In the article, we will show how to solve fractions with simple clear examples. Let's understand what a fraction is and consider solving fractions!
concept fractions is introduced into the course of mathematics starting from the 6th grade of secondary school.
Fractions look like: ±X / Y, where Y is the denominator, it tells how many parts the whole was divided into, and X is the numerator, it tells how many such parts were taken. For clarity, let's take an example with a cake:
In the first case, the cake was cut equally and one half was taken, i.e. 1/2. In the second case, the cake was cut into 7 parts, from which 4 parts were taken, i.e. 4/7.
If the part of dividing one number by another is not a whole number, it is written as a fraction.
For example, the expression 4:2 \u003d 2 gives an integer, but 4:7 is not completely divisible, so this expression is written as a fraction 4/7.
In other words fraction is an expression that denotes the division of two numbers or expressions, and which is written with a slash.
If the numerator is less than the denominator, the fraction is correct, if vice versa, it is incorrect. A fraction can contain an integer.
For example, 5 whole 3/4.
This entry means that in order to get the whole 6, one part of four is not enough.
If you want to remember how to solve fractions for 6th grade you need to understand that solving fractions basically comes down to understanding a few simple things.
- A fraction is essentially an expression for a fraction. That is numeric expression what part is given value from one whole. For example, the fraction 3/5 expresses that if we divide something whole into 5 parts and the number of parts or parts of this whole is three.
- A fraction can be less than 1, for example 1/2 (or essentially half), then it is correct. If the fraction is greater than 1, for example 3/2 (three halves or one and a half), then it is incorrect and to simplify the solution, it is better for us to select the whole part 3/2= 1 whole 1/2.
- Fractions are the same numbers as 1, 3, 10, and even 100, only the numbers are not whole, but fractional. With them, you can perform all the same operations as with numbers. Counting fractions is not more difficult, and further on concrete examples we will show it.
How to solve fractions. Examples.
A variety of arithmetic operations are applicable to fractions.
Bringing a fraction to a common denominator
For example, you need to compare the fractions 3/4 and 4/5.
To solve the problem, we first find the lowest common denominator, i.e. the smallest number that is divisible without remainder by each of the denominators of the fractions
Least common denominator(4.5) = 20
Then the denominator of both fractions is reduced to the smallest common denominator
Answer: 15/20
Addition and subtraction of fractions
If it is necessary to calculate the sum of two fractions, they are first brought to a common denominator, then the numerators are added, while the denominator remains unchanged. The difference of fractions is considered the same way, the only difference is that the numerators are subtracted.
For example, you need to find the sum of fractions 1/2 and 1/3
Now find the difference between the fractions 1/2 and 1/4
Multiplication and division of fractions
Here the solution of fractions is simple, everything is quite simple here:
- Multiplication - numerators and denominators of fractions are multiplied among themselves;
- Division - first we get a fraction, the reciprocal of the second fraction, i.e. swap its numerator and denominator, after which we multiply the resulting fractions.
For example:
On this about how to solve fractions, all. If you have any questions about solving fractions, something is not clear, then write in the comments and we will answer you.
If you are a teacher, it is possible to download the presentation for elementary school(http://school-box.ru/nachalnaya-shkola/prezentazii-po-matematike.html) will come in handy.
1 What are ordinary fractions. Types of fractions.A fraction always means some part of a whole. The fact is that it is not always possible to convey the quantity in natural numbers, that is, to recalculate: 1,2,3, etc. How, for example, to designate half a watermelon or a quarter of an hour? This is why fractional numbers, or fractions, appeared.
To begin with, it must be said that in general there are two types of fractions: ordinary fractions and decimal fractions. Ordinary fractions are written like this:
Decimals are written differently:
Ordinary fractions consist of two parts: at the top is the numerator, at the bottom is the denominator. The numerator and denominator are separated by a fractional bar. So remember:
Every fraction is part of a whole. The whole is usually taken 1 (unit). The denominator of a fraction shows how many parts the whole is divided into ( 1 ), and the numerator is how many parts were taken. If we cut the cake into 6 identical pieces (in mathematics they say shares ), then each part of the cake will be equal to 1/6. If Vasya ate 4 pieces, then he ate 4/6.
On the other hand, a fractional bar is nothing more than a division sign. Therefore, a fraction is a quotient of two numbers - the numerator and the denominator. In the text of problems or in recipes for dishes, fractions are usually written like this: 2/3, 1/2, etc. Some fractions got own name, for example, 1/2 - "half", 1/3 - "third", 1/4 - "quarter"
Now let's figure out what types of ordinary fractions are.
2 Types of ordinary fractions
There are three types of common fractions: regular, improper, and mixed:
Proper fraction
If the numerator is less than the denominator, then such a fraction is called correct, for example: 
Proper fraction always less than 1.
Improper fraction
If the numerator is greater than or equal to the denominator, the fraction is called wrong, for example:
An improper fraction is greater than one (if the numerator is greater than the denominator) or equal to one (if the numerator is equal to the denominator)
mixed fraction
If a fraction consists of a whole number (whole part) and a proper fraction (fractional part), then such a fraction is called mixed, for example: 
A mixed fraction is always greater than one.
3 Fraction conversions
In mathematics, ordinary fractions often have to be converted, that is, a mixed fraction must be turned into an improper one and vice versa. This is necessary to perform some operations, such as multiplication and division.
So, any mixed fraction can be converted to an improper. To do this, the integer part is multiplied by the denominator and the numerator of the fractional part is added. The resulting amount is taken as the numerator, and the denominator is left the same, for example:
Any improper fraction can be converted into a mixed fraction. To do this, divide the numerator by the denominator (with a remainder). The resulting number will be the integer part, and the remainder will be the numerator of the fractional part, for example:
At the same time, they say: “We singled out the whole part from an improper fraction.”
There is one more rule to remember: Any whole number can be represented as a common fraction with denominator 1, for example:
Let's talk about how to compare fractions.
4 Fraction Comparison
When comparing fractions, there are several options: It is easy to compare fractions with the same denominators, much more difficult if the denominators are different. There is also a comparison of mixed fractions. But don't worry, now we'll take a closer look at each option and learn how to compare fractions.
Comparing fractions with the same denominators
Of two fractions with the same denominator but different numerators, the fraction with the larger numerator is larger, for example:
Comparing fractions with the same numerator
Of two fractions with the same numerators but different denominators, the fraction with the smaller denominator is greater, for example:
Comparing mixed and improper fractions with proper fractions
An improper or mixed fraction is always greater than a proper fraction, for example:
Comparing two mixed fractions
When comparing two mixed fractions, the fraction with the larger integer part is greater, for example:
If the integer parts of mixed fractions are the same, the fraction with the larger fractional part is greater, for example:
Comparing fractions with different numerators and denominators
It is impossible to compare fractions with different numerators and denominators without converting them. First, the fractions must be brought to the same denominator, and then their numerators should be compared. The larger fraction is the one with the larger numerator. But how to bring fractions to the same denominator, we will consider in the next two sections of the article. First, we will consider the basic property of a fraction and the reduction of fractions, and then directly reducing fractions to the same denominator.
5 Basic property of a fraction. Fraction reduction. The concept of GCD.
Remember: You can only add, subtract, and compare fractions that have the same denominators.. If the denominators are different, then first you need to bring the fractions to the same denominator, that is, transform one of the fractions in such a way that its denominator becomes the same as that of the second fraction.
Fractions have one important property, also called basic property of a fraction:
If both the numerator and the denominator of a fraction are multiplied or divided by the same number, then the value of the fraction will not change:
Thanks to this property, we can reduce fractions:
To reduce a fraction means to divide both the numerator and the denominator by the same number.(see example just above). When we reduce a fraction, we can describe our actions as follows:
More often, in a notebook, a fraction is reduced like this:
But remember: only multipliers can be reduced. If the numerator or denominator is the sum or difference, the terms cannot be reduced. Example:
We need to convert the sum to a multiplier first:
Sometimes, when working with big numbers, in order to reduce the fraction, it is convenient to find greatest common factor of numerator and denominator (gcd)
Greatest Common Divisor (GCD) several numbers - this is the largest natural number by which these numbers are divisible without a remainder.
In order to find the GCD of two numbers (for example, the numerator and denominator of a fraction), you need to decompose both numbers into prime factors, note the same factors in both expansions, and multiply these factors. The resulting product will be GCD. For example, we need to reduce a fraction:
Find the GCD of numbers 96 and 36:
The GCD shows us that both the numerator and the denominator have a factor12, and we can easily reduce the fraction. 
Sometimes, to bring fractions to the same denominator, it is enough to reduce one of the fractions. But more often it is necessary to select additional factors for both fractions. Now we will look at how this is done. So:
6 How to bring fractions to the same denominator. Least common multiple (LCM).
When we reduce fractions to the same denominator, we select for the denominator a number that would be divisible by both the first and the second denominator (that is, it would be a multiple of both denominators, in mathematical terms). And it is desirable that this number be as small as possible, so it is more convenient to count. So we have to find the LCM of both denominators.
Least common multiple of two numbers (LCM) is the smallest natural number that is divisible by both of these numbers without a remainder. Sometimes the LCM can be found orally, but more often, especially when working with large numbers, you have to find the LCM in writing, using the following algorithm:
In order to find the LCM of several numbers, you need:
- Decompose these numbers into prime factors
- Take the largest expansion, and write these numbers as a product
- Select in other expansions the numbers that do not occur in the largest expansion (or occur in it a smaller number of times), and add them to the product.
- Multiply all the numbers in the product, this will be the LCM.
For example, let's find the LCM of numbers 28 and 21:
But back to our fractions. After we have selected or calculated in writing the LCM of both denominators, we must multiply the numerators of these fractions by additional multipliers. You can find them by dividing the LCM by the denominator of the corresponding fraction, for example: 
Thus, we reduced our fractions to one denominator - 15.
7 Addition and subtraction of fractions
Adding and subtracting fractions with the same denominators
To add fractions with the same denominators, you need to add their numerators, and leave the denominator the same, for example:
To subtract fractions with the same denominators, subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same, for example: 
Addition and subtraction of mixed fractions with the same denominators
To add mixed fractions, you must separately add their whole parts, and then add their fractional parts, and write the result mixed fraction:
If, when adding the fractional parts, an improper fraction is obtained, we select the integer part from it and add it to the integer part, for example:
Subtraction is carried out in the same way: the integer part is subtracted from the integer, and the fractional part is subtracted from the fractional part: 
If the fractional part of the subtrahend is greater than the fractional part of the minuend, we “take” one from the integer part, turning the minuend into an improper fraction, and then proceed as usual: 
Similarly subtract a fraction from a whole number:
How to add a whole number and a fraction
In order to add a whole number and a fraction, you just need to add this number before the fraction, and you get a mixed fraction, for example:
If we add a whole number and a mixed fraction, we add this number to the integer part of the fraction, for example:
Addition and subtraction of fractions with different denominators.
In order to add or subtract fractions with different denominators, you must first bring them to the same denominator, and then proceed as when adding fractions with the same denominators (add the numerators):
When subtracting, we proceed in the same way:
If we work with mixed fractions, we reduce their fractional parts to the same denominator and then subtract as usual: the integer part from the integer, and the fractional part from the fractional part:
8 Multiplication and division of fractions.
Multiplying and dividing fractions is much easier than adding and subtracting because you don't have to bring them to the same denominator. Remember simple rules multiplication and division of fractions:
Before multiplying numbers in the numerator and denominator, it is desirable to reduce the fraction, that is, to get rid of the same factors in the numerator and denominator, as in our example.
To divide a fraction by a natural number, you need to multiply the denominator by this number, and leave the numerator unchanged:
For example:
Division of a fraction by a fraction
To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor ( reciprocal).What is this reciprocal?
If we flip the fraction, that is, swap the numerator and denominator, we get the reciprocal. The product of a fraction and its reciprocal gives one. In mathematics, such numbers are called mutually reciprocal numbers:
For example, numbers 
are mutually inverse, since
Thus, we return to the division of a fraction by a fraction:
To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor:
For example:
When dividing mixed fractions, in the same way as when multiplying, you must first translate them into improper fractions:
When multiplying and dividing fractions by whole natural numbers, you can also represent these numbers as fractions with a denominator 1 .
And at dividing a whole number by a fraction represent this number as a fraction with a denominator 1 :
The numerator and denominator of a fraction. Types of fractions. Let's continue with fractions. First, a small caveat - we, considering fractions and the corresponding examples with them, for now we will work only with its numerical representation. There are also fractional literal expressions (with and without numbers).However, all the "principles" and rules also apply to them, but we will talk about such expressions separately in the future. I recommend visiting and studying (remembering) the topic of fractions step by step.
The most important thing is to understand, remember and realize that a FRACTION is a NUMBER!!!
Common fraction is a number of the form:
The number located "on top" (in this case m) is called the numerator, the number located below (number n) is called the denominator. Those who have just touched on the topic often get confused - what is the name.
Here's a trick for you, how to remember forever - where is the numerator, and where is the denominator. This technique is associated with verbal-figurative association. Imagine a jar muddy water. It is known that as water settles, clean water remains on top, and turbidity (dirt) settles, remember:
CHISSS melt water ABOVE (CHISSS pourer on top)
mud ZZZNNN th water BOTTOM (ZZZNN Amenator below)
So, as soon as it becomes necessary to remember where the numerator is and where the denominator is, then they immediately visually presented a jar of settled water, in which Pure water, and below dirty water. There are other tricks to remember, if they help you, then good.
Examples of ordinary fractions:
What does the horizontal line between numbers mean? This is nothing more than a division sign. It turns out that a fraction can be considered as an example with the action of division. This action is simply recorded in this form. That is, the top number (numerator) is divided by the bottom number (denominator):
In addition, there is another form of recording - a fraction can be written like this (through a slash):
1/9, 5/8, 45/64, 25/9, 15/13, 45/64 and so on...
We can write the above fractions as follows:
The result of the division, as you know, is the number.
Clarified - FRACTION THIS NUMBER !!!
As you have already noticed, in an ordinary fraction, the numerator may be less than the denominator, may be greater than the denominator, and may be equal to it. Here there are many important points, which are understandable intuitively, without any theoretical frills. For example:
1. Fractions 1 and 3 can be written as 0.5 and 0.01. Let's run a little ahead - these are decimal fractions, we'll talk about them a little lower.
2. Fractions 4 and 6 result in an integer 45:9=5, 11:1 = 11.
3. Fraction 5 as a result gives a unit 155:155 = 1.
What conclusions suggest themselves? The following:
1. The numerator, when divided by the denominator, can give a finite number. It may not work, divide by a column 7 by 13 or 17 by 11 - no way! You can divide indefinitely, but we will also talk about this a little lower.
2. A fraction can result in an integer. Therefore, we can represent any integer as a fraction, or rather an infinite series of fractions, look, all these fractions are equal to 2:
Yet! We can always write any whole number as a fraction - this number itself is in the numerator, one in the denominator:
3. We can always represent a unit as a fraction with any denominator:
*The points indicated are extremely important for working with fractions in calculations and conversions.
Types of fractions.
And now about the theoretical division of ordinary fractions. They are divided into right and wrong.
A fraction whose numerator is less than the denominator is called a proper fraction. Examples:
A fraction whose numerator is greater than or equal to the denominator is called an improper fraction. Examples:
mixed fraction(mixed number).
A mixed fraction is a fraction written as a whole number and a proper fraction and is understood as the sum of this number and its fractional part. Examples:
A mixed fraction can always be represented as an improper fraction and vice versa. Let's go further!
Decimals.
We have already touched on them above, these are examples (1) and (3), now in more detail. Here are examples of decimals: 0.3 0.89 0.001 5.345.
A fraction whose denominator is a power of 10, such as 10, 100, 1000, and so on, is called a decimal. It is not difficult to write the first three indicated fractions as ordinary fractions:
The fourth is a mixed fraction (mixed number):
A decimal fraction has the following notation - withthe integer part began, then the separator of the integer and fractional parts was a dot or a comma and then the fractional part, the number of digits of the fractional part is strictly determined by the dimension of the fractional part: if these are tenths, the fractional part is written as one digit; if thousandths - three; ten-thousandths - four, etc.
These fractions are finite and infinite.
Ending decimal examples: 0.234; 0.87; 34.00005; 5.765.
Examples are endless. For example, pi is infinite. decimal, more - 0.333333333333…... 0.16666666666…. and others. Also the result of extracting the root from the numbers 3, 5, 7, etc. will be an infinite fraction.
The fractional part can be cyclic (there is a cycle in it), the two examples above are exactly the same, more examples:
0.123123123123…... cycle 123
0.781781781718…... cycle 781
0.0250102501…. cycle 02501
They can be written as 0, (123) 0, (781) 0, (02501).
The number Pi is not a cyclic fraction, like, for example, the root of three.
Below in the examples, words such as “turn over” the fraction will sound - this means that the numerator and denominator are interchanged. In fact, such a fraction has a name - the reciprocal fraction. Examples of reciprocal fractions:
Small summary! Fractions are:
Ordinary (correct and incorrect).
Decimals (finite and infinite).
Mixed (mixed numbers).
That's all!
Sincerely, Alexander.
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, the smallest number that 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 different denominators, you should reduce them to a common one and only then perform addition. 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, it is imperative to check whether it is possible to reduce given number 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 main property algebraic fraction, as well as skills in finding the NOC and common divisors 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 making 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 the final result is always written not requiring reduction fractional number.
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 "hammer" 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.
This article is about common fractions. Here we will get acquainted with the concept of a fraction of a whole, which will lead us to the definition of an ordinary fraction. Next, we will dwell on the accepted notation for ordinary fractions and give examples of fractions, say about the numerator and denominator of a fraction. After that, we will give definitions of correct and incorrect, positive and negative fractions, and also consider the position of fractional numbers on the coordinate ray. In conclusion, we list the main actions with fractions.
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Shares of the whole
First we introduce share concept.
Let's assume that we have some object made up of several absolutely identical (that is, equal) parts. For clarity, you can imagine, for example, an apple cut into several equal parts, or an orange, consisting of several equal slices. Each of these equal parts that make up the whole object is called share of the whole or simply shares.
Note that the shares are different. Let's explain this. Let's say we have two apples. Let's cut the first apple into two equal parts, and the second one into 6 equal parts. It is clear that the share of the first apple will be different from the share of the second apple.
Depending on the number of shares that make up the whole object, these shares have their own names. Let's analyze share names. If the object consists of two parts, any of them is called one second part of the whole object; if the object consists of three parts, then any of them is called one third part, and so on.
One second beat has a special name - half. One third is called third, and one quadruple - quarter.
For the sake of brevity, the following share designations. One second share is designated as or 1/2, one third share - as or 1/3; one fourth share - like or 1/4, and so on. Note that the notation with a horizontal bar is used more often. To consolidate the material, let's give one more example: the entry denotes one hundred and sixty-seventh of the whole.
The concept of a share naturally extends from objects to magnitudes. For example, one of the measures of length is the meter. To measure lengths less than a meter, fractions of a meter can be used. So you can use, for example, half a meter or a tenth or thousandth of a meter. Shares of other quantities are applied similarly.
Common fractions, definition and examples of fractions
To describe the number of shares are used common fractions. Let's give an example that will allow us to approach the definition of ordinary fractions.
Let an orange consist of 12 parts. Each share in this case represents one twelfth of a whole orange, that is, . Let's denote two beats as , three beats as , and so on, 12 beats as . Each of these entries is called an ordinary fraction.
Now let's give a general definition of common fractions.
The voiced definition of ordinary fractions allows us to bring examples of common fractions: 5/10 , , 21/1 , 9/4 , . And here are the records 
do not fit the voiced definition of ordinary fractions, that is, they are not ordinary fractions.
Numerator and denominator
For convenience, in ordinary fractions we distinguish numerator and denominator.
Definition.
Numerator ordinary fraction (m / n) is a natural number m.
Definition.
Denominator ordinary fraction (m / n) is a natural number n.
So, the numerator is located above the fraction bar (to the left of the slash), and the denominator is below the fraction bar (to the right of the slash). For example, let's take an ordinary fraction 17/29, the numerator of this fraction is the number 17, and the denominator is the number 29.
It remains to discuss the meaning contained in the numerator and denominator of an ordinary fraction. The denominator of the fraction shows how many shares one item consists of, the numerator, in turn, indicates the number of such shares. For example, the denominator 5 of the fraction 12/5 means that one item consists of five parts, and the numerator 12 means that 12 such parts are taken.
Natural number as a fraction with denominator 1
The denominator of a common fraction can be equal to one. In this case, we can assume that the object is indivisible, in other words, it is something whole. The numerator of such a fraction indicates how many whole items are taken. In this way, common fraction of the form m/1 has the meaning of a natural number m . This is how we substantiated the equality m/1=m .
Let's rewrite the last equality like this: m=m/1 . This equality allows us to represent any natural number m as an ordinary fraction. For example, the number 4 is the fraction 4/1, and the number 103498 is the fraction 103498/1.
So, any natural number m can be represented as an ordinary fraction with denominator 1 as m/1 , and any ordinary fraction of the form m/1 can be replaced by a natural number m.
Fraction bar as division sign
The representation of the original object in the form of n shares is nothing more than a division into n equal parts. After the item is divided into n shares, we can divide it equally among n people - each will receive one share.
If we initially have m identical objects, each of which is divided into n shares, then we can equally divide these m objects among n people, giving each person one share from each of the m objects. In this case, each person will have m shares 1/n, and m shares 1/n gives an ordinary fraction m/n. Thus, the common fraction m/n can be used to represent the division of m items among n people.
So we got an explicit connection between ordinary fractions and division (see the general idea of the division of natural numbers). This relationship is expressed as follows: The bar of a fraction can be understood as a division sign, that is, m/n=m:n.
With the help of an ordinary fraction, you can write the result of dividing two natural numbers, for which integer division is not performed. For example, the result of dividing 5 apples by 8 people can be written as 5/8, that is, each will get five eighths of an apple: 5:8=5/8.
Equal and unequal ordinary fractions, comparison of fractions
A fairly natural action is comparison of common fractions, because it is clear that 1/12 of an orange is different from 5/12, and 1/6 of an apple is the same as the other 1/6 of this apple.
As a result of comparing two ordinary fractions, one of the results is obtained: the fractions are either equal or not equal. In the first case we have equal common fractions, and in the second unequal common fractions. Let's give a definition of equal and unequal ordinary fractions.
Definition.
equal, if the equality a d=b c is true.
Definition.
Two common fractions a/b and c/d not equal, if the equality a d=b c is not satisfied.
Here are some examples of equal fractions. For example, the common fraction 1/2 is equal to the fraction 2/4, since 1 4=2 2 (if necessary, see the rules and examples of multiplication of natural numbers). For clarity, you can imagine two identical apples, the first is cut in half, and the second - into 4 shares. It is obvious that two-fourths of an apple is 1/2 a share. Other examples of equal common fractions are the fractions 4/7 and 36/63, and the pair of fractions 81/50 and 1620/1000.
And ordinary fractions 4/13 and 5/14 are not equal, since 4 14=56, and 13 5=65, that is, 4 14≠13 5. Another example of unequal common fractions are the fractions 17/7 and 6/4.
If, when comparing two ordinary fractions, it turns out that they are not equal, then you may need to find out which of these ordinary fractions less another, and which more. To find out, the rule for comparing ordinary fractions is used, the essence of which is to bring the compared fractions to a common denominator and then compare the numerators. Detailed information on this topic is collected in the article comparison of fractions: rules, examples, solutions.
Fractional numbers
Each fraction is a record fractional number. That is, a fraction is just a “shell” of a fractional number, its appearance, and the entire semantic load is contained precisely in a fractional number. However, for brevity and convenience, the concept of a fraction and a fractional number are combined and simply called a fraction. Here it is appropriate to paraphrase a well-known saying: we say a fraction - we mean a fractional number, we say a fractional number - we mean a fraction.
Fractions on the coordinate beam
All fractional numbers corresponding to ordinary fractions have their own unique place on , that is, there is a one-to-one correspondence between fractions and points of the coordinate ray.
In order to get to the point corresponding to the fraction m / n on the coordinate ray, it is necessary to postpone m segments from the origin in the positive direction, the length of which is 1 / n of the unit segment. Such segments can be obtained by dividing a single segment into n equal parts, which can always be done using a compass and ruler.
For example, let's show the point M on the coordinate ray, corresponding to the fraction 14/10. The length of the segment with ends at the point O and the point closest to it, marked with a small dash, is 1/10 of the unit segment. The point with coordinate 14/10 is removed from the origin by 14 such segments.
Equal fractions correspond to the same fractional number, that is, equal fractions are the coordinates of the same point on the coordinate ray. For example, one point corresponds to the coordinates 1/2, 2/4, 16/32, 55/110 on the coordinate ray, since all the written fractions are equal (it is located at a distance of half the unit segment, postponed from the origin in the positive direction).
On a horizontal and right-directed coordinate ray, the point whose coordinate is a large fraction is located to the right of the point whose coordinate is a smaller fraction. Similarly, the point with the smaller coordinate lies to the left of the point with the larger coordinate.
Proper and improper fractions, definitions, examples
Among ordinary fractions, there are proper and improper fractions. This division basically has a comparison of the numerator and denominator.
Let's give a definition of proper and improper ordinary fractions.
Definition.
Proper fraction is an ordinary fraction, the numerator of which is less than the denominator, that is, if m Definition.
Improper fraction is an ordinary fraction in which the numerator is greater than or equal to the denominator, that is, if m≥n, then the ordinary fraction is improper. Here are some examples of proper fractions: 1/4 , , 32 765/909 003 . Indeed, in each of the written ordinary fractions, the numerator is less than the denominator (if necessary, see the article comparison of natural numbers), so they are correct by definition. And here are examples of improper fractions: 9/9, 23/4,. Indeed, the numerator of the first of the written ordinary fractions is equal to the denominator, and in the remaining fractions the numerator is greater than the denominator. There are also definitions of proper and improper fractions based on comparing fractions with one. Definition. correct if it is less than one. Definition. The common fraction is called wrong, if it is either equal to one or greater than 1 . So the ordinary fraction 7/11 is correct, since 7/11<1
, а обыкновенные дроби 14/3
и 27/27
– неправильные, так как 14/3>1 , and 27/27=1 . Let's think about how ordinary fractions with a numerator greater than or equal to the denominator deserve such a name - "wrong". Let's take the improper fraction 9/9 as an example. This fraction means that nine parts of an object are taken, which consists of nine parts. That is, from the available nine shares, we can make up a whole subject. That is, the improper fraction 9/9 essentially gives a whole object, that is, 9/9=1. In general, improper fractions with a numerator equal to the denominator denote one whole object, and such a fraction can be replaced by a natural number 1. Now consider the improper fractions 7/3 and 12/4. It is quite obvious that from these seven thirds we can make two whole objects (one whole object is 3 shares, then to compose two whole objects we need 3 + 3 = 6 shares) and there will still be one third share. That is, the improper fraction 7/3 essentially means 2 items and even 1/3 of the share of such an item. And from twelve quarters we can make three whole objects (three objects with four parts each). That is, the fraction 12/4 essentially means 3 whole objects. The considered examples lead us to the following conclusion: improper fractions can be replaced either by natural numbers, when the numerator is divided entirely by the denominator (for example, 9/9=1 and 12/4=3), or the sum of a natural number and a proper fraction, when the numerator is not evenly divisible by the denominator (for example, 7/3=2+1/3 ). Perhaps this is precisely what improper fractions deserve such a name - “wrong”. Of particular interest is the representation of an improper fraction as the sum of a natural number and a proper fraction (7/3=2+1/3). This process is called the extraction of an integer part from an improper fraction, and deserves a separate and more careful consideration. It is also worth noting that there is a very close relationship between improper fractions and mixed numbers. Each ordinary fraction corresponds to a positive fractional number (see the article positive and negative numbers). That is, ordinary fractions are positive fractions. For example, ordinary fractions 1/5, 56/18, 35/144 are positive fractions. When it is necessary to emphasize the positiveness of a fraction, then a plus sign is placed in front of it, for example, +3/4, +72/34. If you put a minus sign in front of an ordinary fraction, then this entry will correspond to a negative fractional number. In this case, one can speak of negative fractions. Here are some examples of negative fractions: −6/10 , −65/13 , −1/18 . The positive and negative fractions m/n and −m/n are opposite numbers. For example, the fractions 5/7 and −5/7 are opposite fractions. Positive fractions, like positive numbers in general, denote an increase, income, a change in some value upwards, etc. Negative fractions correspond to expense, debt, a change in any value in the direction of decrease. For example, a negative fraction -3/4 can be interpreted as a debt, the value of which is 3/4. On the horizontal and right-directed negative fractions are located to the left of the reference point. The points of the coordinate line whose coordinates are the positive fraction m/n and the negative fraction −m/n are located at the same distance from the origin, but on opposite sides of the point O . Here it is worth mentioning fractions of the form 0/n. These fractions are equal to the number zero, that is, 0/n=0 . Positive fractions, negative fractions, and 0/n fractions combine to form rational numbers. One action with ordinary fractions - comparing fractions - we have already considered above. Four more arithmetic are defined operations with fractions- addition, subtraction, multiplication and division of fractions. Let's dwell on each of them. The general essence of actions with fractions is similar to the essence of the corresponding actions with natural numbers. Let's draw an analogy. Multiplication of fractions can be considered as an action in which a fraction is found from a fraction. To clarify, let's take an example. Suppose we have 1/6 of an apple and we need to take 2/3 of it. The part we need is the result of multiplying the fractions 1/6 and 2/3. The result of multiplying two ordinary fractions is an ordinary fraction (which in a particular case is equal to a natural number). Further we recommend to study the information of the article multiplication of fractions - rules, examples and solutions. Bibliography.Positive and negative fractions
Actions with fractions