Explanations of the decimal fraction are converted. Converting ordinary fractions to finite and infinite periodic fractions

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 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, no action, in general, needs to be done, the result is “seen” immediately, for example, fractions:

*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 - we divide the numerator by the denominator by a corner until the remainder is less than the divisor. Division scheme:

For example:

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

We get the integer part (incomplete quotient) and the remainder. We write down - an integer, then a fraction (there is a remainder in the numerator, and we leave the denominator the same):

3. We translate the decimal into an ordinary one.

Partially in the first paragraph, where we talked about decimal fractions, we have already touched on this. As we hear, so we write. 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 have it, we will translate 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. For the reduction, a separate paragraph awaits you below, where we will analyze everything in detail.

4. Ordinary translate into decimal.

It's not all that simple. For some fractions, you can immediately see and clearly what to do with it so that it becomes decimal, for example:

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

If there is an integer part, then nothing complicated either:

We multiply the fractional part, respectively, by 2, 25, 2 and 5, we 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 you multiply the numerator and denominator by?

Here again, a proven method comes to the rescue - division by a corner, a universal method, you can always use it to convert an ordinary fraction to a decimal:

So you can always determine whether a fraction is converted to a decimal. The fact is that not every ordinary fraction can be converted to decimal, for example, such as 1/9, 3/7, 7/26 are not translated. And what then turns out for a fraction when dividing 1 by 9, 3 by 7, 5 by 11? I answer - infinite decimal (we talked about them in paragraph 1). Let's divide:

That's all! Good luck to you!

Sincerely, Alexander Krutitskikh.

Speaking dry mathematical language, a fraction is a number that is represented as a fraction of a unit. Fractions are widely used in human life: with the help of fractional numbers, we indicate proportions in recipes, set decimal marks in competitions or use them to calculate discounts in stores.

Representation of fractions

There are at least two forms of writing one fractional number: in decimal form or in the form of an ordinary fraction. In decimal form, numbers look like 0.5; 0.25 or 1.375. We can represent any of these values ​​as an ordinary fraction:

• 0,5 = 1/2;
• 0,25 = 1/4;
• 1,375 = 11/8.

And if we easily convert 0.5 and 0.25 from an ordinary fraction to a decimal and vice versa, then in the case of the number 1.375, everything is not obvious. How to quickly convert any decimal number to a fraction? There are three easy ways.

Getting rid of the comma

The simplest algorithm involves multiplying a number by 10 until the comma disappears from the numerator. This transformation is carried out in three steps:

Step 1: To begin with, we will write the decimal number as a fraction “number / 1”, that is, we will get 0.5 / 1; 0.25/1 and 1.375/1.

Step 2: After that, multiply the numerator and denominator of new fractions until the comma disappears from the numerators:

• 0,5/1 = 5/10;
• 0,25/1 = 2,5/10 = 25/100;
• 1,375/1 = 13,75/10 = 137,5/100 = 1375/1000.

Step 3: We reduce the resulting fractions to a digestible form:

• 5/10 = 1 x 5 / 2 x 5 = 1/2;
• 25/100 = 1 x 25 / 4 x 25 = 1/4;
• 1375/1000 = 11 x 125 / 8 x 125 = 11/8.

The number 1.375 had to be multiplied by 10 three times, which is no longer very convenient, but what will we have to do if we need to convert the number 0.000625? In this situation, we use next way fraction conversions.

Getting rid of the comma is even easier

The first method describes in detail the algorithm for "removing" a comma from a decimal fraction, however, we can simplify this process. Again, we follow three steps.

Step 1: We consider how many digits are after the decimal point. For example, the number 1.375 has three such digits, and 0.000625 has six. We will denote this number by the letter n.

Step 2: Now it is enough for us to represent the fraction in the form C/10 n , where C are the significant digits of the fraction (without zeros, if any), and n is the number of digits after the decimal point. For example:

• for the number 1.375 C \u003d 1375, n \u003d 3, the final fraction according to the formula 1375/10 3 \u003d 1375/1000;
• for the number 0.000625 C \u003d 625, n \u003d 6, the final fraction according to the formula 625/10 6 \u003d 625/1000000.

Essentially, 10 n is 1 with n zeros, so you don't have to worry about raising the tens to a power - just specify 1 with n zeros. After that, it is desirable to reduce the fraction so rich in zeros.

Step 3: Reduce the zeros and get the final result:

• 1375/1000 = 11 x 125 / 8 x 125 = 11/8;
• 625/1000000 = 1 x 625/ 1600 x 625 = 1/1600.

The fraction 11/8 is improper fraction, since its numerator is greater than its denominator, which means that we can select the whole part. In this situation, we subtract the whole part of 8/8 from 11/8 and get the remainder 3/8, therefore, the fraction looks like 1 and 3/8.

Transformation by ear

For those who know how to read decimals correctly, it is easiest to convert them by ear. If you read 0.025 not as "zero, zero, twenty-five", but as "25 thousandths", then you will have no problem converting decimal numbers to common fractions.

0,025 = 25/1000 = 1/40

So the correct reading decimal number allows you to immediately write it as an ordinary fraction and reduce it if necessary.

Examples of using fractions in everyday life

At first glance, common fractions are practically not used in everyday life or at work, and it is difficult to imagine a situation where you need to convert a decimal fraction to a common one outside of school problems. Let's look at a couple of examples.

Work

So, you work in a candy store and sell halva by weight. For ease of sale of the product, you divide halva into kilogram briquettes, but few buyers are ready to purchase a whole kilogram. Therefore, you have to divide the treat into pieces every time. And if another buyer asks you for 0.4 kg of halva, you will sell him the right portion without any problems.

0,4 = 4/10 = 2/5

Life

For example, you need to make a 12% solution for painting the model in the shade you need. To do this, you need to mix paint and thinner, but how to do it right? 12% is a decimal fraction of 0.12. We convert the number to an ordinary fraction and get:

0,12 = 12/100 = 3/25

Knowing the fractions, you can mix the components correctly and get the right color.

Conclusion

Fractions are widely used in Everyday life, so if you often need to convert decimals to fractions, you will need an online calculator that can instantly get the result in the form of an already reduced fraction.

Often, children who study at school are interested in why they are in real life Mathematics may be needed, especially those sections that already go much further than simple counting, multiplication, division, summation and subtraction. Many adults also ask this question if their professional activity very far from mathematics and various calculations. However, it should be understood that there are all sorts of situations, and sometimes you can’t do without the very notorious school curriculum that we so dismissively refused in childhood. For example, not everyone knows how to convert a fraction to a decimal fraction, and such knowledge can be extremely useful for the convenience of counting. First, you need to make sure that the fraction you need can be converted to a final decimal. The same goes for percentages, which can also be easily converted to decimals.

Checking an ordinary fraction for the possibility of converting it to a decimal

Before counting anything, you need to make sure that the resulting decimal fraction will be finite, otherwise it will turn out to be infinite and it will simply be impossible to calculate the final version. And infinite fractions can also be periodic and simple, but this is a topic for a separate section.

It is possible to convert an ordinary fraction to its final, decimal version only if its unique denominator can be decomposed only into factors of 5 and 2 ( prime factors). And even if they are repeated an arbitrary number of times.

Let us clarify that both of these numbers are prime, so in the end they can only be divided without a remainder by themselves, or by one. table prime numbers can be found without problems on the Internet, it is not at all difficult, although it has no direct relation to our account.

Consider examples:

The fraction 7/40 lends itself to being converted from a common fraction to its decimal equivalent because its denominator can be easily factored by 2 and 5.

However, if the first option results in a final decimal fraction, then, for example, 7/60 will not give a similar result, since its denominator will no longer be decomposed into the numbers we are looking for, but will have three among the denominator factors.

Converting a fraction to a decimal is possible in several ways.

After it became clear which fractions can be converted from ordinary to decimal, you can proceed, in fact, to the conversion itself. In fact, there is nothing super complicated, even for someone who has school program completely faded from memory.

How to convert fractions to decimals: the easiest method

This way of converting an ordinary fraction into a decimal is indeed the simplest, but many people are not even aware of its mortal existence, since at school all these “common truths” seem unnecessary and not very important. Meanwhile, not only an adult can figure it out, but a child can easily perceive such information.

So, to convert a fraction to a decimal, you need to multiply the numerator, as well as the denominator, by one number. However, everything is not so simple, so as a result, it is in the denominator that it should turn out 10, 100, 1000, 10,000, 100,000 and so on, ad infinitum. Do not forget to check first whether it is possible given fraction convert to decimal.

Consider examples:

Let's say we need to convert the fraction 6/20 to decimal. We check:

After we have made sure that it is still possible to convert a fraction to a decimal fraction, and even a final one, since its denominator is easily decomposed into twos and fives, we should proceed to the translation itself. The best option, logically, to multiply the denominator and get a result of 100 is 5, since 20x5=100.

Can be considered additional example, for clarity:

The second and more popular way convert fractions to decimals

The second option is somewhat more complicated, but it is more popular due to the fact that it is much easier to understand. Everything is transparent and clear here, so let's immediately move on to the calculations.

Worth remembering

In order to correctly transform a simple, that is, fraction to its decimal equivalent, you need to divide the numerator by the denominator. In fact, a fraction is a division, you can’t argue with that.

Let's take a look at an example:

So, first of all, in order to convert the fraction 78/200 into a decimal, you need to divide its numerator, that is, the number 78, by the denominator 200. But the first thing that should become a habit is to check, which was already mentioned above.

After making a check, you need to remember the school and divide the numerator by the denominator with a “corner” or “column”.

As you can see, everything is extremely simple, and you don’t need to be seven spans in the forehead to easily solve such problems. For simplicity and convenience, we also give a table of the most popular fractions that are easy to remember and do not even make efforts to translate them.

How to convert percentages to decimals: there is nothing easier

Finally, the move came to percentages, which, it turns out, as the same school curriculum says, can be converted into a decimal fraction. And here everything will be even much easier, and you should not be afraid. Even those who did not graduate from universities will cope with the task, and the fifth grade of the school skipped at all and does not understand anything in mathematics.

Perhaps you need to start with a definition, that is, to figure out what, in fact, interest is. A percentage is one hundredth of a number, that is, absolutely arbitrary. From a hundred, for example, it will be a unit, and so on.

Thus, to convert percentages to decimals, you simply need to remove the% sign, and then divide the number itself by a hundred.

Consider examples:

Moreover, in order to make a reverse “conversion”, you simply need to do the opposite, that is, the number must be multiplied by a hundred and a percent sign must be assigned to it. In exactly the same way, by applying the knowledge gained, it is also possible to convert an ordinary fraction into a percentage. To do this, it will be enough just to first convert the usual fraction to a decimal, and therefore already convert it to a percentage, and you can also easily perform the reverse action. As you can see, there is nothing super complicated, all this is elementary knowledge that you just need to keep in mind, especially if you are dealing with numbers.

The path of least resistance: convenient online services

It also happens that you don’t feel like counting at all, and there is simply no time. It is for such cases, or for especially lazy users, that there are many convenient and easy-to-use services on the Internet that will allow you to convert ordinary fractions, as well as percentages, into decimal fractions. This is really the path of least resistance, so using such resources is a pleasure.

Useful reference portal "Calculator"

In order to use the "Calculator" service, just follow the link http://www.calc.ru/desyatichnyye-drobi.html and enter the required numbers in the required fields. Moreover, the resource allows you to convert to decimal, both ordinary and mixed fractions.

After a short wait, about three seconds, the service will give the final result.

In the same way, you can convert a decimal fraction to a common fraction.

Online calculator on the "Mathematical resource" Calcs.su

Another very useful service is the fraction calculator on the Mathematical Resource. Here you also don’t have to count anything on your own, just select from the proposed list what you need and go ahead, for orders.

Further, in the field specially reserved for this, you need to enter the required number of percent, which you need to convert to a regular fraction. Moreover, if you need decimal fractions, then you can easily cope with the translation task yourself or use the calculator that is intended for this.

In the end, it’s worth adding that no matter how many newfangled services would be invented, how many resources would not offer you their services, but it won’t hurt to train your head from time to time. Therefore, it is worthwhile to apply the knowledge gained, especially since you can then proudly help your own children, and then grandchildren, do their homework. For those who suffer from eternal lack of time, such online calculators on mathematical portals will come in handy and even help you understand how to convert a common fraction to a decimal.

We have already said that fractions are ordinary and decimal. At the moment, we have studied ordinary fractions a little. We learned that there are regular fractions and improper fractions. We also learned that ordinary fractions can be reduced, added, subtracted, multiplied and divided. And we also learned that there are so-called mixed numbers, which consist of an integer part and a fractional part.

We have not yet fully studied ordinary fractions. There are many subtleties and details that should be discussed, but today we will begin to study decimal fractions, since ordinary and decimal fractions often have to be combined. That is, when solving problems, you have to use both types of fractions.

This lesson may seem complicated and incomprehensible. It's quite normal. These kinds of lessons require that they be studied and not skimmed over.

Lesson content

Expressing quantities in fractional form

Sometimes it is convenient to show something in fractional form. For example, one tenth of a decimeter is written like this:

This expression means that one decimeter was divided into ten parts, and one part was taken from these ten parts:

As you can see in the figure, one tenth of a decimeter is one centimeter.

Consider the following example. Show 6 cm and another 3 mm in centimeters in fractional form.

So, it is required to express 6 cm and 3 mm in centimeters, but in fractional form. We already have 6 whole centimeters:

but there are still 3 millimeters left. How to show these 3 millimeters, while in centimeters? Fractions come to the rescue. 3 millimeters is one third of a centimeter. And the third part of a centimeter is written as cm

A fraction means that one centimeter has been divided by ten equal parts, and three parts were taken from these ten parts (three out of ten).

As a result, we have six whole centimeters and three tenths of a centimeter:

In this case, 6 shows the number of whole centimeters, and the fraction shows the number of fractional centimeters. This fraction is read as "six point and three tenths of a centimeter".

Fractions, in the denominator of which there are numbers 10, 100, 1000, can be written without a denominator. First write the integer part, and then the numerator of the fractional part. The integer part is separated from the numerator of the fractional part by a comma.

For example, let's write without a denominator. To do this, we first write down the whole part. The integer part is the number 6. We write down this number first:

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

And now we write down the numerator of the fractional part. In a mixed number, the numerator of the fractional part is the number 3. We write the three after the decimal point:

Any number that is represented in this form is called decimal.

Therefore, you can show 6 cm and another 3 mm in centimeters using a decimal fraction:

6.3 cm

It will look like this:

In fact, decimals are the same common fractions and mixed numbers. The peculiarity of such fractions is that the denominator of their fractional part contains the numbers 10, 100, 1000 or 10000.

Like a mixed number, a decimal has an integer part and a fractional part. For example, in a mixed number, the integer part is 6 and the fractional part is .

In the decimal fraction 6.3, the integer part is the number 6, and the fractional part is the numerator of the fraction, that is, the number 3.

It also happens that ordinary fractions in the denominator of which the numbers 10, 100, 1000 are given without an integer part. For example, a fraction is given without an integer part. To write such a fraction as a decimal, first write down 0, then put a comma and write down the numerator of the fractional part. A fraction without a denominator would be written like this:

Reads like "zero point five tenths".

Convert mixed numbers to decimals

When we write mixed numbers without a denominator, we are converting them to decimals. When transferring ordinary fractions in decimal fractions, you need to know a few things, which we'll talk about now.

After the integer part is written, it is imperative to count the number of zeros in the denominator of the fractional part, since the number of zeros in the fractional part and the number of digits after the decimal point in the decimal fraction must be the same. What does it mean? Consider the following example:

At first

And you could immediately write down the numerator of the fractional part and the decimal fraction is ready, but you must definitely count the number of zeros in the denominator of the fractional part.

So, we count the number of zeros in the fractional part of the mixed number. The denominator of the fractional part has one zero. So in the decimal fraction after the decimal point 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, the mixed number, when translated into a decimal fraction, becomes 3.2.

This decimal is read like this:

"Three whole two tenths"

"Tenths" because the fractional part of the mixed number contains the number 10.

Example 2 Convert mixed number to decimal.

We write down the whole part and put a comma:

And you could immediately write down the numerator of the fractional part and get the decimal fraction 5.3, but the rule says that after the decimal point there should be as many digits as there are zeros in the denominator of the fractional part of the mixed number. And we see that there are two zeros in the denominator of the fractional part. So in our decimal fraction after the decimal point there should be two digits, not one.

In such cases, the numerator of the fractional part needs to be slightly modified: add a zero before the numerator, that is, before the number 3

Now you can convert this mixed number to a decimal. We write down the whole part and put a comma:

And write the numerator of the fractional part:

The decimal fraction 5.03 reads like this:

"Five point three hundredths"

"Hundredths" because the denominator of the fractional part of the mixed number is the number 100.

Example 3 Convert mixed number to decimal.

From the previous examples, we learned that in order to successfully convert a mixed number to a decimal, the number of digits in the numerator of the fractional part and the number of zeros in the denominator of the fractional part must be the same.

Before converting a mixed number into a decimal fraction, its fractional part needs to be slightly modified, namely, to make sure that the number of digits in the numerator of the fractional part and the number of zeros in the denominator of the fractional part are the same.

First of all, we look at the number of zeros in the denominator of the fractional part. We see that there are three zeros:

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

Now we can turn this mixed number into a decimal. We write down the whole part first and put a comma:

and immediately write down the numerator of the fractional part

3,002

We see that the number of digits after the decimal point and the number of zeros in the denominator of the fractional part of the mixed number are the same.

The decimal 3.002 reads like this:

"Three whole, two thousandths"

"Thousandths" because the denominator of the fractional part of the mixed number is the number 1000.

Converting common fractions to decimals

Ordinary fractions, in which the denominator is 10, 100, 1000 or 10000, can also be converted to decimal fractions. Since an ordinary fraction does not have an integer part, first write down 0, then put a comma and write down the numerator of the fractional part.

Here, too, the number of zeros in the denominator and the number of digits in the numerator must be the same. Therefore, you should be careful.

Example 1

The integer part is missing, so first we write 0 and put a comma:

Now look at the number of zeros in the denominator. We see that there is one zero. And the numerator has one digit. So you can safely continue the decimal fraction by writing the number 5 after the decimal point

In the resulting decimal fraction 0.5, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is correct.

The decimal fraction 0.5 reads like this:

"Zero point, five tenths"

Example 2 Convert common fraction to decimal.

The whole part is missing. We write 0 first and put a comma:

Now look at the number of zeros in the denominator. We see that there are two zeros. And the numerator has only one digit. To make the number of digits and the number of zeros the same, add one zero in the numerator before the number 2. Then the fraction will take the form . Now the number of zeros in the denominator and the number of digits in the numerator are the same. So you can continue the decimal:

In the resulting decimal fraction 0.02, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is correct.

The decimal fraction 0.02 reads like this:

"Zero point, two hundredths."

Example 3 Convert common fraction to decimal.

We write 0 and put a comma:

Now we count the number of zeros in the denominator of the fraction. We see that there are five zeros, and there is only one digit in the numerator. To make the number of zeros in the denominator and the number of digits in the numerator the same, you need to add four zeros in the numerator before the number 5:

Now the number of zeros in the denominator and the number of digits in the numerator are the same. So you can continue the decimal. We write down the numerator of the fraction after the decimal point

In the resulting decimal fraction 0.00005, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is correct.

The decimal fraction 0.00005 reads like this:

"Zero point, five hundred-thousandths."

Convert improper fractions to decimals

An improper fraction is a fraction whose numerator is greater than the denominator. There are improper fractions that have the numbers 10, 100, 1000 or 10000 in the denominator. Such fractions can be converted to decimal fractions. But before converting to a decimal fraction, such fractions must have an integer part.

Example 1

The fraction is an improper fraction. To convert such a fraction to a decimal fraction, you must first select its integer part. We recall how to select the whole part of improper fractions. If you forgot, we advise you to return to and study it.

So, let's select the integer part in the improper fraction. Recall that a fraction means division - in this case dividing the number 112 by the number 10

Let's look at this picture and assemble a new mixed number, like a children's construction set. The number 11 will whole part, the number 2 is the numerator of the fractional part, the number 10 is the denominator of the fractional part.

We got a mixed number. Let's convert it to a decimal. And we already know how to translate such numbers into decimal fractions. First we write down the whole part and put a comma:

Now we count the number of zeros in the denominator of the fractional part. We see that there is one zero. And the numerator of the fractional part has one digit. This means that the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part are the same. This gives us the opportunity to immediately write the numerator of the fractional part after the decimal point:

In the resulting decimal fraction 11.2, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is correct.

This means that an improper fraction, when converted to a decimal fraction, turns into 11.2

Decimal 11.2 reads like this:

"Eleven whole, two tenths."

Example 2 Convert improper fraction to decimal.

This is an improper fraction because the numerator is greater than the denominator. But it can be converted to a decimal fraction, since the denominator is the number 100.

First of all, we select the integer part of this fraction. To do this, divide 450 by 100 by a corner:

Let's collect a new mixed number - we get . And we already know how to translate mixed numbers into decimal fractions.

We write down the whole part and put a comma:

Now we count the number of zeros in the denominator of the fractional part and the number of digits in the numerator of the fractional part. We see that the number of zeros in the denominator and the number of digits in the numerator are the same. This gives us the opportunity to immediately write the numerator of the fractional part after the decimal point:

In the resulting decimal fraction 4.50, the number of digits after the decimal point and the number of zeros in the denominator of the fraction are the same. So the fraction is translated correctly.

So the improper fraction, when translated into a decimal fraction, turns into 4.50

When solving problems, if there are zeros at the end of the decimal fraction, they can be discarded. Let's drop the zero in our answer. Then we get 4.5

This is one of interesting features decimal fractions. It lies in the fact that the zeros that are at the end of the fraction do not give this fraction any weight. In other words, the decimals 4.50 and 4.5 are equal. Let's put an equal sign between them:

4,50 = 4,5

The question arises: why is this happening? After all, it looks like 4.50 and 4.5 different fractions. The whole secret lies in the basic property of the fraction, which we studied earlier. We will try to prove why the decimal fractions 4.50 and 4.5 are equal, but after studying next topic, which is called "converting a decimal to a mixed number."

Decimal to mixed number conversion

Any decimal fraction can be converted back to a mixed number. To do this, it is enough to be able to read decimal fractions. For example, let's convert 6.3 to a mixed number. 6.3 is six whole points and three tenths. We write down six integers first:

and next three tenths:

Example 2 Convert decimal 3.002 to mixed number

3.002 is three integers and two thousandths. Write down three integers first.

and next we write two thousandths:

Example 3 Convert decimal 4.50 to mixed number

4.50 is four point and fifty hundredths. Write down four integers

and next fifty hundredths:

By the way, let's remember last example from the previous thread. We said that the decimals 4.50 and 4.5 are equal. We also said that zero can be discarded. Let's try to prove that decimal 4.50 and 4.5 are equal. To do this, we convert both decimal fractions to mixed numbers.

After converting to a mixed number, the decimal 4.50 becomes , and the decimal 4.5 becomes

We have two mixed numbers and . Convert these mixed numbers to improper fractions:

Now we have two fractions and . It is time to remember the basic property of a fraction, which says that when multiplying (or dividing) the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

Let's divide the first fraction by 10

Received, and this is the second fraction. So and are equal to each other and equal to the same value:

Try dividing 450 by 100 first on a calculator, and then 45 by 10. A funny thing will work out.

Convert decimal to common fraction

Any decimal fraction can be converted back to a common fraction. To do this, again, it is enough to be able to read decimal fractions. For example, let's convert 0.3 to an ordinary fraction. 0.3 is zero and three tenths. We write zero integers first:

and next to three tenths 0 . Zero is traditionally not written down, so the final answer will not be 0, but simply.

Example 2 Convert decimal 0.02 to common fraction.

0.02 is zero and two hundredths. We don’t write down zero, so we immediately write down two hundredths

Example 3 Convert 0.00005 to fraction

0.00005 is zero and five hundred thousandths. Zero is not written down, so we immediately write down five hundred thousandths

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They are used extremely widely, and in the most various fields human activity be it scientific and applied computing, development and operation of various equipment, economic calculation, and so on. in mind different kind reasons often have to be carried out decimal inversion, as well as the process inverse to it. It should be noted that such transformations are produced relatively easily, and in accordance with certain rules and methods that have existed in mathematics for many hundreds of years.

Converting a decimal to a simple fraction

Decimal conversion into fraction "ordinary" is made quite easily and simply. For this, the following technique is used: the number that is located to the right of the decimal point of the original number is taken as the numerator of the new fraction, the number ten is used as the denominator, to a degree equal to the number of digits of the numerator. As for the remaining whole part, it remains unchanged. If the integer part is equal to zero, then after the transformation it is simply omitted.

EXAMPLE 1

Fifty point twenty five hundredths equals fifty point and twenty five divided by one hundred equals fifty point one fourth.

Converting a fraction to a decimal

Converting a fraction to a decimal, in fact, is the inverse converting a decimal to a simple. Its implementation also does not cause any difficulties and is, in fact, quite simple. arithmetic operation. In order to draw simple fraction to decimal you need to divide the numerator by its denominator in accordance with certain rules.

EXAMPLE 1

Need to implement fraction conversion five eighths decimal.

Dividing five by eight gives decimal zero point six hundred twenty-five thousandths.

=

0.625

Rounding the result of converting a fraction to a decimal

It should be noted that, in contrast to such a process as decimal conversion, this procedure can often last indefinitely. In such cases, it is said that the result of the procedure converting a fraction to a decimal may not be accurate. However, practice shows that in the vast majority of cases, obtaining is ideal. exact result and not required. As a rule, the division process ends when the values ​​of those decimal parts that are of practical interest in each particular case have already been obtained in its course.

EXAMPLE 1

It is required to cut a piece of butter weighing one kilogram into nine parts of the same mass. When performing this procedure, it turns out that the mass of each of them is 1/9 of a kilogram. If, according to all the rules, to carry out transformation this ordinary fraction in decimal fraction, it turns out that the mass of each of the resulting parts is equal to zero integers and one in the period of a kilogram.

Rounding is carried out according to the standard rules provided for in arithmetic: if the first of the "discarded" digits has a value of 5 or more, then the last of the significant ones is increased by one. Otherwise, it remains unchanged.

EXAMPLE 2

Convert common fraction one eighth to a decimal.

When dividing one by eight, you get zero point one hundred and twenty-five thousandths, or rounded up - zero point thirteen hundredths.