Algebra and mathematics are complex sciences that are not easily given even to those who devote a lot of time to them. Problems can arise with any tasks. For example, not everyone knows how to convert a decimal fraction to a common fraction.
Fraction Features
To easily translate one type of fraction into another, it is best to understand what it is. They can be called non-integer. It consists of one or more parts of the unit.
First of all, ordinary or so-called simple fractions are distinguished. For any species, the rule is that denominator cannot be zero. If so, then it means that the value is an integer, that is, it cannot be a fraction.
There are several ways to write such a number. A horizontal line or a slash is used, with the second option being printed in three different ways. In school notebooks, as a rule, ordinary fractions are written with a classic horizontal line.
In addition to simple fractions, there are mixed and compound fractions. The former differ in that they also have an integer written at the beginning. Compound numerator and denominator seem to be another fraction too.
How do you convert a decimal to a common fraction?
It is not so difficult to convert a decimal fraction into an ordinary fraction, since, despite external changes, the essence of the number will remain the same. The key difference is that decimals are written using commas, not dashes. Of course, this does not mean that the fraction ½ will equal 1.2.
The 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 is very easy to turn one fraction into another, especially if the non-integer part is tenths, not hundredths or thousandths. The classic example is -0.5. First of all, it should be read correctly, then it will turn out zero point, five tenths. Zero integers cannot be written down in any way, but five tenths easily turn into 5/10. All that remains is to reduce by dividing by five. The result is ½.
Fraction with whole number
It is necessary to consider other examples, with increased complexity. It is 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 whole, twenty-five hundredths. Due to the fact that there are two digits after the sign, they are hundredths.
How to convert a decimal to a common 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 cut. For simplicity, it's realistic to start by dividing by 5, but it's a good idea to use the number 25 right away. The result of the reduction is 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 thousandths. Let's take 4.112 for analysis. Again, the work must begin with a correct reading. It will turn out four whole, one hundred and twelve thousandths. Without difficulty, it will be possible to select the first digit, 4, and then substitute one hundred and twelve thousandths for it. They look like this - 112/100.
It remains only to cut to give a better look. In this particular 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 percentages without much difficulty. To do this, you need to understand that a percentage is one hundredth. In other words, 1% at once can easily be 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 that are written with a comma. With them, the task 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 translate an ordinary fraction, then first it will have to be converted into 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, which are quite a lot on the Internet. By entering the numerator and denominator in the appropriate fields, it will be easy to find out which percentage will turn out of this.
In general, the conversion of fractions to percentages is always tied to multiplying by 100. In order to easily deal with this, you need to understand how to convert an ordinary fraction to a decimal, but, first, you should understand the reverse process.
Video instruction
Here, it would seem, the translation of a decimal fraction into a common one is an elementary topic, but many students do not understand it! Therefore, today we will take a closer look at several algorithms at once, with the help of which you will deal with any fractions in just a second.
Let me remind you that there are at least two forms of writing the same fraction: ordinary and decimal. Decimal fractions are all kinds of constructions like 0.75; 1.33; and even -7.41. And here are examples of ordinary fractions that express the same numbers:
Now let's figure it out: how to switch from decimal to normal? And most importantly: how to do it as quickly as possible?
Basic Algorithm
In fact, there are at least two algorithms. And we will now look at both. Let's start with the first - the simplest and most understandable.
To convert a decimal to a common fraction, you need to follow three steps:
An important note about negative numbers. If in the original example there is a minus sign before the decimal fraction, then at the output there should also be a minus sign before the ordinary fraction. Here are some more examples:
Examples of the transition from decimal notation to ordinary fractions I would like to pay special attention to the last example. As you can see, in the fraction 0.0025 there are many zeros after the decimal point. Because of this, you have to multiply the numerator and denominator by 10 as much as four times. Is it possible to somehow simplify the algorithm in this case?
Of course you can. And now we will consider 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 common fraction from a decimal, you need to do the following:
- Calculate 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 $n$.
- 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 "starting" zeros on the left, if any), and $n$ is the same number of digits after the decimal point that we counted in the first step. In other words, it is necessary to divide the digits of the original fraction by one with $n$ zeros.
- If possible, reduce the resulting fraction.
That's all! 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, only one zero), then we get the number 64. Go to the second step: $((10)^(n))=((10)^(2))=100$, so the denominator is exactly one hundred. Well, then it remains only to reduce the numerator and denominator. :)
One more example:
Here everything is a little more complicated. Firstly, there are already 3 digits 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, then we get this: 0.004 → 0004. Recall 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 an integer part. Therefore, at the output we get an improper fraction 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 it can be done even 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 the correct fraction, then we need to remove the integer part from it for the time of transformation, and then, when we get the result, add it again to the right in front of the fractional bar.
For example, consider the same number: 1.88. Let's score by one (whole part) and look at the fraction 0.88. It is easily converted:
Then we remember about the “lost” unit and add it in front:
\[\frac(22)(25)\to 1\frac(22)(25)\]
That's all! The answer turned out to be the same as after the selection of 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 another technique that helps many.
Transformations by ear
Let's think about what a decimal is. More precisely, how we read it. For example, the number 0.64 - we read it as "zero integer, 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, the key word is "thousandths", i.e. 1000.
Well, what's wrong with that? And the fact that it is these numbers that eventually “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 train yourself - it's very simple. The main thing is to correctly read the original fraction. For example, 2.5 is "2 integers, 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 \u003d 10 3, and 10 \u003d 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 is decomposed only into factors 2 and 5 - it is these factors that must be sought in the numerator, so that in the end everything is reduced.
This lesson is over. Let's move on to a more complex inverse operation - see "
Decimal numbers such as 0.2; 1.05; 3.017 etc. as they are heard, so they are written. Zero point two, we get a fraction. One whole five hundredths, we get a fraction. Three whole seventeen thousandths, we get a fraction. The digits before the decimal point in a decimal number are the integer part of the fraction. The number after the decimal point is the numerator of the future fraction. If there is a one-digit number after the decimal point, the denominator will be 10, if two-digit - 100, three-digit - 1000, etc. Some of the resulting fractions can be reduced. In our examples 
Converting a fraction to a decimal number
This is the reverse of the previous transformation. What is a decimal fraction? Her denominator is always 10, or 100, or 1000, or 10,000, and so on. If your usual fraction has such a denominator, there is no problem. For example, or
If a fraction, for example . In this case, you need to use the basic property of the fraction and convert the denominator to 10 or 100, or 1000 ... In our example, if we multiply the numerator and denominator by 4, we get a fraction that can be written as 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,
Converting a mixed fraction to an improper
A mixed fraction, such as , is easily converted to an improper fraction. To do this, you need to multiply the integer part by the denominator (bottom) and add it to the numerator (top), leaving the denominator (bottom) unchanged. I.e
When converting a mixed fraction to an improper one, you can remember that you can use the addition of fractions
Converting an improper fraction to a mixed one (highlighting the whole part)
An improper fraction can be converted to a mixed fraction by highlighting the whole part. Consider an example, . Determine how many integer times "3" fit in "23". Or we divide 23 by 3 on the calculator, the whole number up to the decimal point is the desired one. This is "7". Next, we determine the numerator of the future fraction: we multiply the resulting "7" by the denominator "3" and subtract the result from the numerator "23". 
How would we find the excess that remains from the numerator "23", if we remove the maximum number of "3". The denominator is left unchanged. Everything is done, write down the result
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 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 quite often have to be combined. That is, when solving problems, you have to work with 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 contentExpressing 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 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 be required to show 6 cm and another 3 mm in centimeters in fractional form.
So, 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. One centimeter is ten millimeters. Three millimeters is three parts out of ten. And three parts out of ten are written as cm
The expression cm means that one centimeter was divided into ten equal parts, and three parts were taken from these ten parts.
As a result, we have six whole centimeters and three tenths of a centimeter:
The number 6 shows the number of whole centimeters, and the fraction shows the number of fractional ones. 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 whole 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. First write down the whole part. The whole part is 6
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 converting ordinary fractions to decimal fractions, there are a few things you need to know, 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:
First we write down the whole part and put a comma:
And you could immediately write down the numerator of the fractional part and the decimal fraction is ready, but you must definitely count how many zeros are contained in the denominator of the fractional part.
So, let's count the number of zeros in the fractional part of the mixed number. We see that there is one zero in the denominator of the fractional part. 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"
"Ten" 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 we can finish the job. We write the numerator of the fractional part after the comma:
5,03
The decimal fraction 5.03 reads like this:
"Five point three hundredths"
"Hundredths" because the denominator of the fractional part of the mixed number contains 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. Let's 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"
"Thousands" because the denominator of the fractional part of the mixed number contains 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:
0,02
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 let's 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 you can continue the decimal. We write down the numerator of the fraction after the decimal point
0,00005
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 in which the denominator contains the numbers 10, 100, 1000 or 10000. Such fractions can be converted to decimals. But before converting to a decimal fraction, such fractions must have an integer part.
Example 1 Convert improper fraction to decimal.
The fraction is incorrect. To convert such a fraction to a decimal, 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 thoroughly.
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. The division must be performed with a remainder:
Let's look at this picture and assemble a new mixed number, like a children's construction set. The quotient 11 will be the integer part, the remainder 2 will be the numerator of the fractional part, the divisor 10 will be 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 let's 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 down the numerator of the fractional part after the decimal point:
This means that an improper fraction, when converted to a decimal, 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 contains the number 100.
First of all, we select the integer part of this fraction. To do this, divide the angle 450 by 100:
Let's collect a new mixed number - we get . Now let's convert it to a decimal. We write down the whole part and put a comma:
Now let's 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:
4,50
So an improper fraction, when converted to decimal, 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 the interesting features of decimals. 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 and you can put an equal sign between them:
4,50 = 4,5
The question arises « why is this happening?» After all, 4.50 and 4.5 look like 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 the next topic, which is called "converting a decimal fraction 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.
A fraction can be converted to an integer or a decimal. An improper fraction, the numerator of which is greater than the denominator and is divisible by it without a remainder, is converted into an integer, for example: 20/5. Divide 20 by 5 and get the number 4. If the fraction is correct, that is, the numerator is less than the denominator, then convert it to a number (decimal fraction). You can learn more about fractions from our section -.
Ways to convert a fraction to a number
- The first way to convert a fraction to a number is suitable for a fraction that can be converted to a number that is a decimal fraction. First, let's find out whether it is possible to convert a given fraction into a decimal fraction. To do this, pay attention to the denominator (the number that is under the line or to the right of the oblique). If the denominator can be decomposed into factors (in our example - 2 and 5), which can be repeated, then this fraction can really be converted into a final decimal fraction. For example: 11/40 =11/(2∙2∙2∙5). This common fraction will be converted into a number (decimal fraction) with a finite number of decimal places. But the fraction 17/60 =17/(5∙2∙2∙3) will be translated into a number with an infinite number of decimal places. That is, when accurately calculating a numerical value, it is quite difficult to determine the final sign after the decimal point, since there are an infinite number of such signs. Therefore, to solve problems, you usually need to round the value to hundredths or thousandths. Further, it is necessary to multiply both the numerator and the denominator by such a number that the denominator will have the numbers 10, 100, 1000, etc. For example: 11/40 = (11∙25)/(40∙25) =275/1000 = 0.275
- The second way to convert a fraction to a number is simpler: you need to divide the numerator by the denominator. To apply this method, we simply perform the division, and the resulting number will be the desired decimal fraction. For example, you need to convert the fraction 2/15 to a number. We divide 2 by 15. We get 0, 1333 ... - an infinite fraction. We write it down like this: 0.13(3). If the fraction is incorrect, that is, the numerator is greater than the denominator (for example, 345/100), then as a result of converting it to a number, an integer numerical value or a decimal fraction with an integer fractional part will be obtained. In our example, this will be 3.45. To convert a mixed fraction like 3 2 / 7 to a number, you must first convert it to an improper fraction: (3∙7+2)/7 =23/7. Next, we divide 23 by 7 and get the number 3.2857143, which we reduce to 3.29.
The easiest way to convert a fraction to a number is to use a calculator or other computing device. We first indicate the numerator of the fraction, then press the button with the "divide" icon and type the denominator. After pressing the "=" key, we get the desired number.
