Translation of fractions in a clear number. Decimal fractions transformation methods fractions

At the very beginning, you still need to know what fraction is and what kinds it happens. And it happens three species. And the first of these is an ordinary fraction, for example ½, 3/7.3 / 432, etc. These numbers can also be recorded using horizontal dash. Both first and second will be the same true. The digit from above is called numeral, and below the denominator. There is even a saying, for those people who constantly confuse these two names. It sounds like this: "ZZZZ ZAPOMNY! Zzzzzzzzzzzzu! " This will help not get confused. Ordinary fraction is just two numbers who are divided into each other. Damn in them and denotes a sign of division. It can be replaced with a colon sign. If there is a question "As a fraction to translate to the number", then it is very simple. It should only be divided into a denominator. And that's it. The fraction is translated.

The second type of frains is called decimal. This is a series of digits with a comma. For example, 0.5, 3.5, etc. called them decimal, only because after the spa, the first digit indicates "dozens", the second ten times more "hundreds" and so on. And the first digit to the comma is called integers. For example, the number 2.4 sounds so, twelve integers and two hundred thirty-four thousandths. Such fractions appear mainly due to the fact that aligning two numbers without the residue does not work. And most ordinary fractions, while they will be translated into numbers, ultimately have the form of a decimal fraction. For example, one second is leveling zero as many as five tenths.

And the final third appearance. These are mixed numbers. An example of this can be given as 2½. Sounds so, two whole and one second. In high school classes, such a type of fractions are no longer used. They will certainly need to lead or in an ordinary look of the fraction, or in decimal. Make it as easy. Just the integer must be multiplied by the denominator and, the resulting designation, folded with numeral. Take our example 2½. Two multiplied by two, it turns out four. Four plus one, equal to five. And the fraction of the form 2½ is formed in 5/2. And five, dividing into two you can get a decimal fraction. 2½ \u003d 5/2 \u003d 2.5. It has already become clear how to translate the fraction in numbers. Just split the numerator to the denominator. If the numbers are large, you can use the calculator.

If it turns out not entire numbers and after the comma there are a lot of digits, then this value can be rounded. Rounded everything is very simple. First, you should decide how the digit needs to be rounded. An example should be considered. A person needs to round the number of zero integers, nine thousand seven hundred fifty six ten thousand or digital value of 0.6. Rounding must be made to hundredths. This means that at the moment up to seven hundredths. After the numbers seven in the fraci goes five. Now you need to use the rules for rounding. Figures more than five are rounded into the biggest side, and smaller to smaller. In the example of a person - five, it stands on the border, but it is believed that rounding occurs in the biggest side. So, all the numbers after the seven we remove and add a unit to it. It turns out 0.8.

Situations also arise when a person needs to quickly translate an ordinary fraction in the number, and there is no calculator. To do this, it is worth applying a division by a column. First of all, it follows to write a numerator and denominator next to each other. There is a corner of division between them, it looks like a letter "t", only lying on the side. For example, you can take a fraction of ten sixth. And so, ten should be divided into six. How many six can fit in the top ten, only one. The unit is recorded under the corner. Ten take up six four. How many six will be in the four, several. So, the comma put in response after the unit, and the four is multiplied by ten. In forty six six. In the answer, six, and from forty takes thirty-six. It turns out again four.

In this example, looped occurred if you continue to do everything exactly the answer is 1.6 (6) the number six continues for infinity, but applying the rounding rule, you can bring the number to 1.7. What is much more convenient. From this we can conclude that not all ordinary fractions can be translated into decimal. In some, looping occurs. But any decimal fraction can be translated into a simple. It will help the elementary rule, both hears and writes. For example, the number 1.5, hears as one twenty-five hundredths. So you need to write down, one whole, twenty-five divided to a hundred. One whole is a hundred, and therefore, a simple fraction will be one hundred and twenty-five per hundred (125/100). Everything is also simple and understandable.

So the most basic rules and transformations were disassembled, which are associated with fractions. All of them are simple, but they should be known. The daily life has long included fracials, especially decimal. It is clearly visible on the price tags in stores. Round prices have no longer anyone who does not write, but with fractions the price seems visually much cheaper. Also, one of the theories says that humanity turned away from Roman numbers and accepted Arab in circulation, only because there were no fractions in Roman. And many scientists agree with this assumption. After all, with fractions you can conduct calculations more accurately. And in our age of space technologies, accuracy in calculations is needed more than ever. So learn the fractions in the school in mathematics is vital for understanding many sciences and technical achievements.

Decimal numbers, such as 0.2; 1.05; 3,017, etc. As heard, they are written. Zero as many as two tenths, we get a fraction. One whole five hundredths, we get a fraction. Three whole seventeen thousandths, we get a fraction. Figures to a comma in decimal number is a whole part of the fraction. The digit after the comma is the numerator of the future fraction. If after the semicolons the unambiguous number - in the denominator will be 10, if two-digit - 100, three-digit - 1000, etc. Some of the resulting fractions can be reduced. In our examples

Fraction conversion to decimal

This is the opposite transformation. Decimal fraction than characteristic? She has 10, or 100, or 1000, or 10,000 and so on. If your usual fraction has such a denominator, there are no problems. For example, or

If fraction, for example. In this case, it is necessary to use the main property of the fraction and convert the denominator to 10 or 100, or 1000 ... In our example, if the numerator and the denominator are 4, we will get a fraction that it is possible to write in the form of a decimal number 0.12.

Some fractions are easier to divide than to convert the denominator. For example,

Some fractions cannot be converted to decimal numbers!
For example,

Conversion of mixed fraction in incorrect

Mixed fraction, for example, easy to convert to the wrong. To do this, it is necessary to multiply by the denominator (bottom) and folded with the numerator (up), the denominator (bottom) is left unchanged. I.e

When converting a mixed fraction in the wrong, you can remember that you can use the addition of fractions

Conversion of incorrect fraction in mixed (allocation of the whole part)

Incorrect fraction can be translated into mixed, highlighting the whole part. Consider an example. We define how many times "3" holds in "23". Or 23 divide on 3 on the calculator, an integer to the comma is the desired. This is "7". Next, we define the numerator of the future fraction: the resulting "7" multiply on the "3" denominator and from the numerator "23" we subtract the obtained. As it were, you find more superfluous, which remains from the numerator "23", if you remove the maximum number of "3". The denominator is left unchanged. Everything is done, write the result

We have already said that fractions are ordinary and decimal. At the moment we have learned the ordinary fractions a little. We learned that ordinary fractions are the right and wrong. We also learned that ordinary fractions can be cut, folded, deduct to multiply and divide. And we learned that there are so-called mixed numbers that consist of an entire fractional part.

We have not yet fully studied ordinary fractions. There are a lot of subtleties and details that should be told, but today we will begin to study decimal The fraci, since ordinary and decimal fractions often have to combine. That is, when solving the tasks, you have to work with both types of fractions.

This lesson may seem complex and incomprehensible. It's quite normal. This kind of lessons require that they are studied, and not viewed superficially.

Design of lesson

Expression of quantities

Sometimes it is convenient to show anything in a fraction. For example, one tenth of the decimeter is written as follows:

This expression means that one decimeter was divided into ten equal parts, and one part was taken from these ten parts. And one part out of ten in this case is equal to one centimeter:

Consider the following example. Let it take to show 6 cm and 3mm in short form centimeters.

So, 6 whole centimeters we already have:

But there are still 3 millimeters. How to show these 3 millimeters, while at centimeters? Fruit come to the rescue. One centimeter is ten millimeters. Three millimeters are three parts of ten. And three parts out of ten are written as cm

The expression will mean that one centimeter was divided into ten equal parts, and three parts took from these ten parts.

As a result, we have six integers and three tenth centimeters:

The number 6 shows the number of integers, and the fraction is the number of fractional. This fraction is read as "Six whole and three tenth centimeters" .

The fraction, in the denominator of which there are numbers 10, 100, 1000 can be recorded without a denominator. First, they write the tile part, and then the numerator of the fractional part. The whole part is separated from the commander of the fractional part of the comma.

For example, write without a denominator. First write the whole part. Whole part is 6

The whole part is recorded. Immediately after writing the whole part, we put a comma:

And now write the numerator of the fractional part. In the mixed number, the knob of the fractional part is the number 3. Record after the semicolons Troika:

Any number that seems to be called decimal fraction.

Therefore, it is possible to show 6 cm and another 3 mm in centimeters using a decimal fraction:

6.3 cm

It will look like this:

In fact, decimal fractions are the same ordinary fractions and mixed numbers. The peculiarity of such fractions is that the denominator of their fractional part contains the numbers 10, 100, 1000 or 10,000.

Like a mixed number, the decimal fraction has a celuce and fractional. For example, in a mixed number, the whole part is 6, and the fractional part is.

In decimal fraction 6.3, the whole part is the number 6, and the fractional part is a fluster numerator, that is, the number 3.

It also happens that ordinary fractions in the denominator of which the numbers 10, 100, 1000 are given without a whole part. For example, the fraction is given without a whole part. To record such a fraction as a decimal, first record 0, then put the comma and write the knob of the fractional part. The fraction without a denominator will be recorded as follows:

Reading as "Zero whole, five tenths".

Translation of mixed numbers in decimal fractions

When we record mixed numbers without a denominator, we thus remarre them in decimal fractions. When translating ordinary fractions to decimal fractions, you need to know a few moments that we will talk about.

After a whole part is recorded, it is necessary to calculate the number of zeros in the protractor of the fractional part, since the number of zeros of the fractional part and the number of numbers after the comma in the decimal fraction should be the same. What does it mean? Consider the following example:

First, write the whole part and put the comma:

And it would be possible to immediately burn the numerator of the fractional part and the decimal fraction is ready, but it is necessary to calculate how many zeros are contained in the prototor of the fractional part.

So, consider the number of zeros in the fractional part of the mixed number. We see that in the denominator of the fractional part one zero. So in the decimal fraction after the comma, there will be one digit and this figure will be the numerator of the fractional part of the mixed number, that is, the number 2

Thus, a mixed number is translated into a decimal fraction in 3.2. This decimal fraction is read like this:

"Three whole, two tenths"

"Tenths" Because in the fractional part of the mixed number contains the number 10.

Example 2. Translate a mixed number in a decimal fraction.

We write down the cenette and put the comma:

And it would be possible to immediately burn the numerator of the fractional part and get the decimal fraction 5.3, but the rule says that after the comma should be as many numbers how many zeros in the denominator of the fractional part of the mixed number. And we see that in the denominator of the fractional part two zero. So in our decimal fraction after the comma should be two digits, and not alone.

In such cases, the fractional part number needs to be changed a little: add zero in front of the numerator, that is, in front of the number 3

Now you can bring the case to the end. We write after the semicolon the fractional part:

5,03

The decimal fraction 5,03 is read like this:

"Five whole, three hundredths"

"Hundredths" Because in the denominator of the fractional part of the mixed number contains the number 100.

Example 3. Translate a mixed number in a decimal fraction.

From previous examples, we learned that for the successful translation of the mixed number in a decimal fraction, the number of numbers in the fractional part numerator and the number of zeros in the valve of the fractional part should be the same.

Before transferring a mixed number to a decimal fraction, its fractional part must be changed slightly, namely to make it so that the number of numbers in the smooth part of the fractional part and the number of zeros in the denominator of the fractional part was the same.

First of all, for example, for the number of zeros in the valve of the fractional part. We see that there are three scratch:

Our task is to organize three digits in the fractional part numerator. We already have one digit - this is the number 2. It remains to add two more numbers. They will be two zero. Add them in front of the number 2. As a result, the number of zeros in the denominator and the number of numbers in the numerator will become the same:

Now you can go to the translation of this mixed number in a decimal fraction. Record first the centellis and put the comma:

and immediately write the fractional part numerator

3,002

We see that the number of numbers after the comma and the number of zeros in the denominator of the fractional part of the mixed number is equally.

Decimal fraction 3,002 is read like this:

"Three whole, two thousandths"

"Thousand" Because in the denominator of the fractional part of the mixed number contains the number 1000.

Translation of ordinary fractions to decimal fractions

Ordinary fractions, in whose denominator numbers 10, 100, 1000 or 10,000 can also be translated into decimal fractions. Since the ordinary fraction is absent, first recorded 0, then put the comma and write the fractional part.

Here also the number of zeros in the denominator and the number of numbers in the numerator should be the same. Therefore, it should be attentive.

Example 1.

The whole part is absent, it means you first write 0 and put the comma:

Now, for example, the number of zeros in the denominator. We see that there is one zero. And in the numerator one digit. So you can safely continue the decimal fraction, writing the number of 5 after the comma

In the resulting decimal fraction 0.5, the number of digits after the semicolons and the number of zeros in the denomoter is the same. So the fraction is translated correctly.

Decimal fraction 0,5 is read like this:

"Zero whole, five tenths"

Example 2. Translate an ordinary fraction in a decimal fraction.

The whole part is absent. We write first 0 and stabilize comma:

Now, for example, the number of zeros in the denominator. We see that there are two zero. And in the numerator only one digit. To make the number of numbers and the number of zeros are the same, add in the numerator in front of the number 2 one zero. Then the fraction will take a look. Now the number of zeros in the denominator and the number of numbers in the numerator is equally. So you can continue the decimal fraction:

0,02

In the resulting decimal fraction 0.02, the number of digits after the semicolons and the number of zeros in the denomoter is the same. So the fraction is translated correctly.

Decimal fraction 0,02 is read like this:

"Zero whole, two hundredths."

Example 3. Translate an ordinary fraction in a decimal fraction.

We write 0 and stave the comma:

Now we calculate the number of zeros in the denoter. We see that there are five zeros, and only one digit in the numerator. To make the number of zeros in the denominator and the number of numbers in the numerator is the same, it is necessary in the numerator in front of the number 5 add four zero:

Now you can continue the decimal fraction. We write after the semicolute the fraction

0,00005

In the resulting decimal fraction 0.00005, the number of numbers after the semicolons and the number of zeros in the denoter is the same. So the fraction is translated correctly.

Decimal fraction 0,00005 is read like this:

"Zero whole, five hundreds".

Translation of incorrect fractions in a decimal fraction

Improper fraction is a fraction that has a numerator more denominator.

There are incorrect fractions, in which the denominator contains numbers 10, 100, 1000 or 10000. Such fractions can be translated into decimal. But before the transfer to the decimal fraction, such fractions need to be allocated to the cen.

Example 1. Translate the wrong fraction in decimal.

The fraction is incorrect. To translate such a fraction in a decimal, you must first allocate a piece of the cen. We remember how to allocate the whole part of the wrong fractions. If you forgot, we advise you to return to and carefully study it.

So, we highlight the whole part in the wrong fraction. Recall that the fraction means division - in this case, the division of the number 112 to the number 10. The division must be performed with the remnant:

Answer on this drawing and collect a new mixed number, like the children's constructor. Private 11 will be a whole part, the residue 2 is a fractional part number, divider 10 - denominator of fractional part:

We got a mixed number. It translates into a decimal fraction. And how to translate such numbers into decimal fractions we already know. First, write the whole part and put the comma:

Now we calculate the number of zeros in the connoisser of the fractional part. We see that there is one zero. And in the numerator of the fractional part one digit. Therefore, the number of zeros in the protractor of the fractional part and the number of numbers in the fractional part number is equally. This gives us the opportunity to immediately record the fractional part number after the comma:

It means that the wrong fraction when transferring to decimal appeals to 11.2

Decimal fraction 11,2 is read like this:

"Eleven entire, two tenths."

Example 2. Translate the wrong fraction into a decimal fraction.

This is the wrong fraction, since the numerator is greater than the denominator. But it can be translated into a decimal fraction, since the denominator contains the number 100.

First of all, we highlight the whole part of this fraction. To do this, we split the corner 450 per 100:

We collect a new mixed number - we get. Now we will transfer it to a decimal fraction. We write a whole part and put the comma:

Now we calculate the number of zeros in the transmoter of the fractional part and the number of numbers in the sluple of the fractional part. We see that the number of zeros in the denominator and the number of numbers in the numerator is equally. This gives us the opportunity to immediately burn the splitter of the fractional part of the comma:

4,50

So the wrong fraction when transferring to decimal appeals to 4.50

When solving tasks, if at the end of the decimal fraction turns out to be zeros, they can be discarded. Let us and we will throw zero in our reply. Then we get 4.5

This is one of the interesting features of decimal fractions. It lies in the fact that the zeros that stand in the late fraction, do not give this fraction of any weight. In other words, decimal fractions 4,50 and 4.5 are equal to and between them can be put a sign of equality:

4,50 = 4,5

The question arises « why does it happen?» After all, on the form of 4.50 and 4.5 different fractions. The whole secret lies in the main property of the fraction, which we studied earlier. We will try to prove why the decimal fractions of 4.50 and 4.5 are equal, but after studying the following theme, which is called the "Decimacy of a decimal fraction in a mixed number".

Translation of decimal fraction in a mixed number

Any decimal fraction can be back translated into a mixed number. To do this, it is enough to be able to read decimal fractions.

For example, we translate 6.3 to a mixed number. 6.3 These are six integers and three tenths. We write down first six integers:

and near three tenths:

Example 2. Translate decimal fraction 3.002 in a mixed number

3.002 These are three whole and two thousandths. Record first three integers

Algebra and mathematics are complex sciences that are easily given even to those who pay a lot of time. Problems may arise with any tasks. For example, not everyone knows how the decimal fraction is translated into an ordinary fraction.

Features of the fraci

To easily translate one kind of fraction in another, it is best to understand what it is. They can be called an intense number. It consists of one or several parts of the unit.

First of all, they allocate ordinary or so-called simple fractions. With respect to any kind, the rule is valid that the denominator cannot be equal to zero. If this is the case, then this means that the meaning is integer, that is, it cannot be a fraction.

There are several types of writing such a number. A horizontal line or inclined trait is used, and the second option may look in print three different ways. In school notebooks, as a rule, ordinary fractions are written with a classic horizontal line.

In addition to simple, mixed and composite fractions are isolated. The first differed in that they also have an integer recorded at the beginning. The compound numerator and the denominator seems to be also another fraction.

How decimal fraction to translate into an ordinary fraction?

The decimal fraction to translate into the usual fraction is not as difficult, since, despite the external changes, the essence of the number will remain the same. The key difference is that decimals are recorded using commas, not thoughts. Of course, this does not mean that the fraction ½ will be 1.2.

The decimal fraction is formed from two components. The first is located before the sign and denotes an integer. The second one, that is after it, these are tenths, hundredths and other numbers. Their name depends on how much they are removed from the comma.

Sometimes it is very simple to turn one fraction into another, especially if the miser part is the tenths, not hundredths or thousands. Classic example -0.5. First of all, it is worth reading it correctly, then it turns out zero as much, five tenths. There will be no zero to write in any way, but five tenths are easily turned into 5/10. All that remains is to reduce by dividing five. Result - ½.

Effect with an integer

Other examples must be considered, with increased complexity. It is worth getting 2.25. As before, first, it is best to properly designate the name of the fraction. This time there are two integers, twenty-five hundredths. Due to the fact that after the sign there are two digits, then they are hundreds.

As a decimal fraction to translate into an ordinary fraction:

• The miser part is written in the form of 25/100.
• It remains to add two integers. They are put at the beginning, and thus the mixed fraction is obtained.
• 25/100 can be reduced. For simplicity, actually start with division by 5, but it is not bad to immediately take advantage of the number 25. As a result of the reduction, it turns out ¼.
• It remains only to sign two integers to ¼. The result is 2 ¼.

Finally, it is worth considering the process of working with thousands. To pars, take 4,112. Read more work to start with sure reading. It turns out four whole, a hundred twelve thousandths. It will be easily possible to highlight the first digit, 4, and then substitute one hundred twelve thousands for it. They look like this - 112/100.

It remains only to cut to give the best view. In this particular example, a common divider is six. The result is a simple fraction 4 14/125.

Translation of fractions in interest

Almost any fraction really without much difficulty translate into interest. To do this, you need to understand that the percentage is one hundred. In other words, 1% is immediately possible to be easily recorded in a fractional form - 1/100 or 0.01.

In the case of other options, you will have to turn to decimal fractions, that is, what they are written through the comma. With them, the task is solved very simply. It is enough to multiply the decimal fraction by 100, and the desired percentage will be.

• 0,27 * 100% = 27%

If it is necessary to carry out the translation of the ordinary fraction, then first it will have to turn it into a decimal.

• For example, 2/5 equals 0.4.
• 0,4 * 100% = 40%.

If the translation process percentages still causes difficulties, then, if you wish, you can use various automatic services that are quite a lot on the Internet. Writing in the appropriate fields, the numerator and denominator will be easy to find out which one will get the percentage.

In general, the translation of fractions in percentages is always tied to multiplication by 100. In order to cope with this, it is necessary to understand how to translate an ordinary fraction in decimal, but for a start, it is worth understanding in the opposite process.

Video instruction

The fraction can be transformed into an integer either in a decimal fraction. Incorrect fraction, the numerator of which is more denominator and is divided into it without a residue, is translated into an integer, for example: 20/5. We divide 20 to 5 and get a number 4. If the fraction is correct, that is, the numerator is less than the denominator, then convert it to the number (decimal fraction). More information about the fractions you can learn from our section -.

Methods for converting fractions in number

• The first method, how to translate the fraction to a number is suitable for the fraction, which can be converted to a decimal fraction. First, find out whether it is possible to translate the specified fraction in the fraction of the decimal. To do this, pay attention to the denominator (a figure that is under the feature or to the right of inclined). If the denominator can be decomposed on multipliers (in our example - 2 and 5), which can be repeated, then this fraction is actually converted to a finite decimal fraction. For example: 11/40 \u003d 11 / (2 ∙ 2 ∙ 2 ∙ 5). This ordinary fraction will be translated into the number (decimal fraction) with a finite number of semicolons. But the fraction 17/60 \u003d 17 / (5 ∙ 2 ∙ 2 ∙ 3) will be translated into a number with an infinite number of semicolons. That is, with accurate calculation of the numerical value, it is quite difficult to determine the final sign after the comma, since such signs are infinite set. Therefore, to solve problems, it is usually necessary to round the value to hundredths or thousands. Next - you need to multiply and the numerator, and the denominator for such a number so that the numbers 10, 100, 1000 and so ones are in the denominator, for example: 11/40 \u003d (11 ∙ 25) / (40 ∙ 25) \u003d 275/1000 \u003d 0.275
• The second way to translate fraction to the number is simpler: it is necessary to divide the numerator to the denominator. To use this method, we simply make a division, and the resulting number and will be the desired decimal fraction. For example, we need to translate the fraction 2/15 to the number. We divide 2 to 15. We get 0, 1333 ... - Infinite fraction. Write as follows: 0.13 (3). If the fraction is wrong, that is, the numerator is greater than the denominator (for example, 345/100), then as a result of the conversion of it, a whole numerical value or a decimal fraction with a whole fractional part is obtained. In our example it will be 3.45. To convert a mixed fraction of such a species, as 3 2/7, in the number, then you must first turn it into the wrong fraction: (3 ∙ 7 + 2) / 7 \u003d 23/7. Next, we divide 23 to 7 and we obtain the number 3,285,7143, which is reduced to 3.29.

The easiest way to translate the fraction is to use a calculator or another computing device. We first point the fluster numerator, then press the button with the "Divide" icon and score the denominator. After pressing the "\u003d" key, we get the desired number.