A decimal fraction consists of two parts, separated by commas. The first part is a whole unit, the second part is tens (if there is one number after the decimal point), hundreds (two numbers after the decimal point, like two zeros in a hundred), thousandths, etc. Let's look at examples decimal: 0, 2; 7, 54; 235.448; 5.1; 6.32; 0.5. These are all decimal fractions. How to convert a decimal fraction to an ordinary fraction?
Example one
We have a fraction, for example, 0.5. As mentioned above, it consists of two parts. The first number, 0, shows how many whole units the fraction has. In our case there are none. The second number shows tens. The fraction even reads zero point five. Decimal number convert to fraction Now it won’t be difficult, we write 5/10. If you see that the numbers have common divisor, you can reduce the fraction. We have this number 5, dividing both sides of the fraction by 5, we get - 1/2.
Example two
Let's take a more complex fraction - 2.25. It reads like this: two point two and twenty-five hundredths. Please note - hundredths, since there are two numbers after the decimal point. Now you can convert it to a common fraction. We write down - 2 25/100. The whole part is 2, the fractional part is 25/100. As in the first example, this part can be shortened. The common factor for the numbers 25 and 100 is the number 25. Note that we always choose the greatest common factor. Dividing both sides of the fraction by GCD, we got 1/4. So 2.25 is 2 1/4.
Example three
And to consolidate the material, let’s take the decimal fraction 4.112 - four point one and one hundred and twelve thousandths. Why thousandths, I think, is clear. Now we write down 4 112/1000. Using the algorithm, we find the gcd of the numbers 112 and 1000. In our case, this is the number 6. We get 4 14/125.
Conclusion
- We break the fraction into whole and fractional parts.
- Let's see how many digits are after the decimal point. If one is tens, two is hundreds, three is thousandths, etc.
- We write the fraction in ordinary form.
- Reduce the numerator and denominator of the fraction.
- We write down the resulting fraction.
- We check and divide top part fractions to the bottom. If there is an integer part, add it to the resulting decimal fraction. The original version turned out great, which means you did everything right.
Using examples, I showed how you can convert a decimal fraction to an ordinary fraction. As you can see, this is very easy and simple to do.
It would seem that converting a decimal fraction into a regular fraction is 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:
An 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 ordinary 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:
- 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 $n$.
- 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.
- 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 you remove the comma and zeros on the left (in in this case— only one zero), then we get the number 64. Let’s move on to the second step: $((10)^(n))=((10)^(2))=100$, so 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 a proper fraction, then we need to remove the whole part from it during the transformation, and then, when we get the result, add it again to the right before the fraction 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, saying 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 is 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 "
A fair number of people ask questions about how to convert a fraction to a decimal fraction. There are several ways. The choice of a specific method depends on the type of fraction that needs to be converted to another form, or more precisely, on the number in its denominator. However, for reliability, it is necessary to indicate that an ordinary fraction is a fraction that is written with a numerator and a denominator, for example, 1/2. More often, the line between the numerator and denominator is drawn horizontally rather than obliquely. The decimal fraction is written ordinary number with a comma: for example, 1.25; 0.35, etc.
So, in order to convert a fraction to a decimal without a calculator you need to:
Pay attention to the denominator of the common fraction. If the denominator can be easily multiplied up to 10 by the same number as the numerator, then you should use this method as the simplest. For example, the common fraction 1/2 is easily multiplied in the numerator and denominator by 5, resulting in the number 5/10, which can already be written as a decimal fraction: 0.5. This rule is based on the fact that a decimal fraction always has a round number in its denominator: 10, 100, 1000 and the like. Therefore, if you multiply the numerator and denominator of a fraction, then it is necessary to achieve exactly the same number in the denominator as a result of the multiplication, regardless of what is obtained in the numerator.
There are ordinary fractions, the calculation of which after multiplication presents certain difficulties. For example, it is quite difficult to determine how much the fraction 5/16 should be multiplied to get one of the above numbers in the denominator. In this case, you should use the usual division, which is done in a column. The answer should be a decimal fraction, which will mark the end of the transfer operation. In the example above, the resulting number is 0.3125. If columnar calculations are difficult, then you can’t do without the help of a calculator.
Finally, there are ordinary fractions that cannot be converted to decimals. For example, when converting the common fraction 4/3, the result is 1.33333, where the three is repeated ad infinitum. The calculator will also not get rid of the repeating three. There are several such fractions, you just need to know them. A way out of the above situation can be rounding, if the conditions of the example or problem being solved allow rounding. If the conditions do not allow this, and the answer must be written exactly in the form of a decimal fraction, it means that the example or problem was solved incorrectly, and you should go back several steps to find the error.
Thus, converting a fraction to a decimal is quite simple, and this task is not difficult to cope with without the help of a calculator. It is even easier to convert decimal fractions into ordinary fractions by performing the reverse steps described in method 1.
Video: 6th grade. Converting a fraction to a decimal.
A fraction is a number that is made up of one or more units. There are three types of fractions in mathematics: common, mixed and decimal.
Common fractions
An ordinary fraction is written as a ratio in which the numerator reflects how many parts are taken from the number, and the denominator shows how many parts the unit is divided into. If the numerator less than the denominator, then we have a proper fraction. For example: ½, 3/5, 8/9.
If the numerator is equal to or greater than the denominator, then we are dealing with an improper fraction. For example: 5/5, 9/4, 5/2 Dividing the numerator can result in a finite number. For example, 40/8 = 5. Therefore, any whole number can be written as an ordinary improper fraction or a series of such fractions. Let's consider records of the same number as a series of different ones.
- Mixed fractions
IN general view a mixed fraction can be represented by the formula: 
Thus, a mixed fraction is written as an integer and an ordinary proper fraction, and such a notation is understood as the sum of the whole and its fractional part.
- Decimals
A decimal is a special type of fraction in which the denominator can be represented as a power of 10. There are infinite and finite decimals. When writing this type of fraction, the whole part is first indicated, then the fractional part is recorded through a separator (period or comma).
The notation of a fractional part is always determined by its dimension. The decimal notation looks like this: 
Rules for converting between different types of fractions
- Translation mixed fraction to ordinary
A mixed fraction can only be converted to an improper fraction. To translate, it is necessary to bring the whole part to the same denominator as the fractional part. In general it will look like this: 
Let's look at the use of this rule using specific examples:
- Converting a common fraction to a mixed fraction
An improper fraction can be converted into a mixed fraction by simple division, resulting in the whole part and the remainder (fractional part).
For example, let's convert the fraction 439/31 to mixed: 
- Converting fractions
In some cases, converting a fraction to a decimal is quite simple. In this case, the basic property of a fraction is applied: the numerator and denominator are multiplied by the same number in order to bring the divisor to a power of 10.
For example:
In some cases, you may need to find the quotient by dividing by corners or using a calculator. And some fractions cannot be reduced to a final decimal. For example, the fraction 1/3 when divided will never give the final result.
Converting a Fraction to a Decimal
Let's say we want to convert the fraction 11/4 to a decimal. The easiest way to do it is this:
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2∙2∙5∙5 |
We succeeded because in this case the decomposition of the denominator into prime factors consists only of twos. We supplemented this expansion with two more fives, took advantage of the fact that 10 = 2∙5, and got a decimal fraction. Such a procedure is obviously possible if and only if the decomposition of the denominator into prime factors contains nothing but twos and fives. If any other prime number is present in the expansion of the denominator, then such a fraction cannot be converted to a decimal. Nevertheless, we will try to do this, but only in a different way, which we will get acquainted with using the example of the same fraction 11/4. Let's divide 11 by 4 using the “corner”:
In the response line we received the whole part (2), and we also have the remainder (3). Previously, we ended the division here, but now we know that we can add a comma and several zeros to the right of the dividend (11), which we will now mentally do. After the decimal point comes the tenths place. We add the zero that appears to the dividend in this digit to the resulting remainder (3):
Now the division can continue as if nothing had happened. You just need to remember to put a comma after the whole part in the answer line:
Now we add a zero to the remainder (2), which is in the hundredths place of the dividend, and complete the division:
As a result, we get, as before,
Let's now try to calculate in exactly the same way what the fraction 27/11 is equal to:
We received the number 2.45 in the answer line, and the number 5 in the remainder line. But we have already encountered such a remnant before. Therefore, we can immediately say that if we continue our division with a “corner”, then the next number in the answer line will be 4, then the number 5 will come, then again 4 and again 5, and so on, ad infinitum:
27 / 11 = 2,454545454545...
We got the so-called periodic a decimal fraction with a period of 45. For such fractions, a more compact notation is used, in which the period is written only once, but it is enclosed in parentheses:
2,454545454545... = 2,(45).
Generally speaking, if you divide one thing into a “corner” natural number on the other hand, writing the answer in the form of a decimal fraction, then only two outcomes are possible: (1) either sooner or later we will get zero in the remainder line, (2) or there will be a remainder that we have already encountered before (the set of possible remainders is limited, because they all obviously less than divisor). In the first case, the result of division is a finite decimal fraction, in the second case - a periodic one.
Convert periodic decimal to fraction
Let us be given a positive periodic decimal fraction with a zero integer part, for example:
a = 0,2(45).
How can I convert this fraction back to a common fraction?
Let's multiply it by 10 k, Where k is the number of digits between the decimal point and the opening parenthesis indicating the beginning of the period. In this case k= 1 and 10 k = 10:
a∙ 10 k = 2,(45).
Multiply the result by 10 n, Where n- the “length” of the period, that is, the number of digits enclosed between parentheses. In this case n= 2 and 10 n = 100:
a∙ 10 k ∙ 10 n = 245,(45).
Now let's calculate the difference
a∙ 10 k ∙ 10 n − a∙ 10 k = 245,(45) − 2,(45).
Since the fractional parts of the minuend and the subtrahend are the same, then the fractional part of the difference is equal to zero, and we come to simple equation relatively a:
a∙ 10 k ∙ (10 n − 1) = 245 − 2.
This equation is solved using the following transformations:
a∙ 10 ∙ (100 − 1) = 245 − 2.
a∙ 10 ∙ 99 = 245 − 2.
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245 − 2 |
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|
10 ∙ 99 |
We deliberately do not complete the calculations yet, so that it is clearly visible how this result can be immediately written down, omitting intermediate arguments. The minuend in the numerator (245) is the fractional part of the number
a = 0,2(45)
if you erase the brackets in her entry. The subtrahend in the numerator (2) is the non-periodic part of the number A, located between the comma and the opening parenthesis. The first factor in the denominator (10) is a unit, to which as many zeros are assigned as there are digits in the non-periodic part ( k). The second factor in the denominator (99) is as many nines as there are digits in the period ( n).
Now our calculations can be completed:
Here the numerator contains the period, and the denominator contains as many nines as there are digits in the period. After reduction by 9, the resulting fraction is equal to
In the same way,