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 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 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, the key word is “thousands”, 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 "
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:
|
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. The zero that appears at the dividend in this digit will be added 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 we divide one natural number by another with a “corner”, 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 such a remainder there, which we have already encountered before (the set of possible remainders is limited, since all of them are obviously smaller than the 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 arrive at a simple equation for 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.
|
245 − 2 |
||
|
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,
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 point five hundredths, we get a fraction. Three point seventeen thousandths, we get the fraction. The numbers before the decimal point are the whole part of the fraction. The number after the decimal point is the numerator of the future fraction. If there is a single-digit number after the decimal point, the denominator will be 10, if there is a two-digit number - 100, a three-digit number - 1000, etc. Some resulting fractions can be reduced. In our examples 
Converting a fraction to a decimal
This is the reverse of the previous transformation. What is the characteristic of a decimal fraction? Its denominator is always 10, or 100, or 1000, or 10000, and so on. If your common fraction has a denominator like this, there's no problem. For example, or
If the fraction is, for example . In this case, it is necessary to use the basic property of a 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 decimals!
For example,
Converting a mixed fraction to an improper fraction
A mixed fraction, for example, can be easily converted to an improper fraction. To do this, you need to multiply the whole part by the denominator (bottom) and add it with the numerator (top), leaving the denominator (bottom) unchanged. That is
When converting a mixed fraction to an improper fraction, you can remember that you can use fraction addition
Converting an improper fraction to a mixed fraction (highlighting the whole part)
An improper fraction can be converted to a mixed fraction by highlighting the whole part. Let's look at an example. We determine how many integer times “3” fits into “23”. Or divide 23 by 3 on a calculator, the whole number 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”. 
It’s as if we find the extra that remains from the numerator “23” if we remove the maximum amount of “3”. We leave the denominator unchanged. Everything is done, write down the result
In this article we will look at how converting fractions to decimals, and also consider the reverse process - converting decimal fractions into ordinary fractions. Here we will outline the rules for converting fractions and provide detailed solutions to typical examples.
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Converting fractions to decimals
Let us denote the sequence in which we will deal with converting fractions to decimals.
First, we'll look at how to represent fractions with denominators 10, 100, 1,000, ... as decimals. This is explained by the fact that decimal fractions are essentially a compact form of writing ordinary fractions with denominators 10, 100, ....
After that, we will go further and show how to write any ordinary fraction (not just those with denominators 10, 100, ...) as a decimal fraction. When ordinary fractions are treated in this way, both finite decimal fractions and infinite periodic decimal fractions are obtained.
Now let's talk about everything in order.
Converting fractions with denominators 10, 100, ... to decimals
Some proper fractions require "preliminary preparation" before converting to decimals. This applies to ordinary fractions, the number of digits in the numerator of which is less than the number of zeros in the denominator. For example, the common fraction 2/100 must first be prepared for conversion to a decimal fraction, but the fraction 9/10 does not need any preparation.
“Preliminary preparation” of proper ordinary fractions for conversion to decimal fractions consists of adding so many zeros to the left in the numerator that the total number of digits there becomes equal to the number of zeros in the denominator. For example, a fraction after adding zeros will look like .
Once you have a proper fraction prepared, you can begin converting it to a decimal.
Let's give rule for converting a proper common fraction with a denominator of 10, or 100, or 1,000, ... into a decimal fraction. It consists of three steps:
- write 0;
- after it we put a decimal point;
- We write down the number from the numerator (along with added zeros, if we added them).
Let's consider the application of this rule when solving examples.
Example.
Convert the proper fraction 37/100 to a decimal.
Solution.
The denominator contains the number 100, which has two zeros. The numerator contains the number 37, its notation has two digits, therefore, this fraction does not need to be prepared for conversion to a decimal fraction.
Now we write 0, put a decimal point, and write the number 37 from the numerator, and we get the decimal fraction 0.37.
Answer:
0,37 .
To strengthen the skills of converting proper ordinary fractions with numerators 10, 100, ... into decimal fractions, we will analyze the solution to another example.
Example.
Write the proper fraction 107/10,000,000 as a decimal.
Solution.
The number of digits in the numerator is 3, and the number of zeros in the denominator is 7, so this common fraction needs to be prepared for conversion to a decimal. We need to add 7-3=4 zeros to the left in the numerator so that the total number of digits there becomes equal to the number of zeros in the denominator. We get.
All that remains is to create the required decimal fraction. To do this, firstly, we write 0, secondly, we put a comma, thirdly, we write the number from the numerator together with zeros 0000107, as a result we have a decimal fraction 0.0000107.
Answer:
0,0000107 .
Improper fractions do not require any preparation when converting to decimals. The following should be adhered to rules for converting improper fractions with denominators 10, 100, ... into decimals:
- write down the number from the numerator;
- We use a decimal point to separate as many digits on the right as there are zeros in the denominator of the original fraction.
Let's look at the application of this rule when solving an example.
Example.
Convert the improper fraction 56,888,038,009/100,000 to a decimal.
Solution.
Firstly, we write down the number from the numerator 56888038009, and secondly, we separate the 5 digits on the right with a decimal point, since the denominator of the original fraction has 5 zeros. As a result, we have the decimal fraction 568880.38009.
Answer:
568 880,38009 .
To convert a mixed number into a decimal fraction, the denominator of the fractional part of which is the number 10, or 100, or 1,000, ..., you can convert the mixed number into an improper ordinary fraction, and then convert the resulting fraction into a decimal fraction. But you can also use the following the rule for converting mixed numbers with a fractional denominator of 10, or 100, or 1,000, ... into decimal fractions:
- if necessary, we perform “preliminary preparation” of the fractional part of the original mixed number by adding the required number of zeros to the left in the numerator;
- write down the integer part of the original mixed number;
- put a decimal point;
- We write down the number from the numerator along with the added zeros.
Let's look at an example in which we complete all the necessary steps to represent a mixed number as a decimal fraction.
Example.
Convert the mixed number to a decimal.
Solution.
The denominator of the fractional part has 4 zeros, but the numerator contains the number 17, consisting of 2 digits, therefore, we need to add two zeros to the left in the numerator so that the number of digits there becomes equal to the number of zeros in the denominator. Having done this, the numerator will be 0017.
Now we write down the integer part of the original number, that is, the number 23, put a decimal point, after which we write the number from the numerator along with the added zeros, that is, 0017, and we get the desired decimal fraction 23.0017.
Let's write down the whole solution briefly: 
.
Of course, it was possible to first represent the mixed number as an improper fraction and then convert it to a decimal fraction. With this approach, the solution looks like this: .
Answer:
23,0017 .
Converting fractions to finite and infinite periodic decimals
You can convert not only ordinary fractions with denominators 10, 100, ... into a decimal fraction, but also ordinary fractions with other denominators. Now we will figure out how this is done.
In some cases, the original ordinary fraction is easily reduced to one of the denominators 10, or 100, or 1,000, ... (see bringing an ordinary fraction to a new denominator), after which it is not difficult to represent the resulting fraction as a decimal fraction. For example, it is obvious that the fraction 2/5 can be reduced to a fraction with a denominator 10, for this you need to multiply the numerator and denominator by 2, which will give the fraction 4/10, which, according to the rules discussed in the previous paragraph, is easily converted to the decimal fraction 0, 4.
In other cases, you have to use another method of converting an ordinary fraction to a decimal, which we now move on to consider.
To convert an ordinary fraction to a decimal fraction, the numerator of the fraction is divided by the denominator, the numerator is first replaced by an equal decimal fraction with any number of zeros after the decimal point (we talked about this in the section equal and unequal decimal fractions). In this case, division is performed in the same way as division by a column of natural numbers, and in the quotient a decimal point is placed when the division of the whole part of the dividend ends. All this will become clear from the solutions to the examples given below.
Example.
Convert the fraction 621/4 to a decimal.
Solution.
Let's represent the number in the numerator 621 as a decimal fraction, adding a decimal point and several zeros after it. First, let's add 2 digits 0, later, if necessary, we can always add more zeros. So, we have 621.00.
Now let's divide the number 621,000 by 4 with a column. The first three steps are no different from dividing natural numbers by a column, after which we arrive at the following picture: 
This is how we get to the decimal point in the dividend, and the remainder is different from zero. In this case, we put a decimal point in the quotient and continue dividing in a column, not paying attention to the commas: 
This completes the division, and as a result we get the decimal fraction 155.25, which corresponds to the original ordinary fraction.
Answer:
155,25 .
To consolidate the material, consider the solution to another example.
Example.
Convert the fraction 21/800 to a decimal.
Solution.
To convert this common fraction to a decimal, we divide with a column of the decimal fraction 21,000... by 800. After the first step, we will have to put a decimal point in the quotient, and then continue the division: 
Finally, we got the remainder 0, this completes the conversion of the common fraction 21/400 to a decimal fraction, and we arrived at the decimal fraction 0.02625.
Answer:
0,02625 .
It may happen that when dividing the numerator by the denominator of an ordinary fraction, we still do not get a remainder of 0. In these cases, division can be continued indefinitely. However, starting from a certain step, the remainders begin to repeat periodically, and the numbers in the quotient also repeat. This means that the original fraction is converted to an infinitely periodic decimal fraction. Let's show this with an example.
Example.
Write the fraction 19/44 as a decimal.
Solution.
To convert an ordinary fraction to a decimal, perform division by column: 
It is already clear that during division the residues 8 and 36 began to be repeated, while in the quotient the numbers 1 and 8 are repeated. Thus, the original common fraction 19/44 is converted into a periodic decimal fraction 0.43181818...=0.43(18).
Answer:
0,43(18) .
To conclude this point, we will figure out which ordinary fractions can be converted into finite decimal fractions, and which ones can only be converted into periodic ones.
Let us have an irreducible ordinary fraction in front of us (if the fraction is reducible, then we first reduce the fraction), and we need to find out which decimal fraction it can be converted into - finite or periodic.
It is clear that if an ordinary fraction can be reduced to one of the denominators 10, 100, 1,000, ..., then the resulting fraction can be easily converted into a final decimal fraction according to the rules discussed in the previous paragraph. But to the denominators 10, 100, 1,000, etc. Not all ordinary fractions are given. Only fractions whose denominators are at least one of the numbers 10, 100, ... can be reduced to such denominators. And what numbers can be divisors of 10, 100, ...? The numbers 10, 100, ... will allow us to answer this question, and they are as follows: 10 = 2 5, 100 = 2 2 5 5, 1,000 = 2 2 2 5 5 5, .... It follows that divisors are 10, 100, 1,000, etc. There can only be numbers whose decompositions into prime factors contain only the numbers 2 and (or) 5.
Now we can make a general conclusion about converting ordinary fractions to decimals:
- if in the decomposition of the denominator into prime factors only the numbers 2 and (or) 5 are present, then this fraction can be converted into a final decimal fraction;
- if, in addition to twos and fives, there are other prime numbers in the expansion of the denominator, then this fraction is converted to an infinite decimal periodic fraction.
Example.
Without converting ordinary fractions to decimals, tell me which of the fractions 47/20, 7/12, 21/56, 31/17 can be converted into a final decimal fraction, and which ones can only be converted into a periodic fraction.
Solution.
The denominator of the fraction 47/20 is factorized into prime factors as 20=2·2·5. This expansion contains only twos and fives, so this fraction can be reduced to one of the denominators 10, 100, 1,000, ... (in this example, to the denominator 100), therefore, can be converted to a final decimal fraction.
The decomposition of the denominator of the fraction 7/12 into prime factors has the form 12=2·2·3. Since it contains a prime factor of 3, different from 2 and 5, this fraction cannot be represented as a finite decimal, but can be converted into a periodic decimal.
Fraction 21/56 – contractile, after contraction it takes the form 3/8. Factoring the denominator into prime factors contains three factors equal to 2, therefore, the common fraction 3/8, and therefore the equal fraction 21/56, can be converted into a final decimal fraction.
Finally, the expansion of the denominator of the fraction 31/17 is 17 itself, therefore this fraction cannot be converted into a finite decimal fraction, but can be converted into an infinite periodic fraction.
Answer:
47/20 and 21/56 can be converted to a finite decimal fraction, but 7/12 and 31/17 can only be converted to a periodic fraction.
Ordinary fractions do not convert to infinite non-periodic decimals
The information in the previous paragraph gives rise to the question: “Can dividing the numerator of a fraction by the denominator result in an infinite non-periodic fraction?”
Answer: no. When converting a common fraction, the result can be either a finite decimal fraction or an infinite periodic decimal fraction. Let us explain why this is so.
From the theorem on divisibility with a remainder, it is clear that the remainder is always less than the divisor, that is, if we divide some integer by an integer q, then the remainder can only be one of the numbers 0, 1, 2, ..., q−1. It follows that after the column has completed dividing the integer part of the numerator of a common fraction by the denominator q, in no more than q steps one of the following two situations will arise:
- or we will get a remainder of 0, this will end the division, and we will get the final decimal fraction;
- or we will get a remainder that has already appeared before, after which the remainders will begin to repeat as in the previous example (since when dividing equal numbers by q, equal remainders are obtained, which follows from the already mentioned divisibility theorem), this will result in an infinite periodic decimal fraction.
There cannot be any other options, therefore, when converting an ordinary fraction to a decimal fraction, an infinite non-periodic decimal fraction cannot be obtained.
From the reasoning given in this paragraph it also follows that the length of the period of a decimal fraction is always less than the value of the denominator of the corresponding ordinary fraction.
Converting decimals to fractions
Now let's figure out how to convert a decimal fraction into an ordinary fraction. Let's start by converting final decimal fractions to ordinary fractions. After this, we will consider a method for inverting infinite periodic decimal fractions. In conclusion, let's say about the impossibility of converting infinite non-periodic decimal fractions into ordinary fractions.
Converting trailing decimals to fractions
Obtaining a fraction that is written as a final decimal is quite simple. The rule for converting a final decimal fraction to a common fraction consists of three steps:
- firstly, write the given decimal fraction into the numerator, having previously discarded the decimal point and all zeros on the left, if any;
- secondly, write one into the denominator and add as many zeros to it as there are digits after the decimal point in the original decimal fraction;
- thirdly, if necessary, reduce the resulting fraction.
Let's look at the solutions to the examples.
Example.
Convert the decimal 3.025 to a fraction.
Solution.
If we remove the decimal point from the original decimal fraction, we get the number 3,025. There are no zeros on the left that we would discard. So, we write 3,025 in the numerator of the desired fraction.
We write the number 1 into the denominator and add 3 zeros to the right of it, since in the original decimal fraction there are 3 digits after the decimal point.
So we got the common fraction 3,025/1,000. This fraction can be reduced by 25, we get 
.
Answer:
.
Example.
Convert the decimal fraction 0.0017 to a fraction.
Solution.
Without a decimal point, the original decimal fraction looks like 00017, discarding the zeros on the left we get the number 17, which is the numerator of the desired ordinary fraction.
We write one with four zeros in the denominator, since the original decimal fraction has 4 digits after the decimal point.
As a result, we have an ordinary fraction 17/10,000. This fraction is irreducible, and the conversion of a decimal fraction to an ordinary fraction is complete.
Answer:
.
When the integer part of the original final decimal fraction is non-zero, it can be immediately converted to a mixed number, bypassing the common fraction. Let's give rule for converting a final decimal fraction to a mixed number:
- the number before the decimal point must be written as an integer part of the desired mixed number;
- in the numerator of the fractional part you need to write the number obtained from the fractional part of the original decimal fraction after discarding all the zeros on the left;
- in the denominator of the fractional part you need to write down the number 1, to which add as many zeros to the right as there are digits after the decimal point in the original decimal fraction;
- if necessary, reduce the fractional part of the resulting mixed number.
Let's look at an example of converting a decimal fraction to a mixed number.
Example.
Express the decimal fraction 152.06005 as a mixed number
Often children who study at school are interested in why they might need mathematics in real life, especially those sections that go much further than simple counting, multiplication, division, addition and subtraction. Many adults also ask this question if their professional activity is very far from mathematics and various calculations. However, it is worth understanding that there are all sorts of situations, and sometimes it is impossible to do without that very notorious school curriculum that we so disdainfully rejected in childhood. For example, not everyone knows how to convert a fraction into a decimal, but such knowledge can be extremely useful for ease of calculation. 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 a fraction to see if it can be converted to a decimal
Before you count 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. Moreover, 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 into its final, decimal version only if its unique denominator can only be expanded 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 be divided without a remainder only by themselves, or by one. A table of 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.
Let's look at examples:
The fraction 7/40 can be converted from a fraction to its decimal equivalent because its denominator can be easily factored into factors of 2 and 5.
However, if the first option results in a final decimal fraction, then, for example, 7/60 will in no way give a similar result, since its denominator will no longer be decomposed into the numbers we are looking for, but will have a three among the denominator factors.
There are several ways to convert a fraction to a decimal
Once it has become clear which fractions can be converted from ordinary to decimal, you can proceed to the conversion itself. In fact, there is nothing super difficult, even for someone whose school curriculum has completely faded from memory.
How to convert fractions to decimals: the easiest method
This method of converting a fraction into a decimal is indeed the simplest, but many people are not even aware of its mortal existence, since at school all these “truths” seem unnecessary and not very important. Meanwhile, not only an adult will be able to figure it out, but a child will also easily perceive such information.
So, to convert a fraction to a decimal, you multiply the numerator, as well as the denominator, by one number. However, everything is not so simple, as a result, it is in the denominator that you should get 10, 100, 1000, 10,000, 100,000 and so on, ad infinitum. Don’t forget to first check whether a given fraction can be converted to a decimal.
Let's look at examples:
Let's say we need to convert the fraction 6/20 to a decimal. We check:
After we are convinced that it is still possible to convert a fraction into a decimal fraction, and even a finite one, since its denominator can easily be decomposed into twos and fives, we should proceed to the translation itself. The best option, logically, to multiply the denominator and get the result 100, is 5, since 20x5=100.
You can consider an additional example for clarity:
The second and more popular method 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 here is transparent and clear, so let’s move on to the calculations right away.
Worth remembering
In order to correctly convert a simple, that is, ordinary fraction into 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 look at the action using an example:
So, the first thing to do is to convert the fraction 78/200 to 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 checking, you need to remember school and divide the numerator by the denominator using a “corner” or “column”.
As you can see, everything is extremely simple, and you don’t need to be a genius to easily solve such problems. For simplicity and convenience, we also provide a table of the most popular fractions that are easy to remember and don’t even make an effort to translate them.
How to convert percentages to decimals: nothing is simpler
Finally, the move has come to percentages, which, it turns out, as the same school curriculum says, can be converted into a decimal fraction. Moreover, everything will be much simpler here, and there is no need to be afraid. Even those who did not graduate from universities, skipped the fifth grade of school and know nothing about mathematics, can cope with the task.
Perhaps we need to start with a definition, that is, understand what interest actually is. A percentage is one hundredth of a number, that is, completely arbitrary. From a hundred, for example, it will be one and so on.
Thus, to convert percentages to a decimal fraction, you simply need to remove the % sign, and then divide the number itself by a hundred.
Let's look at examples:
Moreover, in order to make a reverse “conversion”, you simply need to do everything the other way around, that is, you need to multiply the number by a hundred and attach a percent sign to it. In exactly the same way, by applying the acquired knowledge, you can also convert an ordinary fraction into a percentage. To do this, it will be enough to simply first convert an ordinary fraction into a decimal, and therefore convert it into a percentage, and you can also easily perform the reverse action. As you can see, there is nothing super complicated, all this is basic knowledge that just needs to be kept in mind, especially if you are dealing with numbers.
The path of least resistance: convenient online services
It also happens that you don’t want to count at all, and you simply don’t have the time. It is for such cases, or 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 truly the path of least resistance, so using such resources is a pleasure.
Useful reference portal "Calculator"
In order to use the Calculator service, simply 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 both ordinary and mixed fractions to decimals.
After a short wait, about three seconds, the service will display the final result.
In exactly the same way, you can convert a decimal fraction to a regular 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 yourself, just select from the list provided what you need and go ahead and get the orders.
Next, in the field provided specifically for this, you need to enter the desired number of percentages, which need to be converted into 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 designed for this.
Ultimately, it’s worth adding that no matter how many newfangled services are invented, no matter how many resources offer you their services, it won’t hurt to train your head periodically. Therefore, you should definitely apply the knowledge you have acquired, especially since you will then be able to proudly help your own children and then grandchildren do their homework. For those who suffer from an eternal lack of time, such online calculators on mathematical portals will come in handy and will even help you understand how to convert a fraction to a decimal.
