obtain a whole 6 lacks one part of four.

If you want to remember, how to solve fractions for grade 6you need to understand that decision fractionsMostly, it comes down to understanding several simple things.

• Fraction in essence is the expression of the share. That is, a numerical expression of what part is this value from one whole. For example, the fraction 3/5 expresses that if we shared something for 5 parts and the number of fractions or parts is this whole - three.
• The fraction may be less than 1, for example 1/2 (or in fact half), then it is correct. If the fraction is greater than 1, for example 3/2 (three half or one and a half), then it is incorrect and to simplify the solution, it is better for us to highlight the whole part 3/2 \u003d 1 a whole 1/2.
• The fractions are the same numbers as 1, 3, 10, and even 100, only numbers are not entire and fractional. With them you can perform all the same operations that with numbers. It is not more difficult to consider the fraction, and then on specific examples, we will show it.

How to solve fractions. Examples.

Spectacles apply a variety of arithmetic operations.

Crushing to a common denominator

For example, it is necessary to compare the fractions 3/4 and 4/5.

To solve the task, we first find the smallest common denominator, i.e. The smallest number, which is divided without a residue for each of the signifiers of fractions

The smallest common denominator (4.5) \u003d 20

Then the denominator of both fractions is driven to the smallest common denominator.

Answer: 15/20

Addition and subtraction of fractions

If you need to calculate the amount of two fractions, they first lead to a common denominator, then the numerals are folded, while the denominator will remain unchanged. The difference of fractions is considered as similar, the difference is only that the numerals are deducted.

For example, you need to find the amount of fractions 1/2 and 1/3

Now we find the difference of fractions 1/2 and 1/4

Multiplication and division of fractions

Here the decision frains is simple, everything is quite simple here:

• Multiplication - numerals and denominators of fractions are multiplied with each other;
• Delivery - I first get a fraction, reverse second fraction, i.e. We change its numerator and denominator in places, after which the resulting fraction is changed.

For example:

On this that how to solve the fraci, everything. If you have any questions about solving fractionsSomething is incomprehensible, then write in the comment and we will answer you.

If you are a teacher, it is possible to download a presentation for the elementary school (http://school-box.ru/nachalnaya-po-matematike.html) will be you by the way.

We agree to believe that under "action with fractions" at our lesson will be understood to be actions with ordinary fractions. Ordinary fraction is a fraction with attributes such as a numerator, fractional feature and denominator. This features an ordinary fraction from decimal, which is obtained from the ordinary way of bringing the denominator to the number, multiple 10. The decimal fraction is recorded with a comma separating the whole part from the fractional. We will be talking about actions with ordinary fractions, since it is they who cause the greatest difficulties among students who forgot the fundamentals of this topic traveled in the first half of the school year of mathematics. At the same time, with transformations of expressions in higher mathematics, they are mainly used precisely actions with ordinary fractions. Some cuts of the frains of what stand! Decimal fractions of special difficulties do not cause. So, forward!

Two fractions are called equal, if.

For example, since

The fractions are also equal and (since), and (since).

Obviously, the fractions are equal and. This means that if the numerator and the denominator of this fraction is multiplied or divided into one and the same natural number, then the fraction is equal to this :.

This property is called the main property of the fraction.

The main property of the fraction can be used to change characters in the numerator and denominator of the fraction. If the numerator and denominator of the fraction are multiplied by -1, then we get. This means that the fraction value does not change if simultaneously change the signs in the numerator and the denominator. If you change the sign only in the numerator or only by the denominator, then the fraction will change your sign:

Reducing fractions

Using the main property of the fraction, it is possible to replace this fraction by another fraction equal to this, but with a smaller numerator and denominator. Such a replacement is called the cutting of the fraction.

Let, for example, a fraction is given. Numbers 36 and 48 have the greatest common divisor 12. Then

.

In general, the reduction of the fraction is possible, if the numerator and denominator are not mutually simple numbers. If the numerator and denominator are mutually simple numbers, then the fraction is called non-interpretable.

So, to cut the fraction - this means split the numerator and denominator of the fraction on the general factor. All of the above applies to fractional expressions containing variables.

Example 1. Reduce fraction

Decision. For the decomposition of the numerator on multipliers, submitting previously unrochene - 5 xY. in the form of the amount - 2 xY. - 3xY. , get

To decompose the denominator, we use the formula of the square difference:

As a result

.

Bringing fractions to a common denominator

Let two fractions and. They have different denominators: 5 and 7. Using the main property of the fraction, it is possible to replace these fractions by other equal to them, and such that the fractions obtained will be the same denominators. Multiplying the numerator and denominator of the fraction on 7, we get

Multiplying the numerator and denominator of the fraction on 5, we get

So, fractions are given to a common denominator:

.

But this is not the only solution to the task: for example, these fractions can also be brought to the total denominator 70:

,

and in general, to any denominator who divorced at the same time on 5 and 7.

Consider another example: we give a fraction to the overall denominator. Arguing, as in the previous example, we get

,

.

But in this case, you can bring the fraction to a common denominator, less than the product of the denominators of these fractions. Find the smallest general multiple numbers 24 and 30: NOC (24, 30) \u003d 120.

Since 120: 4 \u003d 5, it is necessary to burn a fraction with a denominator 120, it is necessary to write a numerator, and the denominator is multiplier by 5, this number is called an additional factor. So .

Next, we get 120: 30 \u003d 4. Multiplying the numerator and denominator of the fraction on an additional factor 4, we get .

So, these fractions are shown to the general denominator.

The smallest general multiple denominator of these fractions is the lowest possible common denominator.

For fractional expressions, which include variables, the common denominator is a polynomial, which is divided into the denominator of each fraction.

Example 2. Find a common denominator fraction and.

Decision. The general denominator of these fractions is a polynomial, as it is divided into and on. However, this polynomial is not the only one that can be a common denominator of these fractions. They may also be a polynomial , and polynomial , and polynomial etc. Typically take such a common denominator that any other common denominator is divided into selected without a residue. Such a denominator is called the smallest common denominator.

In our example, the smallest overall denominator is equal. Received:

;

.

We managed to bring the fraction to the smallest general denominator. This happened by multiplying the numerator and denominator of the first fraction on, and the numerator and denominator of the second fraction - on. Polynomials and are called additional multipliers, respectively, for the first and for the second fraction.

Addition and subtraction of fractions

Addition of fractions is determined as follows:

.

For example,

.

If a b. = d. T.

.

This means that in order to add fractions with the same denominator, it is enough to add numerals, and the denominator is left for the same. For example,

.

If the fractions with different denominants are folded, then the fractions usually lead to the smallest common denominator, and then the numerals are folded. For example,

.

Now consider an example of the addition of fractional expressions with variables.

Example 3. Convert to one fraction expression

.

Decision. Find the smallest common denominator. To do this, first decompose the denominators for multipliers.

Calculator fractions Designed to quickly calculate operations with fractions, it will be easy to fold, multiply, divide or deduct.

Modern schoolchildren begin to study fractions already in grade 5, each year the exercises are complicated with them. Mathematical terms and quantities that we learn at school are rarely useful to us in adulthood. However, the fractions, unlike logarithms and degrees, are found in everyday life quite often (distance measurement, weighing goods, etc.). Our calculator is designed for quick operations with fractions.

To begin with, we define what fractions are and what they happen. We call the ratio of one number to another, this is a number consisting of a whole number of units.

Varieties of fractions:

• Ordinary
• Decimal
• Mixed

Example ordinary fractions:

The top value is the numerator, the lower denominator. Dash shows us that the upper number is divided into the lower. Instead of a similar writing format, when the packer is horizontally, you can write differently. You can put an inclined line, for example:

1/2, 3/7, 19/5, 32/8, 10/100, 4/1

Decimal fractions They are the most popular variety of fractions. They consist of a whole part and fractional separated by the comma.

An example of decimal fractions:

0.2, or 6.71 or 0.125

Consist of an integer and fractional part. To find out the meaning of this fraction, you need to fold an integer and fraction.

An example of mixed fractions:

The fraction calculator on our site is able to quickly in the online mode of performing any mathematical transactions:

• Addition
• Subtraction
• Multiplication
• Division

To make the calculation, you need to enter numbers into the field and select the action. The fractions need to fill the numerator and denominator, an integer may not be written (if the fraction is ordinary). Do not forget to click on the "Equal" button.

It is convenient that the calculator immediately provides the process of solving an example with fractions, and not just a ready-made answer. It is thanks to the deployed solution that you can use this material when solving school challenges and for better development of the material passed.

You need to calculate the example:

After entering the indicators in the field form, we get:

To make an independent calculation, enter the data into the form.

Calculator fractions

Enter two fractions:

		+ - * :

Related sections.

Drobi.

Attention!
This topic has additional
Materials in a special section 555.
For those who are strongly "not very ..."
And for those who are "very ...")

The fractions in high schools are not very annoyed. For the time being. So far, do not come up with degrees with rational indicators and logarithms. And here .... You give, you give a calculator, and he all the complete scoreboard does not seem to. I have to think about thinking as in the third grade.

Let's figure out with the fractions finally! Well, how much can you get confused!? Moreover, it's simple and logical. So, what are the fractions?

Types of fractions. Conversion.

The fraraty is three species.

1. Ordinary fractions , eg:

Sometimes instead of horizontal screenshots, they put an inclined line: 1/2, 3/4, 19/5, well, and so on. Here we will often be this writing to use. The upper number is called numerator, Lower - denominator. If you constantly confuse these names (happens ...), tell me with the phrase expression: " ZZZZapumnney! ZZZZnamer - Vni zZZZy! "You look, everything and zzzzozomnikh.)

Chertochka that is horizontal that inclined means division top number (numerator) to the bottom (denominator). And that's all! Instead of a screw, it is quite possible to put a fission sign - two points.

When the division is possible, it should be done. So, instead of fractions "32/8", it is much more pleasant to write the number "4". Those. 32 Just divide by 8.

32/8 = 32: 8 = 4

I'm not talking about the fraction "4/1". Which is also just "4". And if it is not divided by a lot, we leave, in the form of a fraction. Sometimes there is a reverse operation to do. Make an integer fraction. But about this below.

2. Decimal fractions , eg:

It is in this form that will need to record the answers to the tasks "B".

3. Mixed numbers , eg:

Mixed numbers are practically not used in high school. In order to work with them, they must be translated into ordinary fractions. But it's necessary to be able to do! And then there will be such a number in a task and hang ... in an empty place. But we will remember this procedure! Slightly lower.

The most universal Ordinary fractions. With them and start. By the way, if there are all sorts of logarithms, sinuses and other beaks, it does not change anything. In the sense that all actions with fractional expressions are no different from action with ordinary fractions!

The main property of the fraction.

So let's go! To begin with, I will surprise you. All fraction transformation varieties is provided by one-sole property! It is called the main property of the fraci. Remember: if the numerator and denominator of the fraci multiply (divided) per and the same number, the fraction will not change. Those:

It is clear that you can write further before the formation. Sinuses and logarithms let you do not embarrass, we'll figure it out with them. The main thing is to understand that all these diverse expressions are one and the same fraction . 2/3.

And we need it, all these transformations? And how! Now you will see. To begin with, we will use the main property of the fraction for reducing fractions. It would seem that the thing is elementary. We divide the numerator and denominator for the same number and all things! It is impossible to make a mistake! But ... a person is creative creature. Make a mistake everywhere! Especially if you have to reduce the fraction of type 5/10, but a fractional expression with all sorts of beaks.

As properly and quickly cut the fraction, without making any extra work, you can read in a special section 555.

The normal student is not bothering the division of the numerator and the denominator on the same number (or expression)! He simply jumps all the same on top and bottom! Here is a typical mistake, a lap, if you want.

For example, you need to simplify the expression:

There is nothing to think here, you jump up the letter "A" from above and a twice from below! We get:

That's right. But really you divided all Numerator I. all danger on "A". If you are used to simply cross, then, you need, you can cross "A" in expression

and get again

What will be categorically incorrect. Because here all Numerator on "A" already not divide! It is impossible to cut this fraction. By the way, such a reduction is, GM ... a serious challenge to the teacher. This is not forgiven! Remember? When cutting, we need to share all Numerator I. all denominator!

Reducing fractions greatly facilitates life. It turns out somewhere you have fraction, for example 375/1000. And how now to work with her? Without a calculator? Multiply, say, fold, in a square to erect!? And if you don't be lazy, yes, it is accurate enough to cut five, and even five, and even ... while it is reduced, in short. We get 3/8! Much more pleasant, right?

The main property of the fraction allows us to translate ordinary fractions to decimal and vice versa without calculator! This is important to the exam, right?

How to translate fractions from one species to another.

With decimal fractions, everything is simple. As heard, it is written! Let's say 0.25. This is a zero whole, twenty-five hundredths. Yes, we write: 25/100. We reduce (divide the numerator and denominator on 25), we get the usual fraction: 1/4. Everything. It happens, and nothing is reduced. Type 0.3. These are three tenths, i.e. 3/10.

And if integers - not zero? Nothing wrong. We write down the entire fraction without any commas In the numerator, and in the denominator is what hearse. For example: 3.17. These are three integers, seventeen hundredths. We write in the numerator 317, and in the denominator 100. We get 317/100. Nothing is reduced, it means everything. This is the answer. Elementary Watson! Of all the told useful conclusion: any decimal fraction can be turned into an ordinary .

But the inverse transformation, ordinary to decimal, some without a calculator cannot do. But you must! How do you write to write on the exam!? Carefully read and master this process.

Decimal fraction than characteristic? She has in the denominator always It costs 10, or 100, or 1000, or 10,000 and so on. If your usual fraction has such a denominator, there are no problems. For example, 4/10 \u003d 0.4. Or 7/100 \u003d 0.07. Or 12/10 \u003d 1.2. And if in response to the task section "in" turned out 1/2? What will we write in response? There are decimal required ...

Remember the main property of the fraci ! Mathematics favorably allows you to multiply the numerator and denominator for the same number. For any, by the way! In addition to zero, of course. So applies this property to yourself! What can be multiplied by the denominator, i.e. 2 So that it become 10, or 100, or 1000 (smaller better, of course ...)? 5, obviously. Boldly multiply the denominator (this us it is necessary) by 5. But, then the numerator must be multiplied, too, for 5. This is already mathematics Requires! We obtain 1/2 \u003d 1x5 / 2x5 \u003d 5/10 \u003d 0.5. That's all.

However, the denominators are all sorts. Will come, for example, the fraction 3/16. Try, figure out here, on which 16 multiply, so that 100 it happens, or 1000 ... does not work? Then you can simply separate 3 to 16. Behind the lack of a calculator, you will have to divide the corner, on a piece of paper, as in junior grades. We get 0.1875.

And there are completely bad denominants. For example, a fraction of 1/3, well, do not turn into a good decimal. And on the calculator, and on a piece of paper, we will get 0,3333333 ... This means that 1/3 in an exact decimal fraction does not translate. Just as 1/7, 5/6 and so on. Many of them undeveloped. From here another useful conclusion. Not every ordinary fraction is translated into decimal !

By the way, this is useful information for self-test. In the section "B" in response, you need a decimal fraction to record. And you have it, for example, 4/3. This fraction is not translated into decimal. This means that somewhere you made a mistake on the road! Return, check the solution.

So, with ordinary and decimal fractions figured out. It remains to deal with mixed numbers. To work with them, they must be translated into ordinary fractions. How to do it? You can catch a sixth grader and ask him. But not always the sixth grader will be at hand ... you have to. It's not hard. It is necessary a denominator of a fractional part to multiply by a whole part and add a fractional part numerator. It will be a numerator of the usual fraction. And the denominator? The denominator will remain the same. It sounds difficult, but in fact everything is elementary. We look an example.

Let in a challenge you with horror saw a number:

Calmly, without panic, we think. The whole part is 1. one. Fractional part - 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be a denominator of an ordinary fraction. We consider the numerator. 7 Multiply with 1 (whole part) and add 3 (numerator of the fractional part). We get 10. It will be a numerator of an ordinary fraction. That's all. Even easier, it looks in a mathematical record:

Clear? Then secure success! Translate into ordinary fractions. You should work 10/7, 7/2, 23/10 and 21/4.

Reverse operation - Translation of incorrect fraction in a mixed number - in high schools is rarely required. Well, if so ... and if you are not in high schools - you can look into a special section 555. There, by the way, and about the wrong fraraty will learn.

Well, almost everything. You remembered the types of fractions and understood as Translate them from one species to another. The question remains: what for do it? Where and when to apply these deep knowledge?

I answer. Any example itself suggests the necessary actions. If an example was mixed into a bunch of ordinary fractions, decimal, and even mixed numbers, we translate everything into ordinary fractions. It can always be done. Well, if it is written, something like 0.8 + 0.3, then I believe without any translation. Why do we need extra work? We choose that path the solution that is convenient us !

If the task is complete decimal fractions, but um ... angry some, go to ordinary, try! You look, everything will work. For example, it will be in a square to erect a number 0.125. Not so easy if you did not pay off from the calculator! Not only you need to multiply the column, so think, where to insert comma! In mind it will not be exactly! And if you go to an ordinary fraction?

0.125 \u003d 125/1000. Reducing on 5 (this is for starters). We get 25/200. Once again by 5. We get 5/40. Oh, still cuts! Again on 5! We get 1/8. Easily erected into a square (in the mind!) And we get 1/64. Everything!

Let's summarize this lesson.

1. Fruit is three species. Ordinary, decimal and mixed numbers.

2. Decimal fractions and mixed numbers always You can translate into ordinary fractions. Reverse translation not always available.

3. Selecting the type of fractions to work with the task depends on this very task. If there are different types of fractions in one task, the most reliable - go to ordinary fractions.

Now you can take care. To begin with, translate these decimal fractions to ordinary:

3,8; 0,75; 0,15; 1,4; 0,725; 0,012

There must be such answers (in disorder!):

On this and end. In this lesson, we refreed in memory key moments for fractions. It happens, however, it is especially nothing to refreshing ...) If someone who has completely forgotten, or has not mastered ... the one can go into a special section 555. There all the foundations are detailed. Many suddenly understand everything Start. And decide the fraraty with the lea).

If you like this site ...

By the way, I have another couple of interesting sites for you.)

It can be accessed in solving examples and find out your level. Testing with instant check. Learn - with interest!)

You can get acquainted with features and derivatives.

Numerator, and on which they divide - denominator.

To burn a fraction, write its numerator first, then spend a horizontal line under this number, and you write a denominator below the line. The horizontal, separating the numerator and the denominator is called a fractional feature. Sometimes it is depicted in the form of inclined "/" or "/". At the same time, the numerator is written to the left of the line, and the denominator on the right. So, for example, the "two third" fraction will be recorded as 2/3. For clarity, the numerator is usually written at the top of the line, and the denominator is in the bottom, that is, instead of 2/3, you can meet: ⅔.

To calculate the work of fractions, multiply first numerator one drobi. On the other. Write down the result in the numerator of the new drobi.. After that, multiply and denominators. Final value Indicate in the new drobi.. For example, 1/3? 1/5 \u003d 1/15 (1? 1 \u003d 1; 3? 5 \u003d 15).

To divide one fraction to another, multiply first the numerator first to the denominator second. Exercise the same with the second fraction (divider). Or before performing all actions, I first "flip" the divider, if it is so more convenient for you: a denominator should be on the site of the numerator. After that, multiply the denominator denominator to the new valve denominator and multiply the numerals. For example, 1/3: 1/5 \u003d 5/3 \u003d 1 2/3 (1? 5 \u003d 5; 3? 1 \u003d 3).

Sources:

• Basic tasks for fractions

Fractional numbers allow you to express the exact value of the value in different ways. With fractions, you can perform the same mathematical operations as with integers: subtraction, addition, multiplication and division. To learn to decide drobi., I must remember some of their features. They depend on the type drobi., availability of a whole part, a common denominator. Some arithmetic actions after execution require a reduction in the fractional part of the result.

You will need

• - Calculator

Instruction

Carefully look at the numbers. If there are decimal and immunity among fractions, sometimes more conveniently performing actions with decimal, and then transfer them to the wrong appearance. You can translate drobi. In this kind, initially, writing the value after the comma in the numerator and putting 10 to the denominator. If necessary, reduce the fraction, separating the numbers above and below one divider. The fractions in which the whole part is distinguished, bring to the wrong mind, multiplying it on the denominator and adding a numerator to the result. This value will become a new numerator. drobi.. To allocate the whole part of the initially incorrect drobi., It is necessary to divide the numerator to the denominator. Write a whole result from drobi.. And the balance of division will become a new numerator, denominator drobi. It does not change. For fractions with a whole part, it is possible to perform actions separately first for the whole, and then for fractional parts. For example, amount 1 2/3 and 2 ¾ can be calculated:
- Transferring fractions to the wrong appearance:
- 1 2/3 + 2 ¾ \u003d 5/3 + 11/4 \u003d 20/12 + 33/12 \u003d 53/12 \u003d 4 5/12;
- Summation of alone and fractional parts of the components:
- 1 2/3 + 2 ¾ \u003d (1 + 2) + (2/3 + ¾) \u003d 3 + (8/12 + 9/12) \u003d 3 + 17/12 \u003d 3 + 1 5/12 \u003d 4 5 /12.

Rewrite them through the separator ":" and continue the usual division.

To obtain the final result, the resulting fraction is reduced by separating the numerator and the denominator for one integer, the largest possible in this case. At the same time, above and below should be integers.

note

Do not perform arithmetic action with fractions, whose denominators are different. Pick up such a number so that with multiplying the numerator and denominator of each fraction as a result, the denominators of both fractions were equal.

Helpful advice

When recruiting fractional numbers, divisible is written above the feature. This value is denoted as a fluster numerator. Under the line recorded a divider, or a denominator, fraction. For example, a half kilogram of rice in the form of a fraction is recorded as follows: 1 ½ kg of rice. If the denominator of the fraction is 10, such a fraction is called decimal. In this case, the numerator (divisible) is written to the right of the whole part over the comma: 1.5 kg of rice. For the convenience of calculations, such a fraction can always be written in the wrong form: 1 2/10 kg of potatoes. To simplify, you can reduce the values \u200b\u200bof the numerator and the denominator, sharing them by one integer. In this example, it is possible to divide on 2. As a result, 1 1/5 kg of potatoes will be obtained. Make sure that the numbers you are going to perform arithmetic action are presented in one form.