Number in decimal fraction. Converting a decimal fraction to a common fraction and vice versa: rule, examples

Materials on fractions and study sequentially. Below for you detailed information with examples and explanations.

1. Mixed number into a common fraction.Let's write it in general view number:

We remember a simple rule - we multiply the whole part by the denominator and add the numerator, that is:

Examples:

2. On the contrary, an ordinary fraction into a mixed number. *Of course, this can only be done with an improper fraction (when the numerator is greater than the denominator).

With “small” numbers, in general, no actions need to be taken; the result is “visible” immediately, for example, fractions:

*More details:

15:13 = 1 remainder 2

4:3 = 1 remainder 1

9:5 = 1 remainder 4

But if the numbers are more, then you can’t do without calculations. Everything is simple here - divide the numerator by the denominator with a corner until the remainder is less than the divisor. Division scheme:

For example:

*Our numerator is the dividend, the denominator is the divisor.

We get the whole part (incomplete quotient) and the remainder. We write down an integer, then a fraction (the numerator contains the remainder, but the denominator remains the same):

3. Convert decimal to ordinary.

Partially in the first paragraph, where we talked about decimal fractions, we already touched on this. We write it down as we hear it. For example - 0.3; 0.45; 0.008; 4.38; 10.00015

We have the first three fractions without an integer part. And the fourth and fifth ones have it, let’s convert them into ordinary ones, we already know how to do this:

*We see that fractions can also be reduced, for example 45/100 = 9/20, 38/100 = 19/50 and others, but we will not do this here. Regarding reduction, you will find a separate paragraph below, where we will analyze everything in detail.

4. Convert ordinary to decimal.

It's not that simple. With some fractions it is immediately obvious and clear what to do with it so that it becomes a decimal, for example:

We use our wonderful basic property of a fraction - we multiply the numerator and denominator by 5, 25, 2, 5, 4, 2, respectively, and we get:

If available whole part, then nothing complicated either:

We multiply the fractional part by 2, 25, 2 and 5, respectively, and get:

And there are those for which without experience it is impossible to determine that they can be converted into decimals, for example:

What numbers should we multiply the numerator and denominator by?

Here again a proven method comes to the rescue - dividing with a corner, a universal method, used for translation common fraction You can always use decimal:

This way you can always determine whether a fraction is converted to a decimal. The fact is that not every ordinary fraction can be converted to a decimal, for example, such as 1/9, 3/7, 7/26 are not converted. What then is the fraction obtained when dividing 1 by 9, 3 by 7, 5 by 11? My answer is infinite decimal (we talked about them in paragraph 1). Let's divide:

That's all! Good luck to you!

Sincerely, Alexander Krutitskikh.

In this article we will look at how converting fractions to decimals, and also consider reverse process– conversion of decimal fractions to ordinary fractions. Here we will outline the rules for converting fractions and provide detailed solutions to typical examples.

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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 in the numerator that the total number of digits there becomes 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 long division natural numbers, after them we come to 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 other prime numbers in the expansion of the denominator, 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

The information in the previous paragraph gives rise to 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 theorem on divisibility with a remainder, it is clear that the remainder is always less than the 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 equal remainders are obtained on q, 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

At the very beginning, you still need to find out what a fraction is and what types it comes in. And there are three types. And the first of them is an ordinary fraction, for example ½, 3/7, 3/432, etc. These numbers can also be written using a horizontal dash. Both the first and second will be equally true. The number on top is called the numeral, and the number on the bottom is called the denominator. There is even a saying for those people who constantly confuse these two names. It goes like this: “Zzzzz remember! Zzzz denominator - downzzzz! " This will help you avoid getting confused. A common fraction is just two numbers that are divisible by each other. The dash in them indicates the division sign. It can be replaced with a colon. If the question is “how to convert a fraction into a number,” then it is very simple. You just need to divide the numerator by the denominator. That's all. The fraction has been translated.

The second type of fraction is called decimal. This is a series of numbers followed by a comma. For example, 0.5, 3.5, etc. They were called decimal only because after the sung number the first digit means “tens”, the second is ten times more than “hundreds”, and so on. And the first digits before the decimal point are called integers. For example, the number 2.4 sounds like this, twelve point two and two hundred thirty-four thousandths. Such fractions appear mainly due to the fact that dividing two numbers without a remainder does not work. And most fractions, when converted to numbers, end up as decimals. For example, one second is equal to zero point five.

And the final third view. This mixed numbers. An example of this can be given as 2½. It sounds like two wholes and one second. In high school, this type of fractions is no longer used. They will probably need to be converted either to ordinary fraction form or to decimal form. It's just as easy to do this. You just need to multiply the integer by the denominator and add the resulting notation to the numeral. Let's take our example 2½. Two multiplied by two equals four. Four plus one equals five. And a fraction of the shape 2½ is formed into 5/2. And five, divided by two, can be obtained as a decimal fraction. 2½=5/2=2.5. It has already become clear how to convert fractions into numbers. You just need to divide the numerator by the denominator. If the numbers are large, you can use a calculator.

If it does not produce whole numbers and there are a lot of digits after the decimal point, then this value can be rounded. Everything is rounded up very simply. First you need to decide what number you need to round to. An example should be considered. A person needs to round the number zero point zero, nine thousand seven hundred fifty-six ten thousandths, or to the digital value of 0.6. Rounding must be done to the nearest hundredth. This means that in this moment up to seven hundredths. After the number seven in the fraction there is five. Now we need to use the rules for rounding. Numbers greater than five are rounded up, and numbers smaller than five are rounded down. In the example, the person has five, she is on the border, but it is considered that rounding occurs upward. This means that we remove all the numbers after seven and add one to it. It turns out 0.8.

Situations also arise when a person needs to quickly convert a common fraction into a number, but there is no calculator nearby. To do this, use column division. The first step is to write the numerator and denominator next to each other on a piece of paper. A dividing corner is placed between them; it looks like the letter “T”, only lying on its side. For example, you can take the fraction ten sixths. And so, ten should be divided by six. How many sixes can fit in a ten, only one. The unit is written under the corner. Ten subtract six equals four. How many sixes will there be in a four, several. This means that in the answer a comma is placed after the one, and the four is multiplied by ten. At forty-six sixes. Six is ​​added to the answer, and thirty-six is ​​subtracted from forty. That turns out to be four again.

In this example, a loop has occurred, if you continue to do everything exactly the same, you will get the answer 1.6(6). The number six continues to infinity, but by applying the rounding rule, you can bring the number to 1.7. Which is much more convenient. From this we can conclude that not all ordinary fractions can be converted to decimals. In some there is a cycle. But any decimal fraction can be converted into a simple fraction. An elementary rule will help here: as it is heard, so it is written. For example, the number 1.5 is heard as one point twenty-five hundredths. So you need to write it down, one whole, twenty-five divided by one hundred. One whole number is one hundred, which means that the simple fraction will be one hundred and twenty-five times one hundred (125/100). Everything is also simple and clear.

So the most basic rules and transformations that are associated with fractions have been discussed. They are all simple, but you should know them. IN daily life Fractions, especially decimals, have long been included. This is clearly visible on price tags in stores. It’s been a long time since anyone writes round prices, but with fractions the price seems visually much cheaper. Also, one of the theories says that humanity turned away from Roman numerals and adopted Arabic ones, only because Roman ones did not have fractions. And many scientists agree with this assumption. After all, with fractions you can make calculations more accurately. And in our age space technology, accuracy in calculations is needed more than ever. So studying fractions in school mathematics is vital for understanding many sciences and technological advances.

A fraction is a number that is made up of one or more units. There are three types of fractions in mathematics: common, mixed and decimal.

• 
  Common fractions

An ordinary fraction is written as a ratio in which the numerator reflects how many parts are taken from the number, and the denominator shows how many parts the unit is divided into. If the numerator is less than the denominator, then we have a proper fraction. For example: ½, 3/5, 8/9.

If the numerator is equal to or greater than the denominator, then we are dealing with an improper fraction. For example: 5/5, 9/4, 5/2 Dividing the numerator can result in a finite number. For example, 40/8 = 5. Therefore, any whole number can be written as an ordinary improper fraction or a series of such fractions. Let's consider the entries of the same number in the form of a number of different ones.

• Mixed fractions

In general, a mixed fraction can be represented by the formula:

Thus, a mixed fraction is written as an integer and an ordinary proper fraction, and such a notation is understood as the sum of the whole and its fractional part.

• Decimals

A decimal is a special type of fraction in which the denominator can be represented as a power of 10. There are infinite and finite decimals. When writing this type of fraction, the whole part is first indicated, then the fractional part is recorded through a separator (period or comma).

The notation of a fractional part is always determined by its dimension. The decimal notation looks like this:

Rules for converting between different types of fractions

• Converting a mixed fraction to a common fraction

A mixed fraction can only be converted to an improper fraction. To translate, it is necessary to bring the whole part to the same denominator as the fractional part. In general it will look like this:
Let's look at the use of this rule using specific examples:

• Converting a common fraction to a mixed fraction

An improper fraction can be converted into a mixed fraction by simple division, which results in the integer part and the remainder (fractional part).

For example, let's convert the fraction 439/31 to mixed:
​​

• Converting fractions

In some cases, converting a fraction to a decimal is quite simple. In this case, the basic property of a fraction is applied: the numerator and denominator are multiplied by the same number in order to bring the divisor to a power of 10.

For example:

In some cases, you may need to find the quotient by dividing by corners or using a calculator. And some fractions cannot be reduced to a final decimal. For example, the fraction 1/3 when divided will never give the final result.

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 a typical mistake, a blunder, if you will, lurks.

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. Fraction- 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. There, by the way, about improper fractions you'll find out.

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 on key points about 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).

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