Converting a decimal fraction into an ordinary fraction and vice versa: rules, examples. Converting decimal numbers to fractions and vice versa - online calculator Converting decimals to mixed numbers

This would seem to be the translation decimal in the usual - an elementary topic, but many students do not understand it! Therefore, today we will take a detailed look at several algorithms at once, with the help of which you will understand any fractions in just a second.

Let me remind you that there are at least two forms of writing the same fraction: common and decimal. Decimal fractions are all kinds of constructions of the form 0.75; 1.33; and even −7.41. Here are examples of ordinary fractions that express the same numbers:

Now let's figure it out: how to move from decimal notation to regular notation? And most importantly: how to do this as quickly as possible?

Basic algorithm

In fact, there are at least two algorithms. And we'll look at both now. Let's start with the first one - the simplest and most understandable.

To convert a decimal to a fraction, you need to follow three steps:

Important note about negative numbers. If in the original example there is a minus sign in front of the decimal fraction, then in the output there should also be a minus sign in front of the common fraction. Here are some more examples:

Examples of transition from decimal notation of fractions to ordinary ones

I would like to pay special attention to the last example. As you can see, the fraction 0.0025 contains many zeros after the decimal point. Because of this, you have to multiply the numerator and denominator by 10 as many as four times. Is it possible to somehow simplify the algorithm in this case?

Of course you can. And now we will look at an alternative algorithm - it is a little more difficult to understand, but after a little practice it works much faster than the standard one.

Faster way

This algorithm also has 3 steps. To get a fraction from a decimal, do the following:

1. Count how many digits are after the decimal point. For example, the fraction 1.75 has two such digits, and 0.0025 has four. Let's denote this quantity by the letter $n$.
2. Rewrite the original number as a fraction of the form $\frac(a)(((10)^(n)))$, where $a$ are all the digits of the original fraction (without the “starting” zeros on the left, if any), and $n$ is the same number of digits after the decimal point that we calculated in the first step. In other words, you need to divide the digits of the original fraction by one followed by $n$ zeros.
3. If possible, reduce the resulting fraction.

That's it! At first glance, this scheme is more complicated than the previous one. But in fact it is both simpler and faster. Judge for yourself:

As you can see, in the fraction 0.64 there are two digits after the decimal point - 6 and 4. Therefore $n=2$. If we remove the comma and zeros on the left (in this case, just one zero), we get the number 64. Let’s move on to the second step: $((10)^(n))=((10)^(2))=100$, Therefore, the denominator is exactly one hundred. Well, then all that remains is to reduce the numerator and denominator. :)

Another example:

Here everything is a little more complicated. Firstly, there are already 3 numbers after the decimal point, i.e. $n=3$, so you have to divide by $((10)^(n))=((10)^(3))=1000$. Secondly, if we remove the comma from the decimal notation, we get this: 0.004 → 0004. Remember that the zeros on the left must be removed, so in fact we have the number 4. Then everything is simple: divide, reduce and get the answer.

Finally, the last example:

The peculiarity of this fraction is the presence of a whole part. Therefore, the output we get is an improper fraction of 47/25. You can, of course, try to divide 47 by 25 with a remainder and thus again isolate the whole part. But why complicate your life if this can be done at the stage of transformation? Well, let's figure it out.

What to do with the whole part

In fact, everything is very simple: if we want to get correct fraction, then it is necessary to remove the whole part from it for the duration of the transformations, and then, when we get the result, add it again to the right before the fractional line.

For example, consider the same number: 1.88. Let's score by one (the whole part) and look at the fraction 0.88. It can be easily converted:

Then we remember about the “lost” unit and add it to the front:

\[\frac(22)(25)\to 1\frac(22)(25)\]

That's it! The answer turned out to be the same as after selecting the whole part last time. A couple more examples:

\[\begin(align)& 2.15\to 0.15=\frac(15)(100)=\frac(3)(20)\to 2\frac(3)(20); \\& 13.8\to 0.8=\frac(8)(10)=\frac(4)(5)\to 13\frac(4)(5). \\\end(align)\]

This is the beauty of mathematics: no matter which way you go, if all the calculations are done correctly, the answer will always be the same. :)

In conclusion, I would like to consider one more technique that helps many.

Transformations “by ear”

Let's think about what a decimal even is. More precisely, how we read it. For example, the number 0.64 - we read it as "zero point 64 hundredths", right? Well, or just “64 hundredths”. The key word here is “hundredths”, i.e. number 100.

What about 0.004? This is “zero point 4 thousandths” or simply “four thousandths”. One way or another, keyword- “thousandths”, i.e. 1000.

So what's the big deal? And the fact is that it is these numbers that ultimately “pop up” in the denominators at the second stage of the algorithm. Those. 0.004 is “four thousandths” or “4 divided by 1000”:

Try to practice yourself - it's very simple. The main thing is to read the original fraction correctly. For example, 2.5 is “2 whole, 5 tenths”, so

And some 1.125 is “1 whole, 125 thousandths”, so

In the last example, of course, someone will object that it is not obvious to every student that 1000 is divisible by 125. But here you need to remember that 1000 = 10 3, and 10 = 2 ∙ 5, therefore

\[\begin(align)& 1000=10\cdot 10\cdot 10=2\cdot 5\cdot 2\cdot 5\cdot 2\cdot 5= \\& =2\cdot 2\cdot 2\cdot 5\ cdot 5\cdot 5=8\cdot 125\end(align)\]

Thus, any power of ten can be decomposed only into factors 2 and 5 - it is these factors that need to be looked for in the numerator so that in the end everything is reduced.

This concludes the lesson. Let's move on to a more complex reverse operation - see "

Algebra and mathematics are complex sciences that are not easy even for those who devote a lot of time to them. Problems can arise with any task. For example, not everyone knows how to convert a decimal fraction into a fraction.

Features of fractions

To easily convert one type of fraction to another, it is best to understand what it is. They can be called a non-integer number. It consists of one or more parts of the unit.

First of all, ordinary or so-called simple fractions are distinguished. For any type, the rule is that the denominator cannot be zero. If this is true, then this means that the value is an integer, that is, it cannot be a fraction.

There are several types of writing this number. A horizontal line or a slash is used, and the latter option can appear in print in three different ways. In school notebooks, as a rule, common fractions recorded with a classic horizontal line.

In addition to simple fractions, mixed and compound fractions are distinguished. The first ones differ in that they also have an integer written at the beginning. In composites, the numerator and denominator seem to also be another fraction.

How to convert a decimal fraction to a fraction?

Converting a decimal fraction into a regular fraction is not so difficult, since, despite external changes, the essence of the number will remain the same. The key difference is that decimals are written using commas, not a dash. Of course, this does not mean that the fraction ½ will equal 1.2.

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

Sometimes it's very easy to convert one fraction into another, especially if the non-integer part is tenths rather than hundredths or thousandths. Classic example–0.5. First of all, you should read it correctly, then you will get zero point five. There is no way to write zero whole numbers, but five tenths easily turns into 5/10. All that remains is to make the reduction by dividing by five. The result is ½.

Fraction with a whole number

It is necessary to consider other examples with increased complexity. It's worth taking 2.25. As before, to begin with, it is best to correctly indicate the name of the fraction. This time there are two point twenty five hundredths. Due to the fact that there are two digits after the sign, they are hundredths.

How to convert a decimal fraction to a fraction:

• The non-integer part is written as 25/100.
• It remains to add two integers. They are placed at the beginning, and thus a mixed fraction is obtained.
• 25/100 can be reduced. For simplicity, it's practical to start by dividing by 5, but it's a good idea to go straight to 25. The reduction results in ¼.
• All that remains is to sign two integers to ¼. The result is 2 ¼.

Finally, it is worth considering the process of working with thousandths. For analysis, let's take 4.112. Again, the work must begin with the correct reading. It turns out to be four point one hundred twelve thousandths. You can easily isolate the first digit, 4, and then substitute one hundred and twelve thousandths to it. They look like this - 112/100.

All that remains is to cut it to give best view. In this specific example the common divisor is six. The result is a simple fraction 4 14/125.

Converting fractions to percentages

Almost any fraction can be easily converted into a percentage. To do this, you need to understand that percent is one hundredth. In other words, 1% can immediately be easily written in fractional form - 1/100 or 0.01.

In the case of other options, you will have to turn to decimal fractions, that is, those written separated by commas. With them the problem is solved very simply. It is enough to multiply the decimal fraction by 100, and you will get the desired percentage.

• 0,27 * 100% = 27%

If it is necessary to convert an ordinary fraction, then first it will have to be converted to a decimal.

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

If the process of converting to percentages still causes difficulties, then, if desired, you can use various automatic services, of which there are quite a few on the Internet. By entering the numerator and denominator in the appropriate fields, you can easily find out what the percentage will be.

In general, converting fractions to percentages always involves multiplying by 100. In order to easily cope with this, you need to understand how to convert a common fraction to a decimal, but first, it’s worth understanding the reverse process.

Video instructions

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 the second will be the same right. 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 confuses 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. These are 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 at the 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, you should 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 of 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.

Simple fractions are not always easy to use. You can’t insert them into a report or statement, and even modern ones computer programs they are not always friendly with such numbers. Converting a fraction to (or to a decimal) is not difficult.

You will need

• piece of paper, pen, calculator

Instructions

Converting a fraction to a number means dividing the numerator by the denominator. The numerator is the top part of the fraction, the denominator is the bottom. If you have a calculator at hand, then press the buttons and the task is completed. The result will be either a whole number or a decimal fraction. A decimal fraction may have a long remainder after the decimal point. In this case, the fraction must be rounded to the specific digit you need, using the rounding rules (numbers up to 5 are rounded down, from 5 inclusive and more - up).

If you don’t have a calculator at hand, you will have to divide into a column. Write the numerator of the fraction next to the denominator, with a corner between them indicating division. For example, convert the fraction 10/6 to a number. First, divide 10 by 6. You get 1. Write the result in a corner. Multiply 1 by 6, you get 6. Subtract 6 from 10. You get a remainder of 4. The remainder must be divided by 6 again. Add the number 0 to 4, and divide 40 by 6. You get 6. Write 6 in the result, after the decimal point. Multiply 6 by 6. You get 36. Subtract 36 from 40. The remainder is again 4. You don’t need to continue further, since it becomes obvious that the result will be the number 1.66(6). Round this fraction to the digit you need. For example, 1.67. This is the final result.