We come across fractions in life much earlier than we begin studying them at school. If we cut a whole apple in half, we get ½ of the fruit. Let's cut it again - it will be ¼. These are fractions. And everything seemed simple. For an adult. For the child (and this topic begin to study at the end of primary school) abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must clearly explain what a proper and improper fraction, common and decimal are, what operations can be performed with them and, most importantly, what all this is needed for.

What are fractions?

Introducing a new topic at school begins with ordinary fractions. They are easily recognized by the horizontal line separating the two numbers - above and below. The top one is called the numerator, the bottom one is the denominator. There is also a lowercase option for writing improper and proper ordinary fractions - through a slash, for example: ½, 4/9, 384/183. This option is used when the line height is limited and it is not possible to use a “two-story” entry form. Why? Yes, because it is more convenient. We'll see this a little later.

In addition to ordinary fractions, there are also decimal fractions. It is very simple to distinguish them: if in one case a horizontal or slash is used, in the other a comma is used to separate sequences of numbers. Let's look at an example: 2.9; 163.34; 1.953. We intentionally used a semicolon as a separator to delimit the numbers. The first of them will read like this: “two point nine.”

New concepts

Let's return to ordinary fractions. They come in two types.

The definition of a proper fraction is as follows: it is a fraction whose numerator is less than its denominator. Why is it important? We'll see now!

You have several apples, halved. Total - 5 parts. How would you say: do you have “two and a half” or “five and a half” apples? Of course, the first option sounds more natural, and we will use it when talking with friends. But if we need to calculate how many fruits each person will get, if there are five people in the company, we will write down the number 5/2 and divide it by 5 - from a mathematical point of view, this will be more clear.

So, for naming proper and improper fractions, the rule is this: if a whole part can be distinguished in a fraction (14/5, 2/1, 173/16, 3/3), then it is improper. If this cannot be done, as in the case of ½, 13/16, 9/10, it will be correct.

The main property of a fraction

If the numerator and denominator of a fraction are simultaneously multiplied or divided by the same number, its value does not change. Imagine: they cut the cake into 4 equal parts and gave you one. They cut the same cake into eight pieces and gave you two. Does it really matter? After all, ¼ and 2/8 are the same thing!

Reduction

Authors of problems and examples in mathematics textbooks often seek to confuse students by offering fractions that are cumbersome to write but can actually be abbreviated. Here is an example of a proper fraction: 167/334, which, it would seem, looks very “scary”. But we can actually write it as ½. The number 334 is divisible by 167 without a remainder - after performing this operation, we get 2.

Mixed numbers

An improper fraction can be represented as a mixed number. This is when the whole part is brought forward and written at the level of the horizontal line. In fact, the expression takes the form of a sum: 11/2 = 5 + ½; 13/6 = 2 + 1/6 and so on.

To take out the whole part, you need to divide the numerator by the denominator. Write the remainder of the division on top, above the line, and the whole part - before the expression. Thus, we get two structural parts: whole units + proper fraction.

You can also carry out the inverse operation - to do this, you need to multiply the integer part by the denominator and add the resulting value to the numerator. Nothing complicated.

Multiplication and division

Oddly enough, multiplying fractions is easier than adding. All that is required is to extend the horizontal line: (2/3) * (3/5) = 2*3 / 3*5 = 2/5.

With division, everything is also simple: you need to multiply the fractions crosswise: (7/8) / (14/15) = 7*15 / 8*14 = 15/16.

Adding Fractions

What to do if you need to perform addition or they have different numbers in the denominator? It will not work to do the same as with multiplication - here you should understand the definition of a proper fraction and its essence. It is necessary to bring the terms to a common denominator, that is, the lower part of both fractions must have the same numbers.

To do this, you should use the basic property of a fraction: multiply both parts by the same number. For example, 2/5 + 1/10 = (2*2)/(5*2) + 1/10 = 5/10 = ½.

How to choose which denominator to reduce the terms to? It should be minimum number, a multiple of both numbers in the denominators of the fractions: for 1/3 and 1/9 it will be 9; for ½ and 1/7 - 14, because there is no smaller value divisible by 2 and 7 without a remainder.

Usage

What are improper fractions used for? After all, it is much more convenient to immediately select the whole part, get a mixed number - and be done with it! It turns out that if you need to multiply or divide two fractions, it is more profitable to use irregular ones.

Let's take next example: (2 + 3/17) / (37 / 68).

It would seem that there is nothing to cut at all. But what if we write the addition result in the first parentheses as an improper fraction? Look: (37/17) / (37/68)

Now everything falls into place! Let's write the example in such a way that everything becomes obvious: (37*68) / (17*37).

Let's cancel 37 in the numerator and denominator and finally divide the top and bottom by 17. Do you remember the basic rule for proper and improper fractions? We can multiply and divide them by any number as long as we do it for the numerator and denominator at the same time.

So, we get the answer: 4. The example looked complicated, but the answer contains only one number. This happens often in mathematics. The main thing is not to be afraid and follow simple rules.

Common Mistakes

When implementing, a student can easily make one of the common mistakes. Usually they occur due to inattention, and sometimes due to the fact that the material studied has not yet been properly stored in the head.

Often the sum of numbers in the numerator makes you want to reduce its individual components. Let’s say in the example: (13 + 2) / 13, written without parentheses (with a horizontal line), many students, due to inexperience, cross out 13 above and below. But this should not be done under any circumstances, because this is a gross mistake! If instead of addition there was a multiplication sign, we would get the number 2 in the answer. But when performing addition, no operations with one of the terms are allowed, only with the entire sum.

Guys also often make mistakes when dividing fractions. Let's take two proper irreducible fractions and divide by each other: (5/6) / (25/33). The student can mix it up and write the resulting expression as (5*25) / (6*33). But this would happen with multiplication, but in our case everything will be somewhat different: (5*33) / (6*25). We reduce what is possible, and the answer will be 11/10. We write the resulting improper fraction as a decimal - 1.1.

Brackets

Remember that in any mathematical expression the order of operations is determined by the precedence of the operation signs and the presence of parentheses. All other things being equal, the order of actions is counted from left to right. This is also true for fractions - the expression in the numerator or denominator is calculated strictly according to this rule.

After all, this is the result of dividing one number by another. If they are not evenly divided, it becomes a fraction - that's all.

How to write a fraction on a computer

Since standard tools do not always allow creating a fraction consisting of two “tiers,” students sometimes resort to various tricks. For example, they copy the numerators and denominators into the Paint graphic editor and glue them together, drawing a horizontal line between them. Of course, there is a simpler option, which, by the way, provides a lot of additional features, which will be useful to you in the future.

Open Microsoft Word. One of the panels at the top of the screen is called “Insert” - click it. On the right, on the side where the close and minimize window icons are located, there is a “Formula” button. This is exactly what we need!

If you use this function, a rectangular area will appear on the screen in which you can use any mathematical symbols that are not on the keyboard, as well as write fractions in classic look. That is, dividing the numerator and denominator with a horizontal line. You might even be surprised that such a proper fraction is so easy to write.

Learn math

If you are in grades 5-6, then soon knowledge of mathematics (including the ability to work with fractions!) will be required in many school subjects. In almost any problem in physics, when measuring the mass of substances in chemistry, in geometry and trigonometry, you cannot do without fractions. Soon you will learn to calculate everything in your mind, without even writing down expressions on paper, but more and more complex examples. Therefore, learn what a proper fraction is and how to work with it, keep up with curriculum, do your homework on time and you will succeed.

Fractions

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

Fractions are not much of a nuisance in high school. For the time being. Until you encounter degrees with rational indicators yes logarithms. And there... You press and press the calculator, and it shows a full display of some numbers. You have to think with your head like in the third grade.

Let's finally figure out fractions! Well, how much can you get confused in them!? Moreover, it’s all simple and logical. So, what are the types of fractions?

Types of fractions. Transformations.

There are three types of fractions.

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...), say to yourself the phrase: " Zzzzz remember! Zzzzz denominator - look zzzzz uh!" Look, everything will be zzzz remembered.)

The dash, either horizontal or inclined, means division the top number (numerator) to the bottom (denominator). That's all! Instead of a dash, it is quite possible to put a division sign - two dots.

When complete division is possible, this 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 even talking about the fraction "4/1". Which is also just "4". And if it’s not completely divisible, we leave it as a fraction. Sometimes you have to do the opposite operation. Convert a whole number into a fraction. But more on that later.

2. Decimals , For example:

It is in this form that you will need 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 translated into common fractions. But you definitely need to be able to do this! Otherwise you will come across such a number in a problem and freeze... Out of nowhere. But we will remember this procedure! A little lower.

Most versatile common fractions. Let's start with them. By the way, if a fraction contains all sorts of logarithms, sines and other letters, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!

The main property of a fraction.

So, let's go! To begin with, I will surprise you. The whole variety of fraction transformations is provided by one single property! That's what it's called main property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction does not change. Those:

It is clear that you can continue to write until you are blue in the face. Don’t let sines and logarithms confuse you, we’ll deal with them further. The main thing is to understand that all these various expressions are the same fraction . 2/3.

Do we need it, all these transformations? And how! Now you will see for yourself. To begin with, let's use the basic property of a fraction for reducing fractions. It would seem like an elementary thing. Divide the numerator and denominator by the same number and that's it! It's impossible to make a mistake! But... man is a creative being. You can make a mistake anywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.

How to correctly and quickly reduce fractions without doing extra work can be read in the special Section 555.

A normal student doesn't bother dividing the numerator and denominator by the same number (or expression)! He simply crosses out everything that is the same above and below! This is where it lurks typical mistake, a blooper, if you will.

For example, you need to simplify the expression:

There’s nothing to think about here, cross out the letter “a” on top and the two on the bottom! We get:

Everything is correct. But really you divided all numerator and all the denominator is "a". If you are used to just crossing out, then in a hurry you can cross out the “a” in the expression

and get it again

Which would be categorically untrue. Because here all the numerator on "a" is already not shared! This fraction cannot be reduced. By the way, such a reduction is, um... a serious challenge for the teacher. This is not forgiven! Do you remember? When reducing, you need to divide all numerator and all denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. How can I continue to work with her now? Without a calculator? Multiply, say, add, square!? And if you’re not too lazy, and carefully cut it down by five, and by another five, and even... while it’s being shortened, in short. Let's get 3/8! Much nicer, right?

The main property of a fraction allows you to convert ordinary fractions to decimals and vice versa without a calculator! This is important for the Unified State Exam, right?

How to convert fractions from one type to another.

With decimal fractions everything is simple. As it is heard, so it is written! Let's say 0.25. This is zero point twenty five hundredths. So we write: 25/100. We reduce (we 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 the integers are not zero? It's OK. We 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 point 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 that has been said, a useful conclusion: any decimal fraction can be converted to a common fraction .

But some people cannot do the reverse conversion from ordinary to decimal without a calculator. And it is necessary! How will you write down the answer on the Unified State Exam!? Read carefully and master this process.

What is the characteristic of a decimal fraction? Her denominator is Always costs 10, or 100, or 1000, or 10000 and so on. If your common fraction has a denominator like this, there's no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. What if the answer to the task in section “B” turned out to be 1/2? What will we write in response? Decimals are required...

Let's remember main property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. Anything, by the way! Except zero, of course. So let’s use this property 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...)? At 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 = 1x5/2x5 = 5/10 = 0.5. That's all.

However, all sorts of denominators come across. You will come across, for example, the fraction 3/16. Try and figure out what to multiply 16 by to make 100, or 1000... Doesn’t it work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide with a corner, on a piece of paper, as they taught in elementary school. We get 0.1875.

And there are also very bad denominators. For example, there is no way to turn the fraction 1/3 into a good decimal. Both on the calculator and on a piece of paper, we get 0.3333333... This means that 1/3 is an exact decimal fraction does not translate. Same as 1/7, 5/6 and so on. There are many of them, untranslatable. This brings us to another useful conclusion. Not every fraction can be converted to a decimal !

By the way, this helpful information for self-test. In section "B" you must write down a decimal fraction in your answer. And you got, for example, 4/3. This fraction does not convert to a decimal. This means you made a mistake somewhere along the way! Go back and check the solution.

So, we figured out ordinary and decimal fractions. All that remains is to deal with mixed numbers. To work with them, they must be converted into ordinary fractions. How to do it? You can catch a sixth grader and ask him. But a sixth grader won’t always be at hand... You’ll have to do it yourself. It is not difficult. You need to multiply the denominator of the fractional part by the whole part and add the numerator of the fractional part. This will be the numerator of the common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but in reality everything is simple. Let's look at an example.

Suppose you were horrified to see the number in the problem:

Calmly, without panic, we think. The whole part is 1. Unit. 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 a common fraction. That's all. It looks even simpler in mathematical notation:

Is it clear? Then secure your success! Convert to ordinary fractions. You should get 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction to a mixed number - is rarely required in high school. Well, if so... And if you are not in high school, you can look into the special Section 555. By the way, you will also learn about improper fractions there.

Well, that's practically all. You remembered the types of fractions and understood How transfer 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 are mixed together, we convert everything into ordinary fractions. It can always be done. Well, if it says something like 0.8 + 0.3, then we count it that way, without any translation. Why do we need extra work? We choose the solution that is convenient us !

If the task is all decimal fractions, but um... some kind of evil ones, go to ordinary ones and try it! Look, everything will work out. For example, you will have to square the number 0.125. It’s not so easy if you haven’t gotten used to using a calculator! Not only do you have to multiply numbers in a column, you also have to think about where to insert the comma! It definitely won’t work in your head! What if we move on to an ordinary fraction?

0.125 = 125/1000. We reduce it by 5 (this is for starters). We get 25/200. Once again by 5. We get 5/40. Oh, it's still shrinking! Back to 5! We get 1/8. We easily square it (in our minds!) and get 1/64. All!

Let's summarize this lesson.

1. There are three types of fractions. Common, decimal and mixed numbers.

2. Decimals and mixed numbers Always can be converted to ordinary fractions. Reverse transfer not always available.

3. The choice of the type of fractions to work with a task depends on the task itself. In the presence of different types fractions in one task, the most reliable thing is to move on to ordinary fractions.

Now you can practice. First, convert these decimal fractions to ordinary fractions:

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

You should get answers like this (in a mess!):

Let's wrap this up. In this lesson we refreshed our memory key points by fractions. It happens, however, that there is nothing special to refresh...) If someone has completely forgotten, or has not yet mastered it... Then you can go to a special Section 555. All the basics are covered in detail 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. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

In mathematics, a fraction is a number made up of one or more units. That is, a fraction represents some part of one whole. For example, if an object is divided into 4 equal parts and taken 1 of them, we get the fraction 1/4, where 3 is the numerator, 4 is the denominator, and the result of such division (0.25) is the quotient. Various fractions are used in the school curriculum; what they are called depends on their type.

Common, decimal and periodic fractions

According to the recording method, ordinary and decimal fractions are distinguished. In the first case, the fraction is also called a simple fraction. It consists of two natural numbers separated by a horizontal or slash, as in the image below.

A decimal is an ordinary fraction with a denominator of one followed by zeros, an example of such a fraction is shown in the following figure. However, such fractions are usually written without a denominator, and a comma (0.3) is used to indicate a part of the whole. In this case, as many numbers are indicated after the decimal point as there are zeros in the denominator of the simple fraction.

The part of the decimal fraction written before the positional point is called the whole part of the fraction, after it - decimals. Moreover, the number of decimal places can be either finite (2.3) or infinite (2.333333).

In the latter case we're talking about about periodic fractions, since repeating numbers are called periods. In writing, it is customary to enclose the period in brackets, for example, 2,(3). This entry reads like this: two integers and three in a period. However, periodic fractions can be rounded, then they are often called round fractions, although in mathematics it would be more correct to say a rounded fraction.

Proper, improper and mixed fractions

A fraction is called proper when the modulus of the numerator is less than the modulus of the denominator (1/3, 2/5, 7/8), otherwise the fraction is called an improper fraction (3/2, 9/7, 13/5). Fractions where the numerator and denominator are equal are also classified as improper fractions.

At the same time, any improper fraction can be represented as a mixed fraction; an example of such a fraction is given below.

Here 1 is the integer part of the mixed number, and 1/2 is the fractional part. To convert a mixed number into a fraction, you need to multiply the whole part by the denominator and add the numerator to the resulting value. As a result of such actions, the numerator of an ordinary fraction is found, while the denominator remains the same.

Reducible and irreducible fractions

When the numerator and denominator of a fraction can be divided by the same number (except for one), the fraction is called reducible, in any other case - irreducible. For example:

• 3/9 is a reducible fraction, since both the numerator and denominator can be divided by 3;
• 3/5 is an irreducible fraction, since both numbers are prime, i.e. are divisible only by themselves and 1;
• 2/7 is an irreducible fraction, since there is no common number that can divide both the numerator and the denominator.

Composite and reciprocal fractions

Often schoolchildren do not understand which fraction is called a reciprocal and which is a composite. It turns out that everything is quite simple. If we take the fraction 7/8 and swap the numerator and denominator, we get the fraction 8/7. It is these fractions (7/8 and 8/7) that are called reciprocal. Moreover, it should be noted that the product of such fractions is always equal to 1.

Compound fractions include expressions that include several features of the fraction. Examples of such fractions are given below.

In addition, a distinction is made between positive and negative fractions. To indicate the latter, a “-” sign is placed before the fraction. In this case, the “+” sign is usually not indicated, as with positive numbers.

Fractions of a unit and is represented as \frac(a)(b).

Numerator of fraction (a)- the number located above the fraction line and showing the number of shares into which the unit was divided.

Fraction denominator (b)- the number located under the line of the fraction and showing how many parts the unit is divided into.

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The main property of a fraction

If ad=bc then two fractions \frac(a)(b) And \frac(c)(d) are considered equal. For example, the fractions will be equal \frac35 And \frac(9)(15), since 3 \cdot 15 = 15 \cdot 9 , \frac(12)(7) And \frac(24)(14), since 12 \cdot 14 = 7 \cdot 24 .

From the definition of equality of fractions it follows that the fractions will be equal \frac(a)(b) And \frac(am)(bm), since a(bm)=b(am) - clear example application of the associative and commutative properties of multiplication of natural numbers in action.

Means \frac(a)(b) = \frac(am)(bm)- this is what it looks like main property of a fraction.

In other words, we get a fraction equal to the given one by multiplying or dividing the numerator and denominator of the original fraction by the same natural number.

Reducing a fraction is the process of replacing a fraction in which the new fraction is equal to the original one, but with a smaller numerator and denominator.

It is customary to reduce fractions based on the basic property of the fraction.

For example, \frac(45)(60)=\frac(15)(20)(numerator and denominator are divided by the number 3); the resulting fraction can again be reduced by dividing by 5, that is \frac(15)(20)=\frac 34.

Irreducible fraction is a fraction of the form \frac 34, where the numerator and denominator are mutually prime numbers. The main purpose of reducing a fraction is to make the fraction irreducible.

Reducing fractions to a common denominator

Let's take two fractions as an example: \frac(2)(3) And \frac(5)(8) with different denominators 3 and 8. In order to bring these fractions to a common denominator, we first multiply the numerator and denominator of the fraction \frac(2)(3) by 8. We get the following result: \frac(2 \cdot 8)(3 \cdot 8) = \frac(16)(24). Then we multiply the numerator and denominator of the fraction \frac(5)(8) by 3. As a result we get: \frac(5 \cdot 3)(8 \cdot 3) = \frac(15)(24). So, the original fractions are reduced to a common denominator 24.

Arithmetic operations on ordinary fractions

Addition of ordinary fractions

a) If the denominators are the same, the numerator of the first fraction is added to the numerator of the second fraction, leaving the denominator the same. As you can see in the example:

\frac(a)(b)+\frac(c)(b)=\frac(a+c)(b);

b) For different denominators, fractions are first reduced to a common denominator, and then the numerators are added according to rule a):

\frac(7)(3)+\frac(1)(4)=\frac(7 \cdot 4)(3)+\frac(1 \cdot 3)(4)=\frac(28)(12) +\frac(3)(12)=\frac(31)(12).

Subtracting fractions

a) If the denominators are the same, subtract the numerator of the second fraction from the numerator of the first fraction, leaving the denominator the same:

\frac(a)(b)-\frac(c)(b)=\frac(a-c)(b);

b) If the denominators of the fractions are different, then first the fractions are brought to a common denominator, and then the actions are repeated as in point a).

Multiplying common fractions

Multiplying fractions obeys the following rule:

\frac(a)(b) \cdot \frac(c)(d)=\frac(a \cdot c)(b \cdot d),

that is, they multiply the numerators and denominators separately.

For example:

\frac(3)(5) \cdot \frac(4)(8) = \frac(3 \cdot 4)(5 \cdot 8)=\frac(12)(40).

Dividing fractions

Fractions are divided in the following way:

\frac(a)(b) : \frac(c)(d)= \frac(ad)(bc),

that is, a fraction \frac(a)(b) multiplied by a fraction \frac(d)(c).

Example: \frac(7)(2) : \frac(1)(8)=\frac(7)(2) \cdot \frac(8)(1)=\frac(7 \cdot 8)(2 \cdot 1 )=\frac(56)(2).

Reciprocal numbers

If ab=1 , then the number b is reciprocal number for the number a.

Example: for the number 9 the reciprocal is \frac(1)(9), because 9\cdot\frac(1)(9)=1, for the number 5 - \frac(1)(5), because 5\cdot\frac(1)(5)=1.

Decimals

Decimal called a proper fraction whose denominator is 10, 1000, 10\,000, ..., 10^n.

For example: \frac(6)(10)=0.6;\enspace \frac(44)(1000)=0.044.

Irregular numbers with a denominator of 10^n or mixed numbers are written in the same way.

For example: 5\frac(1)(10)=5.1;\enspace \frac(763)(100)=7\frac(63)(100)=7.63.

Any ordinary fraction with a denominator that is a divisor of a certain power of 10 is represented as a decimal fraction.

Example: 5 is a divisor of 100, so it is a fraction \frac(1)(5)=\frac(1 \cdot 20)(5 \cdot 20)=\frac(20)(100)=0.2.

Arithmetic operations on decimals

Adding Decimals

To add two decimal fractions, you need to arrange them so that there are identical digits under each other and a comma under the comma, and then add the fractions like ordinary numbers.

Subtracting Decimals

It is performed in the same way as addition.

Multiplying Decimals

When multiplying decimal numbers it is enough to multiply the given numbers, not paying attention to commas (as integers), and in the resulting answer, a comma on the right separates as many digits as there are after the decimal point in both factors in total.

Let's multiply 2.7 by 1.3. We have 27 \cdot 13=351 . We separate two digits on the right with a comma (the first and second numbers have one digit after the decimal point; 1+1=2). As a result, we get 2.7 \cdot 1.3=3.51.

If the resulting result contains fewer digits than need to be separated by a comma, then the missing zeros are written in front, for example:

To multiply by 10, 100, 1000, you need to move the decimal point 1, 2, 3 digits to the right (if necessary, a certain number of zeros are assigned to the right).

For example: 1.47\cdot 10\,000 = 14,700.

Decimal division

Dividing a decimal fraction by a natural number is done in the same way as dividing a natural number by a natural number. The comma in the quotient is placed after the division of the whole part is completed.

If the integer part of the dividend is less than the divisor, then the answer is zero integers, for example:

Let's look at dividing a decimal by a decimal. Let's say we need to divide 2.576 by 1.12. First of all, let's multiply the dividend and divisor of the fraction by 100, that is, move the decimal point to the right in the dividend and divisor by as many digits as there are in the divisor after the decimal point (in in this example by two). Then you need to divide the fraction 257.6 by the natural number 112, that is, the problem is reduced to the case already considered:

It happens that the final decimal fraction is not always obtained when dividing one number by another. The result is an infinite decimal fraction. In such cases, we move on to ordinary fractions.

2.8: 0.09= \frac(28)(10) : \frac (9)(100)= \frac(28 \cdot 100)(10 \cdot 9)=\frac(280)(9)= 31\frac(1)(9).

1 What are ordinary fractions? Types of fractions.
A fraction always means some part of a whole. The fact is that quantity cannot always be expressed in natural numbers, that is, recalculated: 1,2,3, etc. How, for example, do you designate half a watermelon or a quarter of an hour? This is why fractions or numbers 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:
Decimal fractions 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 fraction line. So remember:

Any fraction is part of a whole. Usually taken as a whole 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 equal parts (in mathematics they say shares ), then each part of the cake will be equal to 1/6. If Vasya ate 4 pieces, it means he ate 4/6.

On the other hand, a slash is nothing more than a division sign. Therefore, a fraction is the quotient of two numbers - the numerator and the denominator. In the text of problems or in recipes, fractions are usually written like this: 2/3, 1/2, etc. Some fractions have their own names, for example, 1/2 - “half”, 1/3 - “third”, 1/4 - “quarter”
Now let’s figure out what types of ordinary fractions there are.

2 Types of ordinary fractions

There are three types of common fractions: proper, improper and mixed:

Proper fraction

If the numerator is less than the denominator, then such a fraction is called correct, For example: A proper fraction is always less than 1.

Improper fraction

If the numerator is greater than the denominator or equal to the denominator, such a 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 (integer 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 converted into an improper fraction and vice versa. This is necessary to perform certain operations, such as multiplication and division.

So, any mixed fraction can be converted to an improper fraction. To do this, the whole 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 have isolated the whole part from the improper fraction.”

One more rule to remember: Any integer can be represented as a fraction with a denominator of 1, For example:

Let's talk about how to compare fractions.

4 Comparison of fractions

When comparing fractions, there can be several options: It is easy to compare fractions with the same denominators, but it is much more difficult if the denominators are different. And there is also a comparison of mixed fractions. But don't worry, now we'll look at each option in detail and learn how to compare fractions.

Comparing fractions with the same denominators

Of two fractions with the same denominators but different numerators, the fraction with the larger numerator is greater, for example:

Comparing fractions with the same numerators

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 larger than a proper fraction, for example:

Comparing two mixed fractions

When comparing two mixed fractions, the fraction whose whole part is larger is greater, for example:

If the whole parts of mixed fractions are the same, the fraction whose fractional part is larger is greater, for example:

Comparing fractions with different numerators and denominators

You cannot compare fractions with different numerators and denominators without converting them. First, the fractions must be reduced to the same denominator, and then their numerators must be compared. The greater is the fraction whose numerator is larger. But we will look at how to reduce fractions to the same denominator in the next two sections of the article. First we will look at the basic property of fractions and reducing fractions, and then directly reducing fractions to the same denominator.

5 The main property of a fraction. Reducing fractions. The concept of GCD.

Remember: You can only add and subtract and compare fractions that have the same denominators. If the denominators are different, then you first need to bring the fractions to the same denominator, that is, transform one of the fractions so that its denominator becomes the same as that of the second fraction.

Fractions have one thing important property, also called the main 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 does not change:

Thanks to this property we can reduce fractions:

To reduce a fraction is to divide both the numerator and the denominator by the same number.(see example just above). When we reduce a fraction, we can write our actions like this:

More often in notebooks the fraction is abbreviated as follows:

But remember: you can only reduce factors. If the numerator or denominator contains a sum or a difference, you cannot reduce the terms. Example:

You must first convert the sum to a multiplier:

Sometimes, when working with large numbers, in order to reduce a fraction, it is convenient to find greatest common divisor of numerator and denominator (GCD)

Greatest Common Divisor (GCD) several numbers 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 factor both numbers into prime factors, mark the same factors in both factorizations, and multiply these factors. The resulting product will be the GCD. For example, we need to reduce a fraction:

Let's find the gcd of numbers 96 and 36:

GCD shows us that both the numerator and the denominator have a factor of 12, 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 reduce fractions to the same denominator. Least common multiple (LCM).

When we reduce fractions to the same denominator, we select a number for the denominator that would be divisible by both the first and 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, it is more convenient to count. Thus, we must 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:

1. Factor these numbers into prime factors
2. Take the largest expansion and write these numbers as a product
3. Select in other decompositions the numbers that do not appear in the largest decomposition (or occur fewer times in it), and add them to the product.
4. Multiply all the numbers in the product, this will be the LCM.

For example, let's find the LCM of the numbers 28 and 21:

However, let's return to our fractions. After we have found or written calculated 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 the same denominator - 15.

7 Adding and subtracting fractions

Adding and subtracting fractions with like denominators

To add fractions with the same denominators, you need to add their numerators, but leave the denominator the same, for example:

To subtract fractions with the same denominators, you need to subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same, for example:

Adding and subtracting mixed fractions with like denominators

To add mixed fractions, you need to separately add their whole parts, and then add their fractional parts, and write the result as a mixed fraction:

If, when adding fractional parts, you get an improper fraction, select the whole part from it and add it to the whole part, for example:

Subtraction is carried out in a similar way: the integer part is subtracted from the whole part, 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 “borrow” one from the whole part, turning the minuend into an improper fraction, and then proceed as usual:

Likewise subtract a fraction from a whole number:

How to add a whole number and a fraction

To add a whole number and a fraction, you simply add that number before the fraction to create a mixed fraction, for example:

If we adding a whole number and a mixed fraction, we add this number to the whole part of the fraction, for example:

Adding and subtracting 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 whole part from the whole part, and the fractional part from the fractional part:

8 Multiplying and dividing fractions.

Multiplying and dividing fractions is much easier than adding and subtracting because you don't have to reduce them to the same denominator. Remember simple rules multiplying and dividing fractions:

Before multiplying the numbers in the numerator and denominator, it is advisable to reduce the fraction, that is, 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:

Dividing a fraction by a fraction

To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor (the reciprocal fraction). What kind of reciprocal fraction is this?

If we flip the fraction, that is, we swap the numerator and denominator, we get a reciprocal fraction. The product of a fraction and its inverse gives one. In mathematics, such numbers are called reciprocals:

For example, numbers - are mutually inverse, since

Thus, let's return to dividing 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, just like when multiplying, you must first convert them to improper fractions:

When multiplying and dividing fractions by whole natural numbers, you can also represent these numbers as fractions with a denominator 1 .

And when dividing a whole number by a fraction represent this number as a fraction with a denominator 1 :