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 come across powers with rational exponents and 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 converted into ordinary 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 doesn't share! 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 can be turned into ordinary .

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 common 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 finish here. 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 this article we will look at how converting fractions to decimals, and also consider the reverse process - converting decimal fractions into ordinary fractions. Here we will outline the rules for converting fractions and give detailed solutions typical examples.

Page navigation.

Converting fractions to decimals

Let us denote the sequence in which we will deal with converting fractions to decimals.

First, we'll look at how to represent fractions with denominators 10, 100, 1,000, ... as decimals. This is explained by the fact that decimal fractions are essentially a compact form of writing ordinary fractions with denominators 10, 100, ....

After that, we will go further and show how to write any ordinary fraction (not just those with denominators 10, 100, ...) as a decimal fraction. When ordinary fractions are treated in this way, both finite decimal fractions and infinite periodic decimal fractions are obtained.

Now let's talk about everything in order.

Converting common fractions with denominators 10, 100, ... to decimals

Some proper fractions require "preliminary preparation" before being converted to decimals. This applies to ordinary fractions, the number of digits in the numerator of which is less than the number of zeros in the denominator. For example, the common fraction 2/100 must first be prepared for conversion to a decimal fraction, but the fraction 9/10 does not need any preparation.

“Preliminary preparation” of proper ordinary fractions for conversion to decimal fractions consists of adding so many zeros to the left of the numerator that total digits became equal to the number of zeros in the denominator. For example, a fraction after adding zeros will look like .

Once you have a proper fraction prepared, you can begin converting it to a decimal.

Let's give rule for converting a proper common fraction with a denominator of 10, or 100, or 1,000, ... into a decimal fraction. It consists of three steps:

• write 0;
• after it we put a decimal point;
• We write down the number from the numerator (along with added zeros, if we added them).

Let's consider the application of this rule when solving examples.

Example.

Convert the proper fraction 37/100 to a decimal.

Solution.

The denominator contains the number 100, which has two zeros. The numerator contains the number 37, its notation has two digits, therefore, this fraction does not need to be prepared for conversion to a decimal fraction.

Now we write 0, put a decimal point, and write the number 37 from the numerator, and we get the decimal fraction 0.37.

Answer:

0,37 .

To strengthen the skills of converting proper ordinary fractions with numerators 10, 100, ... into decimal fractions, we will analyze the solution to another example.

Example.

Write the proper fraction 107/10,000,000 as a decimal.

Solution.

The number of digits in the numerator is 3, and the number of zeros in the denominator is 7, so this common fraction needs to be prepared for conversion to a decimal. We need to add 7-3=4 zeros to the left in the numerator so that the total number of digits there becomes equal to the number of zeros in the denominator. We get.

All that remains is to create the required decimal fraction. To do this, firstly, we write 0, secondly, we put a comma, thirdly, we write the number from the numerator together with zeros 0000107, as a result we have a decimal fraction 0.0000107.

Answer:

0,0000107 .

Improper fractions do not require any preparation when converting to decimals. The following should be adhered to rules for converting improper fractions with denominators 10, 100, ... into decimals:

• write down the number from the numerator;
• We use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.

Let's look at the application of this rule when solving an example.

Example.

Convert the improper fraction 56,888,038,009/100,000 to a decimal.

Solution.

Firstly, we write down the number from the numerator 56888038009, and secondly, we separate the 5 digits on the right with a decimal point, since the denominator of the original fraction has 5 zeros. As a result, we have the decimal fraction 568880.38009.

Answer:

568 880,38009 .

To convert a mixed number into a decimal fraction, the denominator of the fractional part of which is the number 10, or 100, or 1,000, ..., you can convert the mixed number into an improper ordinary fraction, and then convert the resulting fraction into a decimal fraction. But you can also use the following the rule for converting mixed numbers with a fractional denominator of 10, or 100, or 1,000, ... into decimal fractions:

• if necessary, we perform “preliminary preparation” of the fractional part of the original mixed number by adding the required number of zeros to the left in the numerator;
• write down the integer part of the original mixed number;
• put a decimal point;
• We write down the number from the numerator along with the added zeros.

Let's look at an example in which we complete all the necessary steps to represent a mixed number as a decimal fraction.

Example.

Convert the mixed number to a decimal.

Solution.

The denominator of the fractional part has 4 zeros, and the numerator contains the number 17, consisting of 2 digits, therefore, we need to add two zeros to the left in the numerator so that the number of digits there becomes equal to the number of zeros in the denominator. Having done this, the numerator will be 0017.

Now we write down the integer part of the original number, that is, the number 23, put a decimal point, after which we write the number from the numerator along with the added zeros, that is, 0017, and we get the desired decimal fraction 23.0017.

Let's write down the whole solution briefly: .

Of course, it was possible to first represent the mixed number as an improper fraction and then convert it to a decimal fraction. With this approach, the solution looks like this: .

Answer:

23,0017 .

Converting fractions to finite and infinite periodic decimals

You can convert not only ordinary fractions with denominators 10, 100, ... into a decimal fraction, but also ordinary fractions with other denominators. Now we will figure out how this is done.

In some cases, the original ordinary fraction is easily reduced to one of the denominators 10, or 100, or 1,000, ... (see bringing an ordinary fraction to a new denominator), after which it is not difficult to represent the resulting fraction as a decimal fraction. For example, it is obvious that the fraction 2/5 can be reduced to a fraction with a denominator 10, for this you need to multiply the numerator and denominator by 2, which will give the fraction 4/10, which, according to the rules discussed in the previous paragraph, is easily converted to the decimal fraction 0, 4 .

In other cases, you have to use another method of converting an ordinary fraction to a decimal, which we now move on to consider.

To convert an ordinary fraction to a decimal fraction, the numerator of the fraction is divided by the denominator, the numerator is first replaced by an equal decimal fraction with any number of zeros after the decimal point (we talked about this in the section equal and unequal decimal fractions). In this case, division is performed in the same way as division by a column of natural numbers, and in the quotient a decimal point is placed when the division of the whole part of the dividend ends. All this will become clear from the solutions to the examples given below.

Example.

Convert the fraction 621/4 to a decimal.

Solution.

Let's represent the number in the numerator 621 as a decimal fraction, adding a decimal point and several zeros after it. First, let's add 2 digits 0, later, if necessary, we can always add more zeros. So, we have 621.00.

Now let's divide the number 621,000 by 4 with a column. The first three steps are no different from dividing natural numbers by a column, after which we arrive at the following picture:

This is how we get to the decimal point in the dividend, and the remainder is different from zero. In this case, we put a decimal point in the quotient and continue dividing in a column, not paying attention to the commas:

This completes the division, and as a result we get the decimal fraction 155.25, which corresponds to the original ordinary fraction.

Answer:

155,25 .

To consolidate the material, consider the solution to another example.

Example.

Convert the fraction 21/800 to a decimal.

Solution.

To convert this common fraction to a decimal, we divide with a column of the decimal fraction 21,000... by 800. After the first step, we will have to put a decimal point in the quotient, and then continue the division:

Finally, we got the remainder 0, this completes the conversion of the common fraction 21/400 to a decimal fraction, and we arrived at the decimal fraction 0.02625.

Answer:

0,02625 .

It may happen that when dividing the numerator by the denominator of an ordinary fraction, we still do not get a remainder of 0. In these cases, division can be continued indefinitely. However, starting from a certain step, the remainders begin to repeat periodically, and the numbers in the quotient also repeat. This means that the original fraction is converted to an infinite periodic decimal fraction. Let's show this with an example.

Example.

Write the fraction 19/44 as a decimal.

Solution.

To convert an ordinary fraction to a decimal, perform division by column:

It is already clear that during division the residues 8 and 36 began to be repeated, while in the quotient the numbers 1 and 8 are repeated. Thus, the original common fraction 19/44 is converted into a periodic decimal fraction 0.43181818...=0.43(18).

Answer:

0,43(18) .

To conclude this point, we will figure out which ordinary fractions can be converted into finite decimal fractions, and which ones can only be converted into periodic ones.

Let us have an irreducible ordinary fraction in front of us (if the fraction is reducible, then we first reduce the fraction), and we need to find out which decimal fraction it can be converted into - finite or periodic.

It is clear that if an ordinary fraction can be reduced to one of the denominators 10, 100, 1,000, ..., then the resulting fraction can be easily converted into a final decimal fraction according to the rules discussed in the previous paragraph. But to the denominators 10, 100, 1,000, etc. Not all ordinary fractions are given. Only fractions whose denominators are at least one of the numbers 10, 100, ... can be reduced to such denominators. And what numbers can be divisors of 10, 100, ...? The numbers 10, 100, ... will allow us to answer this question, and they are as follows: 10 = 2 5, 100 = 2 2 5 5, 1,000 = 2 2 2 5 5 5, .... It follows that the divisors are 10, 100, 1,000, etc. There can only be numbers whose decompositions into prime factors contain only the numbers 2 and (or) 5.

Now we can make a general conclusion about converting ordinary fractions to decimals:

• if in the decomposition of the denominator into prime factors only the numbers 2 and (or) 5 are present, then this fraction can be converted into a final decimal fraction;
• if, in addition to twos and fives, there are others in the expansion of the denominator prime numbers, then this fraction is converted to an infinite decimal periodic fraction.

Example.

Without converting ordinary fractions to decimals, tell me which of the fractions 47/20, 7/12, 21/56, 31/17 can be converted into a final decimal fraction, and which ones can only be converted into a periodic fraction.

Solution.

The denominator of the fraction 47/20 is factorized into prime factors as 20=2·2·5. In this expansion there are only twos and fives, so this fraction can be reduced to one of the denominators 10, 100, 1,000, ... (in this example, to the denominator 100), therefore, can be converted to a final decimal fraction.

The decomposition of the denominator of the fraction 7/12 into prime factors has the form 12=2·2·3. Since it contains a prime factor of 3, different from 2 and 5, this fraction cannot be represented as a finite decimal, but can be converted into a periodic decimal.

Fraction 21/56 – contractile, after contraction it takes the form 3/8. Factoring the denominator into prime factors contains three factors equal to 2, therefore, the common fraction 3/8, and therefore the equal fraction 21/56, can be converted into a final decimal fraction.

Finally, the expansion of the denominator of the fraction 31/17 is 17 itself, therefore this fraction cannot be converted into a finite decimal fraction, but can be converted into an infinite periodic fraction.

Answer:

47/20 and 21/56 can be converted to a finite decimal fraction, but 7/12 and 31/17 can only be converted to a periodic fraction.

Ordinary fractions do not convert to infinite non-periodic decimals

Information previous paragraph raises the question: “Can dividing the numerator of a fraction by the denominator result in an infinite non-periodic fraction?”

Answer: no. When converting a common fraction, the result can be either a finite decimal fraction or an infinite periodic decimal fraction. Let us explain why this is so.

From the divisibility theorem with remainder it is clear that the remainder is always less than divisor, that is, if we divide some integer by an integer q, then the remainder can only be one of the numbers 0, 1, 2, ..., q−1. It follows that after the column has completed dividing the integer part of the numerator of an ordinary fraction by the denominator q, in no more than q steps one of the following two situations will arise:

• or we will get a remainder of 0, this will end the division, and we will get the final decimal fraction;
• or we will get a remainder that has already appeared before, after which the remainders will begin to repeat as in the previous example (since when dividing equal numbers by q, equal remainders are obtained, which follows from the already mentioned divisibility theorem), this will result in an infinite periodic decimal fraction.

There cannot be any other options, therefore, when converting an ordinary fraction to a decimal fraction, an infinite non-periodic decimal fraction cannot be obtained.

From the reasoning given in this paragraph it also follows that the length of the period of a decimal fraction is always less than the value of the denominator of the corresponding ordinary fraction.

Converting decimals to fractions

Now let's figure out how to convert a decimal fraction into an ordinary fraction. Let's start by converting final decimal fractions to ordinary fractions. After this, we will consider a method for inverting infinite periodic decimal fractions. In conclusion, let's say about the impossibility of converting infinite non-periodic decimal fractions into ordinary fractions.

Converting trailing decimals to fractions

Obtaining a fraction that is written as a final decimal is quite simple. The rule for converting a final decimal fraction to a common fraction consists of three steps:

• firstly, write the given decimal fraction into the numerator, having previously discarded the decimal point and all zeros on the left, if any;
• secondly, write one into the denominator and add as many zeros to it as there are digits after the decimal point in the original decimal fraction;
• thirdly, if necessary, reduce the resulting fraction.

Let's look at the solutions to the examples.

Example.

Convert the decimal 3.025 to a fraction.

Solution.

If we remove the decimal point from the original decimal fraction, we get the number 3,025. There are no zeros on the left that we would discard. So, we write 3,025 in the numerator of the desired fraction.

We write the number 1 into the denominator and add 3 zeros to the right of it, since in the original decimal fraction there are 3 digits after the decimal point.

So we got the common fraction 3,025/1,000. This fraction can be reduced by 25, we get .

Answer:

.

Example.

Convert the decimal fraction 0.0017 to a fraction.

Solution.

Without a decimal point, the original decimal fraction looks like 00017, discarding the zeros on the left we get the number 17, which is the numerator of the desired ordinary fraction.

We write one with four zeros in the denominator, since the original decimal fraction has 4 digits after the decimal point.

As a result, we have an ordinary fraction 17/10,000. This fraction is irreducible, and the conversion of a decimal fraction to an ordinary fraction is complete.

Answer:

.

When the integer part of the original final decimal fraction is non-zero, it can be immediately converted to a mixed number, bypassing the common fraction. Let's give rule for converting a final decimal fraction to a mixed number:

• the number before the decimal point must be written as an integer part of the desired mixed number;
• in the numerator of the fractional part you need to write the number obtained from the fractional part of the original decimal fraction after discarding all the zeros on the left;
• in the denominator of the fractional part you need to write down the number 1, to which add as many zeros to the right as there are digits after the decimal point in the original decimal fraction;
• if necessary, reduce the fractional part of the resulting mixed number.

Let's look at an example of converting a decimal fraction to a mixed number.

Example.

Express the decimal fraction 152.06005 as a mixed number

If we need to divide 497 by 4, then when dividing we will see that 497 is not evenly divisible by 4, i.e. the remainder of the division remains. In such cases it is said that it is completed division with remainder, and the solution is written as follows:
497: 4 = 124 (1 remainder).

The division components on the left side of the equality are called the same as in division without a remainder: 497 - dividend, 4 - divider. The result of division when divided with a remainder is called incomplete private. In our case, this is the number 124. And finally, the last component, which is not in ordinary division, is remainder. In cases where there is no remainder, one number is said to be divided by another without a trace, or entirely. It is believed that with such a division the remainder is zero. In our case, the remainder is 1.

The remainder is always less than the divisor.

Division can be checked by multiplication. If, for example, there is an equality 64: 32 = 2, then the check can be done like this: 64 = 32 * 2.

Often in cases where division with a remainder is performed, it is convenient to use the equality
a = b * n + r,
where a is the dividend, b is the divisor, n is the partial quotient, r is the remainder.

The quotient of natural numbers can be written as a fraction.

The numerator of a fraction is the dividend, and the denominator is the divisor.

Since the numerator of a fraction is the dividend and the denominator is the divisor, believe that the line of a fraction means the action of division. Sometimes it is convenient to write division as a fraction without using the ":" sign.

The quotient of the division of natural numbers m and n can be written as a fraction \(\frac(m)(n) \), where the numerator m is the dividend, and the denominator n is the divisor:
\(m:n = \frac(m)(n) \)

The following rules are true:

To get the fraction \(\frac(m)(n)\), you need to divide the unit into n equal parts (shares) and take m such parts.

To get the fraction \(\frac(m)(n)\), you need to divide the number m by the number n.

To find a part of a whole, you need to divide the number corresponding to the whole by the denominator and multiply the result by the numerator of the fraction that expresses this part.

To find a whole from its part, you need to divide the number corresponding to this part by the numerator and multiply the result by the denominator of the fraction that expresses this part.

If both the numerator and denominator of a fraction are multiplied by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a \cdot n)(b \cdot n) \)

If both the numerator and denominator of a fraction are divided by the same number (except zero), the value of the fraction will not change:
\(\large \frac(a)(b) = \frac(a: m)(b: m) \)
This property is called main property of a fraction.

The last two transformations are called reducing a fraction.

If fractions need to be represented as fractions with the same denominator, then this action is called bringing fractions to a common denominator.

Proper and improper fractions. Mixed numbers

You already know that a fraction can be obtained by dividing a whole into equal parts and taking several such parts. For example, the fraction \(\frac(3)(4)\) means three-quarters of one. In many of the problems in the previous paragraph, fractions were used to represent parts of a whole. Common sense suggests that the part should always be less than the whole, but then what about fractions such as, for example, \(\frac(5)(5)\) or \(\frac(8)(5)\)? It is clear that this is no longer part of the unit. This is probably why fractions whose numerator is greater than or equal to the denominator are called improper fractions. The remaining fractions, i.e. fractions whose numerator is less than the denominator, are called correct fractions.

As you know, any common fraction, both proper and improper, can be thought of as the result of dividing the numerator by the denominator. Therefore, in mathematics, unlike ordinary language, the term “improper fraction” does not mean that we did something wrong, but only that the numerator of this fraction is greater than or equal to the denominator.

If a number consists of an integer part and a fraction, then such fractions are called mixed.

For example:
\(5:3 = 1\frac(2)(3) \) : 1 is the integer part, and \(\frac(2)(3) \) is the fractional part.

If the numerator of the fraction \(\frac(a)(b) \) is divisible by a natural number n, then in order to divide this fraction by n, its numerator must be divided by this number:
\(\large \frac(a)(b) : n = \frac(a:n)(b) \)

If the numerator of the fraction \(\frac(a)(b)\) is not divisible by a natural number n, then to divide this fraction by n, you need to multiply its denominator by this number:
\(\large \frac(a)(b) : n = \frac(a)(bn) \)

Note that the second rule is also true when the numerator is divisible by n. Therefore, we can use it when it is difficult to determine at first glance whether the numerator of a fraction is divisible by n or not.

Actions with fractions. Adding fractions.

You can perform arithmetic operations with fractional numbers, just like with natural numbers. Let's look at adding fractions first. It's easy to add fractions with like denominators. Let us find, for example, the sum of \(\frac(2)(7)\) and \(\frac(3)(7)\). It is easy to understand that \(\frac(2)(7) + \frac(2)(7) = \frac(5)(7) \)

To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.

Using letters, the rule for adding fractions with like denominators can be written as follows:
\(\large \frac(a)(c) + \frac(b)(c) = \frac(a+b)(c) \)

If you need to add fractions with different denominators, then they must first be brought to a common denominator. For example:
\(\large \frac(2)(3)+\frac(4)(5) = \frac(2\cdot 5)(3\cdot 5)+\frac(4\cdot 3)(5\cdot 3 ) = \frac(10)(15)+\frac(12)(15) = \frac(10+12)(15) = \frac(22)(15) \)

For fractions, as for natural numbers, the commutative and associative properties of addition are valid.

Adding mixed fractions

Notations such as \(2\frac(2)(3)\) are called mixed fractions. In this case, the number 2 is called whole part mixed fraction, and the number \(\frac(2)(3)\) is its fractional part. The entry \(2\frac(2)(3)\) is read as follows: “two and two thirds.”

When dividing the number 8 by the number 3, you can get two answers: \(\frac(8)(3)\) and \(2\frac(2)(3)\). They express the same fractional number, i.e. \(\frac(8)(3) = 2 \frac(2)(3)\)

Thus, the improper fraction \(\frac(8)(3)\) is represented as a mixed fraction \(2\frac(2)(3)\). In such cases they say that from an improper fraction highlighted the whole part.

Subtracting fractions (fractional numbers)

Subtraction of fractional numbers, like natural numbers, is determined on the basis of the action of addition: subtracting another from one number means finding a number that, when added to the second, gives the first. For example:
\(\frac(8)(9)-\frac(1)(9) = \frac(7)(9) \) since \(\frac(7)(9)+\frac(1)(9 ) = \frac(8)(9)\)

The rule for subtracting fractions with like denominators is similar to the rule for adding such fractions:
To find the difference between fractions with the same denominators, you need to subtract the numerator of the second from the numerator of the first fraction, and leave the denominator the same.

Using letters, this rule is written like this:
\(\large \frac(a)(c)-\frac(b)(c) = \frac(a-b)(c) \)

Multiplying fractions

To multiply a fraction by a fraction, you need to multiply their numerators and denominators and write the first product as the numerator, and the second as the denominator.

Using letters, the rule for multiplying fractions can be written as follows:
\(\large \frac(a)(b) \cdot \frac(c)(d) = \frac(a \cdot c)(b \cdot d) \)

Using the formulated rule, you can multiply a fraction by a natural number, by mixed fraction, and also multiply mixed fractions. To do this, you need to write a natural number as a fraction with a denominator of 1, a mixed fraction - as an improper fraction.

The result of multiplication should be simplified (if possible) by reducing the fraction and isolating the whole part of the improper fraction.

For fractions, as for natural numbers, the commutative and combinative properties of multiplication, as well as the distributive property of multiplication relative to addition, are valid.

Division of fractions

Let's take the fraction \(\frac(2)(3)\) and “flip” it, swapping the numerator and denominator. We get the fraction \(\frac(3)(2)\). This fraction is called reverse fractions \(\frac(2)(3)\).

If we now “reverse” the fraction \(\frac(3)(2)\), we will get the original fraction \(\frac(2)(3)\). Therefore, fractions such as \(\frac(2)(3)\) and \(\frac(3)(2)\) are called mutually inverse.

For example, the fractions \(\frac(6)(5) \) and \(\frac(5)(6) \), \(\frac(7)(18) \) and \(\frac (18)(7)\).

Using letters, reciprocal fractions can be written as follows: \(\frac(a)(b) \) and \(\frac(b)(a) \)

It is clear that the product of reciprocal fractions is equal to 1. For example: \(\frac(2)(3) \cdot \frac(3)(2) =1 \)

Using reciprocal fractions, you can reduce division of fractions to multiplication.

The rule for dividing a fraction by a fraction is:
To divide one fraction by another, you need to multiply the dividend by the reciprocal of the divisor.

Using letters, the rule for dividing fractions can be written as follows:
\(\large \frac(a)(b) : \frac(c)(d) = \frac(a)(b) \cdot \frac(d)(c) \)

If the dividend or divisor is natural number or a mixed fraction, then, in order to use the rule for dividing fractions, it must first be represented as an improper fraction.

Already in primary school students encounter fractions. And then they appear in every topic. You cannot forget actions with these numbers. Therefore, you need to know all the information about ordinary and decimal fractions. These concepts are not complicated, the main thing is to understand everything in order.

Why are fractions needed?

The world around us consists of entire objects. Therefore, there is no need for shares. But everyday life constantly pushes people to work with parts of objects and things.

For example, chocolate consists of several pieces. Consider a situation where his tile is formed by twelve rectangles. If you divide it into two, you get 6 parts. It can easily be divided into three. But it will not be possible to give five people a whole number of chocolate slices.

By the way, these slices are already fractions. And their further division leads to the appearance of more complex numbers.

What is a "fraction"?

This is a number made up of parts of a unit. Outwardly, it looks like two numbers separated by a horizontal or slash. This feature is called fractional. The number written at the top (left) is called the numerator. What is at the bottom (right) is the denominator.

Essentially, the slash turns out to be a division sign. That is, the numerator can be called the dividend, and the denominator can be called the divisor.

What fractions are there?

In mathematics there are only two types: ordinary and decimal fractions. Schoolchildren become acquainted with the first ones in elementary school, calling them simply “fractions.” The latter will be learned in 5th grade. That's when these names appear.

Common fractions are all those that are written as two numbers separated by a line. For example, 4/7. A decimal is a number in which the fractional part has a positional notation and is separated from the whole number by a comma. For example, 4.7. Students need to clearly understand that the two examples given are completely different numbers.

Every simple fraction can be written as a decimal. This statement is almost always true in reverse. There are rules that allow you to write a decimal fraction as a common fraction.

What subtypes do these types of fractions have?

It's better to start in chronological order, as they are being studied. Common fractions come first. Among them, 5 subspecies can be distinguished.

Correct. Its numerator is always less than its denominator.

Wrong. Its numerator is greater than or equal to its denominator.

Reducible/irreducible. It may turn out to be either right or wrong. Another important thing is whether the numerator and denominator have common factors. If there are, then it is necessary to divide both parts of the fraction by them, that is, reduce it.

Mixed. An integer is assigned to its usual regular (irregular) fractional part. Moreover, it is always on the left.

Composite. It is formed from two fractions divided by each other. That is, it contains three fractional lines at once.

Decimal fractions have only two subtypes:

finite, that is, one whose fractional part is limited (has an end);

infinite - a number whose digits after the decimal point do not end (they can be written endlessly).

How to convert a decimal fraction to a common fraction?

If this is a finite number, then an association is applied based on the rule - as I hear, so I write. That is, you need to read it correctly and write it down, but without a comma, but with a fractional bar.

As a hint about the required denominator, you need to remember that it is always one and several zeros. You need to write as many of the latter as there are digits in the fractional part of the number in question.

How to convert decimal fractions into ordinary fractions if their integer part is missing, that is, equal to zero? For example, 0.9 or 0.05. After applying the specified rule, it turns out that you need to write zero integers. But it is not indicated. All that remains is to write down the fractional parts. The first number will have a denominator of 10, the second will have a denominator of 100. That is, the given examples will have the following numbers as answers: 9/10, 5/100. Moreover, it turns out that the latter can be reduced by 5. Therefore, the result for it needs to be written as 1/20.

How can you convert a decimal fraction into an ordinary fraction if its integer part is different from zero? For example, 5.23 or 13.00108. In both examples, the whole part is read and its value is written. In the first case it is 5, in the second it is 13. Then you need to move on to the fractional part. The same operation is supposed to be carried out with them. The first number appears 23/100, the second - 108/100000. The second value needs to be reduced again. The answer gives the following mixed fractions: 5 23/100 and 13 27/25000.

How to convert an infinite decimal fraction to an ordinary fraction?

If it is non-periodic, then such an operation will not be possible. This fact is due to the fact that each decimal fraction is always converted to either a finite or a periodic fraction.

The only thing you can do with such a fraction is round it. But then the decimal will be approximately equal to that infinite. It can already be turned into an ordinary one. But the reverse process: converting to decimal will never give the initial value. That is, infinite non-periodic fractions are not converted into ordinary fractions. This needs to be remembered.

How to write an infinite periodic fraction as an ordinary fraction?

In these numbers, there are always one or more digits after the decimal point that are repeated. They are called a period. For example, 0.3(3). Here "3" is in the period. They are classified as rational because they can be converted into ordinary fractions.

Those who have encountered periodic fractions know that they can be pure or mixed. In the first case, the period starts immediately from the comma. In the second, the fractional part begins with some numbers, and then the repetition begins.

The rule by which you need to write an infinite decimal as a common fraction will be different for the two types of numbers indicated. It is quite easy to write pure periodic fractions as ordinary fractions. As with finite ones, they need to be converted: write down the period in the numerator, and the denominator will be the number 9, repeated as many times as the number of digits the period contains.

For example, 0,(5). The number does not have an integer part, so you need to immediately start with the fractional part. Write 5 as the numerator and 9 as the denominator. That is, the answer will be the fraction 5/9.

The rule on how to write an ordinary decimal periodic fraction that is mixed.

Look at the length of the period. That's how many 9s the denominator will have.

Write down the denominator: first nines, then zeros.

To determine the numerator, you need to write down the difference of two numbers. All numbers after the decimal point will be minified, along with the period. Deductible - it is without a period.

For example, 0.5(8) - write the periodic decimal fraction as a common fraction. The fractional part before the period contains one digit. So there will be one zero. There is also only one number in the period - 8. That is, there is only one nine. That is, you need to write 90 in the denominator.

To determine the numerator, you need to subtract 5 from 58. It turns out 53. For example, you would have to write the answer as 53/90.

How are fractions converted to decimals?

The simplest option is a number whose denominator is the number 10, 100, etc. Then the denominator is simply discarded, and a comma is placed between the fractional and integer parts.

There are situations when the denominator easily turns into 10, 100, etc. For example, the numbers 5, 20, 25. It is enough to multiply them by 2, 5 and 4, respectively. You just need to multiply not only the denominator, but also the numerator by the same number.

For all other cases, a simple rule is useful: divide the numerator by the denominator. In this case, you may get two possible answers: a finite or a periodic decimal fraction.

Operations with ordinary fractions

Addition and subtraction

Students become acquainted with them earlier than others. And first for fractions same denominators, and then different. General rules can be reduced to such a plan.

Find the least common multiple of the denominators.

Write additional factors for all ordinary fractions.

Multiply the numerators and denominators by the factors specified for them.

Add (subtract) the numerators of the fractions and leave the common denominator unchanged.

If the numerator of the minuend is less than the subtrahend, then we need to find out whether we have a mixed number or a proper fraction.

In the first case, you need to borrow one from the whole part. Add the denominator to the numerator of the fraction. And then do the subtraction.

In the second, it is necessary to apply the rule of subtracting a larger number from a smaller number. That is, from the module of the subtrahend, subtract the module of the minuend, and in response put a “-” sign.

Look carefully at the result of addition (subtraction). If you get an improper fraction, then you need to select the whole part. That is, divide the numerator by the denominator.

Multiplication and division

To perform them, fractions do not need to be reduced to a common denominator. This makes it easier to perform actions. But they still require you to follow the rules.

When multiplying fractions, you need to look at the numbers in the numerators and denominators. If any numerator and denominator have a common factor, then they can be reduced.

Multiply the numerators.

Multiply the denominators.

If the result is a reducible fraction, then it must be simplified again.

When dividing, you must first replace division with multiplication, and the divisor (second fraction) with the reciprocal fraction (swap the numerator and denominator).

Then proceed as with multiplication (starting from point 1).

In tasks where you need to multiply (divide) by a whole number, the latter should be written as an improper fraction. That is, with a denominator of 1. Then act as described above.



Operations with decimals

Addition and subtraction

Of course, you can always convert a decimal into a fraction. And act according to the plan already described. But sometimes it is more convenient to act without this translation. Then the rules for their addition and subtraction will be exactly the same.

Equalize the number of digits in the fractional part of the number, that is, after the decimal point. Add the missing number of zeros to it.

Write the fractions so that the comma is below the comma.

Add (subtract) like natural numbers.

Remove the comma.

Multiplication and division

It is important that you do not need to add zeros here. Fractions should be left as they are given in the example. And then go according to plan.

To multiply, you need to write the fractions one below the other, ignoring the commas.

Multiply like natural numbers.

Place a comma in the answer, counting from the right end of the answer as many digits as they are in the fractional parts of both factors.

To divide, you must first transform the divisor: make it a natural number. That is, multiply it by 10, 100, etc., depending on how many digits are in the fractional part of the divisor.

Multiply the dividend by the same number.

Divide a decimal fraction by a natural number.

Place a comma in your answer at the moment when the division of the whole part ends.



What if one example contains both types of fractions?

Yes, in mathematics there are often examples in which you need to perform operations on ordinary and decimal fractions. In such tasks there are two possible solutions. You need to objectively weigh the numbers and choose the optimal one.

First way: represent ordinary decimals

It is suitable if division or translation results in finite fractions. If at least one number gives a periodic part, then this technique is prohibited. Therefore, even if you don’t like working with ordinary fractions, you will have to count them.

Second way: write decimal fractions as ordinary

This technique turns out to be convenient if the part after the decimal point contains 1-2 digits. If there are more of them, you may end up with a very large common fraction and decimal notation will make the task faster and easier to calculate. Therefore, you always need to soberly evaluate the task and choose the simplest solution method.

Very often in school curriculum Mathematicians children are faced with the problem of how to convert a fraction to a decimal. In order to convert a common fraction to a decimal, let us first remember what a common fraction and a decimal are. An ordinary fraction is a fraction of the form m/n, where m is the numerator and n is the denominator. Example: 8/13; 6/7, etc. Fractions are divided into regular, improper and mixed numbers. Proper fraction– this is when the numerator is less than the denominator: m/n, where m 3. An improper fraction can always be represented as a mixed number, namely: 4/3 = 1 and 1/3;

Converting a fraction to a decimal

Now let's look at how to convert a mixed fraction to a decimal. Any ordinary fraction, whether proper or improper, can be converted to a decimal. To do this, you need to divide the numerator by the denominator. Example: simple fraction (proper) 1/2. Divide numerator 1 by denominator 2 to get 0.5. Let's take the example of 45/12; it is immediately clear that this is an irregular fraction. Here the denominator is less than the numerator. Converting an improper fraction to a decimal: 45: 12 = 3.75.

Converting mixed numbers to decimals

Example: 25/8. First we turn the mixed number into an improper fraction: 25/8 = 3x8+1/8 = 3 and 1/8; then divide the numerator equal to 1 by the denominator equal to 8, using a column or on a calculator and get a decimal fraction equal to 0.125. The article provides the easiest examples of conversion to decimal fractions. Having understood the translation technique into simple examples, you can easily solve the most difficult of them.